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abstract: 'We prove quantitative convergence rates at which discrete Langevin-like processes converge to the invariant distribution of a related stochastic differential equation. We study the setup where the additive noise can be non-Gaussian and state-dependent and the potential function can be non-convex. We show that the key properties of these processes depend on the potential function and the second moment of the additive noise. We apply our theoretical findings to studying the convergence of Stochastic Gradient Descent (SGD) for non-convex problems and corroborate them with experiments using SGD to train deep neural networks on the CIFAR-10 dataset.'
bibliography:
- 'references.bib'
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[Introduction]{} Stochastic Gradient Descent (SGD) is one of the workhorses of modern machine learning. In many nonconvex optimization problems, such as training deep neural networks, SGD is able to produce solutions with good generalization error; indeed, there is evidence that the generalization error of an SGD solution can be significantly better than that of Gradient Descent (GD) [@keskar2016large; @jastrzkebski2017three; @he2019control]. This suggests that, to understand the behavior of SGD, it is not enough to consider the limiting cases such as small step size or large batch size where it degenerates to GD. In this paper, we take an alternate view of SGD as a sampling algorithm, and aim to understand its convergence to an appropriate stationary distribution.
There has been rapid recent progress in understanding the finite-time behavior of MCMC methods, by comparing them to stochastic differential equations (SDEs), such as the Langevin diffusion. It is natural in this context to think of SGD as a discrete-time approximation of an SDE. There are, however, two significant barriers to extending previous analyses to the case of SGD. First, these analysis are often restricted to isotropic Gaussian noise, whereas the noise in SGD can be far from Gaussian. Second, the noise depends significantly on the current state (the optimization variable). For instance, if the objective is an average over training data with a nonnegative loss, as the objective approaches zero the variance of minibatch SGD goes to zero. Any attempt to cast SGD as an SDE must be able to handle this kind of noise.
This motivates the study of Langevin MCMC-like methods that have a state-dependent noise term: $$\begin{aligned}
w_{(k+1)\delta} = w_{k\delta} - \delta \nabla U(w_{k\delta}) + \sqrt{\delta} \xi(w_{k\delta}, \eta_k),
\numberthis \label{e:discrete-clt}\end{aligned}$$ where $ w_t \in \Re^d $ is the state variable at time $t$, $\delta$ is the step size, $U:\Re^d\to\Re$ is a (possibly nonconvex) potential, $\xi:\Re^d \times \Omega \to \Re^d$ is the [*noise function*]{}, and $\eta_k$ are sampled i.i.d. according to some distribution over $\Omega$ (for example, in minibatch SGD, $\Omega$ is the set of subsets of indices in the training sample). Throughout this paper, we assume that $\Ep{\eta}{\xi(x,\eta)}=0$ for all $x$. We define a matrix-valued function $M(\cdot):\Re^d \to \Re^{d\times d}$ to be the square root of the covariance matrix of $\xi$; i.e., for all $x$, $M(x) := \sqrt{\Ep{\eta}{\xi(x,\eta) \xi(x,\eta)^T}}$, where for a positive semidefinite matrix $G$, $A=\sqrt{G}$ is the unique positive semidefinite matrix such that $A^2 = G$.
In studying the generalization behavior of SGD, earlier work [@jastrzkebski2017three; @he2019control] propose that be approximated by the stochastic process $y_{(k+1)\delta} = y_{k\delta} - \delta \nabla U(y_{k\delta}) + \sqrt{\delta} M(y_{k\delta}) \theta_k$ where $\theta_k\sim \N(0,I)$, or, equivalently: $$\begin{aligned}
&d y_t = -\nabla U(y_{k\delta}) dt + M(y_{k\delta}) dB_t
\numberthis \label{e:discrete-langevin}\\
&\quad \text{for } t\in[k\delta, (k+1)\delta],\end{aligned}$$ with $B_t$ denoting standard Brownian motion [@karatzas1998brownian]. Specifically, the non-Gaussian noise $\xi(\cdot,\eta)$ is approximated by a Gaussian variable $M(\cdot) \theta$ with the same covariance, via an assumption that the minibatch size is large and an appeal to the central limit theorem.
The process in can be seen as the Euler-Murayama discretization of the following SDE: $$\begin{aligned}
\numberthis \label{e:exact-sde}
d x_t = -\nabla U(x_t) dt + M(x_t) dB_t.\end{aligned}$$ We let $p^*$ denote the invariant distribution of .
We prove quantitative bounds on the discretization error between , and , as well as convergence rates of and to $p^*$. Our bounds are in Wasserstein-1 distance (denoted by $W_1(\cdot, \cdot)$ in the following). We present the full theorem statements in Section \[s:main\_results\], and summarize our contributions below:
1. In Theorem \[t:main\_gaussian\], we bound the discretization error between and . Informally, Theorem \[t:main\_gaussian\] states: $$\begin{aligned}
&1.\ \textit{If $x_0 = y_0$, then for all $k$, $W_1(x_{k\delta}, y_{k\delta}) = O(\sqrt{\delta})$ }; \\
&2.\ \textit{For $n \geq \tilde{O}\lrp{\frac{1}{\delta}}$, $W_1(p^*, \Law(y_{n\delta})) = O(\sqrt{\delta})$},\end{aligned}$$ where $\Law(\cdot)$ denotes the distribution of a random vector. This is a crucial intermediate result that allows us to prove the convergence of to . We highlight that the variable diffusion matrix: 1) leads to a very large discretization error, due to the scaling factor of $\sqrt{\delta}$ in the $M(y_{k\delta}) \theta_k$ noise term, and 2) makes the stochastic process non-contractive (this is further compounded by the non-convex drift). Our convergence proof relies on a carefully constructed Lyapunov function together with a specific coupling. Remarkably, the $\epsilon$ dependence in our iteration complexity is the same as that in Langevin MCMC with constant isotropic diffusion [@durmus2016high].
2. In Theorem \[t:main\_nongaussian\], we bound the discretization error between and . Informally, Theorem \[t:main\_nongaussian\] states: $$\begin{aligned}
&1.\ \textit{If $x_0 = w_0$, then for all $k$, $W_1(x_{k\delta}, w_{k\delta}) = O(\delta^{1/8})$ }; \\
&2.\ \textit{For $n \geq \tilde{O}\lrp{\frac{1}{\delta}}$, $W_1(p^*, \Law(w_{n\delta})) = O(\delta^{1/8})$}.\end{aligned}$$ Notably, the noise in each step of may be far from Gaussian, but for sufficiently small step size, is nonetheless able to approximate . This is a weaker condition than earlier work, which must assume that the batch size is sufficiently large so that CLT ensures that the per-step noise is approximately Gaussian.
3. Based on Theorem 2, we predict that for sufficiently small $\delta$, two different processes of the form will have similar distributions if their noise terms $\xi$ have the same covariance matrix, as that leads to the same limiting SDE . In Section \[s:sgd\], we evaluate this claim empirically: we design a family of SGD-like algorithms and evaluate their test error at convergence. We observe that the noise covariance alone is a very strong predictor for the test error, regardless of higher moments of the noise. This corroborates our theoretical prediction that the noise covariance approximately determines the distribution of the solution. This is also in line with, and extends upon, observations in earlier work that the ratio of batch size to learning rate correlates with test error [@jastrzkebski2017three; @he2019control].
[Related Work]{} Previous work has drawn connections between SGD noise and generalization [@mandt2016variational; @jastrzkebski2017three; @he2019control; @hoffer2017train; @keskar2016large]. Notably, @mandt2016variational [@he2019control; @jastrzkebski2017three] analyze favorable properties of SGD noise by arguing that in the neighborhood of a local minimum, is roughly the discretization of an Ornstein-Uhlenbeck (OU) process, and so the distribution of $y_{k\delta}$ approximates is approximately Gaussian. However, empirical results [@keskar2016large; @hoffer2017train] suggest that SGD generalizes better by finding better local minima, which may require us to look beyond the “OU near local minimum” assumption to understand the global distributional properties of SGD. Indeed, @hoffer2017train suggest that SGD performs a random walk on a random loss landscape, @kleinberg2018alternative propose that SGD noise helps smoooth out “sharp minima.” @jastrzkebski2017three further note the similarity between and an Euler-Murayama approximation of . @chaudhari2018stochastic also made connections between SGD and SDE. Our work tries to make these connections rigorous, by quantifying the error between , and , without any assumptions about being close to an OU process or being close to a local minimum.
Our work builds on a long line of work establishing the convergence rate of Langevin MCMC in different settings [@dalalyan2017theoretical; @durmus2016high; @ma2018sampling; @gorham2016measuring; @cheng2018sharp; @erdogdu2018global; @li2019stochastic]. We will discuss our rates in relation to some of this work in detail following our presentation of Theorem \[t:main\_gaussian\]. We note here that some of the techniques used in this paper were first used by @eberle2011reflection [@gorham2016measuring], who analyzed the convergence of to $p^*$ without log-concavity assumptions. @erdogdu2018global studied processes of the form as an approximation to under a distant-dissipativity assumption, which is similar to the assumptions made in this paper. For the sequence , they prove an $O(1/\epsilon^2)$ iteration complexity to achieve $\epsilon$ integration error for any loss $f$ which has bounded derivatives up to fourth order. In comparison, we prove $W_1$ convergence between $\Law(y_{k\delta})$ and $p^*$, which is equivalent to $\sup_{\lrn{\nabla f}_{\infty} \leq 1} \lrabs{\E{f(y_{k\delta})} - \Ep{y\sim p^*}{f(y)}}$, also with rate $\tilde{O}(1/\epsilon^2)$. By smoothing the $W_1$ test function to have bounded derivatives up to fourth order, we believe that the results by @erdogdu2018global can imply a qualitatively similar result to Theorem \[t:main\_gaussian\], but with a worse dimension and $\epsilon$ dependence.
In concurrent work by @li2019stochastic, the authors study a process based on a stochastic Runge-Kutta discretization scheme of . They prove an $\tilde{O}\lrp{\frac{d}{\epsilon^{-2/3}}}$ iteration complexity to achieve $\epsilon$ error in $W_2$ for an algorithm based on Runge-Kutta discretization of . They make a strong assumption of *uniform dissipativity* (essentially assuming that the process is uniformly contractive), which is much stronger than the assumptions in this paper, and may be violated in the settings of interest considered in this paper.
There has been a number of work [@chen2016statistical; @li2018statistical; @anastasiou2019normal] which establish CLT results for SGD with very small step size (rescaled to have constant variance). These work generally focus on the setting of “OU process near a local minimum”, in which the diffusion matrix is constant.
Finally, a number of authors have studied the setting of heavy-tailed gradient noise in neural network training. [@zhang2019adam] showed that in some cases, the heavy-tailed noise can be detrimental to training, and a clipped version of SGD performs much better. [@simsekli2019tail] argue that when the SGD noise is heavy-tailed, it should not be modelled as a Gaussian random variable, but instead as an $\alpha$-stable random variable, and propose a Generalized Central Limit Theorem to analyze the convergence in distribution. Our paper does not handle the setting of heavy-tailed noise; our theorems require that the norm of the noise term uniformly bounded, which will be satisfied, for example, if gradients are explicitly clipped at a threshold, or if the optimization objective has Lipschitz gradients and the SGD iterates stay within a bounded region.
[Motivating Example]{} It is generally difficult to write down the invariant distribution of . In this section, we consider a very simple one-dimensional setting which does admit an explicit expression for $p^*$, and serves to illustrate some remarkable properties of anisotropic diffusion matrices.
Let us define $D(x) := M^2(x)$. Our analysis will be based on the Fokker-Planck equation, which states that $p^*$ is the invariant distribution of if $$\begin{aligned}
0 = \div(p^*(x) \nabla U(x)) + \div\lrp{p^*(x) \Gamma(x) + D(x) \nabla p^*(x)},
\numberthis \label{e:fokker_planck}
\end{aligned}$$ where $\Gamma(x)$ is a vector whose $i^{th}$ coordinate equals $\sum_{j=1}^d \frac{\del}{\del x_j} \lrb{D(x)}_{i,j}$. In the one-dimensional setting, we can explicitly write down the density of $p^*(x)$. Note that in this case, $\Gamma(x) = \nabla D(x)$. Let $V(x) := \int_{0}^{x} \lrp{\frac{\nabla U(x)}{D(x)} + \frac{\nabla D(X)}{D(x)}} dx = \int_{0}^{x} \lrp{\frac{\nabla U(x)}{D(x)}} dx + \log D(x) - \log D(0)$. We can verify that $p^*(x) \propto e^{-V(x)}$ satisfies .
For a concrete example, let the potential $U(x)$ and the diffusion function $M(x)$ be defined as $$\begin{aligned}
& U(x) := \threecase{\frac{1}{2} x^2}{x\in[-1,4]}{\frac{1}{2} (x+2)^2 - 1}{x\leq -1}{\frac{1}{2} (x-8)^2 -16}{x\geq 4} \\
& M(x) = \threecase
{\frac{1}{2} (x+2)}{x\in[-2,8]}
{1}{x \leq -2}
{6}{x \geq 8}.
\end{aligned}$$
[0.23]{} 
[0.23]{} 
[0.23]{} 
[0.23]{} 
We plot $U(x)$ in Figure \[fig:1d\_ux\]. Note that $U(x)$ has two local minima: a shallow minimum at $x=-2$ and a deeper minimum at $x=8$. A plot of $M(x)$ can be found in Figure \[fig:1d\_mx\]. $M(x)$ is constructed to have increasing magnitude at larger values of $x$. This has the effect of biasing the invariant distribution towards smaller values of $x$.
We plot $V(x)$ in Figure \[fig:1d\_vx\]. Remarkably, $V(x)$ has only one local minimum at $x=-2$. The larger minimum of $U(x)$ at $x=8$ has been smoothed over by the effect of the large diffusion $M(x)$. This is very different from when the noise is homogeneous (e.g., $M(x)=I$), in which case $p^*(x) \propto e^{-U(x)}$. We also simulate (using ) for the given $U(x)$ and $M(x)$ for 1000 samples (each simulated for 1000 steps), and plot the histogram in Figure \[fig:1d\_histo\].
[Assumptions and Definitions]{} In this section, we state the assumptions and definitions that we need for our main results in Theorem \[t:main\_gaussian\] and Theorem \[t:main\_nongaussian\]. \[ss:ass\]
We assume that $U(x)$ satisfies \[ass:U\_properties\]
1. The function $U(x)$ is continuously-differentiable on $\mathbb{R}^d$ and has Lipschitz continuous gradients; that is, there exists a positive constant $L\geq0$ such that\
for all $x,y\in \mathbb{R}^d$, $
\lVert \nabla U(x) - \nabla U(y) \rVert_2 \le L \lVert x-y\rVert_2.
$
2. $U$ has a stationary point at zero: $\nabla U(0) = 0.$
3. There exists a constant $m>0, \LR, R$ such that for all $\lrn{x-y}_2 \geq R$, $$\label{e:convexity_outside_ball}
\lin{ \nabla U(x) - \nabla U(y),x-y} \geq m \lrn{x-y}_2^2.$$ and for all $\lrn{x-y}_2 \leq R$, $
\lVert \nabla U(x) - \nabla U(y) \rVert_2 \le \LR \lVert x-y\rVert_2.
$
The assumption in roughly states that “$U(x)$ is convex outside a ball of radius $R$”. This assumption, and minor variants, is common in the nonconvex sampling literature [@eberle2011reflection; @eberle2016reflection; @cheng2018sharp; @ma2018sampling; @erdogdu2018global; @gorham2016measuring].
\[ass:xi\_properties\] We make the following assumptions on $\xi$ and $M$:
1. For all $x$, $\E{\xi(x,\eta)}=0$.
2. For all $x$, $\lrn{\xi(x,\eta)}_2 \leq \beta$ almost surely.
3. For all $x,y$, $\lrn{\xi(x,\eta) - \xi(y,\eta)}_2\leq L_{\xi} \lrn{x-y}_2$ almost surely.
4. There is a positive constant $\cm$ such that for all $x$, $2 \cm I \prec M(x)$.
We discuss these assumptions in a specific setting in Section \[ss:example\_ass\].
For convenience we define a matrix-valued function $N(\cdot) : \Re^d \to \Re^{d\times d}$: $$\begin{aligned}
N(x) := \sqrt{M(x)^2 - \cm^2 I}.
\numberthis
\label{d:N}
\end{aligned}$$ Under Assumption \[ass:U\_properties\], we can prove that $N(x)$ and $M(x)$ are bounded and Lipschitz (see Lemma \[l:M\_is\_regular\] and \[l:N\_is\_regular\] in Appendix \[ss:mnregularity\]). These properties will be crucial in ensuring convergence. Given an arbitrary sample space $\Omega$ and any two distribution $p\in \Pspace\lrp{\Omega}$ and $q\in \Pspace\lrp{\Omega}$, a joint distribution $\zeta \in \Pspace\lrp{\Omega \times \Omega}$ is a *coupling* between $p$ and $q$ if its marginals are equal to $p$ and $q$ respectively.
For a matrix, we use $\lrn{G}_2$ to denote the operator norm: $
\lrn{G}_2 = \sup_{v\in \Re^d, \|v\|_2 =1} \lrn{Gv}_2.
$.
Finally, we define a few useful constants which will be used throughout the paper: $$\begin{aligned}
&\LN := \frac{4\beta L_\xi}{\cm}, \ \ \aq:=\frac{\LR+L_N^2}{2\cm^2}, \\ &\Rq:=\max\lrbb{R,{\frac{16\beta^2 L_N}{m\cdot \cm}}} \\
&\lambda :=\min\lrbb{\frac{m}{2}, \frac{2\cm^2}{32\Rq^2}}\exp\lrp{-\frac{7}{3}\aq\Rq^2}.
\numberthis \label{d:constants}
\end{aligned}$$
$\LN$ is the smoothness parameter of the matrix $N(x)$, and we show in Lemma \[l:N\_is\_regular\] that $\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \LN^2\lrn{x-y}_2^2$. The constants $\aq$ and $\Rq$ are used to define a Lyapunov function $q$ in Appendix \[ss:defining-q\]. A key step in our proof uses the fact that, under the dynamics , $q$ contracts at a rate of $e^{-\lambda}$, plus discretization error.
[Main Results]{}\[s:main\_results\] In this section, we present our main convergence results beginning with convergence under Gaussian noise and proceeding to the non-Gaussian case.
\[t:main\_gaussian\] Let $x_t$ and $y_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{x_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{y_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. Let $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\twocase{\frac{\lambda^2 \hat{\epsilon}^2}{512 \beta^2\lrp{L^2 + \LN^4}\exp\lrp{\frac{14\aq\Rq^2}{3}}}}{\frac{2\lambda \hat{\epsilon}}{(L^2+\LN^4)\exp\lrp{\frac{7\aq\Rq^2}{3}}\sqrt{R^2 + \beta^2/m}}}.
\end{aligned}$$
If we assume that $x_0 = y_0$, then there exists a coupling between $x_t$ and $y_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{x_{k\delta} - y_{k\delta}}_2} \leq \hat{\epsilon}
\end{aligned}$$
Alternatively, if we assume $n \geq \frac{ 3\aq\Rq^2}{\delta} \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^y_{n\delta}} \leq 2\hat{\epsilon}
\end{aligned}$$ where $p^y_t := \Law(y_t)$.
Note that $m,L,R$ are from Assumption \[ass:U\_properties\], $L_N$ is from , $\cm, \beta,L_\xi$ are from Assumption \[ass:xi\_properties\]).
Finding a suitable $y_0$ can be done very quickly using gradient descent wrt $U(\cdot)$. The convergence rate to the ball of radius $R$ is very fast, due to Assumption \[ass:U\_properties\].3.
After some algebraic simplifications, we see that for a sufficiently small $\hat{\epsilon}$, achieving $W_1(p^y_{n\delta}, p^*) \leq \hat{\epsilon}$ requires number of steps $$\begin{aligned}
n = \tilde{O}\left(\frac{\beta^2}{\hat{\epsilon}^2} \cdot \exp\left(\frac{14}{3} \cdot \lrp{\frac{\LR}{\cm^2} + \frac{16\beta^2 L_\xi^2}{\cm^4}} \right. \right.\\
\left.\left. \cdot\max\lrbb{R^2, \frac{2^{12} \beta^6 L_\xi^2}{m^2 \cm^4}}\right)\right).\end{aligned}$$
The convergence rate contains a term $e^{R^2}$; this term is also present in all of the work cited in the previous section under Remark 1. Given our assumptions, including the “convexity outside a ball of radius $R$” assumption, this dependence is unavoidable as it describes the time to transit between two modes of the invariant distribution. It can be verified to be tight by considering a simple double-well potential.
As illustrated in Section \[ss:example\_ass\], the $m$ from Assumption \[ass:xi\_properties\].3 should be thought of as a regularization term which can be set arbitrarily large. In the following discussion, we will assume that $ \max\lrbb{R^2, \frac{\beta^6 L_\xi^2}{m^2 \cm^4}}$ is dominated by the $R^2$ term.
To gain intuition about this term, let’s consider what it looks like under a sequence of increasingly weaker assumptions:
**a. Strongly convex, constant noise**: $U(x)$ $m$-strongly convex, $L$-smooth, $\xi(x,\eta)\sim \N(0,I)$ for all $x$. (In reality we need to consider a truncated Gaussian so as not to violate Assumption \[ass:xi\_properties\].2, but this is a minor issue). In this case, $L_\xi=0$, $\cm = 1$, $R=0$, $\beta = \tilde{O}(\sqrt{d})$, so $k=O(\frac{d}{\hat{\epsilon}^2})$. This is the same rate as obtained by @durmus2016high. We remark that @durmus2016high obtain a $W_2$ bound which is stronger than our $W_1$ bound.
**b. Non-convex, constant noise**: $U(x)$ not strongly convex but satisfies Assumption \[ass:U\_properties\], and $\xi(x,\eta) \sim \N(0,I)$. In this case, $L_\xi=0$, $\cm=1$, $\beta = \tilde{O}(\sqrt{d})$ This is the setting studied by @cheng2018sharp and @ma2018sampling. The rate we recover is $k=\tilde{O}\lrp{\frac{d}{\hat{\epsilon^2}} \cdot \exp\lrp{\frac{14}{3}LR^2}}$, which is in line with @cheng2018sharp, and is the best $W_1$ rate obtainable from @ma2018sampling.
**c. Non-convex, state-dependent noise**: $U(x)$ satisfies Assumption \[ass:U\_properties\], and $\xi$ satisfies Assumption \[ass:xi\_properties\]. To simplify matters, suppose the problem is rescaled so that $\cm =1$. Then the main additional term compared to setting b. above is $\exp\lrp{\frac{64\beta^2 L_\xi^2 R^2}{\cm^4}}$. This suggests that the effect of a $L_\xi$-Lipschitz noise can play a similar role in hindering mixing as a $\LR$-Lipschitz nonconvex drift.
When the dimension is high, computing $M(y_k)$ can be difficult, but if for each $x$, one has access to samples whose covariance is $M(x)$, then one can approximate $M(y_k) \theta_k$ via the central limit theorem by drawing a sufficiently large number of samples. The proof of Theorem \[t:main\_gaussian\] can be readily modified to accommodate this (see Appendix \[ss:simlutating\_discrete\_sde\]).
We now turn to the non-Gaussian case.
\[t:main\_nongaussian\] Let $x_t$ and $w_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{x_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{w_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. Let $\epsilon:= \frac{\lambda}{16 (L+\LN^2)} \exp\lrp{-\frac{7\aq\Rq^2}{3}} \hat{\epsilon}$. Let $T:= \min\lrbb{\frac{1}{16L}, \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$ and let $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \frac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}.
\end{aligned}$$
If we assume that $x_0 = w_0$, then there exists a coupling between $x_t$ and $w_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{x_{k\delta} - w_{k\delta}}_2} \leq \hat{\epsilon}.
\end{aligned}$$
Alternatively, if we assume that $n \geq \frac{3\aq\Rq^2}{\delta} \cdot \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^w_{n\delta}} \leq 2\hat{\epsilon},
\end{aligned}$$ where $p^w_t := \Law(w_t)$.
To achieve $W_1(p^*, p^w_{n\delta}) \leq \hat{\epsilon}$, the number of steps needed is of order $n= \tilde{O}\lrp{\frac{1}{\hat{\epsilon}^8} \cdot e^{29 \aq \Rq^2}}$. The $\hat{\epsilon}$ dependency is considerably worse than in Theorem \[t:main\_gaussian\]. This is because we need to take many steps of in order to approximate a single step of . For details, see the coupling construction in equations – of Appendix \[s:nongaussianproof\].
[Application to Stochastic Gradient Descent]{}\[s:sgd\] In this section, we will cast SGD in the form of . We consider an objective of the form $$\begin{aligned}
\numberthis
\label{e:example_U}
U(w) = \frac{1}{n} \sum_{i=1}^n U_i(w).\end{aligned}$$ We reserve the letter $\eta$ to denote a random minibatch from $\lrbb{1, \ldots, n}$, sampled with replacement, and define $\zeta(w,\eta)$ as follows: $$\label{e:def_zeta}
\zeta(w,\eta) := \nabla U(w) - \frac{1}{\lrabs{\eta}} \sum_{i\in \eta} \nabla U_i(w)$$ For a sample of size one, i.e. $\lrabs{\eta}=1$, we define $$\begin{aligned}
H(w) := \E{\zeta(w,\eta) \zeta(w,\eta)^T }
\numberthis\label{e:m0}\end{aligned}$$ as the covariance matrix of the difference between the true gradient and a single sampled gradient at $w$. A standard run of SGD, with minibatch size $b := \lrabs{\eta_k} $, then has the following form: $$\begin{aligned}
w_{k+1} &= w_k - \delta \frac{1}{b} \sum_{i\in\eta_k} \nabla U_{i}(w_k)\\
&= w_k - \delta \nabla U(w_k) + \sqrt{\delta}\lrp{\sqrt{\delta} \zeta (w_k, \eta_k)}.
\numberthis \label{e:sgd}\end{aligned}$$ We refer to an SGD algorithm with step size $\delta$ and minibatch size $b$ a $(\delta, b)$-SGD. Notice that $\eqref{e:sgd}$ is in the form of , with $\xi(w,\eta) = \sqrt{\delta} \zeta (w,\eta)$. The covariance matrix of the noise term is $$\label{e:noise_cov_sgd}
\E{\xi (w,\eta)\xi (w,\eta)^T} = \frac{\delta}{b} H(w).$$ Because the magnitude of the noise covariance scales with $\sqrt{\delta}$, it follows that as $\delta \to 0$, converges to deterministic gradient flow. However, the loss of randomness as $\delta\to 0$ is not desirable as it has been observed that as SGD approaches GD, through either small step size or large batch size, the generalization error goes up [@jastrzkebski2017three; @he2019control; @keskar2016large; @hoffer2017train]; this is also consistent with our experimental observations in Section \[ss:acc\_rel\_var\].
Therefore, a more meaningful way to take the limit of SGD is to hold the noise term constant in . More specifically, we define the ***constant-noise limit*** of as $$d x_t = - \nabla U(x_t) dt + M(x_t) dB_t,
\numberthis
\label{e:sgd-limit}$$ where $M(x) := \sqrt{\frac{\delta}{b}H(x)}$. Note that this is in the form of , with noise covariance $M(x_t)^2$ matching that of SGD in . Using Theorem \[t:main\_nongaussian\], we can bound the $W_1$ distance between the SGD iterates $w_{k}$ from , and the continuous-time SDE $x_t$ from .
Importance of Noise Covariance {#ss:noise_covariance}
------------------------------
We highlight the fact that the limiting SDE of a discrete process, $$\label{e:disc-mcmc-new}
w_{k+1} = w_k - s\nabla U(w_k) + \sqrt{s} \xi(w_k, \eta_k),$$ depends only on the covariance matrix of $\xi$. More specifically, as long as $\xi$ satisfies $\sqrt{\E{\xi(w, \eta)\xi(w, \eta)^T}} = M(w)$, will have as its limiting SDE, *regardless of higher moments of $\xi$*. This fact, combined with Theorem \[t:main\_nongaussian\], means that in the limit of $\delta \to 0$ and $k\to \infty$, the distribution of $w_k$ will be determined by the covariance of $\xi$ alone. An immediate consequence is the following: *at convergence, the test performance of any Langevin MCMC-like algorithm is almost entirely determined by the covariance of its noise term.*
Returning to the case of SGD algorithms, since the noise covariance is $M(x)^2 = \frac{\delta}{b} H(x)$ (see ), we know that the ratio of step size $\delta$ to batch size $b$ is an important quantity which can dictate the test error of the algorithm; this observation has been made many times in prior work [@jastrzkebski2017three; @he2019control], and our results in this paper are in line with these observations. Here, we move one step further, and provide experimental evidence to show that more fundamentally, it is the noise covariance in the constant-noise limit that controls the test error.
To verify this empirically, we propose the following algorithm called *large-noise SGD*.
\[def:large\_noise\_sgd\] An $(s, \sigma, b_1, b_2)$-large-noise SGD is an algorithm that aims to minimize using the following updates: $$\begin{aligned}
\numberthis\label{e:sgd-noisy}
& w_{k+1} = w_k - \frac{s}{b_1} \sum_{i \in \eta_k} \nabla U_i(w_k) \\
& \qquad + \frac{\sigma \sqrt{s}}{b_2} \left( \sum_{i\in \eta_k'}\nabla U_i(w_k) - \sum_{i\in\eta_k''}\nabla U_i(w_k) \right),\end{aligned}$$ where $\eta_k$, $\eta_k'$, and $\eta_k''$ are minibatches of sizes $b_1$, $b_2$, and $b_2$, sampled uniformly at random from $\{1, \ldots, n\}$ with replacement. The three minibatches are sampled independently and are also independent of other iterations.
Intuitively, an $(s, \sigma, b_1, b_2)$-large-noise SGD should be considered as an SGD algorithm with step size $s$ and minibatch size $b_1$ and an additional noise term. The noise term computes the difference of two independent and unbiased estimates of the full gradient $\nabla U(w_k)$, each using a batch of $b_2$ data points. Using the definition of $\zeta$ in , we can verify that the update is equivalent to $$\begin{aligned}
\numberthis\label{e:sgd-noisy-2}
& w_{k+1} = w_k - s\nabla U(w_k) + s\zeta(w_k, \eta_k) \\
&\qquad + \sigma \sqrt{s}(\zeta(w_k, \eta_k'') - \zeta(w_k, \eta_k')),\end{aligned}$$ which is in the form of , with $$\begin{aligned}
\xi(w, \tilde{\eta}) = \sqrt{s} \zeta(w, \eta) + \sigma \lrp{\zeta(w, \eta'')-\zeta(w, \eta')},
\numberthis
\label{e:sgd-noisy-xi}\end{aligned}$$ where $\tilde{\eta} = (\eta, \eta', \eta'')$, and $|\eta| = b_1$, $|\eta'|=|\eta''|=b_2$. Further, the noise covariance matrix is $$\label{e:noise_cov_large_noise_sgd}
\E{\xi(w, \tilde{\eta}) \xi(w, \tilde{\eta})^T} = (\frac{s}{b_1} + \frac{2\sigma^2}{b_2}) H(w).$$ Therefore, if we have $$\label{e:var_matching}
\frac{s}{b_1} + \frac{2\sigma^2}{b_2} = \frac{\delta}{b},$$ then an $(s, \sigma, b_1, b_2)$-large-noise SGD should have the same noise covariance as a $(\delta, b)$-SGD *(but very different higher noise moments due to the injected noise)*, and based on our theory, the large-noise SGD should have similar test error to that of the SGD algorithm, even if the step size and batch size are different. In Section \[subsection:experiments\], we verify this experimentally. We stress that we are not proposing the large-noise SGD as a practical algorithm. The reason that this algorithm is interesting is that it gives us a family of $\lrp{w_k}_{k=1,2,\ldots}$ which converges to , and is implementable in practice. Thus this algorithm helps us uncover the importance of noise covariance (and the unimportance of higher noise moments) in Langevin MCMC-like algorithms. We also remark that @hoffer2017train proposed a different way of injecting noise, multiplying the sampled gradient with a suitably scaled Gaussian noise.
[Satisfying the Assumptions]{}\[ss:example\_ass\]
Before presenting the experimental results, we remark on a particular way that a function $U(w)$ defined in , along with the stochastic sequence $w_k$ defined in , can satisfy the assumptions in Section \[ss:ass\].
Suppose first that we shift the coordinate system so that $\nabla U(0)=0$. Let us additionally assume that for each $i$, $U_i(w)$ has the form $$\begin{aligned}
U_i(w) = U_i'(w) + V(w),\end{aligned}$$ where $V(w):= m \lrp{\|x\|_2-R/2}^2$ is a $m$-strongly convex regularizer outside a ball of radius $R$, and each $U_i'(w)$ has $\LR$-Lipschitz gradients. Suppose further that $m \geq 4 \cdot \LR$. These additional assumptions make sense when we are only interested in $U(w)$ over $B_R(0)$, so $V(w)$ plays the role of a barrier function that keeps us within $B_R(0)$. Then, it can immediately be verified that $U(w)$ satisfies Assumption \[ass:U\_properties\] with $L=m+\LR$.
The noise term $\xi$ in satisfies Assumption \[ass:xi\_properties\].1 by definition, and satisfies Assumption \[ass:xi\_properties\].3 with $L_\xi = \lrp{\sqrt{s} + 2\sigma} L$. Assumption \[ass:xi\_properties\].2 is satisfied if $\zeta(w,\eta)$ is bounded for all $w$, i.e. the sampled gradient does not deviate from the true gradient by more than a constant. We will need to assume directly Assumption \[ass:xi\_properties\].4, as it is a property of the distribution of $\nabla U_i(w)$ for $i=1, \ldots, n$.
![Relationship between test accuracy and the noise covariance of SGD algorithm. In each plot, the dots with the same color correspond to SGD runs with the same batch size but different step sizes.[]{data-label="fig:const_lr_acc_vs_var"}](sgd_runs.pdf){width="1\linewidth"}
[Experiments]{}\[subsection:experiments\]
{width="1.0\linewidth"}
{width="1.0\linewidth"}
In this section, we present experimental results that validate the importance of noise covariance in predicting the test error of Langevin MCMC-like algorithms. In all experiments, we use two different neural network architectures on the CIFAR-10 dataset [@krizhevsky2009learning] with the standard test-train split. The first architecture is a simple convolutional neural network, which we call CNN in the following,[^1] and the other is the VGG19 network [@simonyan2014very]. To make our experiments consistent with the setting of SGD, we do not use batch normalization or dropout, and use constant step size. In all of our experiments, we run SGD algorithm $2000$ epochs such that the algorithm converges sufficiently. Since in most of our experiments, the accuracies on the training dataset are almost $100\%$, we use the test accuracy to measure the generalization performance.
Recall that according to and , for both SGD and large-noise SGD, the noise covariance is a scalar multiple of $H(w)$. For simplicity, in the following, we will slightly abuse our terminology and call this scalar the *noise covariance*; more specifically, for $(\delta, b)$-SGD, the noise covariance is $\delta / b$, and for an $(s, \sigma, b_1, b_2)$-large-noise SGD, the noise covariance is $\frac{s}{b_1} + \frac{2\sigma^2}{b_2}$.
[Accuracy vs Noise Covariance]{} \[ss:acc\_rel\_var\]
In our first experiment, we focus on the SGD algorithm, and show that there is a positive correlation between the noise covariance and the final test accuracy of the trained model. One major purpose of this experiment is to establish baselines for our experiments on large-noise SGD.
We choose constant step size $\delta$ from $$\{0.001, 0.002, 0.004, 0.008, 0.016, 0.032, 0.064, 0.128\}$$ and minibatch size $b$ from $\{32, 64, 128, 256, 512\}$. For each (step size, batch size) pair, we plot its final test accuracy against its noise covariance in Figure \[fig:const\_lr\_acc\_vs\_var\]. From the plot, we can see that higher noise covariance leads to better final test accuracy, and there is a linear trend between the test accuracy and the logarithm. We also highlight the fact that conditioned on the noise covariance, the test accuracy is not significantly correlated with either the step size or the minibatch size. In other words, similar to the observations in prior work [@jastrzkebski2017three; @he2019control], there is a strong correlation between relative variance of an SGD sequence and its test accuracy, regardless of the combination of minibatch size and step size.
[Large-Noise SGD]{} \[ss:inject\]
In this section, we implement and examine the performance of the large-noise SGD algorithm proposed in . We select a subset of SGD runs with relatively small noise covariance in the experiment in the previous section (we call them *baseline SGD runs*), and implement large-noise SGD by injecting noise. Our goal is to see, for a particular noise covariance, whether large-noise SGD has test accuracy that is similar to SGD, *in spite of significant differences in third-and-higher moments of the noise in large-noise SGD compared to standard SGD*. Our first experiment is to add noise with the same minibatch size to the $(\delta, b)$ baseline SGD run such that the new noise covariance matches that of an $(8\delta, b)$-SGD (an SGD run with larger step size). In other words, we implement $(\delta, \sqrt{7\delta / 2}, b, b)$-large-noise SGD, whose noise covariance is $8$ times of that of the baseline. Our results are shown in Figure \[fig:lr\_matching\]. Our second experiment is similar: we add noise with minibatch size $128$ to the $(\delta, b)$ baseline SGD run with $b \in \{256, 512\}$ such that the new noise covariance matches that of a $(\delta, 128)$-SGD (an SGD run with smaller batch size). More specifically, we implement $(\delta, \sqrt{\frac{1}{2} (1-\frac{128}{b})\delta }, b, 128)$-large-noise SGD runs. The results are shown in Figure \[fig:batch\_matching\]. In these figures, each $\times$ denotes a baseline SGD run, with step size specified in the legend and minibatch size specified by plot title. For each baseline SGD run, we have a corresponding large-noise SGD run, denoted by $\diamond$ with the same color. As mentioned, these $\diamond$ runs are designed to match the noise covariance of SGD with larger step size or smaller batch size. In addition to $\times$ and $\diamond$, we also plot using a small teal marker all the other runs from Section \[ss:acc\_rel\_var\]. This helps highlight the linear trend between the logarithm of noise covariance and test accuracy that we observed in Section \[ss:acc\_rel\_var\].
As can be seen, the (noise variance, test accuracy) values for the $\diamond$ runs fall close to the linear trend. More specifically, a run of large-noise SGD produces similar test accuracy to vanilla SGD runs with the same noise variance. We highlight two potential implications: First, just like in Section \[ss:acc\_rel\_var\], we observe that the test accuracy strongly correlates with relative variance, even for noise of the form , which can have rather different higher moments than $\zeta$ (standard SGD noise); Second, since the $\diamond$ points fall close to the linear trend, we hypothesize that the constant-noise limit SDE should also have similar test error. If true, then this implies that we only need to study the potential $U(x)$ and noise covariance $M(x)$ to explain the generalization properties of SGD.
Appendix {#appendix .unnumbered}
========
[Proofs for Convergence under Gaussian Noise (Theorem \[t:main\_gaussian\])]{}
[Proof Overview]{} The main proof of Theorem \[t:main\_gaussian\] is contained in Appendix \[ss:proof:t:main\_gaussian\].
Here, we outline the steps of our proof:
1. In Appendix \[ss:coupling\_construction\], we construct a coupling between $\eqref{e:exact-sde}$ and $\eqref{e:discrete-langevin}$ over a single step (i.e. for $t\in[k\delta, (k+1)\delta]$, for some $k$ and $\delta$).
2. Appendix \[ss:step\_gaussian\], we prove Lemma \[l:gaussian\_contraction\], which shows that under the coupling constructed in Step 1, a Lyapunov function $f(x_T - y_T)$ contracts exponentially with rate $\lambda$, plus a discretization error term. The function $f$ is defined in Appendix \[s:defining-q\], and sandwiches $\lrn{x_T - y_T}_2$. In Corollary \[c:main\_gaussian:1\], we apply the results of Lemma \[l:gaussian\_contraction\] recursively over multiple steps to give a bound on $f(x_{k\delta}-y_{k\delta})$ for all $k$, and for sufficiently small $\delta$.
3. Finally, in Appendix \[ss:proof:t:main\_gaussian\], we prove Theorem \[t:main\_gaussian\] by applying the results of Corollary \[c:main\_gaussian:1\], together with the fact that $f(z)$ upper bounds $\lrn{z}_2$ up to a constant factor.
[A coupling construction]{} \[ss:coupling\_construction\] In this subsection, we will study the evolution of and over a small time interval. Specifically, we will study $$\begin{aligned}
\numberthis
\label{e:marginal_x}
d x_t =& - \nabla U(x_t) dt + M(x_t) dB_t\\
\numberthis
\label{e:marginal_y}
d y_t =& - \nabla U(y_0) dt + M(y_0) dB_t
\end{aligned}$$ One can verify that is equivalent to , and is equivalent to a single step of (i.e. over an interval $t\leq \delta$).
We first give the explicit coupling between and : ( A similar coupling in the continuous-time setting is first seen in [@gorham2016measuring] in their proof of contraction of .)
Given arbirary $(x_0,y_0)$, define $(x_t,y_t)$ using the following coupled SDE: $$\begin{aligned}
\numberthis \label{e:coupled_2_processes}
x_t =& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(x_s) dW_s\\
y_t =& y_0 + \int_0^t -\nabla U(y_0) dt + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^T} dV_s + \int_0^t N(y_0) dW_s
\end{aligned}$$
Where $dV_t$ and $dW_t$ are two independent standard Brownian motion, and $$\begin{aligned}
\gamma_t := \frac{x_t-y_t}{\|x_t-y\|_2} \cdot \ind{\|x_t-y_t\|_2 \in [2\epsilon, \Rq)}
\numberthis \label{d:gammat}
\end{aligned}$$
By Lemma \[l:marginal\_of\_coupling\], we show that has the same distribution as $x_t$ in , and has the same distribution as $y_t$ in . Thus, for any $t$, the process $(x_t,y_t)$ defined by is a valid coupling for and .
[One step contraction]{} \[ss:step\_gaussian\]
\[l:gaussian\_contraction\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameters $\epsilon$ satisfying $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$. Let $x_t$ and $y_t$ be as defined in . If we assume that $\E{\lrn{y_0}_2^2} \leq 8\lrp{R^2 + \beta^2/m}$ and $T\leq \min\lrbb{\frac{\epsilon^2}{\beta^2}, \frac{\epsilon}{6 L\sqrt{R^2 + \beta^2/m}}}$, then $$\begin{aligned}
\E{f(x_T - y_T)}
\leq e^{-\lambda T} \E{f(x_0 - y_0)} + 3T (L+\LN^2) \epsilon
\end{aligned}$$
For ease of reference: $m,L,\LR, R$ are from Assumption \[ass:U\_properties\], $\cm, \beta$ are from Assumption \[ass:xi\_properties\], $\aq, \Rq, \LN, \lambda$ are defined in .
For notational convenience, for the rest of this proof, let us define $z_t := x_t - y_t$ and $\nabla_t := \nabla U(x_t) - \nabla U(y_t)$, $\Delta_t := \nabla U(y_0) - \nabla U(y_t)$ $N_t := N(x_t) - N(y_t)$.
It follows from that $$\begin{aligned}
d z_t = - \nabla_t dt + \Delta_t dt + 2 \cm \gamma_t \gamma_t^T dV_t + \lrp{N_t + N(y_t) - N(y_0)} dW_t
\numberthis \label{e:coupled_difference_sde}
\end{aligned}$$
Using Ito’s Lemma, the dynamics of $f(z_t)$ is given by $$\begin{aligned}
&d f(z_t)\\
=& {\lin{\nabla f(z_t), dz_t}}
+ {2\cm^2\tr\lrp{\nabla^2 f(z_t) \lrp{\gamma_t \gamma_t^T}}} dt
+ {\frac{1}{2}\tr\lrp{\nabla^2 f(z_t) \lrp{N_t+ N(y_t) - N(y_0)}^2}} dt\\
=& \underbrace{-\lin{\nabla f(z_t), \nabla_t}}_{\circled{1}} dt + \underbrace{\lin{\nabla f(z_t), \Delta_t}}_{\circled{2}} dt + \underbrace{\lin{\nabla f(z_t), 2 \cm \gamma_t \gamma_t^T dV_t + \lrp{N_t + N(y_t) - N(y_0)} dW_t }}_{\circled{3}}\\
&\quad + \underbrace{2\cm^2\tr\lrp{\nabla^2 f(z_t) \lrp{\gamma_t \gamma_t^T}}}_{\circled{4}} dt
+ \underbrace{\frac{1}{2}\tr\lrp{\nabla^2 f(z_t) \lrp{N_t+ N(y_t) - N(y_0)}^2}}_{\circled{5}} dt
\numberthis \label{e:t:asldka}
\end{aligned}$$
$\circled{3}$ goes to $0$ when we take expectation, so we will focus on $\circled{1}, \circled{2}, \circled{4}, \circled{5}$. We will consider 3 cases
**Case 1: $\|z_t\|_2 \leq 2\epsilon$**\
From item 1(c) of Lemma \[l:fproperties\], $\lrn{\nabla f(z)}_2 \leq 1$. Using Assumption \[ass:U\_properties\].1, $\lrn{\nabla_t} \leq L\lrn{z_t}_2$, so that $$\begin{aligned}
\circled{1} \leq \lrn{\nabla_t}_2 \leq L \lrn{z_t}_2 \leq 2 L \epsilon
\end{aligned}$$
Also by Cauchy Schwarz, $$\begin{aligned}
\circled{2}
=& \lin{\nabla f(z_t), \Delta_t}
\leq \lrn{\Delta_t}_2 \leq L \lrn{y_t - y_0}_2
\end{aligned}$$
Since $\gamma_t = 0$ in this case by definition in , $\circled{4}=0$.
Using Lemma \[l:fproperties\].2.c. $ \lrn{\nabla^2 f(z_t)}_2 \leq \frac{2}{\epsilon}$, so that $$\begin{aligned}
\circled{5}
\leq& \frac{1}{\epsilon} \lrp{\tr\lrp{N_t^2 + N(y_t) - N(y_0)}^2}\\
\leq& \frac{2}{\epsilon} \lrp{\tr\lrp{N_t^2} + \tr\lrp{\lrp{N(y_t) - N(y_0)}^2}}\\
\leq& \frac{2L_N^2}{\epsilon} \lrp{\lrn{z_t}_2^2 + \lrn{y_t - y_0}_2^2}\\
\leq& 4L_N^2 \epsilon + \frac{2L_N^2}{\epsilon} \|y_t-y_0\|_2^2
\end{aligned}$$
Where the second inequality is by Young’s inequality, the third inequality is by item 2 of Lemma \[l:N\_is\_regular\], the fourth inequality is by our assumption that $\lrn{z_t}_2 \leq 2\epsilon$.
Summing these, $$\begin{aligned}
\circled{1} + \circled{2} + \circled{4} + \circled{5}
\leq 4 \lrp{L + L_N^2}\epsilon + L \lrn{y_t - y_0}_2 + \frac{2L_N^2}{\epsilon}\|y_t-y_0\|_2^2
\end{aligned}$$
**Case 2: $\|z_t\|_2\in (2\epsilon, \Rq)$**\
In this case, $\gamma_t = \frac{z_t}{\|z_t\|_2}$. Let $q$ be as defined in and $g$ be as defined in Lemma \[l:gproperties\]. By items 1(b) and 2(b) of Lemma \[l:fproperties\] and items 1(b) and 2(b) of Lemma \[l:gproperties\], $$\begin{aligned}
\nabla f(z_t)
=& q'(g(z_t)) \nabla g(z_t) \\
=& q'(g(z_t)) \frac{z_t}{\lrn{z_t}_2}\\
\nabla^2 f(z_t)
=& q''(g(z_t))\nabla g(z_t) \nabla g(z_t)^T + q'(g(z_t))\nabla^2 g(z_t)\\
=& q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}
\end{aligned}$$
Once again, by Assumption \[ass:U\_properties\].3, $$\begin{aligned}
\circled{1} \leq q'(g(z_t))\lrn{\nabla_t}_2 \leq q'(g(z_t)) \cdot \LR \cdot \|z_t\|_2 \leq& L \cdot q'(g(z_t)) g(z_t) + 2 L \epsilon
\end{aligned}$$ Where the last inequality uses Lemma \[l:gproperties\].4. We can also verify that $$\begin{aligned}
\circled{2} \leq L \lrn{y_t - y_0}_2
\end{aligned}$$
Using the expression for $\nabla^2 f(z_t)$, $$\begin{aligned}
\circled{4}
= 2\cm^2 \tr\lrp{\nabla^2 f(z_t) \gamma_t \gamma_t^T} = 2\cm^2 \cdot q''(g(z_t))
\end{aligned}$$
Finally, $$\begin{aligned}
\circled{5}
=& \frac{1}{2}\tr\lrp{\nabla^2 f(z_t) \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
=& \frac{1}{2}\tr\lrp{\lrp{q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{1}{2}\tr\lrp{\lrp{ q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{q'(g(z_t))}{\|z_t\|_2}\cdot \lrp{\tr\lrp{N_t^2} + \tr\lrp{\lrp{N(y_t) - N(y_0)}^2}}\\
\leq& q'(g(z_t)) \cdot L_N^2 \|z_t\|_2 + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon}\\
\leq& q'(g(z_t)) \cdot L_N^2 g(z_t) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 L_N^2 \epsilon
\end{aligned}$$
The above uses multiples times the fact that $0 \leq q' \leq 1$ and $q'' \leq 0$ (proven in items 3 and 4 of Lemma \[l:qproperties\]). The second inequality is by Young’s inequality, the third inequality is by item 2 of Lemma \[l:N\_is\_regular\], the fourth inequality uses item 4 of Lemma \[l:gproperties\].
Summing these, $$\begin{aligned}
\circled{1} + \circled{2} + \circled{4} + \circled{5}
\leq& \lrp{\LR + L_N^2} q'(g(z_t)) g(z_t) + 2\cm^2 q''(g(z_t)) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 \lrp{L + \LN^2}\epsilon\\
\leq& - \frac{2\cm^2\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{32\Rq^2} q(g(z_t)) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 \lrp{L + \LN^2}\epsilon\\
\leq& - \lambda q(g(z_t)) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 (L+L_N^2) \epsilon\\
=& - \lambda f(z_t) + \frac{L_N^2\|y_t-y_0\|_2^2}{2\epsilon} + 2 (L+L_N^2) \epsilon + L \lrn{y_t - y_0}_2
\end{aligned}$$
Where the last inequality follows from Lemma \[l:qproperties\].\[f:contraction\] and the definition of $\lambda$ in .
**Case 3: $\|z_t\|_2 \geq \Rq$**\
In this case, $\gamma_t=0$. Similar to case 2, $$\begin{aligned}
\nabla f(z_t) = q'(g(z_t)) \frac{z_t}{\|z_t\|_2}
\end{aligned}$$
Thus by Assumption \[ass:U\_properties\].3, $$\begin{aligned}
\circled{1}
=& \lin{q'(g(z_t)) \frac{z_t}{\|z_t\|_2}, - \nabla_t}\\
\leq& -m q'(g(z_t)) \lrn{z_t}_2
\end{aligned}$$ Where the inequality is by Assumption \[ass:U\_properties\].3.
For identical reasons as in Case 1, $\circled{2}\leq \LR \lrn{y_t-y_0}_2$, and $\circled{4} = 0$. Finally, $$\begin{aligned}
\circled{5}
=& \frac{1}{2}\tr\lrp{\nabla^2 f(z_t) \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
=& \frac{1}{2}\tr\lrp{\lrp{q''(g(z_t)) \frac{z_t z_t^T}{\|z_t\|_2^2} + q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{1}{2}\tr\lrp{\lrp{ q'(g(z_t)) \frac{1}{\|z_t\|_2} \lrp{I - \frac{z_tz_t^T}{\|z_t\|_2^2}}} \lrp{N_t+ N(y_t) - N(y_0)}^2}\\
\leq& \frac{q'(g(z_t))}{\|z_t\|_2}\cdot \lrp{\tr\lrp{N_t^2} + \tr\lrp{\lrp{N(y_t) - N(y_0)}^2}}
\end{aligned}$$
Where the first inequality is because $q''\leq 0$ from item 4 of Lemma \[l:qproperties\], the second inequality is by Young’s inequality. (These steps are identical to Case 2). Continuing from above, and using item 2 and 3 of Lemma \[l:N\_is\_regular\], $$\begin{aligned}
\circled{5}
\leq& q'(g(z_t)) \cdot \lrp{\frac{8\beta^2 \LN}{\cm} + \frac{\LN^2\|y_t-y_0\|_2^2}{\epsilon}}\\
\leq& q'(g(z_t)) \cdot \lrp{\frac{m}{2} \|z_t\|_2}
+ q'(g(z_t)) \cdot \lrp{ \frac{\LN^2\|y_t-y_0\|_2^2}{\epsilon}}
\end{aligned}$$ Where the second inequality is by our definition of $\Rq$ in the Lemma statement, which ensures that $\frac{8\beta^2 \LN}{\cm} \leq \frac{m}{2} \Rq \leq \frac{m}{2} \|z_t\|_2 $.
Thus $$\begin{aligned}
& \circled{1} + \circled{2} + \circled{4} + \circled{5}\\
\leq& -m q'(g(z_t)) \|z_t\|_2 + \LR \lrn{y_t - y_0}_2 + \frac{m}{2} q'(g(z_t)) \|z_t\|_2 + q'(g(z_t)) \cdot \lrp{ \frac{\LN^2\|y_t-y_0\|_2^2}{\epsilon}}\\
\leq& -\frac{m}{2} q'(g(z_t)) \|z_t\|_2 + \frac{\LN^2}{\epsilon} \lrn{y_t-y_0}_2^2 + L \lrn{y_t - y_0}_2\\
\leq& - \lambda f(z_t) + \frac{\LN^2}{\epsilon} \lrn{y_t-y_0}_2^2 + L \lrn{y_t - y_0}_2
\end{aligned}$$
where the second inequality uses $q'\leq 1$ from item 3 of Lemma \[l:qproperties\], the third inequality uses our definition of $\lambda$ in .
Combining the three cases, can be upper bounded with probability 1: $$\begin{aligned}
&d f(z_t)
\leq - \lambda f(z_t) + \frac{\LN^2}{\epsilon} \lrn{y_t-y_0}_2^2 + L \lrn{y_t - y_0}_2 + \lin{\nabla f(z_t), 2 \cm \gamma_t \gamma_t^T dV_t + \lrp{N_t + N(y_t) - N(y_0)} dW_t }
\end{aligned}$$
To simplify notation, let us define $G_t \in \Re^{1\times 2d}$ as $G_t:= \lrb{\nabla f(z_t)^T 2\cm \gamma_t \gamma_t^T, \nabla f(z_t)^T \lrp{N_t + N(y_t) - N(y_0)}}$, and let $A_t$ be a $2d$-dimensional Brownian motion from concatenating $A_t = \cvec{V_t}{W_t}$. Thus $$\begin{aligned}
d f(z_t) \leq -\lambda f(z_t) dt + \lrp{\frac{L_N^2}{\epsilon}\lrn{y_t - y_0}_2^2 + L \lrn{y_t - y_0}_2} + G_t dA_t.
\end{aligned}$$ We will study the Lyapunov function $$\begin{aligned}
\mathcal{L}_t:= f(z_t) - \int_0^t e^{-\lambda(t-s)}\lrp{\frac{L_N^2}{\epsilon}\lrn{y_s - y_0}_2^2 + L \lrn{y_s - y_0}_2} ds - \int_0^t e^{-\lambda(t-s)} G_s dA_s.
\end{aligned}$$ By taking derivatives, we see that $$\begin{aligned}
d\mathcal{L}_t
\leq& -\lambda f(z_t) dt + \lrp{\frac{L_N^2}{\epsilon}\lrn{y_t - y_0}_2^2 + L \lrn{y_t - y_0}_2}dt + G_t dA_t\\
&\qquad + \lambda \lrp{\int_0^t e^{-\lambda(t-s)}\lrp{\frac{L_N^2}{\epsilon}\lrn{y_s - y_0}_2^2 + L \lrn{y_s - y_0}_2} ds} dt - \lrp{\frac{L_N^2}{\epsilon}\lrn{y_t - y_0}_2^2 + L \lrn{y_t - y_0}_2} dt\\
&\qquad + \lambda \lrp{\int_0^t e^{-\lambda(t-s)} G_s dA_s} dt - G_t dA_t\\
=& -\lambda \mathcal{L}_t dt
\end{aligned}$$
We can then apply Gronwall’s Lemma to $\mathcal{L}_t$, so that $$\begin{aligned}
\mathcal{L}_T \leq e^{-\lambda T} \mathcal{L}_0,
\end{aligned}$$ which is equivalent to $$\begin{aligned}
f(z_T) - \int_0^T e^{-\lambda(T-s)}\lrp{\frac{L_N^2}{\epsilon}\lrn{y_s - y_0}_2^2 + L \lrn{y_s - y_0}_2} ds - \int_0^T e^{-\lambda(t-s)} G_s dA_s \leq e^{-\lambda T} f(z_0).
\end{aligned}$$ Observe that $G_s$ is measurable wrt the natural filtration generated by $A_s$, so that $\int_0^T e^{-\lambda (T-s)} G_s dA_s$ is a martingale. Thus taking expectations, $$\begin{aligned}
\E{f(z_T)} \leq e^{-\lambda T} \E{f(z_0)} + \int_0^T \frac{L_N^2}{\epsilon} \E{\lrn{y_s - y_0}_2^2} + L \E{\lrn{y_s - y_0}_2} ds
\end{aligned}$$ By Lemma \[l:divergence\_yt\], $\E{\lrn{y_t - y_0}_2^2} \leq t^2 L^2 \E{\lrn{y_0}_2^2} + t\beta^2$, so that $$\begin{aligned}
& \int_0^T \frac{L_N^2}{\epsilon} \E{\lrn{y_s - y_0}_2^2} ds \leq \frac{T^3 L_N^2 L^2}{\epsilon}\E{\lrn{y_0}_2^2} + \frac{T^2L_N^2}{\epsilon}\beta^2\\
& L \E{\lrn{y_s - y_0}_2} \leq T^2 L^2 \sqrt{\E{\lrn{y_0}_2^2}} + T^{3/2} L \beta
\end{aligned}$$ Furthermore, using our assumption in the Lemma statement that $T \leq \min\lrbb{\frac{\epsilon^2}{\beta^2}, \frac{\epsilon}{{6L \sqrt{R^2 + \beta^2/m}}}}$ and $\E{\lrn{y_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$, we can verify that $$\begin{aligned}
& \int_0^T \frac{L_N^2}{\epsilon} \E{\lrn{y_s - y_0}_2^2} ds \leq \frac{1}{4} TL_N^2 \epsilon + TL_N^2 \epsilon\\
& L \E{\lrn{y_s - y_0}_2} \leq \frac{1}{2} TL\epsilon + TL\epsilon
\end{aligned}$$ Combining the above gives $$\begin{aligned}
\E{f(z_T)} \leq e^{-\lambda T} \E{f(z_0)} + 3 T\lrp{L+L_N^2} \epsilon
\end{aligned}$$
\[c:main\_gaussian:1\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$.
Let $\delta\leq \min\lrbb{\frac{\epsilon^2}{\beta^2}, \frac{\epsilon}{8 L\sqrt{R^2 + \beta^2/m}}}$, and let $\bx_t$ and $\by_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\by_0}_2^2}\leq R^2 + \beta^2/m$. Then there exists a coupling between $\bx_t$ and $\by_t$ such that $$\begin{aligned}
\E{f(\bx_{i\delta} - \by_{i\delta})} \leq e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
From Lemma \[l:energy\_x\] and \[l:energy\_y\], our initial conditions imply that for all $t$, $\E{\|\bx_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}$ and $\E{\|\by_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}$.
Consider an arbitrary $k$, and for $t\in[k\delta,(k+1)\delta)$, define $$\begin{aligned}
x_t := \bx_{k\delta+t} \quad \text{and} \quad
y_t := \by_{k\delta+t}
\end{aligned}$$ Under this definition, $x_t$ and $y_t$ have dynamics described in and . Thus the coupling in , which describes a coupling between $x_t$ and $y_t$, equivalently describes a coupling between $\bx_t$ and $\by_t$ over $t\in[k\delta, (k+1)\delta)$.
We now apply Lemma \[l:gaussian\_contraction\]. Given our assumed bound on $\delta$ and our proven bounds on $\E{\lrn{\bx_t}_2^2}$ and $\E{\lrn{\by_t}_2^2}$, $$\begin{aligned}
& \E{f(\bx_{(k+1)\delta} - \by_{(k+1)\delta})}\\
=& \E{f(x_{\delta} - y_{\delta})}\\
\leq& e^{-\lambda \delta} \E{f(x_0 - y_0)} + 6\delta (L+\LN^2) \epsilon\\
=& e^{-\lambda \delta} \E{f(\bx_{k\delta} - \by_{k\delta})} + 6\delta (L+\LN^2) \epsilon
\end{aligned}$$ Applying the above recursively gives, for any $i$ $$\begin{aligned}
\E{f(\bx_{i\delta} - \by_{i\delta})} \leq e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
[Proof of Theorem \[t:main\_gaussian\]]{}\[ss:proof:t:main\_gaussian\]
For ease of reference, we re-state Theorem \[t:main\_gaussian\] below as Theorem \[t:main\_gaussian:restated\] below. We make a minor notational change: using the letters $\bx_t$ and $\by_t$ in Theorem \[t:main\_gaussian:restated\], instead of the letters $x_t$ and $y_t$ in Theorem \[t:main\_gaussian\]. This is to avoid some notation conflicts in the proof.
\[Equivalent to Theorem \[t:main\_gaussian\]\] \[t:main\_gaussian:restated\] Let $\bx_t$ and $\by_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\by_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. Let $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\twocase{\frac{\lambda^2 \hat{\epsilon}^2}{512 \beta^2\lrp{L^2 + \LN^4}\exp\lrp{\frac{14\aq\Rq^2}{3}}}}{\frac{2\lambda \hat{\epsilon}}{(L^2+\LN^4)\exp\lrp{\frac{7\aq\Rq^2}{3}}\sqrt{R^2 + \beta^2/m}}}.
\end{aligned}$$
If we assume that $\bx_0 = \by_0$, then there exists a coupling between $\bx_t$ and $\by_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{\bx_{k\delta} - \by_{k\delta}}_2} \leq \hat{\epsilon}
\end{aligned}$$
Alternatively, if we assume $k \geq \frac{ 3\aq\Rq^2}{ \delta} \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^y_{k\delta}} \leq 2\hat{\epsilon}
\end{aligned}$$ where $p^y_t := \Law(\by_t)$.
Let $\epsilon := \frac{\lambda}{16 (L+\LN^2)} \exp\lrp{-\frac{7\aq\Rq^2}{3}} \hat{\epsilon}$. Let $f$ be defined as in Lemma \[l:fproperties\] with the parameter $\epsilon$.
$$\begin{aligned}
& \E{\lrn{\bx_{i\delta} - \by_{i\delta}}_2}\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\E{f(\bx_{i\delta} - \by_{i\delta})} + 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon} + 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \frac{16\lrp{L+\LN^2}}{\lambda}\exp\lrp{\frac{7\aq\Rq^2}{3}} \cdot \epsilon
\numberthis \label{e:t:asdkja:1}\\
=& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}e^{-\lambda i\delta} \E{f(\bx_{0} - \by_{0})} + \hat{\epsilon}
\end{aligned}$$
where the first inequality is by item 4 of Lemma \[l:fproperties\], the second inequality is by Corollary \[c:main\_gaussian:1\] (notice that $\delta$ satisfies the requirement on $T$ in Theorem \[t:main\_gaussian\], for the given $\epsilon$). The third inequality uses the fact that $1\leq L/m \leq \frac{\lrp{L+\LN^2}}{\lambda}$.
The first claim follows from substituting $\bx_0 = \by_0$ into , so that the first term is $0$, and using the definition of $\epsilon$, so that the second term is $0$.
For the second claim, let $\bx_0 \sim p^*$, the invariant distribution of . From Lemma \[l:energy\_x\], we know that $\bx_0$ satisfies the required initial conditions in this Lemma. Continuing from , $$\begin{aligned}
& \E{\lrn{\bx_{i\delta} - \by_{i\delta}}_2}\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{2e^{-\lambda i\delta} \E{\lrn{\bx_0}_2^2 + \lrn{\by_0}_2^2} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon} + \epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{2e^{-\lambda i\delta} \lrp{R^2 + \beta^2/m}} + \frac{16}{\lambda} \exp\lrp{2\frac{7\aq\Rq^2}{3}}\lrp{L + \LN^2} \epsilon\\
=& 4\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{e^{-\lambda i\delta} \lrp{R^2 + \beta^2/m}} + \hat{\epsilon}
\end{aligned}$$ By our assumption that $i\geq \frac{1}{\delta} \cdot 3\aq\Rq^2 \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, the first term is also bounded by $\hat{\epsilon}$, and this proves our second claim.
[Simulating the SDE]{} \[ss:simlutating\_discrete\_sde\] One can verify that the SDE in can be simulated (at discrete time intervals) as follows: $$\begin{aligned}
y_{(k+1) \delta} = y_{k\delta} - \delta \nabla U(y_{k\delta}) + \sqrt{\delta} M(y_{k\delta}) \theta_k
\end{aligned}$$ Where $\theta_k \sim \N(0,I)$. This however requires access to $M(y_{k,\delta})$, which may be difficult to compute.
If for any $y$, one is able to draw samples from some distribution $p_y$ such that
1. $\Ep{\xi \sim p_y}{\xi}=0$
2. $\Ep{\xi\sim p_y}{\xi \xi^T}=M(y)$
3. $\lrn{\xi}_2 \leq \beta$ almost surely, for some $\beta$.
then one might sample a noise that is $\delta$ close to $M(y_{k\delta}) \theta_k$ through Theorem \[t:zhai\].
Specifically, if one draws $n$ samples $\xi_1...\xi_n\overset{iid}{\sim} p_y$, and let $S_n := \frac{1}{\sqrt{n}}\sum_{i=1}^n \xi_i$, Theorem \[t:zhai\] guarantees that\
$W_2 \lrp{S_n, M(y) \theta} \leq \frac{6\sqrt{d}\beta\sqrt{\log n}}{\sqrt{n}}$. We remark that the proof of Theorem \[t:main\_gaussian\] can be modified to accommodate for this sampling error. The number of samples needed to achieve $\epsilon$ accuracy will be on the order of $n \approxeq O(\delta \epsilon)^{-2} = O(\epsilon^{-6})$.
[Proofs for Convergence under Non-Gaussian Noise (Theorem \[t:main\_nongaussian\])]{} \[s:nongaussianproof\]
[Proof Overview]{} The main proof of Theorem \[t:main\_nongaussian\] is contained in Appendix \[ss:proof:t:main\_nongaussian\].
Here, we outline the steps of our proof:
1. In Appendix \[ss:4\_coupling\], we construct a coupling between $\eqref{e:exact-sde}$ and $\eqref{e:discrete-clt}$ over an epoch which consists of an interval $[k\delta , (k+n)\delta)$ for some $k$. The coupling in consists of four processes $(x_t,y_t,v_t,w_t)$, where $y_t$ and $v_t$ are auxiliary processes used in defining the coupling. Notably, the process $(x_t,y_t)$ has the same distribution over the epoch as .
2. In Appendix \[ss:epoch\_nongaussian\], we prove Lemma \[l:non\_gaussian\_contraction\_stationary\] and Lemma \[l:non\_gaussian\_contraction\_anisotropic\], which, combined with Lemma \[l:gaussian\_contraction\] from Appendix \[ss:step\_gaussian\], show that under the coupling constructed in Step 1, a Lyapunov function $f(x_T - w_T)$ contracts exponentially with rate $\lambda$, plus a discretization error term. In Corollary \[c:main\_nongaussian:1\], we apply the results of Lemma \[l:gaussian\_contraction\], Lemma \[l:non\_gaussian\_contraction\_stationary\] and Lemma \[l:non\_gaussian\_contraction\_anisotropic\] recursively over multiple steps to give a bound on $f(x_{k\delta}-w_{k\delta})$ for all $k$, and for sufficiently small $\delta$.
3. Finally, in Appendix \[ss:proof:t:main\_nongaussian\], we prove Theorem \[t:main\_nongaussian\] by applying the results of Corollary \[c:main\_nongaussian:1\], together with the fact that $f(z)$ upper bounds $\lrn{z}_2$ up to a constant.
[Constructing a Coupling]{} \[ss:4\_coupling\] In this subsection, we construct a coupling between and , given arbitrary initialization $(x_0,w_0)$. We will consider a finite time $T=n\delta$, which we will refer to as an *epoch*.
1. Let $V_t$ and $W_t$ be two independent Brownian motion.
2. Using $V_t$ and $W_t$, define $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_x}
x_t =& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(w_0) dW_s
\end{aligned}$$
3. Using the same $V_t$ and $W_t$ in , we will define $y_t$ as $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_y}
y_t =& w_0 + \int_0^t -\nabla U(w_0) ds + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^t} dV_s + \int_0^T N(x_s) dW_s
\end{aligned}$$
Where $\gamma_t := \frac{x_t - y_t}{\|x_t-y_t\|_2} \cdot \ind{\|x_t-y_t\|_2 \in [2\epsilon, \Rq)}$. The coupling $(x_t,y_t)$ defined in and is identical to the coupling in (with $y_0 = w_0$).
4. We now define a process $v_{k\delta}$ for $k=0...n$: $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_v}
v_{k\delta} =& w_0 + \sum_{i=0}^{k-1} - \delta \nabla U(w_0) + {\sqrt{\delta}} \sum_{i=0}^{k-1} \xi(w_0,\eta_i)
\end{aligned}$$ where marginally, the variables $\lrp{\eta_0...\eta_{n-1}}$ are drawn $i.i.d$ from the same distribution as in .
Notice that $y_T - w_0 - T \nabla U(w_0) = \int_0^T \cm dB_t + \int_0^T N(w_0) dW_t$, so that $\Law(y_T - w_0 - T \nabla U(w_0)) = \N(0, T M(w_0)^2)$. Notice also that $v_T - w_0 - T\nabla U(w_0) = \sqrt{\delta} \sum_{i=0}^{n-1} \xi(w_0, \eta_i)$. By Corollary \[c:clt\_sum\], $W_2(y_T - w_0 - T \nabla U(w_0), v_T - w_0 - T \nabla U(w_0)) = 6 \sqrt{d\delta}\beta \sqrt{\log n}$. Let the joint distribution between and be the one induced by the optimal coupling between $y_T - w_0 - T \nabla U(w_0)$ and $v_T - w_0 - T \nabla U(w_0)$, so that $$\begin{aligned}
& \sqrt{\E{\lrn{y_T - v_T}_2^2}} \\
=& \sqrt{\E{\lrn{y_T - T \nabla U(w_0) - v_T + T \nabla U(w_0)}_2^2}} \\
=& W_2(y_T - w_0 - T\nabla U(w_0), v_T - w_0 - T\nabla U(w_0)) \\
\leq& 6 \sqrt{d\delta}\beta \sqrt{\log n}
\numberthis \label{e:yt-vt}
\end{aligned}$$ where the last inequality is by Corollary \[c:clt\_sum\].
5. Given the sequence $\lrp{\eta_0...\eta_{n-1}}$ from , we can define $$\begin{aligned}
\numberthis \label{e:coupled_4_processes_w}
w_{k\delta} =& w_0 + \sum_{i=0}^{k-1} -\delta\nabla U(w_{i\delta}) + {\sqrt{\delta}} \sum_{i=0}^{k-1} \xi(w_{i\delta},\eta_i)
\end{aligned}$$ specifically, $(w_0 ... w_{n\delta})$ in and $(v_0...v_{n\delta})$ in are coupled through the shared $(\eta_0...\eta_{n-1})$ variables.
For convenience, we will let $v_t := v_{i \delta}$ and $w_t := w_{i\delta}$, where $i$ is the unique integer satisfying $t\in[i\delta, (i+1)\delta)$.
We can verify that, marginally, the process $x_t$ in has the same distribution as , using the proof as Lemma \[l:marginal\_of\_coupling\]. It is also straightforward to verify that $w_{k\delta}$, as defined in , has the same marginal distribution as , due to the definition of $\eta_i$ in .
[One Epoch Contraction]{} \[ss:epoch\_nongaussian\] In Lemma \[l:non\_gaussian\_contraction\_stationary\], we prove a discretization error bound between $f(x_T - y_T)$ and $f(x_T - v_T)$, for the coupling defined in , and .
In Lemma \[l:non\_gaussian\_contraction\_anisotropic\], we prove a discretization error bound between $f(x_T - v_T)$ and $f(x_T - w_T)$, for the coupling defined in , and .
\[l:non\_gaussian\_contraction\_stationary\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$. Let $x_t$, $y_t$ and $v_t$ be as defined in , , . Let $n$ be any integer and $\delta$ be any step size, and let $T:= n\delta$.
If $\E{\lrn{x_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$, $\E{\lrn{y_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T\leq \min\lrbb{\frac{1}{16L}, \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}}$ and $$\begin{aligned}
\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \frac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}
\end{aligned}$$ Then $$\begin{aligned}
& \E{f(x_T - v_T)} - \E{f(x_T - y_T)} \leq 4TL\epsilon
\end{aligned}$$
By Taylor’s Theorem, $$\begin{aligned}
& \E{f(x_T - v_T)}\\
=& \E{f(x_T - y_T)+ \lin{\nabla f(x_T - y_T), y_T - v_T} + \int_0^1\int_0^s \lin{\nabla^2 f(x_T - y_T + s(y_T-v_T)), (y_T - v_T) (y_T - v_T)^T} ds dt}\\
=& \E{f(x_T - y_T)+ \underbrace{\lin{\nabla f(x_0 - y_0), y_T - v_T}}_{\circled{1}} + \underbrace{\lin{\nabla f(x_T - y_T) - \nabla f(x_0 - y_0), y_T - v_T}}_{\circled{2}} }\\
&\quad + \E{\underbrace{\int_0^1\int_0^s \lin{\nabla^2 f(x_T - y_T + s(y_T-v_T)), (y_T - v_T) (y_T - v_T)^T} ds dt}_{\circled{3}}}
\end{aligned}$$
We will bound each of the terms above separately. $$\begin{aligned}
& \E{\circled{1}}\\
=& \E{\lin{\nabla f(x_0 - y_0), y_T - v_T}}\\
=& \E{\lin{\nabla f(x_0 - y_0), n \delta \nabla U(y_0) - n \delta \nabla U(v_0) + \int_0^T -\nabla U(w_0) dt + \int_0^T \cm dV_t + \int_0^T N(w_0) dW_t + \sum_{i=0}^{n-1} \sqrt{\delta} \xi(v_0,\eta_i)}}\\
=& \E{\lin{\nabla f(x_0 - y_0), n \delta \nabla U(y_0) - n \delta \nabla U(v_0) }}\\
=& 0
\end{aligned}$$ where the third equality is because $\int_0^T dB_t$, $\int_0^T dW_t$ and $\sum_{k=1}^T \xi(v_0,\eta_i)$ have zero mean conditioned on the information at time $0$, and the fourth equality is because $y_0 = v_0$ by definition in and . $$\begin{aligned}
& \E{\circled{2}}\\
=& \E{\lin{\nabla f(x_T - y_T) - \nabla f(x_0 - y_0), y_T - v_T}}\\
\leq& \sqrt{\E{\lrn{\nabla f(x_T - y_T) - \nabla f(x_0 - y_0)}_2^2} }\sqrt{\E{\lrn{y_T - v_T}_2^2}}\\
\leq& \frac{2}{\epsilon}\sqrt{2\E{\lrn{x_T - x_0}_2^2 + \lrn{y_T - y_0}_2^2}} \sqrt{\E{\lrn{y_T - v_T}_2^2}}\\
\leq& \frac{2}{\epsilon}\sqrt{\lrp{32T\beta^2 + 4T\beta^2}} \cdot \lrp{6 \sqrt{d\delta}\beta {\log n}}\\
\leq& \frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta} {\log n}}
\end{aligned}$$
Where the second inequality is by $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\] and Young’s inequality. The third inequality is by Lemma \[l:divergence\_xt\] and Lemma \[l:divergence\_yt\] and .
Finally, we can bound $$\begin{aligned}
& \E{\circled{3}}\\
\leq& \int_0^1\int_0^s \E{\lrn{\nabla^2 f(x_T - y_T + s(y_T-v_T))}_2 \lrn{y_T - v_T}_2^2} ds dt\\
\leq& \frac{2}{\epsilon} \E{\lrn{y_T - v_T}_2^2}\\
\leq& \frac{72 d \delta \beta^2 \log^2 n}{\epsilon}
\end{aligned}$$ Where the second inequality is by $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\], the third inequality is by .
Summing these 3 terms, $$\begin{aligned}
&\E{f(x_T - v_T) - f(x_T - y_T)} \\
\leq& \frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta} \sqrt{\log n}} + \frac{36 d \delta \beta^2 \log n}{\epsilon}\\
=& \frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta}\sqrt{\log \frac{T}{\delta}}} + \frac{36 d \delta \beta^2 \log \frac{T}{\delta}}{\epsilon }
\end{aligned}$$
Let us bound the first term. We apply Lemma \[l:xlogxbound\] (with $x = \frac{T}{\delta}$ and $c = \frac{\epsilon^4}{2^{14} d\beta^4}$), which shows that $$\begin{aligned}
\frac{T}{\delta} \geq \frac{2^{14} d\beta^4}{\epsilon^4} \log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}
\quad \Rightarrow \quad
\frac{T}{\delta} \frac{1}{\log \frac{T}{\delta}} \geq \frac{2^{14} d\beta^4}{\epsilon^4L^2}
\quad \Leftrightarrow \quad
\frac{128}{\epsilon} \sqrt{T}\beta^2 \cdot \lrp{ \sqrt{d\delta}{\log \frac{T}{\delta}}} \leq TL\epsilon
\end{aligned}$$
For the second term, we can again apply Lemma \[l:xlogxbound\] ($x = \frac{T}{\delta}$ and $c = \frac{\epsilon^2L}{36 d\beta^2}$), which shows that $$\begin{aligned}
\frac{T}{\delta} \geq \frac{36 d\beta^2}{\epsilon^2L} \log \lrp{ \frac{36 d\beta^2}{\epsilon^2L} }
\quad \Rightarrow \quad
\frac{T}{\delta} \frac{1}{\log \frac{T}{\delta}} \geq \frac{36 d\beta^2}{\epsilon^2L}
\quad \Rightarrow \quad
\frac{36 d \delta \beta^2 \log \frac{T}{\delta}}{\epsilon } \leq TL\epsilon
\end{aligned}$$ The above imply that $$\begin{aligned}
\E{f(x_T - v_T) - f(x_T - y_T)} \leq 2TL\epsilon
\end{aligned}$$
\[l:non\_gaussian\_contraction\_anisotropic\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon\leq \frac{\Rq}{\aq\Rq^2 + 1}$. Let $x_t$, $v_t$ and $w_t$ be as defined in , , . Let $n$ be an integer and $\delta$ be a step size, and let $T:= n\delta$.
If we assume that $\E{\lrn{x_0}_2^2}$, $\E{\lrn{v_0}_2^2}$, and $\E{\lrn{w_0}_2^2}$ are each upper bounded by $8 \lrp{R^2 + \beta^2/m}$ and that $T \leq \min\lrbb{ \frac{1}{16L}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$, then $$\begin{aligned}
& \E{f(x_T - w_T)} - \E{f(x_T - v_T)} \leq 4T(L+\LN^2)\epsilon
\end{aligned}$$
For sufficiently small $\epsilon$, our assumption on $T$ boils down to $T = o(\epsilon^4)$
First, we can verify using Taylor’s theorem that for any $x,y$, $$\begin{aligned}
f(y) =& f(x) + \lin{\nabla f(x), y-x} + \int_0^1\int_0^s \lin{\nabla^2 f(x + s(y-x)), (y-x) (y-x)^T} ds dt
\numberthis \label{e:taylor1}\\
\nabla f(y) =& \nabla f(x) + \lin{\nabla^2 f(x), y-x} + \int_0^1\int_0^s \lin{\nabla^3 f(x + s(y-x)), (y-x) (y-x)^T} ds dt
\numberthis \label{e:taylor2}
\end{aligned}$$
Thus $$\begin{aligned}
& \E{f(x_T - w_T)}\\
=& \E{f(x_T - v_T)+ \lin{\nabla f(x_T - v_T), v_T - w_T} + \int_0^1\int_0^s \lin{\nabla^2 f(x_T - v_T + s(v_T-w_T)), (v_T - w_T) (v_T - w_T)^T} ds dt}\\
=& \E{f(x_T - v_T)+ \underbrace{\lin{\nabla f(x_0 - v_0), v_T - w_T}}_{\circled{1}} + \underbrace{\lin{\nabla f(x_T - v_T) - \nabla f(x_0 - v_0), v_T - w_T}}_{\circled{2}}}\\
&\quad + \E{\underbrace{\int_0^1\int_0^s \lin{\nabla^2 f(x_T - v_T + s(v_T-w_T)), (v_T - w_T) (v_T - w_T)^T} ds dt}_{\circled{3}}}
\end{aligned}$$ Recall from and that $$\begin{aligned}
v_{n\delta} =& w_0 + \sum_{i=0}^{n-1} \delta \nabla U(w_0) + {\sqrt{\delta}} \sum_{i=0}^{n-1} \xi(w_0,\eta_i)\\
w_{n\delta} =& w_0 + \sum_{i=0}^{n-1} \delta \nabla U(w_{i\delta}) + {\sqrt{\delta}} \sum_{i=0}^{n-1} \xi(w_{i\delta},\eta_i)
\end{aligned}$$ Note that conditioned on the randomness up to time $0$, $\E{\sum_{i=0}^{n-1} \xi(w_0,\eta_i)} = \E{\sum_{i=0}^{n-1} \xi(w_{i\delta},\eta_i)} = 0$, so that $$\begin{aligned}
& \E{\circled{1}}\\
=& \E{\lin{\nabla f(x_0 - v_0), v_T - w_T}}\\
=& \delta \E{\lin{\nabla f(x_0 - v_0), \sum_{i=0}^{n-1} \nabla U(w_{0}) - \nabla U(w_{i\delta})}} + \sqrt{\delta} \E{\lin{\nabla f(x_0 - v_0), \sum_{i=0}^{n-1} \xi(w_{0},\eta_i) - \sum_{i=0}^{n-1} \xi(w_{i\delta},\eta_i)}}\\
=& \delta \E{\lin{\nabla f(x_0 - v_0), \sum_{i=0}^{n-1} \nabla U(w_{0}) - \nabla U(w_{i\delta})}}\\
\leq& \delta \sum_{i=0}^{n-1} L \E{\lrn{w_0 - w_{i\delta}}_2}\\
\leq& T L \sqrt{32 T\beta^2} \leq 8 T^{3/2} L \beta
\end{aligned}$$
where the third equality is becayse $\xi(\cdot, \eta_i)$ has $0$ mean conditioned on the randomness at time $0$, and the second inequality is by Lemma \[l:divergence\_wt\].
Next, $$\begin{aligned}
& \E{\circled{2}}\\
=& \E{\lin{\nabla f(x_T - v_T) - \nabla f(x_0 - v_0), v_T - w_T}}\\
\leq& \E{\lrn{\nabla f(x_T - v_T) - \nabla f(x_0 - v_0)}_2 \lrn{v_T - w_T}}\\
\leq& \frac{4}{\epsilon} \sqrt{\E{\lrn{x_T - x_0}_2^2 + \lrn{v_T - v_0}_2^2}} \cdot \sqrt{\E{\lrn{v_T - w_T}_2^2}}\\
\leq& \frac{4}{\epsilon} \sqrt{16T\beta^2 + 2T\beta^2} \cdot \sqrt{32 \lrp{T^2 L^2 + TL_\xi^2} T\beta^2}\\
\leq& \frac{128}{\epsilon} T \beta^2 \lrp{\sqrt{T} L_\xi + TL}
\end{aligned}$$ where the second inequality is because $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\] and by Young’s inequality. The third inequality is by Lemma \[l:divergence\_xt\], Lemma \[l:divergence\_vt\] and Lemma \[l:vt-wt\].
Finally, $$\begin{aligned}
& \E{\circled{3}}\\
=& \E{\int_0^1\int_0^s \lin{\nabla^2 f(x_T - v_T + s(v_T-w_T)), (v_T - w_T) (v_T - w_T)^T} ds dt}\\
\leq& \int_0^1 \int_0^s \E{\lrn{\nabla^2 f(x_T - v_T + s(v_T-w_T))}_2\lrn{v_T - w_T}_2^2} ds \\
\leq& \frac{1}{\epsilon} \E{\lrn{v_T - w_T}_2^2}\\
\leq& \frac{32}{\epsilon} \lrp{T^2 L^2 + TL_\xi^2} T\beta^2
\end{aligned}$$ wehere the second inequality is because $\lrn{\nabla^2 f}_2 \leq \frac{2}{\epsilon}$ from item 2(c) of Lemma \[l:fproperties\] and by Young’s inequality. The third inequality is by Lemma \[l:vt-wt\].
Summing the above, $$\begin{aligned}
&\E{f(x_T - w_T) - f(x_T - v_T)} \\
\leq& 8T^{3/2} L\beta + \frac{128}{\epsilon} T \beta^2 \lrp{\sqrt{T} L_\xi + TL} + \frac{32}{\epsilon} \lrp{T^2 L^2 + TL_\xi^2} T\beta^2\\
\leq& T^{3/2} \epsilon
\end{aligned}$$ where the last inequality is by our assumption on $T$, specifically, $$\begin{aligned}
&T \leq \frac{\epsilon^2}{128\beta^2}
\Rightarrow T^{3/2} L\beta \leq TL\epsilon\\
&T \leq \frac{\epsilon^2}{128\beta^2} \Rightarrow \frac{128}{\epsilon} T^2L \beta^2 \leq TL \epsilon\\
&T \leq \frac{\epsilon}{32\sqrt{L} \beta} \Rightarrow \frac{32}{\epsilon}(T^3 L^2 \beta^2) \leq TL\epsilon\\
& T \leq \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2} \Rightarrow \frac{128}{\epsilon} T^{3/2}\beta^2 L_\xi \leq T\LN^2 \epsilon\\
& T \leq \frac{\epsilon^2}{128\beta^2} \Rightarrow T \leq \frac{\epsilon^2}{128\cm^2} \Rightarrow \frac{32}{\epsilon} T^2L_\xi^2\beta^2 \leq T\LN^2\epsilon
\end{aligned}$$ where the last line uses the fact that $\beta \geq \cm^2$.
\[c:main\_nongaussian:1\] Let $f$ be as defined in Lemma \[l:fproperties\] with parameter $\epsilon$ satisfying $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$.\
Let $T= \min\lrbb{\frac{1}{16L}, \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$ and let $\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \frac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}$, assume additionally that $n=T/\delta$ is an integer.\
Let $\bx_t$ and $\bw_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\bw_0}_2^2}\leq R^2 + \beta^2/m$. Then there exists a coupling between $\bx_t$ and $\bw_t$ such that $$\begin{aligned}
\E{f(\bx_{i\delta} - \bw_{i\delta})} \leq e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
From Lemma \[l:energy\_x\] and \[l:energy\_w\], our initial conditions imply that for all $t$, $\E{\|\bx_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}$ and $\E{\|\bw_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}$.
Consider an arbitrary $k$, and for $t\in[0,T)$, define $$\begin{aligned}
x_t := \bx_{kT+t} \quad \text{and} \quad
w_t := \bw_{kT+t} \numberthis \label{e:t:kasjnd}
\end{aligned}$$ Notice that as described above, $x_t$ and $w_t$ have dynamics described in and . Let $x_t,w_t$ have joint distribution as described in and , and let $(y_t,v_t)$ be the processes defined in and . Notice that the joint distribution between $x_t$ and $w_t$ equivalently describes a coupling between $\bx_t$ and $\bw_t$ over $t\in[kT, (k+1)T)$.
First, notice that the processes $\eqref{e:coupled_4_processes_x}$ and $\eqref{e:coupled_4_processes_y}$ have the same distribution as . We can thus apply Lemma \[l:gaussian\_contraction\]: $$\begin{aligned}
\E{f(x_{T} - y_{T})}
\leq& e^{-\lambda T} \E{f(x_0 - y_0)} + 6T (L+\LN^2) \epsilon
\end{aligned}$$
By Lemma \[l:non\_gaussian\_contraction\_stationary\], $$\begin{aligned}
\E{f(x_T - v_T)} - \E{f(x_T - y_T)}
\leq 4TL\epsilon
\end{aligned}$$ By Lemma \[l:non\_gaussian\_contraction\_anisotropic\], $$\begin{aligned}
\E{f(x_T - w_T)} - \E{f(x_T - v_T)}
\leq 4T(L+\LN^2)\epsilon
\end{aligned}$$
Summing the above three equations, $$\begin{aligned}
\E{f(x_T - w_T)}
\leq e^{-\lambda \delta} \E{f(x_0 - w_0)} + 14T (L+\LN^2)
\end{aligned}$$ Where we use the fact that $y_0 = w_0$ by construction in .
Recalling , this is equivalent to $$\begin{aligned}
\E{f(\bx_{(k+1)T} - \bw_{(k+1)T})}
\leq e^{-\lambda \delta} \E{f(\bx_{kT} - \bw_{kT})} + 14T (L+\LN^2)
\end{aligned}$$
Applying the above recursively gives, for any $i$ $$\begin{aligned}
\E{f(\bx_{iT} - \bw_{iT})} \leq e^{-\lambda iT} \E{f(\bx_{0} - \bw_{0})} + \frac{14}{\lambda} \lrp{L + \LN^2} \epsilon
\end{aligned}$$
[Proof of Theorem \[t:main\_nongaussian\]]{} \[ss:proof:t:main\_nongaussian\] For ease of reference, we re-state Theorem \[t:main\_nongaussian\] below as Theorem \[t:main\_nongaussian:restated\] below. We make a minor notational change: using the letters $\bx_t$ and $\by_t$ in Theorem \[t:main\_nongaussian:restated\], instead of the letters $x_t$ and $y_t$ in Theorem \[t:main\_nongaussian\]. This is to avoid some notation conflicts in the proof.
\[Equivalent to Theorem \[t:main\_nongaussian\]\] \[t:main\_nongaussian:restated\] Let $\bx_t$ and $w_t$ have dynamics as defined in and respectively, and suppose that the initial conditions satisfy $\E{\lrn{\bx_0}_2^2}\leq R^2 + \beta^2/m$ and $\E{\lrn{\bw_0}_2^2}\leq R^2 + \beta^2/m$. Let $\hat{\epsilon}$ be a target accuracy satisfying $\hat{\epsilon} \leq \lrp{\frac{16\lrp{L + \LN^2}}{\lambda}} \cdot \exp\lrp{7\aq\Rq/3} \cdot \frac{\Rq}{\aq\Rq^2 + 1}$. Let $\epsilon:= \frac{\lambda}{16 (L+\LN^2)} \exp\lrp{-\frac{7\aq\Rq^2}{3}} \hat{\epsilon}$. Let $T:= \min\lrbb{\frac{1}{16L}, \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}, \frac{\epsilon}{32\sqrt{L} \beta}, \frac{\epsilon^2}{128\beta^2}, \frac{\epsilon^4 \LN^2}{2^{14}\beta^2 \cm^2}}$ and let $\delta$ be a step size satisfying $$\begin{aligned}
\delta \leq \min\lrbb{\frac{T\epsilon^2L}{36 d\beta^2\log \lrp{ \frac{36 d\beta^2}{\epsilon^2L}}}, \frac{T\epsilon^4L^2} {2^{14} d\beta^4\log\lrp{\frac{2^{14} d\beta^4}{\epsilon^4L^2}}}}.
\end{aligned}$$
If we assume that $\bx_0 = \bw_0$, then there exists a coupling between $\bx_t$ and $\bw_t$ such that for any $k$, $$\begin{aligned}
\E{\lrn{\bx_{k\delta} - \bw_{k\delta}}_2} \leq \hat{\epsilon}.
\end{aligned}$$
Alternatively, if we assume that $k \geq \frac{3\aq\Rq^2}{ \delta} \cdot \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, then $$\begin{aligned}
W_1\lrp{p^*, p^w_{k\delta}} \leq 2\hat{\epsilon},
\end{aligned}$$ where $p^w_t := \Law(\bw_t)$.
Let $f$ be defined as in Lemma \[l:fproperties\] with parameter $\epsilon$. $$\begin{aligned}
& \E{\lrn{\bx_{i\delta} - \bw_{i\delta}}_2}\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\E{f(\bx_{i\delta} - \bw_{i\delta})} + 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon} + 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \frac{16\lrp{L+\LN^2}}{\lambda}\exp\lrp{\frac{7\aq\Rq^2}{3}} \cdot \epsilon
\numberthis \label{e:t:asdkjas:1}\\
=& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}e^{-\lambda i\delta} \E{f(\bx_{0} - \bw_{0})} + \hat{\epsilon}
\end{aligned}$$ where the first inequality is by item 4 of Lemma \[l:fproperties\], the second inequality is by Corollary \[c:main\_nongaussian:1\] (notice that $\delta$ satisfies the requirement on $T$ in Theorem \[t:main\_gaussian\], for the given $\epsilon$). The third inequality uses the fact that $1\leq L/m \leq \frac{\lrp{L+\LN^2}}{\lambda}$.
The first claim follows from substituting $\bx_0 = \bw_0$ into , so that the first term is $0$, and using the definition of $\epsilon$, so that the second term is $0$.
For the second claim, let $\bx_0 \sim p^*$, the invariant distribution of . From Lemma \[l:energy\_x\], we know that $\bx_0$ satisfies the required initial conditions in this Lemma. Continuing from , $$\begin{aligned}
& \E{\lrn{\bx_{i\delta} - \bw_{i\delta}}_2}\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{2e^{-\lambda i\delta} \E{\lrn{\bx_0}_2^2 + \lrn{\bw_0}_2^2} + \frac{6}{\lambda} \lrp{L + \LN^2} \epsilon} + \epsilon\\
\leq& 2\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{2e^{-\lambda i\delta} \lrp{R^2 + \beta^2/m}} + \frac{16}{\lambda} \exp\lrp{2\frac{7\aq\Rq^2}{3}}\lrp{L + \LN^2} \epsilon\\
=& 4\exp\lrp{\frac{7\aq\Rq^2}{3}}\lrp{e^{-\lambda i\delta} \lrp{R^2 + \beta^2/m}} + \hat{\epsilon}
\end{aligned}$$ By our assumption that $i\geq \frac{1}{ \delta} \cdot 3\aq\Rq^2 \log \frac{R^2 + \beta^2/m}{\hat{\epsilon}}$, the first term is also bounded by $\hat{\epsilon}$, and this proves our second claim.
[Coupling Properties]{} \[s:coupling\_properties\]
\[l:marginal\_of\_coupling\] Consider the coupled $(x_t,y_t)$ in . Let $p_t$ denote the distribution of $x_t$, and $q_t$ denote the distribution of $y_t$. Let $p_t'$ and $q_t'$ denote the distributions of and .
If $p_0 = p_0'$ and $q_0 = q_0'$, then $p_t = p_t'$ and $q_t=q_t'$ for all $t$.
Consider the coupling in , reproduced below for ease of reference: $$\begin{aligned}
x_t =& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(x_s) dW_s\\
y_t =& y_0 + \int_0^t -\nabla U(y_0) dt + \int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^T} dV_s + \int_0^t N(y_0) dW_s
\end{aligned}$$
Let us define the stochastic process $A_t := \int_0^t M(x_s)^{-1} \cm dV_s + \int_0^t M(x_s)^{-1} N(x_s) dW_s$. We can verify using Levy’s characterization that $A_t$ is a standard Brownian motion: first, since $V_t$ and $W_t$ are Brownian motions, and $N(x)$ is differentiable with bounded derivatives, we know that $A_t$ has continuous sample paths. We now verify that $A_t^i A_t^j - \ind{i=j} t$ is a martingale.
Notice that $d A_t = \cm dV_t + M(x_s)^{-1} N(x_s) dW_s$. Then $$\begin{aligned}
d A_t^i A_t^j
=& d A_t^T \lrp{e_i e_j^T} A_t\\
=& A_t \lrp{e_i e_j^T} \lrp{\cm dV_t + M(x_s)^{-1} N(x_s) dW_s}^T + \lrp{\cm dV_t + M(x_s)^{-1} N(x_s) dW_s} \lrp{e_j e_i^T} a_t^T\\
&\qquad + \frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{c_m^2 M(x_s)^{-2} + M(x_s)^{-1} N(x_s)^2 M(x_s)^{-1}}} dt
\end{aligned}$$ where the second inequality is by Ito’s Lemma applied to $f(A_t) = A_t^T e_j e_j^T A_t$. Taking expectations, $$\begin{aligned}
d \E{A^i_t A^j_t}
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{c_m^2 M(x_s)^{-2} + M(x_s)^{-1} N(x_s) N(x_s)^T \lrp{M(x_s)^{-1}}^T}}}dt\\
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{M(x_s)^{-1} \lrp{c_m^2 I + N(x_s)^2} M(x_s)^{-1}}}}dt\\
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}\lrp{M(x_s)^{-1} \lrp{M(x_s)^2} M(x_s)^{-1}}}}dt\\
=& \E{\frac{1}{2} \tr\lrp{ \lrp{e_i e_j^T + e_j e_i^T}}}dt\\
=& \ind{i=j} dt
\end{aligned}$$
This verifies that $A_t^i A_t^j - \ind{i=j} t$ is a martingale, and hence by Levy’s characterization, $A_t$ is a standard Brownian motion. In turn, we verify that by definition of $A_t$, $$\begin{aligned}
x_t
=& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t \cm dV_s + \int_0^t N(x_s) dW_s \\
=& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t M(x_s) \lrp{M(x_s)^{-1}\lrp{\cm dV_s + N(x_s) dW_s}}\\
=& x_0 + \int_0^t -\nabla U(x_s) ds + \int_0^t M(x_s) dA_s
\end{aligned}$$ Since we showed that $A_t$ is a standard Brownian motion, we verify that $x_t$ as defined in has the same distribution as .
On the other hand, we can verify that $A'_t := \int_0^T (I - 2\gamma_s \gamma_s^T) V_s$ is a standard Brownian motion by the reflection principle. Thus $$\begin{aligned}
\int_0^t \cm \lrp{I - 2\gamma_s \gamma_s^T} dV_s + \int_0^t N(y_0) dW_s \sim \N(0, \lrp{c_m^2 I + N(y_0)^2}) = \N(0, M(y_0)^2)
\end{aligned}$$ where the equality is by definition of $N$ in .
It follows immediately that $y_t$ in has the same distribution as $y_t$ in .
[Energy Bounds]{} \[ss:energy\_bounds\]
\[l:energy\_x\] Consider $x_t$ as defined in . If $x_0$ satisfies $\E{\|x_0\|_2^2} \leq R^2 + \frac{\beta^2}{m}$, then Then for all $t$, $$\begin{aligned}
\E{\|x_t\|_2^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}\\
\end{aligned}$$ We can also show that $$\begin{aligned}
\Ep{p^*}{\lrn{x}_2^2} \leq 4\lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
We consider the potential function $a(x) = \lrp{\|x\|_2 - R}_+^2$ We verify that $$\begin{aligned}
\nabla a(x) =& (\|x\|_2 - R)_+ \frac{x}{\|x\|_2}\\
\nabla^2 a(x) =& \ind{\|x\|_2 \geq R} \frac{xx^T}{\|x\|_2^2} + \frac{(\|x\|_2 - R)_+}{\|x\|_2}\lrp{I - \frac{xx^T}{\|x\|_2^2}}
\end{aligned}$$ Observe that
1. $\lrn{\nabla^2 a(x)}_2 \leq 2 \ind{\|x\|_2 \geq R} \leq 2$
2. $\lin{\nabla a(x), - \nabla U(x)} \leq -m a(x)$. This can be verified by considering 2 cases. If $\|x\|_2 \leq R$, then $\nabla a(x) = 0$ and $a(x) = 0$. If $\|x\|_2 \geq R$, then by Assumption \[ass:U\_properties\], $$\begin{aligned}
\lin{\nabla a(x), -\nabla U(x)}
\leq - m \lrp{\|x\|_2 - R}_+ \|w\|_2
\leq - m \lrp{\|x\|_2 - R}_+^2 = -m \cdot a(x)
\end{aligned}$$
3. $a(x) \geq \frac{1}{2}\|x\|_2^2 - 2R^2$. One can first verify that $a(x) \geq (\lrn{x}_2 - R)^2 - R^2$. Next, by Young’s inequality, $(\lrn{x}_2 - R)^2 = \lrn{x}_2^2 + R^2 - 2\lrn{x}_2 R \geq \lrn{x}_2^2 + R^2 - \frac{1}{2} \lrn{x}_2^2 - 2R^2 = \frac{1}{2} \lrn{x}_2^2 - R^2$.
Therefore, $$\begin{aligned}
& \ddt \E{a(x_t)}
= \E{\lin{\nabla a(x_t), -\nabla U(x_t) dt}} + \frac{1}{2}\E{\tr\lrp{M(x_t)^2 \nabla^2 a(x)}}
\leq -m\E{a(x_t)} + \beta^2\\
\Rightarrow \qquad &
\ddt \lrp{\E{a(x_t)} - \frac{\beta^2}{m}} \leq - m \lrp{\E{a(x_t)} - \frac{\beta^2}{m}}\\
\Rightarrow \qquad &
\ddt \lrp{\E{a(x_t)} - R^2 - \frac{\beta^2}{m}} \leq - m \lrp{\E{a(x_t)} - R^2 - \frac{\beta^2}{m}}
\end{aligned}$$
Thus if $\E{\|x_0\|_2^2} \leq R^2 + \frac{\beta^2}{m}$, then $\E{a(x_0)}\leq R^2 - \frac{\beta^2}{m}$, then $ \lrp{\E{a(x_0)} - R^2 - \frac{\beta^2}{m}} \leq 0$, and $\lrp{\E{a(x_t)} - R^2 + \frac{\beta^2}{m}} \leq e^{-mt} \cdot 0 \leq 0$ for all $t$. This implies that, for all $t$, $$\begin{aligned}
\E{\|x_t\|_2^2} \leq \E{2 a(x_t) + 4R^2} \leq 6\lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
For our second claim that $\Ep{p^*}{\lrn{x}_2^2} \leq R^2 + \frac{\beta^2}{m}$, we can use the fact that if $x_0 \sim p^*$, then $\E{a(x_t)}$ does not change as $p^*$ is invariant, so that $$\begin{aligned}
0 = \ddt \E{a(x_t)} \leq -m\E{a(x_t)} + \beta^2
\end{aligned}$$ Thus $$\begin{aligned}
\E{a(x_t)} \leq \frac{\beta^2}{m}
\end{aligned}$$ Again, $$\begin{aligned}
\Ep{p^*}{\lrn{x}_2^2} = \E{\lrn{x_t}_2^2} \leq 2 \E{a(x_t)} + 4 R^2 \leq 4 \lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
\[l:energy\_y\] Let the sequence $y_{k\delta}$ be as defined in . Assuming that $\delta \leq m/(16L^2)$ and $\E{\|y_0\|_2^2} \leq 2 \lrp{R^2 + \frac{\beta^2}{m}}$ Then for all $k$, $$\begin{aligned}
\E{\|y_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
Let $a(w) := \lrp{\|w\|_2 - R}_+^2$. We can verify that $$\begin{aligned}
\nabla a(w) =& \lrp{\|w\|_2 - R}_+\frac{w}{\|w\|_2}\\
\nabla^2 a(w) =& \ind{\|w\|_2 \geq R}\frac{ww^T}{\|w\|_2^2} + \lrp{\|w\|_2 - R}_+ \frac{1}{\|w\|_2} \lrp{I - \frac{ww^T}{\|w\|_2^2}}
\end{aligned}$$ Observe that
1. $\lrn{\nabla^2 a(w)}_2 \leq 2 \ind{\|w\|_2 \geq R} \leq 2$
2. $\lin{\nabla a(w), - \nabla U(w)} \leq -m a(w)$.
3. $a(w) \geq \frac{1}{2}\|w\|_2^2 - 2R^2$.
The proofs are identical to the proof at the start of Lemma \[l:energy\_w\], so we omit them here.
Using Taylor’s Theorem, and taking expectation of $y_{(k+1)\delta}$ conditioned on $y_{k\delta}$, $$\begin{aligned}
&\E{a(y_{(k+1)\delta})}\\
=& \E{a(y_{k\delta})} + \E{\lin{\nabla a(y_{k\delta}), y_{(k+1)\delta} - y_{k\delta}}}\\
&\quad + \E{\int_0^1 \int_0^t \lin{\nabla^2 a(y_{k\delta} + s(y_{(k+1)\delta} - y_{k\delta}), (y_{(k+1)\delta} - y_{k\delta})(y_{(k+1)\delta} - y_{k\delta})^T} dt ds}\\
\leq& \E{a(y_{k\delta})} + \E{\lin{\nabla a(y_{k\delta}), y_{(k+1)\delta} - y_{k\delta}}} + \E{\lrn{(y_{(k+1)\delta} - y_{k\delta})}_2^2 ds}\\
\leq& \E{a(y_{k\delta})} + \E{\lin{\nabla a(y_{k\delta}), - \delta \nabla U(y_{k\delta})}} + 2\delta^2 \lrn{\nabla U(y_{k\delta})}_2^2 + 2\delta \E{\tr\lrp{M(y_{k\delta})^2}}\\
\leq& \E{a(y_{k\delta})} -m\delta \E{a(y_{k\delta})} + 2\delta^2 \E{\lrn{\nabla U(y_{k\delta})}_2^2} + 2\delta \E{\tr\lrp{M(y_{k\delta})^2}}\\
\leq& \E{a(y_{k\delta})} -m\delta \E{a(y_{k\delta})} + 2\delta^2L^2 \E{\lrn{y_{k\delta}}_2^2} + 2\delta \beta^2\\
\leq& \E{a(y_{k\delta})} -m\delta \E{a(y_{k\delta})} + 4\delta^2L^2 \E{a(y_{k\delta})} + 8\delta^2 L^2 R^2 + 2\delta \beta^2\\
\leq& (1-m\delta/2) \E{a(y_{k\delta})} + {m\delta} R^2 + 2\delta \beta^2
\end{aligned}$$ Where the first inequality uses the upper bound on $\lrn{\nabla^2 a(y)}_2$ above, the second inequality uses the fact that $y_{(k+1)\delta} \sim \N\lrp{y_{k\delta} - \delta \nabla U(y_{k\delta}), \delta M(y_{k\delta})^2}$, the third inequality uses claim 2. at the start of this proof, the fourth inequality uses item 2 of Assumption \[ass:xi\_properties\]. The fifth inequality uses claim 3. above, the sixth inequality uses our assumption that $\delta \leq \frac{m}{16L^2}$.
Taking expectation wrt $y_{k\delta}$, $$\begin{aligned}
& \E{a(y_{(k+1)\delta})} \leq \E{a(y_{k})}-m\delta \lrp{\E{a(y_{k\delta})} - 2R^2 + 2\beta^2/m}\\
\Rightarrow \qquad &
\E{a(y_{(k+1)\delta})} - (2R^2/2 + 2\beta^2/m) \leq (1-m\delta) \lrp{\E{a(y_{k\delta})} - (2R^2 + 2\beta^2/m}
\end{aligned}$$
Thus, if $\E{\|y_0\|_2^2} \leq 2R^2 + 2\beta^2/m$, then $\E{a(y_0)} - \lrp{2R^2 + 2\beta^2/m} \leq 0 $, then $\E{a(y_{k\delta})} - \lrp{2R^2 + 2\beta^2/m}\leq 0$ for all $k$, which implies that $$\begin{aligned}
\E{\lrn{y_{k\delta}}_2^2} \leq 2\E{a(y_{k\delta})} + 4 R^2 \leq 8\lrp{R^2 + \beta^2/m}
\end{aligned}$$ for all $k$.
\[l:energy\_w\] Let the sequence $w_{k\delta}$ be as defined in . Assuming that $\delta \leq m/(16L^2)$ and $\E{\|w_0\|_2^2} \leq 2 \lrp{R^2 + \frac{\beta^2}{m}}$ Then for all $k$, $$\begin{aligned}
\E{\|w_{k\delta}\|_2^2} \leq 8 \lrp{R^2 + \frac{\beta^2}{m}}
\end{aligned}$$
The proof is almost identical to that of Lemma \[l:energy\_y\]. Let $a(w) := \lrp{\|w\|_2 - R}_+^2$. We can verify that $$\begin{aligned}
\nabla a(w) =& \lrp{\|w\|_2 - R}_+\frac{w}{\|w\|_2}\\
\nabla^2 a(y) =& \ind{\|w\|_2 \geq R}\frac{ww^T}{\|w\|_2^2} + \lrp{\|w\|_2 - R}_+ \frac{1}{\|w\|_2} \lrp{I - \frac{ww^T}{\|w\|_2^2}}
\end{aligned}$$ Observe that
1. $\lrn{\nabla^2 a(w)}_2 \leq 2 \ind{\|w\|_2 \geq R} \leq 2$
2. $\lin{\nabla a(w), - \nabla U(w)} \leq -m a(w)$.
3. $a(w) \geq \frac{1}{2}\|w\|_2^2 - 2R^2$.
The proofs are identical to the proof at the start of Lemma \[l:energy\_w\], so we omit them here.
Using Taylor’s Theorem, and taking expectation of $w_{(k+1)\delta}$ conditioned on $w_{k\delta}$, $$\begin{aligned}
&\E{a(w_{(k+1)\delta})}\\
=& \E{a(w_{k\delta})} + \E{\lin{\nabla a(w_{k\delta}), w_{(k+1)\delta} - w_{k\delta}}}\\
&\quad + \E{\int_0^1 \int_0^t \lin{\nabla^2 a(w_{k\delta} + s(w_{(k+1)\delta} - w_{k\delta}), (w_{(k+1)\delta} - w_{k\delta})(w_{(k+1)\delta} - w_{k\delta})^T} dt ds}\\
\leq& \E{a(w_{k\delta})} + \E{\lin{\nabla a(w_{k\delta}), w_{(k+1)\delta} - w_{k\delta}}} + \E{\lrn{(w_{(k+1)\delta} - w_{k\delta})}_2^2 ds}\\
\leq& \E{a(w_{k\delta})} + \E{\lin{\nabla a(w_{k\delta}), - \delta \nabla U(w_{k\delta})}} + 2\delta^2 \lrn{\nabla U(w_{k\delta})}_2^2 + 2\delta \E{\lrn{\xi(w_{k\delta},\eta_k)}_2^2}\\
\leq& \E{a(w_{k\delta})} -m\delta \E{a(w_{k\delta})} + 2\delta^2 \E{\lrn{\nabla U(w_{k\delta})}_2^2} + 2\delta \E{\lrn{\xi(w_{k\delta},\eta_k)}_2^2}\\
\leq& \E{a(w_{k\delta})} -m\delta \E{a(w_{k\delta})} + 2\delta^2L^2 \E{\lrn{w_{k\delta}}_2^2} + 2\delta \beta^2\\
\leq& \E{a(w_{k\delta})} -m\delta \E{a(w_{k\delta})} + 2\delta^2L^2 a(w_{k\delta}) + 2\delta^2 L^2 R^2 + 2\delta \beta^2\\
\leq& (1-m\delta/2)a(w_{k\delta}) + {m\delta} R^2 + 2\delta \beta^2
\end{aligned}$$
Where the first inequality uses the upper bound on $\lrn{\nabla^2 a(y)}_2$ above, the second inequality uses the fact that $w_{(k+1)\delta} = \lrp{y_{k\delta} - \delta \nabla U(y_{k\delta}) = \xi(w_{k\delta}, \eta_k)}$, and $\E{ \xi(w_{k\delta}, \eta_k) | w_{k\delta}} = 0$, the third inequality uses claim 2. at the start of this proof, the fourth inequality uses item 2 of Assumption \[ass:xi\_properties\]. The fifth inequality uses claim 3. above, the sixth inequality uses our assumption that $\delta \leq \frac{m}{16L^2}$.
Taking expectation wrt $w_{k\delta}$, $$\begin{aligned}
& \E{a(w_{(k+1)\delta})} \leq \E{a(w_{k})}-m\delta \lrp{\E{a(w_{k\delta})} - 2R^2 + 2\beta^2/m}\\
\Rightarrow \qquad &
\E{a(w_{(k+1)\delta})} - (2R^2/2 + 2\beta^2/m) \leq (1-m\delta) \lrp{\E{a(w_{k\delta})} - (2R^2 + 2\beta^2/m}
\end{aligned}$$
Thus, if $\E{\|w_0\|_2^2} \leq 2R^2 + 2\beta^2/m$, then $\E{a(w_0)} - \lrp{2R^2 + 2\beta^2/m} \leq 0 $, then $\E{a(w_{k\delta})} - \lrp{2R^2 + 2\beta^2/m}\leq 0$ for all $k$, which implies that $$\begin{aligned}
\E{\lrn{w_{k\delta}}_2^2} \leq 2\E{a(w_{k\delta})} + 4 R^2 \leq 8\lrp{R^2 + \beta^2/m}
\end{aligned}$$ for all $k$.
[Divergence Bounds]{}
\[l:divergence\_xt\] Let $x_t$ be as defined in (or equivalently or ), initialized at $x_0$. Then for any $T\leq \frac{1}{16L}$, $$\begin{aligned}
\E{\lrn{x_T - x_0}_2^2} \leq 8 \lrp{T\beta^2 + T^2 L^2 \E{\|x_0\|_2^2}}
\end{aligned}$$ If we additionally assume that $\E{\lrn{x_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{x_T - x_0}_2^2} \leq 16 T\beta^2
\end{aligned}$$
By Ito’s Lemma, $$\begin{aligned}
& \ddt \E{\lrn{x_t}_2^2} \\
=& 2\E{\lin{\nabla U(x_t), x_t - x_0}} + \E{ \tr\lrp{M(x_t)^2}}\\
\leq& 2L \E{\lrn{x_t}_2 \lrn{x_t - x_0}_2} + \beta^2\\
\leq& 2L \E{\lrn{x_t - x_0}_2^2} + 2L\E{\lrn{x_0}_2\lrn{x_t - x_0}_2} + \beta^2\\
\leq& 2L \E{\lrn{x_t - x_0}_2^2} + L^2 T\E{\lrn{x_0}_2^2} + \frac{1}{T} \E{\lrn{x_t - x_0}_2^2} + \beta^2\\
\leq& \frac{2}{T} \E{\lrn{x_t - x_0}_2^2} + \lrp{L^2 T \E{\lrn{x_0}_2^2} + \beta^2}
\end{aligned}$$ where the first inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\], the second inequality is by triangle inequality, the third inequality is by Young’s inequality, the last inequality is by our assumption on $T$.
Applying Gronwall’s inequality for $t\in[0,T]$, $$\begin{aligned}
& \lrp{\E{\lrn{x_t - x_0}_2^2} + L^2 T^2 \E{\lrn{x_0}_2^2} + T \beta^2} \\
\leq& e^{2}\lrp{\E{\lrn{x_0 - x_0}} + L^2 T^2 \E{\lrn{x_0}_2^2} + T \beta^2}\\
\leq& 8 L^2 T^2 \E{\lrn{x_0}_2^2} + T \beta^2
\end{aligned}$$ This concludes our proof.
\[l:divergence\_yt\] Let $y_t$ be as defined in (or equivalently or ), initialized at $y_0$. Then for any $T$, $$\begin{aligned}
\E{\lrn{y_T - y_0}_2^2} \leq T^2 L^2 \E{\lrn{y_0}_2^2} + T\beta^2
\end{aligned}$$ If we additionally assume that $\E{\lrn{y_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{y_T - y_0}_2^2} \leq 2 T\beta^2
\end{aligned}$$
Notice from the definition in that $y_T - y_0\sim \N \lrp{-T \nabla U(y_0), T M(y_0)^2}$, the conclusion immediately follows from where the inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\], and the fact that $$\begin{aligned}
\tr\lrp{M(x)^2} = \tr\lrp{\E{\xi(x,\eta) \xi(x,\eta)^T}} = \E{\lrn{\xi(x,\eta)}_2^2}
\end{aligned}$$
\[l:divergence\_vt\] Let $v_t$ be as defined in , initialized at $v_0$. Then for any $T=n\delta$, $$\begin{aligned}
\E{\lrn{v_T - v_0}_2^2} \leq T^2 L^2 \E{\lrn{v_0}_2^2} + T\beta^2
\end{aligned}$$ If we additionally assume that $\E{\lrn{v_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{v_T - v_0}_2^2} \leq 2 T\beta^2
\end{aligned}$$
From , $$\begin{aligned}
v_T - v_0 = - T \nabla U(v_0) + \sqrt{\delta} \sum_{i=0}^{n-1} \xi(v_0, \eta_i)
\end{aligned}$$ Conditioned on the randomness up to time $i$, $\E{\xi(v_0,\eta_{i+1})}=0$. Thus $$\begin{aligned}
& \E{\lrn{v_T - v_0}_2^2}\\
=& T^2\E{\lrn{\nabla U(v_0)}_2^2} + \delta \sum_{i=0}^{n-1} \E{\lrn{\xi(v_0,\eta_i)}_2^2}\\
\leq& T^2 L^2 \E{\lrn{v_0}_2^2} + T \beta^2
\end{aligned}$$ where the inequality is by item 1 of Assumption \[ass:U\_properties\] and item 2 of Assumption \[ass:xi\_properties\].
\[l:divergence\_wt\] Let $w_t$ be as defined in , initialized at $w_0$. Then for any $T=n\delta$ such that $T \leq \frac{1}{2L}$, $$\begin{aligned}
\E{\lrn{w_T - w_0}_2^2} \leq 16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}
\end{aligned}$$ If we additionally assume that $\E{\lrn{w_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{w_T - w_0}_2^2} \leq 32 T\beta^2
\end{aligned}$$
$$\begin{aligned}
& \E{\lrn{w_{(k+1)\delta} - w_0}_2^2}\\
=& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) + \sqrt{\delta} \xi\lrp{w_{k\delta}, \eta_k} - w_0}_2^2}\\
=& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) - w_0}_2^2} + \delta \E{\lrn{\xi\lrp{w_{k\delta}, \eta_k}}_2^2} \numberthis \label{e:t:sfddd}
\end{aligned}$$
We can bound $\delta \E{\lrn{\xi\lrp{w_{k\delta}, \eta_k}}_2^2} \leq \delta \beta^2$ by item 2 of Assumption \[ass:xi\_properties\]. $$\begin{aligned}
& \E{\lrn{w_{k\delta} - \delta \nabla U(w_{k\delta}) - w_0}_2^2}\\
\leq& \E{\lrp{\lrn{w_{k\delta} - w_0 - \delta \lrp{\nabla U(w_{k\delta}) - \nabla U(w_0)}}_2 + \delta \lrn{\nabla U(w_0)}_2}^2}\\
\leq& \lrp{1 + \frac{1}{n}}\E{\lrn{w_{k\delta} - w_0 - \delta \lrp{\nabla U(w_{k\delta}) - \nabla U(w_0)}}_2^2 }\\
&\quad + (1+n)\delta^2 \E{\lrn{\nabla U(w_0)}_2^2}\\
\leq& \lrp{1 + \frac{1}{n}} \lrp{1 + \delta L}^2 \E{\lrn{w_{k\delta} - w_0}_2^2} + 2n\delta^2 L^2 \E{\lrn{w_0}_2^2}\\
\leq& e^{1/n + 2\delta L }\E{\lrn{w_{k\delta} - w_0}_2^2} + 2n\delta^2 L^2 \E{\lrn{w_0}_2^2}
\end{aligned}$$ where the first inequality is by triangle inequality, the second inequality is by Young’s inequality, the third inequality is by item 1 of Assumption \[ass:U\_properties\].
Inserting the above into gives $$\begin{aligned}
\E{\lrn{w_{(k+1)\delta} - w_0}_2^2} \leq e^{1/n + 2\delta L }\E{\lrn{w_{k\delta} - w_0}_2^2} + 2n\delta^2 L^2 \E{\lrn{w_0}_2^2} + \delta \beta^2
\end{aligned}$$
Applying the above recursively for $k=1...n$, we see that $$\begin{aligned}
& \E{\lrn{w_{n\delta} - w_0}_2^2}\\
\leq& \sum_{k=0}^{n-1} e^{(n-k) \cdot (1/n + 2\delta L)} \cdot \lrp{2n\delta^2 L^2 \E{\lrn{w_0}_2^2} + \delta \beta^2}\\
\leq& 16 \lrp{n^2 \delta^2 L^2 \E{\lrn{w_0}_2^2} + n\delta \beta^2}\\
=& 16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}
\end{aligned}$$
[Discretization Bounds]{} \[ss:discretization\_bonuds\]
\[l:vt-wt\] Let $v_{k\delta}$ and $w_{k\delta}$ be as defined in and . Then for any $\delta,n$, such that $T:= n\delta \leq \frac{1}{16L}$, $$\begin{aligned}
\E{\lrn{v_{T} - w_{T}}_2^2} \leq 8 \lrp{2T^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2} + T L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}}
\end{aligned}$$ If we additionally assume that $\E{\lrn{v_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$, $\E{\lrn{w_0}_2^2} \leq 8 \lrp{R^2 + \beta^2/m}$ and $T \leq \frac{\beta^2}{8L^2\lrp{R^2 + \beta^2/m}}$, then $$\begin{aligned}
\E{\lrn{v_T - w_T}_2^2} \leq 32 \lrp{T^2 L^2 + TL_\xi^2} T\beta^2
\end{aligned}$$
Using the fact that conditioned on the randomness up to step $k$, $\E{\xi(v_0,\eta_{k+1}) - \xi(w_{k\delta}, \eta_{k+1})}=0$, we can show that for any $k\leq n$, $$\begin{aligned}
& \E{\lrn{v_{(k+1)\delta} - w_{(k+1)\delta}}_2^2}\\
=& \E{\lrn{v_{k\delta} - \delta \nabla U(v_0) - w_{k\delta} +\delta \nabla U(w_{k\delta}) + \sqrt{\delta} \xi(w_0, \eta_k) - \sqrt{\delta} \xi(w_{k\delta}, \eta_k)}_2^2}\\
=& \E{\lrn{v_{k\delta} - \delta \nabla U(v_0) - w_{k\delta} +\delta \nabla U(w_{k\delta})}_2^2} + \delta \E{\lrn{\xi(w_0, \eta_k) - \xi(w_{k\delta}, \eta_k)}_2^2}
\numberthis \label{e:t:mkqwm}
\end{aligned}$$ where the first inequality is by (Assumption on smoothness of U and xi).
Using (smoothness of xi), and Lemma \[l:divergence\_vt\], we can bound $$\begin{aligned}
& \delta \E{\lrn{\xi(w_0, \eta_k) - \xi(w_{k\delta}, \eta_k)}_2^2}\\
\leq& \delta L_\xi^2 \E{\lrn{w_{k\delta} - w_0}_2^2}\\
\leq& \delta L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}
\end{aligned}$$
We can also bound $$\begin{aligned}
& \E{\lrn{v_{k\delta} - \delta \nabla U(v_0) - w_{k\delta} +\delta \nabla U(w_{k\delta})}_2^2}\\
\leq& \lrp{1 + \frac{1}{n}}\E{\lrn{v_{k\delta} - \delta \nabla U(v_{k\delta}) - w_{k\delta} +\delta \nabla U(w_{k\delta})}_2^2} + (1+n) \delta^2 \E{\lrn{\nabla U(v_{k\delta}) - \nabla U(v_0)}_2^2}\\
\leq& \lrp{1+ \frac{1}{n}}\lrp{1+ \delta L}^2 \E{\lrn{v_{k\delta} - w_{k\delta}}_2^2} + 2n\delta^2 L^2 \E{\lrn{v_{k\delta} - v_0}_2^2}\\
\leq& e^{1/n + 2\delta L}E{\lrn{v_{k\delta} - w_{k\delta}}_2^2} + 2n\delta^2 L^2 \E{\lrn{v_{k\delta} - v_0}_2^2}\\
\leq& e^{1/n + 2\delta L}E{\lrn{v_{k\delta} - w_{k\delta}}_2^2} + 2n\delta^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2}
\end{aligned}$$ where the first inequality is by Young’s inequality and the second inequality is by item 1 of Assumption \[ass:U\_properties\], the fourth inequality uses Lemma \[l:divergence\_vt\].
Substituting the above two equation blocks into , and applying recursively for $k=0...n-1$ gives $$\begin{aligned}
& \E{\lrn{v_{T} - w_{T}}_2^2} \\
=& \E{\lrn{v_{n\delta} - w_{n\delta}}_2^2} \\
\leq& e^{1+2n\delta L} \lrp{2n^2\delta^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2} + n\delta L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}}\\
\leq& 8 \lrp{2T^2 L^2 \lrp{T^2L^2 \E{\lrn{v_0}_2^2} + T\beta^2} + T L_\xi^2 \lrp{16 \lrp{T^2 L^2 \E{\lrn{w_0}_2^2} + T \beta^2}}}
\end{aligned}$$
the last inequality is by noting that $T = n\delta \leq \frac{1}{4L}$.
[Regularity of $M$ and $N$]{}\[ss:mnregularity\]
[\[l:M\_is\_regular\]]{} $$\begin{aligned}
&1.\ \tr\lrp{M(x)^2} \leq \beta^2\\
&2.\ \tr\lrp{(M(x)^2 - M(y)^2)^2} \leq 16 \beta^2 L_\xi^2 \|x-y\|_2^2\\
&3.\ \tr\lrp{(M(x)^2 - M(y)^2)^2} \leq 32\beta^3 L_\xi \|x-y\|_2
\end{aligned}$$
In this proof, we will use the fact that $\xi(\cdot,\eta)$ is $L_\xi$-Lipschitz from Assumption \[ass:xi\_properties\].
The first property is easy to see: $$\begin{aligned}
& \tr\lrp{M(x)^2}\\
=& \tr\lrp{\Ep{\eta}{\xi(x,\eta) \xi(x,\eta)^T}}\\
=& \Ep{\eta}{\tr\lrp{{\xi(x,\eta) \xi(x,\eta)^T}}}\\
=& \Ep{\eta}{\lrn{\xi(x,\eta)}_2^2}\\
\leq& \beta^2
\end{aligned}$$
We now prove the second and third claims. Consider a fixed $x$ and fixed $y$, let $u_{\eta} := \xi(x,\eta)$, $v_{\eta} := \xi(y,\eta)$. Then
$$\begin{aligned}
& \tr\lrp{\lrp{M(x)^2 - M(y)^2}^2 }\\
=& \tr\lrp{\lrp{\Ep{\eta}{u_\eta u_\eta^T - v_\eta v_\eta^T}}^2}\\
=& \tr\lrp{\Ep{\eta, \eta'}{\lrp{u_\eta u_\eta^T - v_\eta v_\eta^T} \lrp{u_{\eta'}u_{\eta'}^T - v_{\eta'} v_{\eta'}^T}}}\\
=& \Ep{\eta,\eta'}{\tr\lrp{\lrp{u_\eta u_\eta^T - v_\eta v_\eta^T} \lrp{u_{\eta'}u_{\eta'}^T - v_{\eta'} v_{\eta'}^T}}}
\end{aligned}$$
For any fixed $\eta$ and $\eta'$, let’s further simplify notation by letting $u,u',v,v'$ denote $u_\eta, u_{\eta'}, v_\eta, v_{\eta'}$. Thus $$\begin{aligned}
&\tr\lrp{\lrp{uu^T - vv^T} \lrp{u'u'^T - v'v'^T}}\\
=& \tr\lrp{ \lrp{(u-v)v^T + v(u-v)^T + (u-v)(u-v)^T} \lrp{(u'-v')v'^T + v'(u'-v')^T + (u'-v')(u'-v')^T} }\\
=& \tr\lrp{ (u-v) v^T (u'-v') v'^T} + \tr\lrp{ (u-v) v^T v'(u'-v')^T } + \tr\lrp{ (u-v) v^T (u'-v')(u'-v')^T }\\
&\quad + \tr\lrp{ v(u-v)^T (u'-v') v'^T} + \tr\lrp{ v(u-v)^T v'(u'-v')^T } + \tr\lrp{ v(u-v)^T (u'-v')(u'-v')^T }\\
&\quad + \tr\lrp{ (u-v)(u-v)^T (u'-v') v'^T} + \tr\lrp{ (u-v)(u-v)^T v'(u'-v')^T }\\
&\quad + \tr\lrp{ (u-v)(u-v)^T (u'-v')(u'-v')^T }\\
\leq& \min\lrbb{16 \beta^2 L_\xi^2 \lrn{x-y}_2^2, 32\beta^3 L_{\xi}\|x-y\|_2}
\end{aligned}$$
Where the last inequality uses Assumption \[ass:xi\_properties\].2 and \[ass:xi\_properties\].3; in particular, $\lrn{v}_2\leq \beta$ and $\lrn{u-v}_2 \leq \min\lrbb{2\beta, L_\xi \|x-y\|_2}$. This proves 2. and 3. of the Lemma statement.
\[l:N\_is\_regular\] Let $N(x)$ be as defined in and $\LN$ be as defined in . Then $$\begin{aligned}
1.\ &\tr\lrp{N(x)^2} \leq \beta^2\\
2.\ &\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \LN^2\lrn{x-y}_2^2\\
3.\ &\tr\lrp{\lrp{N(x) - N(y)}^2} \leq \frac{8 \beta^2}{\cm} \cdot \LN \lrn{x-y}_2.
\end{aligned}$$
The first inequality holds because $N(x)^2 := M(x)^2 - \cm^2 I$, and then applying Lemma \[l:M\_is\_regular\].1, and the fact that $\tr\lrp{M(x)^2 - \cm^2 I } \leq \tr\lrp{M(x)^2}$ by Assumption \[ass:xi\_properties\].4.
The second inequality is a immediate consequence of Lemma \[l:eldan-matrix\], Lemma \[l:M\_is\_regular\].2, and the fact that $\lambda_{min} \lrp{N(x)^2} = \lambda_{min} \lrp{M(x)^2 - \cm^2} \geq \cm^2$ by Assumption \[ass:xi\_properties\].4.
The proof for the third inequality is similar to the second inequality, and follows from Lemma \[l:M\_is\_regular\] and Lemma \[l:eldan-matrix\].
\[l:eldan-matrix\] Let $A$, $B$ be positive definite matrices. Then $$\begin{aligned}
\tr\lrp{\lrp{\sqrt{A} - \sqrt{B}}^2} \leq \tr\lrp{(A-B)^2 A^{-1}}
\end{aligned}$$
[Defining $f$ and related inequalities]{} \[s:defining-q\]
In this section, we define the Lyapunov function $f$ which is central to the proof of our main results. Here, we give an overview of the various functions defined in this section:
1. $g(z): \Re^d \to \Re^+$: A smoothed version of $\lrn{z}_2$, with bounded derivatives up to third order.
2. $q(r): \Re^+ \to \Re^+$: A concave potential function, similar to the one defined in [@eberle2016reflection], which has bounded derivatives up to third order everywhere except at $r=0$.
3. $f(z) = q(g(z)): \Re^d \to \Re^+$, a concave function which upper and lower bounds $\lrn{z}_2$ within a constant factor, has bounded derivatives up to third order everywhere.
\[l:fproperties\] Let $\epsilon$ satisfy $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$. We define the function $$\begin{aligned}
f(z) := q(g(z))
\end{aligned}$$ Where $q$ is as defined in Appendix \[ss:defining-q\], and $g$ is as defined in Lemma \[l:gproperties\] (with parameter $\epsilon$). Then
1. 1. $\nabla f(z) = q'(g(z)) \cdot \nabla g(z)$
2. For $\lrn{z}_2 \geq 2\epsilon$, $\nabla f(z) = q'(g(z)) \frac{z}{\|z\|_2}$
3. For all $z$, $\lrn{\nabla f(z)}_2 \leq 1$.
2. 1. $\nabla^2 f(z) = q''(g(z)) \nabla g(z) \nabla g(z)^T + q'(g(z)) \nabla^2 g(z)$
2. For $r\geq 2\epsilon$, $\nabla^2 f(z) = q''(g(z)) \frac{zz^T}{\|z\|_2^2} + q'(g(z)) \frac{1}{\|z\|_2} \lrp{I - \frac{zz^T}{\|z\|_2^2}}$
3. For all $z$, $\lrn{\nabla^2 f(z)}_2 \leq \frac{2}{\epsilon}$
4. For all $z,v$, $v^T \nabla^2 f(z) v \leq\frac{q'(g(z))}{\|z\|_2}$
3. For any $z$, $\lrn{\nabla^3 f(z)}_2 \leq \frac{9}{\epsilon^2}$
4. For any $z$, $f(z) \in \lrb{\frac{1}{2}\exp\lrp{-\frac{7\aq\Rq^2}{3}} g(\|z\|_2), g(\|z\|_2)} \in \lrb{\frac{1}{2}\exp\lrp{-\frac{7\aq\Rq^2}{3}} (\|z\|_2 - 2\epsilon), \|z\|_2}$
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1. 1. chain rule
2. Use definition of $\nabla g(z)$ from Lemma \[l:gproperties\].
3. By definition, $\nabla f(z) = q'(g(z)) \nabla g(z)$. From Lemma \[l:qproperties\], $\lrabs{q'(g(z))} \leq 1$. By definition, $\nabla g(z) = h'(\lrn{z}_2) \frac{z}{\lrn{z}_2}$. Our conclusion follows from $h' \leq 1$ using item 2 of Lemma \[l:hproperties\].
2. 1. chain rule
2. by item 2 b) of Lemma \[l:gproperties\]
3. by item 1 c) and item 2 d) of Lemma \[l:gproperties\], and item 3 and item 4 of Lemma \[l:qproperties\], and our assumption that $\epsilon\leq \frac{\Rq}{\aq + \Rq^2 + 1}$.
4. by item 4 of Lemma \[l:qproperties\]), and items 2 c) and 2 d) of Lemma \[l:gproperties\], and our expression for $\nabla^2 f(z)$ established in item 2 a).
3. It can be verified that $$\begin{aligned}
\nabla^3 f(z) =& q'''(g(z)) \cdot \nabla g(z)^{\bo 3} + q''(g(z)) \nabla g(z) \bo \nabla^2 g(z) + q''(g(z)) \nabla^2 g(z) \bo \nabla g(z) \\
&\quad + q''(g(z)) \nabla g(z) \bo \nabla^2 g(z) + q'(g(z)) \nabla^3 g(z)
\end{aligned}$$
Thus $$\begin{aligned}
\lrn{\nabla^3 f(z)}_2
\leq& \lrabs{q'''(g(z))} \lrn{\nabla g(z)}_2^3 + 3 q''(g(z)) \lrn{\nabla g(z)}_2 \lrn{\nabla^2 g(z)}_2 + q'(g(z)) \lrn{\nabla^3 g(z)}\\
\leq& 5\lrp{\aq + \frac{1}{\Rq^2}} \lrp{\aq\Rq^2 + 1} + 3\lrp{\frac{5\aq\Rq}{4} + \frac{4}{\Rq}}\cdot \frac{1}{\epsilon} + \frac{1}{\epsilon^2}\\
\leq& \frac{9}{\epsilon^2}
\end{aligned}$$ Where the first inequality uses Lemma \[l:qproperties\] and Lemma \[l:gproperties\], and the second inequality assumes that $\epsilon \leq \frac{\Rq}{\aq\Rq^2 + 1}$
4. $$\begin{aligned}
f(z) \in \lrb{\frac{1}{2}\exp\lrp{-\frac{7\aq\Rq^2}{3}} g(\|z\|_2), g(\|z\|_2)} \in \lrb{\frac{1}{2}\exp\lrp{-\frac{7\aq\Rq^2}{3}} (\|z\|_2 - 2\epsilon), \|z\|_2}
\end{aligned}$$
The first containment is by Lemma \[l:qproperties\].\[f:q(r)\_bounds\]: $\frac{1}{2}\exp\lrp{-\frac{7\aq\Rq^2}{3}}\cdot g(z) \leq q(g(z)) \leq g(z)$. THe second containment is by Lemma \[l:gproperties\].4: $g(\|z\|_2) \in [\|z\|_2-2\epsilon, \|z\|_2]$.
\[l:hproperties\] Given a parameter $\epsilon$, define $$\begin{aligned}
h(r) := \threecase
{\frac{r^3}{6\epsilon^2}}{r\in [0,\epsilon]}
{\frac{\epsilon}{6} + \frac{r-\epsilon}{2} + \frac{(r-\epsilon)^2}{2\epsilon} - \frac{(r-\epsilon)^3}{6\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{r }{r\geq 2\epsilon}
\end{aligned}$$
1. The derivatives of $h$ are as follows: $$\begin{aligned}
h'(r) =& \threecase
{\frac{r^2}{2\epsilon^2}}{r\in [0,\epsilon]}
{\frac{1}{2} + \frac{r-\epsilon}{\epsilon} - \frac{(r-\epsilon)^2}{2\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{1}{r\geq 2\epsilon}\\
h''(r) =& \threecase
{\frac{r}{\epsilon^2}}{r\in [0,\epsilon]}
{\frac{1}{\epsilon} - \frac{r-\epsilon}{\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{0}{r\geq 2\epsilon}\\
h'''(r) =& \threecase
{\frac{1}{\epsilon^2}}{r\in [0,\epsilon]}
{-\frac{1}{\epsilon^2}}{r\in[\epsilon, 2\epsilon]}
{0}{r\geq 2\epsilon}
\end{aligned}$$
2. 1. $h'$ is positive, motonically increasing.
2. $h'(0)=0$, $h'(r) =1$ for $r\geq \epsilon$
3. $\frac{h'(r)}{r}\leq \min\lrbb{\frac{1}{\epsilon}, \frac{1}{r}}$ for all $r$
3. 1. $h''(r)$ is positive
2. $h''(r) = 0$ for $r=0$ and $r\geq 2\epsilon$
3. $h''(r) \leq \frac{1}{\epsilon}$
4. $\frac{h''(r)}{r} \leq \frac{1}{\epsilon^2}$
4. $\lrabs{h'''(r)} \leq \frac{1}{\epsilon^2}$
5. $r-2\epsilon\leq h(r) \leq r$
The claims can all be verified with simple algebra.
\[l:gproperties\] Given a parameter $\epsilon$, let us define $$\begin{aligned}
g(z) := h(\|z\|_2)
\end{aligned}$$ Where $h$ is as defined in Lemma \[l:hproperties\] (using parameter $\epsilon$). Then
1. 1. $\nabla g(z) = h'(\|z\|_2) \frac{z}{\|z\|_2}$
2. For $\|z\|_2 \geq 2\epsilon$, $\nabla g(z) = \frac{z}{\|z\|_2}$.
3. For any $\|z\|_2$, $\lrn{\nabla g(z)}_2\leq 1$
2. 1. $\nabla^2 g(z) = h''(\|z\|_2)\frac{zz^T}{\|z\|_2^2} + h'(\|z\|_2) \frac{1}{\|z\|_2}\lrp{I - \frac{z z^T}{\|z\|_2^2}}$
2. For $\lrn{z}_2\geq 2\epsilon$, $\nabla^2 g(z) = \frac{1}{\|z\|_2}\lrp{I - \frac{z z^T}{\|z\|_2^2}}$.
3. For $\lrn{z}_2\geq 2\epsilon$, $\lrn{\nabla^2 g(z)}_2 = \frac{1}{\|z\|_2}$
4. For all $z$, $\lrn{\nabla^2 g(z)}_2 \leq \frac{1}{\epsilon}$
3. $\lrn{\nabla^3 g(z)}_2 \leq \frac{5}{\epsilon^2}$
4. $\|z\|_2 - 2\epsilon \leq g(z) \leq \|z\|_2$.
All the properties can be verified with algebra. We provide a proof for $3.$ since it is a bit involved.
Let us define the functions $\kappa^1(z) = \nabla(\|z\|_2), \kappa^2(z) = \nabla^2(\|z\|_2), \kappa^3(z) = \nabla^3(\|z\|_2)$. Specifically, $$\begin{aligned}
\kappa^1(z) =& \frac{z}{\|z\|_2}\\
\kappa^2(z) =& \frac{1}{\|z\|_2}\lrp{I - \frac{zz^T}{\|z\|_2^2}}\\
\kappa^3(z) =& -\frac{1}{\|z\|_2^2} \frac{z}{\|z\|_2} \bo \lrp{I - \frac{zz^T}{\|z\|_2^2}} + \frac{1}{\|z\|_2} \lrp{\frac{z}{\|z\|_2} \bo \kappa^2(z) + \kappa^2(z) \bo \frac{z}{\|z\|_2}}
\end{aligned}$$
It can be verified that $$\begin{aligned}
\lrn{\kappa^2(z)}_2 =& \frac{1}{\|z\|_2}\\
\lrn{\kappa^3(z)}_2 =& \frac{1}{\|z\|_2^2}
\end{aligned}$$
It can be verified that $\nabla^2 g(z)$ has the following form: $$\begin{aligned}
&\nabla^3 g(z) = h'''(\|z\|_2) \lrp{\kappa^1(z)}^{\bo 3} + h''(\|z\|_2) \kappa^1(z) \bo \kappa^2(z) + h''(\|z\|_2) \kappa^2(z) \bo \kappa^1(z) \\
&\quad + h'(\|z\|_2) \kappa^3(z) + h''(\|z\|_2) \kappa^1(z) \bo \kappa^2(z)
\end{aligned}$$
Thus $$\begin{aligned}
\lrn{\nabla^3 g(z)}_2 \leq \lrabs{h'''(\|z\|_2)} + 3\frac{h''(\|z\|_2)}{\|z\|_2} + \frac{h'(\|z\|_2)}{\|z\|_2^2} \leq \frac{5}{\epsilon^2}
\end{aligned}$$
Where we use properties of $h$ from Lemma \[l:hproperties\].
The last claim follows immediately from Lemma \[l:hproperties\].4.
[Defining q]{} \[ss:defining-q\]
In this section, we define the function $q$ that is used in Lemma \[l:fproperties\]. Our construction is a slight modification to the original construction in [@eberle2011reflection].
Let $\aq$ and $\Rq$ be as defined in . We begin by defining auxiliary functions $\psi(r)$, $\Psi(r)$ and $\nu(r)$, all from $ \Re^+$ to $\Re$: $$\begin{aligned}
\label{d:psietal}
&\psi(r) := e^{- \aq \tau(r)}\,, \qquad
&\Psi(r) := \int_0^r \psi(s) ds\,, \qquad \nu(r) := 1- \frac{1}{2}
\frac{\int_0^{r}\frac{ \mu(s) \Psi(s)}{\psi(s)} ds}{\int_0^{4\Rq}\frac{ \mu(s) \Psi(s)}{\psi(s)}ds}\,,
\end{aligned}$$
Where $\tau(r)$ and $\mu(r)$ are as defined in Lemma \[l:tau\] and Lemma \[l:mu\] with $\R = \Rq$.
Finally we define $q$ as $$\label{d:f}
q(r) := \int_0^r \psi(s) \nu(s) ds.$$ We now state some useful properties of the distance function $q$.
\[l:qproperties\] The function $q$ defined in has the following properties.
1. For all $r\leq \Rq $, $q''(r) + \aq q'(r) \cdot r \leq - \frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{32\Rq^2} q(r)$ \[f:contraction\]
2. For all $r$, $\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\cdot r \leq q(r) \leq r$ \[f:q(r)\_bounds\]
3. For all $r$, $\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\leq q'(r) \leq 1$ \[f:q’(r)\_bounds\]
4. For all $r$, $q''(r) \leq 0$ and $\lrabs{q''(r)} \leq \lrp{\frac{5\aq\Rq}{4} + \frac{4}{\Rq}}$ \[f:q”(r)\_bounds\]
5. For all $r$, $\lrabs{q'''(r)} \leq 5\aq + 2\aq\lrp{\aq\Rq^2 + 1} + \frac{2(\aq\Rq^2 + 1)}{\Rq^2}$ \[f:q”’(r)\_bounds\]
**Proof of \[f:contraction\]** It can be verified that $$\begin{aligned}
\psi'(r)
=& \psi(r) (-\aq \tau'(r))\\
\psi''(r)
=& \psi(r) \lrp{\lrp{\aq \tau'(r)}^2 + \aq \tau''(r)}\\
\nu'(r)=& -\frac{1}{2} \frac{\frac{\mu(r)\Psi(r)}{\psi(r)}}{\int_0^{4\Rq} \frac{\mu(s)\Psi(s)}{\psi(s)}ds}
\end{aligned}$$
For $r\in[0,\Rq]$, $\tau'(r) = r$, so that $\psi'(r) = \psi(r)(-\aq r)$. Thus $$\begin{aligned}
q'(r)
=& \psi(r) \nu(r)\\
q''(r)
=& \psi'(r) \nu(r) + \psi(r) \nu'(r)\\
=& \psi(r) \nu(r) (-\aq r) + \psi(r) \nu'(r)\\
=& -\aq r \nu'(r) + \psi(r) \nu'(r)\\
q''(r) + \aq r q'(r)
=& \psi(r) \nu'(r)\\
=& - \frac{1}{2}\frac{\mu(r)\Psi(r)}{\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}\\
=& - \frac{1}{2}\frac{\Psi(r)}{\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}
\end{aligned}$$ Where the last equality is by definition of $\mu(r)$ in Lemma \[l:mu\] and the fact that $r\leq \Rq$.
We can upper bound $$\begin{aligned}
\int_0^{4\Rq} \frac{\mu(s) \Psi(s)}{\psi(s)} ds
\leq \int_0^{4\Rq} \frac{\Psi(s)}{\psi(s)} ds
\leq \frac{\int_0^{4\Rq} s ds}{\psi(4\Rq)}
= \frac{16\Rq^2}{\psi(4\Rq)}
\leq& 16 \Rq^2 \cdot \exp\lrp{\frac{7\aq\Rq^2}{3}}
\end{aligned}$$ Where the first inequality is by Lemma \[l:mu\], the second inequality is by the fact that $\psi(s)$ is monotonically decreasing, the third inequality is by Lemma \[l:tau\].
Thus $$\begin{aligned}
q''(r) + \aq r q'(r)
\leq& -\frac{1}{2} \lrp{\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{16\Rq^2}} \Psi(r)\\
\leq& -\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{32\Rq^2} q(r)
\end{aligned}$$ Where the last inequality is by $\Psi(r) \geq q(r)$.
**Proof of \[f:q(r)\_bounds\]** Notice first that $\nu(r) \geq \frac{1}{2}$ for all $r$. Thus $$\begin{aligned}
q(r)
:=& \int_0^r \psi(s) \nu(s) ds\\
\geq& \frac{1}{2} \int_0^r \psi(s) ds\\
\geq& \frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\cdot r
\end{aligned}$$ Where the last inequality is by Lemma \[l:tau\].
**Proof of \[f:q’(r)\_bounds\]** By definition of $f$, $q'(r) = \psi(r) \nu(r) $, and $$\begin{aligned}
\frac{\exp\lrp{-\frac{7\aq\Rq^2}{3}}}{2}\leq \psi(r)\nu(r)\leq 1
\end{aligned}$$ Where we use Lemma \[l:tau\] and the fact that $\nu(r) \in [1/2,1]$
**Proof of \[f:q”(r)\_bounds\]** Recall that $$\begin{aligned}
q''(r) = \psi'(r) \nu(r) + \psi(r) \nu'(r)
\end{aligned}$$
That $q''\leq 0$ can immediately be verified from the definitions of $\psi$ and $\nu$.
Thus $$\begin{aligned}
\lrabs{q''(r)}
\leq& \lrabs{\psi'(r) \nu(r)} + \lrabs{\psi(r) \nu'(r)}\\
\leq& \aq \tau'(r) + \lrabs{\psi(r) \nu'(r)}
\end{aligned}$$ From Lemma \[l:tau\], we can upperbound $\tau'(r) \leq \frac{5\Rq}{4}$. In addition, $\Psi(r) = \int_0^r \psi(s) \geq r \psi(r)$, so that $$\begin{aligned}
\numberthis \label{e:l:psi(r)/psi(r)}
\frac{\Psi(r)}{\psi(r)} \geq r
\end{aligned}$$ (Recall again that $\psi(s)$ is monotonically decreasing). Thus $\Psi(r)/r \geq r$ for all $r$. In addition, using the fact that $\psi(r) \leq 1$, $$\begin{aligned}
\numberthis \label{e:l:psi(r)_upperbound}
\Psi(r) = \int_0^r \psi(s)ds \leq r
\end{aligned}$$
Combining the previous expressions, $$\begin{aligned}
\lrabs{\psi(r) \nu'(r)}
=& \lrabs{ \frac{1}{2}\frac{\mu(r)\Psi(r)}{\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}}\\
\leq& \lrabs{\frac{1}{2}\frac{\mu(r) r}{\int_0^{\Rq}\frac{\Psi(s)}{\psi(s)} ds}}\\
\leq& \lrabs{\frac{1}{2} \frac{4\Rq}{\int_0^{\Rq} s ds}}\\
\leq& \frac{4}{\Rq}
\end{aligned}$$ Where the first inequality are by definition of $\mu(r)$ and , and the second inequality is by and the fact that $\mu(r) = 0$ for $r\geq 4\Rq$. Combining with our bound on $\psi'(r) \nu(r)$ gives the desired bound.
**Proof of \[f:q”’(r)\_bounds\]** $$\begin{aligned}
q'''(r) = \psi''(r) \nu(r) + 2\psi'(r) \nu'(r) + \psi(r) \nu''(r)
\end{aligned}$$ We first bound the middle term: $$\begin{aligned}
\lrabs{\psi'(r) \nu'r)}
=& \lrabs{\psi(r)(\aq \tau'(r)) \nu'r)}\\
\leq& \aq \lrabs{\tau'(r)}\lrabs{\psi(r) \nu'r)}\\
\leq& \frac{5\aq\Rq}{4} \cdot \frac{4}{\Rq}\\
\leq& 5\aq
\end{aligned}$$ Where the second last line follows form Lemma \[l:tau\] and our proof of \[f:q”(r)\_bounds\].
Next, $$\begin{aligned}
\psi''(r) = \psi(r) \lrp{\aq^2 \tau'(r)^2 - \aq \tau''(r)}
\end{aligned}$$ Thus applying Lemma \[l:tau\].1 and Lemma \[l:tau\].3, $$\begin{aligned}
\lrabs{\psi''(r) \nu(r)} \leq& 2\aq^2\Rq^2 + \aq
\end{aligned}$$
Finally, $$\begin{aligned}
\nu''(r)
=& \frac{1}{2\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}\cdot \frac{d}{dr} {\mu(r)\Psi(r)/\psi(r)}
\end{aligned}$$
Expanding the numerator, $$\begin{aligned}
\frac{d}{dr} \frac{\mu(r) \Psi(r)}{\psi(r)}
=& \mu'(r) \frac{\Psi(r)}{\psi(r)} + \mu(r) - \mu(r) \frac{\Psi(r) \psi'(r)}{\psi(r)^2}\\
=& \mu'(r) \frac{\Psi(r)}{\psi(r)} + \mu(r) + \mu(r) \frac{\Psi(r) \psi(r)\aq \tau'(r)}{\psi(r)^2}
\end{aligned}$$
Thus $$\begin{aligned}
\psi(r) \nu''(r) = \frac{1}{2\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}\cdot\lrp{\mu'(r) \Psi(r) + \mu(r) \psi(r) + \mu(r) \Psi(r) \aq \tau'(r)}
\end{aligned}$$ Using the same argument as from the proof of \[f:q”(r)\_bounds\], we can bound $$\begin{aligned}
\frac{1}{2\int_0^{4\Rq}\frac{\mu(s)\Psi(s)}{\psi(s)} ds}
\leq& \frac{1}{2\int_0^\Rq s ds}\\
\leq& \frac{1}{\Rq^2}
\end{aligned}$$ Finally, from Lemma \[l:mu\], $\lrabs{\mu'(r)}\leq \frac{\pi}{6\Rq}$, so $$\begin{aligned}
\lrabs{\psi(r) \nu''(r)}\leq& \frac{\pi/6 + 1 + 5\aq \Rq^2/4}{\Rq^2}\\
\leq& \frac{2(\aq\Rq^2 + 1)}{\Rq^2}
\end{aligned}$$
\[l:tau\] Let $\tau(r): [0,\infty) \to \Re$ be defined as $$\begin{aligned}
\tau(r)=\fourcase
{\frac{r^2}{2}}{r\leq \R}
{\frac{\R^2}{2} + \R (r-\R) + \frac{(r-\R)^2}{2}- \frac{(r-\R)^3}{3\R}}{r\in[\R,2\R]}
{\frac{5\R^2}{3} + \R(r-2\R) - \frac{(r-2\R)^2}{2} + \frac{(r-2\R)^3}{12\R}}{r\in [2\R,4\R]}
{\frac{7\R^2}{3}}{r\geq 4\R]}
\end{aligned}$$ Then
1. $\tau'(r) \in [0, \frac{5\R}{4}]$, with maxima at $r= \frac{3\R}{2}$. $\tau'(r) = 0$ for $r\in \lrbb{0}\bigcup [4\R, \infty)$
2. As a consequence of 1, $\tau(r)$ is monotonically increasing
3. $\tau''(r) \in [-1,1]$
We provide the derivatives of $\tau$ below. The claims in the Lemma can then be immediately verified. $$\begin{aligned}
\tau'(r) =&
\fourcase
{r}{r\leq \R}
{\R + (r-\R) - \frac{(r-\R)^2}{\R}}{r\in[\R,2\R]}
{\R - (r-2\R) + \frac{(r-2\R)^2}{4\R} }{r\in [2\R,4\R]}
{0}{r\geq 4\R]}
\end{aligned}$$
$$\begin{aligned}
\tau''(r) =&
\fourcase
{1}{r\leq \R}
{1-\frac{2(r-\R)}{\R}}{r\in[\R,2\R]}
{-1 + \frac{r-2\R}{2\R}}{r\in [2\R,4\R]}
{0}{r\geq 4\R]}
\end{aligned}$$
\[l:mu\] Let $$\begin{aligned}
\mu(r) := \threecase{1}{r \leq \R}{\frac{1}{2} + \frac{1}{2} \cos \lrp{\frac{\pi (r-\R)}{3\R}}}{r\in[\R,4\R]}{0}{r\geq 4\R}
\end{aligned}$$ Then $$\begin{aligned}
\mu'(r) := \threecase{0}{r \leq \R}{-\frac{\pi}{6\R} \sin \lrp{\frac{\pi (r-\R)}{\R}}}{r\in[\R,4\R]}{0}{r\geq 4\R}
\end{aligned}$$ Furthermore, $\mu'(r) \in [-\frac{\pi}{6\R},0]$
This Lemma can be easily verified by algebra.
[Miscellaneous]{} The following Theorem, taken from [@eldan2018clt], establishes a quantitative CLT.
\[t:zhai\] Let $X_1...X_n$ be random vectors with mean 0, covariance $\Sigma$, and $\lrn{X_i}\leq \beta$ almost surely for each $i$. Let $S_n= \frac{1}{\sqrt{n}}\sum_{i=1}^n X_i$, and let $Z$ be a Gaussian with covariance $\Sigma$, then $$\begin{aligned}
W_2(S_n, Z)\leq \frac{6\sqrt{d}\beta\sqrt{\log n}}{\sqrt{n}}\end{aligned}$$
\[c:clt\_sum\] Let $X_1...X_n$ be random vectors with mean 0, covariance $\Sigma$, and $\lrn{X_i}\leq \beta$ almost surely for each $i$. let $Y$ be a Gaussian with covariance $n\Sigma$. Then $$\begin{aligned}
W_2\lrp{\sum_i X_i, Y}\leq 6\sqrt{d}\beta \sqrt{\log n}\end{aligned}$$
This is simply taking the result of Theorem \[t:zhai\] and scaling the inequality by $\sqrt{n}$ on both sides.
The following Lemma is taken from [@cheng2019quantitative] and included here for completeness.
\[l:xlogxbound\] For any $c> 0$, $x> 3 \max\lrbb{\frac{1}{c} \log \frac{1}{c},0}$, the inequality $$\begin{aligned}
\frac{1}{c}\log(x) \leq x\end{aligned}$$ holds.
We will consider two cases:
**Case 1**: If $c\geq \frac{1}{e}$, then the inequality $$\log(x) \leq c x$$ is true for all $x$.
**Case 2**: $c \leq \frac{1}{e}$.
In this case, we consider the Lambert W function, defined as the inverse of $f(x) = x e^x$. We will particularly pay attention to $W_{-1}$ which is the lower branch of $W$. (See Wikipedia for a description of $W$ and $W_{-1}$).
We can lower bound $W_{-1}(-c)$ using Theorem 1 from [@chatzigeorgiou2013bounds]: $$\begin{aligned}
& \forall u>0,\quad W_{-1} (-e^{-u-1}) > -u-\sqrt{2u} -1\\
\text{equivalently}\quad &\forall c\in (0,1/e),\quad -W_{-1} (-c) < \log\lrp{\frac{1}{c}} + 1 + \sqrt{2\lrp{\log\lrp{\frac{1}{c}}-1}}-1 \\
&\qquad \qquad\qquad\qquad\qquad\ \ = \log\lrp{\frac{1}{c}} + \sqrt{2\lrp{\log\lrp{\frac{1}{c}}-1}}\\
&\qquad \qquad\qquad\qquad\qquad\ \ \leq 3\log \frac{1}{c}\end{aligned}$$
Thus by our assumption, $$\begin{aligned}
& x\geq 3\cdot \frac{1}{c}\log\lrp{\frac{1}{c}}\\
\Rightarrow & x\geq \frac{1}{c}\lrp{-W_{-1}(-c)}\end{aligned}$$
then $W_{-1}(-c)$ is defined, so $$\begin{aligned}
&x \geq \frac{1}{c}\max\lrbb{-W_{-1} (-c), 1}\\
\Rightarrow & (-cx) e^{-cx} \geq -c\\
\Rightarrow & x e^{-cx} \leq 1\\
\Rightarrow & \log(x) \leq cx\end{aligned}$$
The first implication is justified as follows: $W_{-1}^{-1}: [-\frac{1}{\epsilon}, \infty) \to (-\infty, -1)$ is monotonically decreasing. Thus its inverse $W_{-1}^{-1}(y) = ye^y$, defined over the domain $(-\infty, -1)$ is also monotonically decreasing. By our assumption, $-cx \leq -3 \log \frac{1}{c} \leq -3$, thus $-cx \in (-\infty, -1]$, thus applying $W_{-1}^{-1}$ to both sides gives us the first implication.
Experiment Details {#apx:experiments}
==================
In this section, we provide additional details of our experiments. In particular, we explain the CNN architecture that we use in our experiments. Denote a convolutional layer with $p$ input filters and $q$ output filters by $\mathsf{conv}(p, q)$, a fully connected layer with q outputs by $\mathsf{fully\_connect}(q)$, and a max pooling operation with stride 2 as $\mathsf{pool2}$. Let $\mathsf{ReLU}(x) = \max\{x, 0\}$. Then the CNN architecture in our paper is the following: $$\begin{aligned}
&\mathsf{conv}(3, 32) \Rightarrow \mathsf{ReLU} \Rightarrow \mathsf{conv}(32, 64) \Rightarrow \mathsf{ReLU} \Rightarrow \mathsf{pool2} \Rightarrow \mathsf{conv}(64, 128) \Rightarrow \mathsf{ReLU} \Rightarrow
\mathsf{conv}(128, 128) \\
&\Rightarrow \mathsf{ReLU} \Rightarrow \mathsf{pool2} \Rightarrow \mathsf{conv}(128, 256) \Rightarrow \mathsf{ReLU} \Rightarrow \mathsf{conv}(256, 256) \Rightarrow \mathsf{ReLU} \Rightarrow
\mathsf{pool2} \Rightarrow \mathsf{fully\_connect}(1024) \\
&\Rightarrow \mathsf{ReLU} \Rightarrow \mathsf{fully\_connect}(512) \Rightarrow \mathsf{ReLU} \Rightarrow \mathsf{fully\_connect}(10).\end{aligned}$$
[^1]: We provide details of this CNN architecture in Appendix \[apx:experiments\].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A Delaunay polytope $P$ is said to be [*extreme*]{} if the only (up to isometries) affine bijective transformations $f$ of ${\ensuremath{\mathbb{R}}}^n$, for which $f(P)$ is again a Delaunay polytope, are the homotheties. This notion was introduced in [@DGL92]; also some examples in dimension $1$, $6$, $7$, $15$, $16$, $22$, $23$ were constructed and it was proved that in dimension less than $6$ there are no extreme Delaunay polytopes, except the segment. In this note, for every $n\geq 6$ we build an extreme Delaunay polytope $ED_n$ of dimension $n$.'
author:
- |
Mathieu DUTOUR\
LIGA, ENS/CNRS, Paris and\
Hebrew University, Jerusalem[^1]\
title: Infinite serie of extreme Delaunay polytope
---
Let $L\subset {\ensuremath{\mathbb{R}}}^n$ be a $n$-dimensional lattice and let $S=S(c,r)$ be a sphere in ${\ensuremath{\mathbb{R}}}^n$ with center $c$ and radius $r$. Then, $S$ is said to be an [*empty sphere*]{} in $L$ if the following two conditions hold:
$\Vert v-c\Vert\geq r$ for all $v\in L$ and the set $S\cap L$ has affine rank $n+1$.
The polytope $P$, which is defined as the convex hull of the set $P=S\cap L$, is called a [*Delaunay polytope*]{}.
Denote by $D_n$ the root lattice defined as $$D_n=\{(x_1, \dots, x_n)\in{\ensuremath{\mathbb{Z}}}^n\mbox{~with~}\sum_{i=1}^n x_i\mbox{~even}\}\;.$$ The Delaunay polytopes of this lattice are:
- [*Half-cube*]{} $\frac{1}{2}H_n=\{x\in \{0,1\}^n\mbox{~with~}\sum_{i=1}^n x_i\mbox{~even}\}$; it has $2^{n-1}$ vertices.
- [*Cross-polytope*]{} $C_n=\{e_1\pm e_i\mbox{~with~}1\leq i\leq n\}$ with $e_i=(0,\dots, 1,\dots,0)$; it has $2n$ vertices.
Define the polytope $P_n$ to be the convex hull of the three following layers of points:
- a vertex $V=(\frac{1}{2},\dots, \frac{1}{2},1)$,
- $2^{n-2}$ vertices $(x_1,\dots, x_{n-1},0)$ with $x_i\in\{0,1\}$ and $\sum_{i=1}^{n-1} x_i$ even,
- $2(n-1)$ vertices $V_{j,\pm}=(\frac{1}{2},\dots, \frac{1}{2},-1)\pm e_j$ with $1\leq j\leq n-1$.
It is easy to see, that, denoting by $L_n$ the lattice generated by all $(x,0)$ with $x\in D_{n-1}$ and $(\frac{1}{2},\dots, \frac{1}{2},1)$, the third layer belongs to $L_n$ if and only if $n$ is even.
\[TheoremInfiniteSeq\] Let $n$ be even and greater than $6$.
\(i) $P_n$ is an extreme Delaunay polytope; the corresponding quadratic form is $q(x)=x_1^2+\dots+x_{n-1}^2+\frac{n-3}{4}x_n^2$.
\(ii) The center of the circumscribing sphere is $(0,\dots,0,\frac{-1}{n-3})$, it radius is $\frac{n-2}{\sqrt{n-3}}$.
\(iii) $P_6$=Schläfli polytope; its group of isometries has size $51840$ and is transitive on vertices. If $n>6$, then its group of isometries is $Sym(n-1)\times 2^{n-2}$ and there are three orbits of vertices.
[[**Proof.**]{} ]{}In order to prove extremality, we will use geometry of numbers, i.e. in order to show that the only transformations preserving the property of being Delaunay are modulo isometries and translations the homotheties, we will determine the set of quadratic forms $q$ such that $P_n$ is a Delaunay polytope for $q$.
The restriction of the quadratic form $q$ to the plane $x_n=0$ left us with the $2^{n-2}$ vertices $(x_1, \dots, x_{n-1}, 0)$ with $x_i\in \{0,1\}$ and $\sum_{i} x_i$ even. The rank of Half-cube is $n$ (see [@DL] and [@Du]), we can write $q$ restricted to the plane $x_n=0$ as $\sum_{i=1}^{n-1} \alpha_i x_i^2$ with $\alpha_i>0$. We write the form $q$ in the following way: $$q(x)=\sum_{i=1}^{n-1}\alpha_i (x_i+\beta_i x_n)^2+\gamma x_n^2\;.$$ Denote by $S(c,r)$ the empty sphere of center $c$ and radius $r$ circumscribing $\frac{1}{2}H_{n-1}$ and $V$. Denoting $c=(h_1, \dots, h_{n-1}, h_n)$, one obtains the equations: $$\left\lbrace\begin{array}{l}
\sum_{i=1}^{n-1}\alpha_i (h_i-x_i+\beta_i h_n)^2+\gamma h_n^2=r^2,\\
\sum_{i=1}^{n-1}\alpha_i (h_i-\frac{1}{2}+\beta_i (h_n-1))^2+\gamma (h_n-1)^2=r^2\;.
\end{array}\right.$$ The first equation is satisfied for all vectors $x_i$ with $\sum_{i=1}^{n-1}x_i$ even and $x_i\in\{0,1\}$. This implies, since $n\geq 6$ the relation $h_i+\beta_i h_n=\frac{1}{2}$. So, the equations reduce to: $$\left\lbrace\begin{array}{l}
\frac{1}{4}\sum_{i=1}^{n-1}\alpha_i+\gamma h_n^2=r^2,\\
\frac{n-2}{2}\sum_{i=1}^{n-1}\alpha_i \beta_i^2+\gamma (h_n-1)^2=r^2\;.
\end{array}\right.$$ Consider now the $2(n-1)$ vertices $V_{j,\pm}=(\frac{1}{2},\dots, \frac{1}{2},-1)\pm e_j$ with $1\leq j\leq n-1$.
Since $V_{j, +}$, and $V_{j, -}$ are on the sphere $S(c, r)$, one obtains: $$\left\lbrace\begin{array}{c}
\alpha_j(1-\beta_j)^2+\sum_{i\not=j} \alpha_i\beta_i^2+\gamma (h_n+1)^2=r^2,\\
\alpha_j(1+\beta_j)^2+\sum_{i\not=j} \alpha_i\beta_i^2+\gamma (h_n+1)^2=r^2\;.
\end{array}\right.$$ Above two equations yield $\beta_j=0$. If all $V_{j, \pm}$ are on the sphere, then $\alpha_j=\alpha$ is independent of $j$ and all equations simplify to: $$\alpha+\gamma (h_n+1)^2=r^2\mbox{~,~~}\frac{n-1}{4}\alpha+\gamma h_n^2=r^2\mbox{~~and~~}\gamma (h_n-1)^2=r^2\;,$$ whose solution is: $$h_n=\frac{1}{3-n}, \;\;\gamma=\alpha\frac{n-3}{4},\;\; r=\frac{n-2}{\sqrt{n-3}}\sqrt{\alpha}\;\;.$$ So, $q$ is equal to $\alpha(\sum_{i=1}^{n-1} x_i^2+\frac{n-3}{4} x_n^2)$.
Now, let us prove that this is a Delaunay polytope. If $v$ is an interior point of $P_{n}$, then it belongs to one of the three layers $x_n=-1$, $0$, $1$. But each one of the corresponding sections (i.e. Cross-polytope, Half-cube or point) is a Delaunay polytope; so, there is no such $v$.
Let us find the symmetry group. Pairwise distance of adjacent vertices of Half-cube is $\sqrt{2}$. Distance between $V$ and vertices of Half-cube is $\sqrt{\frac{n-2}{2}}$. So, if $n>6$, then any isometry preserving $P_n$ must leave the vertex $V$ invariant; hence, if $n>6$, then the symmetry group is that of Half-cube, i.e. $Sym(n-1)\times 2^{n-2}$.
Standart computation shows that $P_6$ is the Schläfli polytope, whose symmetry group has size $51840$. [$\Box$ ]{}
Define $ED_n$ to be $P_n$ if $n$ is even and the polytope obtained from $P_{n-1}$ by using Lemma 15.3.7 is $f$ is odd. Remark that $ED_6$ (respectively $ED_7$) is Schläfli (respectively, Gosset) polytope.
If $n\geq 6$, then the polytope $ED_n$ is extreme.
[[**Proof.**]{} ]{}If $n$ is even, this follows from \[TheoremInfiniteSeq\], while if $n$ is odd, this follows from Lemma 15.3.7. [$\Box$ ]{}
Remark, that $ED_6$ is [*unique*]{} extreme Delaunay polytope in dimension $6$ (see [@Hyp7]); [*all*]{} $6241$ 6-dimensional Delaunay polytopes were listed in [@Du].
[99]{}
M.Deza and M.Laurent, [*Geometry of cuts and metrics*]{}, Springer–Verlag, Berlin 1997.
M.Dutour, [*The six-dimensional Delaunay polytopes*]{}, submitted, <http://www.arxiv.org/abs/math.MG/0212353>.
M.Deza and M.Dutour, [*The hypermetric cone on seven vertices*]{}, submitted, <http://www.arxiv.org/abs/math.MG/0108177>.
M. Deza, V.P. Grishukhin, and M. Laurent, [*Extreme hypermetrics and L-polytopes*]{}, in G.Halász et al. eds [*Sets, Graphs and Numbers, Budapest (Hungary), 1991*]{}, [**60**]{} [*Colloquia Mathematica Societatis János Bolyai*]{}, (1992) 157–209.
[^1]: Research financed by EC’s IHRP Programme, within the Research Training Network “Algebraic Combinator ics in Europe,” grant HPRN-CT-2001-00272.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we investigate the modeling power of contextualized embeddings from pre-trained language models, e.g. BERT, on the E2E-ABSA task. Specifically, we build a series of simple yet insightful neural baselines to deal with E2E-ABSA. The experimental results show that even with a simple linear classification layer, our BERT-based architecture can outperform state-of-the-art works. Besides, we also standardize the comparative study by consistently utilizing a hold-out development dataset for model selection, which is largely ignored by previous works. Therefore, our work can serve as a BERT-based benchmark for E2E-ABSA.[^1]'
author:
- |
Xin Li[^1^]{}, Lidong Bing[^2^]{}, Wenxuan Zhang[^1^]{} and Wai Lam[^1^]{}\
[[^1^]{}Department of Systems Engineering and Engineering Management\
The Chinese University of Hong Kong, Hong Kong]{}\
[[^2^]{}R&D Center Singapore, Machine Intelligence Technology, Alibaba DAMO Academy]{}\
[`{lixin,wxzhang,wlam}@se.cuhk.edu.hk`]{}\
[`[email protected]`]{}\
bibliography:
- 'emnlp-ijcnlp-2019.bib'
title: 'Exploiting BERT for End-to-End Aspect-based Sentiment Analysis[^2]'
---
Introduction
============
Aspect-based sentiment analysis (ABSA) is to discover the users’ sentiment or opinion towards an aspect, usually in the form of explicitly mentioned aspect terms [@mitchell-etal-2013-open; @zhang-etal-2015-neural] or implicit aspect categories [@wang-etal-2016-attention], from user-generated natural language texts [@liu2012sentiment]. The most popular ABSA benchmark datasets are from SemEval ABSA challenges [@pontiki-etal-2014-semeval; @pontiki-etal-2015-semeval; @pontiki-etal-2016-semeval] where a few thousand review sentences with gold standard aspect sentiment annotations are provided.
Table \[tab:problem\_settings\] summarizes three existing research problems related to ABSA. The first one is the original ABSA, aiming at predicting the sentiment polarity of the sentence towards the given aspect. Compared to this classification problem, the second one and the third one, namely, Aspect-oriented Opinion Words Extraction (AOWE) [@fan-etal-2019-target] and End-to-End Aspect-based Sentiment Analysis (E2E-ABSA) [@ma-etal-2018-joint; @schmitt-etal-2018-joint; @li2019unified; @li2017learning; @li2019learning], are related to a sequence tagging problem. Precisely, the goal of AOWE is to extract the aspect-specific opinion words from the sentence given the aspect. The goal of E2E-ABSA is to jointly detect aspect terms/categories and the corresponding aspect sentiments.
Many neural models composed of a task-agnostic pre-trained word embedding layer and task-specific neural architecture have been proposed for the original ABSA task (i.e. the aspect-level sentiment classification) [@tang-etal-2016-aspect; @wang-etal-2016-attention; @chen-etal-2017-recurrent-attention; @liu-zhang-2017-attention; @ma2017interactive; @ma2018targeted; @majumder-etal-2018-iarm; @li-etal-2018-transformation; @he-etal-2018-exploiting; @xue-li-2018-aspect; @wang-etal-2018-target; @fan-etal-2018-multi; @huang-carley-2018-parameterized; @lei2019human; @li2019exploiting; @zhang2019aspect][^3], but the improvement of these models measured by the accuracy or F1 score has reached a bottleneck. One reason is that the task-agnostic embedding layer, usually a linear layer initialized with Word2Vec [@mikolov2013distributed] or GloVe [@pennington-etal-2014-glove], only provides context-independent word-level features, which is insufficient for capturing the complex semantic dependencies in the sentence. Meanwhile, the size of existing datasets is too small to train sophisticated task-specific architectures. Thus, introducing a context-aware word embedding[^4] layer pre-trained on large-scale datasets with deep LSTM [@mccann2017learned; @peters-etal-2018-deep; @howard-ruder-2018-universal] or Transformer [@radford2018improving; @radford2019language; @devlin-etal-2019-bert; @lample2019cross; @yang2019xlnet; @dong2019unified] for fine-tuning a lightweight task-specific network using the labeled data has good potential for further enhancing the performance.
@xu-etal-2019-bert [@sun-etal-2019-utilizing; @song2019attentional; @yu2019adapting; @rietzler2019adapt; @huang2019syntax; @hu2019learning] have conducted some initial attempts to couple the deep contextualized word embedding layer with downstream neural models for the original ABSA task and establish the new state-of-the-art results. It encourages us to explore the potential of using such contextualized embeddings to the more difficult but practical task, i.e. E2E-ABSA (the third setting in Table \[tab:problem\_settings\]).[^5] Note that we are not aiming at developing a task-specific architecture, instead, our focus is to examine the potential of contextualized embedding for E2E-ABSA, coupled with various simple layers for prediction of E2E-ABSA labels.[^6]
In this paper, we investigate the modeling power of BERT [@devlin-etal-2019-bert], one of the most popular pre-trained language model armed with Transformer [@vaswani2017attention], on the task of E2E-ABSA. Concretely, inspired by the investigation of E2E-ABSA in @li2019unified, which predicts aspect boundaries as well as aspect sentiments using a single sequence tagger, we build a series of simple yet insightful neural baselines for the sequence labeling problem and fine-tune the task-specific components with BERT or deem BERT as feature extractor. Besides, we standardize the comparative study by consistently utilizing the hold-out development dataset for model selection, which is ignored in most of the existing ABSA works [@tay2018learning].
Model
=====
In this paper, we focus on the aspect term-level End-to-End Aspect-Based Sentiment Analysis (E2E-ABSA) problem setting. This task can be formulated as a sequence labeling problem. The overall architecture of our model is depicted in Figure \[fig:architecture\]. Given the input token sequence $\mathrm{\bf x} = \{x_1, \cdots, x_T \}$ of length $T$, we firstly employ BERT component with $L$ transformer layers to calculate the corresponding contextualized representations $H^{L} = \{h^L_1, \cdots, h^L_T\} \in \mathbb{R}^{T \times \mathrm{dim}_h}$ for the input tokens where $\mathrm{dim}_h$ denotes the dimension of the representation vector. Then, the contextualized representations are fed to the task-specific layers to predict the tag sequence $\mathrm{\bf y} = \{y_1, \cdots, y_T\}$. The possible values of the tag $y_t$ are `B`-{`POS,NEG,NEU`}, `I`-{`POS,NEG,NEU`}, `E`-{`POS,NEG,NEU`}, `S`-{`POS,NEG,NEU`} or `O`, denoting the beginning of aspect, inside of aspect, end of aspect, single-word aspect, with positive, negative or neutral sentiment respectively, as well as outside of aspect.
![Overview of the designed model.[]{data-label="fig:architecture"}](architecture_crop.pdf){width="50.00000%"}
BERT as Embedding Layer
-----------------------
Compared to the traditional Word2Vec- or GloVe-based embedding layer which only provides a single context-independent representation for each token, the BERT embedding layer takes the sentence as input and calculates the token-level representations using the information from the entire sentence. First of all, we pack the input features as $H^0 = \{e_1,\cdots,e_T\}$, where $e_t$ ($t\in[1, T]$) is the combination of the token embedding, position embedding and segment embedding corresponding to the input token $x_t$. Then $L$ transformer layers are introduced to refine the token-level features layer by layer. Specifically, the representations $H^{l} = \{h^l_1, \cdots, h^l_T\}$ at the $l$-th ($l \in [1, L]$) layer are calculated below: $$\label{eq:bert}
H^l = \text{Transformer}_l(H^{l-1})$$ We regard $H^L$ as the contextualized representations of the input tokens and use them to perform the predictions for the downstream task.
Design of Downstream Model
--------------------------
After obtaining the BERT representations, we design a neural layer, called E2E-ABSA layer in Figure \[tab:problem\_settings\], on top of BERT embedding layer for solving the task of E2E-ABSA. We investigate several different design for the E2E-ABSA layer, namely, linear layer, recurrent neural networks, self-attention networks, and conditional random fields layer.
#### Linear Layer
The obtained token representations can be directly fed to linear layer with softmax activation function to calculate the token-level predictions: $$P(y_t|x_t) = \text{softmax}(W_o h^L_t + b_o)$$ where $W_o \in \mathbb{R}^{\mathrm{dim}_h \times |\mathcal{Y}|}$ is the learnable parameters of the linear layer.
#### Recurrent Neural Networks
Considering its sequence labeling formulation, Recurrent Neural Networks (RNN) [@elman1990finding] is a natural solution for the task of E2E-ABSA. In this paper, we adopt GRU [@cho-etal-2014-learning], whose superiority compared to LSTM [@hochreiter1997long] and basic RNN has been verified in @jozefowicz2015empirical. The computational formula of the task-specific hidden representation $h^{\mathcal{T}}_t \in \mathbb{R}^{\mathrm{dim}_h}$ at the $t$-th time step is shown below: $$\small
\begin{split}
\begin{bmatrix}
r_t \\
z_t
\end{bmatrix}
&= \sigma(\textsc{Ln}(W_x h^L_t)+\textsc{Ln}(W_h h^{\mathcal{T}}_{t-1})) \\
n_t &= \text{tanh}(\textsc{Ln}(W_{xn} h^L_t)+r_t * \textsc{Ln}(W_{hn} h^{\mathcal{T}}_{t-1})) \\
h^{\mathcal{T}}_{t} &= (1-z_t) * n_t + z_t * h^{\mathcal{T}}_{t-1}
\end{split}$$ where $\sigma$ is the sigmoid activation function and $r_t$, $z_t$, $n_t$ respectively denote the reset gate, update gate and new gate. $W_x,W_h \in \mathbb{R}^{2\mathrm{dim}_h \times \mathrm{dim}_h}$, $W_{xn}, W_{hn} \in \mathbb{R}^{\mathrm{dim}_h \times \mathrm{dim}_h}$ are the parameters of GRU. Since directly applying RNN on the output of transformer, namely, the BERT representation $h^L_t$, may lead to unstable training [@chen-etal-2018-best; @liu2019fine], we add additional layer-normalization [@ba2016layer], denoted as $\textsc{Ln}$, when calculating the gates. Then, the predictions are obtained by introducing a softmax layer: $$\label{output}
p(y_t|x_t) = \text{softmax}(W_o h^{\mathcal{T}}_t + b_o)$$
#### Self-Attention Networks
With the help of self attention [@cheng-etal-2016-long; @lin2017structured], Self-Attention Network [@vaswani2017attention; @shen2018disan] is another effective feature extractor apart from RNN and CNN. In this paper, we introduce two SAN variants to build the task-specific token representations $H^{\mathcal{T}}=\{h^{\mathcal{T}}_1,\cdots,h^{\mathcal{T}}_T\}$. One variant is composed of a simple self-attention layer and residual connection [@he2016deep], dubbed as “SAN”. The computational process of SAN is below: $$\begin{split}
H^{\mathcal{T}} &= \textsc{Ln}(H^{L}+\textsc{Slf-Att} (Q, K, V)) \\
Q, K, V &= H^{L} W^{Q}, H^{L}W^{K}, H^{L} W^{V}
\end{split}$$ where <span style="font-variant:small-caps;">Slf-Att</span> is identical to the self-attentive scaled dot-product attention [@vaswani2017attention]. Another variant is a transformer layer (dubbed as “TFM”), which has the same architecture with the transformer encoder layer in the BERT. The computational process of TFM is as follows: $$\begin{split}
\hat{H}^{L} &= \textsc{Ln}(H^{L}+\textsc{Slf-Att} (Q, K, V)) \\
H^{\mathcal{T}} &= \textsc{Ln}(\hat{H}^{L}+\textsc{Ffn}(\hat{H}^{L}))
\end{split}$$ where <span style="font-variant:small-caps;">Ffn</span> refers to the point-wise feed-forward networks [@vaswani2017attention]. Again, a linear layer with softmax activation is stacked on the designed SAN/TFM layer to output the predictions (same with that in Eq(\[output\])).
[ll|c|c|c|c]{} & Train & Dev & Test & Total\
& \# sent & 2741 & 304 & 800 & 4245\
& \# aspect & 2041 & 256 & 634 & 2931\
& \# sent & 3490 & 387 & 2158 & 6035\
& \# aspect & 3893 & 413 & 2287 & 6593\
#### Conditional Random Fields
Conditional Random Fields (CRF) [@lafferty2001conditional] is effective in sequence modeling and has been widely adopted for solving the sequence labeling tasks together with neural models [@huang2015bidirectional; @lample-etal-2016-neural; @ma-hovy-2016-end]. In this paper, we introduce a linear-chain CRF layer on top of the BERT embedding layer. Different from the above mentioned neural models maximizing the token-level likelihood $p(y_t|x_t)$, the CRF-based model aims to find the globally most probable tag sequence. Specifically, the sequence-level scores $s(\mathrm{\bf x},\mathrm{\bf y})$ and likelihood $p(\mathrm{\bf y}|\mathrm{\bf x})$ of $\mathrm{\bf y} = \{y_1,\cdots,y_T\}$ are calculated as follows: $$ \begin{split}
s(\mathrm{\bf x},\mathrm{\bf y}) &= \sum^T_{t=0} M^A_{y_t,y_{t+1}} + \sum^T_{t=1} M^P_{t, y_t} \\
p(\mathrm{\bf y}|\mathrm{\bf x}) &= \text{softmax}(s(\mathrm{\bf x},\mathrm{\bf y}))
\end{split}$$ where $M^A \in \mathbb{R}^{|\mathcal{Y}| \times |\mathcal{Y}|}$ is the randomly initialized transition matrix for modeling the dependency between the adjacent predictions and $M^P \in \mathbb{R}^{T \times |\mathcal{Y}|}$ denote the emission matrix linearly transformed from the BERT representations $H^L$. The softmax here is conducted over all of the possible tag sequences. As for the decoding, we regard the tag sequence with the highest scores as output: $$\mathrm{\bf y}^* = \arg\max_{\mathrm{\bf y}} s(\mathrm{\bf x},\mathrm{\bf y})$$ where the solution is obtained via Viterbi search.
![Performances on the Dev set of `REST`.[]{data-label="fig:overfit"}](overfit_v4.pdf){width="40.00000%"}
Experiment
==========
Dataset and Settings
--------------------
We conduct experiments on two review datasets originating from SemEval [@pontiki-etal-2014-semeval; @pontiki-etal-2015-semeval; @pontiki-etal-2016-semeval] but re-prepared in @li2019unified. The statistics are summarized in Table \[tab:dataset\]. We use the pre-trained “bert-base-uncased” model[^7], where the number of transformer layers $L=12$ and the hidden size $\mathrm{dim}_h$ is 768. For the downstream E2E-ABSA component, we consistently use the single-layer architecture and set the dimension of task-specific representation as $\mathrm{dim}_h$. The learning rate is 2e-5. The batch size is set as 25 for `LAPTOP` and 16 for `REST`. We train the model up to 1500 steps. After training 1000 steps, we conduct model selection on the development set for very 100 steps according to the micro-averaged F1 score. Following these settings, we train 5 models with different random seeds and report the average results.
We compare with **Existing Models**, including tailor-made E2E-ABSA models [@li2019unified; @luo-etal-2019-doer; @he-etal-2019-interactive], and competitive **LSTM-CRF** sequence labeling models [@lample-etal-2016-neural; @ma-hovy-2016-end; @liu2018empower].
Main Results
------------
From Table \[tab:main\_results\], we surprisingly find that only introducing a simple token-level classifier, namely, BERT-Linear, already outperforms the existing works without using BERT, suggesting that BERT representations encoding the associations between arbitrary two tokens largely alleviate the issue of context independence in the linear E2E-ABSA layer. It is also observed that slightly more powerful E2E-ABSA layers lead to much better performance, verifying the postulation that incorporating context helps to sequence modeling.
Over-parameterization Issue
---------------------------
Although we employ the smallest pre-trained BERT model, it is still over-parameterized for the E2E-ABSA task (110M parameters), which naturally raises a question: does BERT-based model tend to overfit the small training set? Following this question, we train BERT-GRU, BERT-TFM and BERT-CRF up to 3000 steps on `REST` and observe the fluctuation of the F1 measures on the development set. As shown in Figure \[fig:overfit\], F1 scores on the development set are quite stable and do not decrease much as the training proceeds, which shows that the BERT-based model is exceptionally robust to overfitting.
Finetuning BERT or Not
----------------------
We also study the impact of fine-tuning on the final performances. Specifically, we employ BERT to calculate the contextualized token-level representations but kept the parameters of BERT component unchanged in the training phase. Figure \[fig:finetune\] illustrate the comparative results between the BERT-based models and those keeping BERT component fixed. Obviously, the general purpose BERT representation is far from satisfactory for the downstream tasks and task-specific fine-tuning is essential for exploiting the strengths of BERT to improve the performance.
![Effect of fine-tuning BERT.[]{data-label="fig:finetune"}](finetuning_v3.pdf){width="40.00000%"}
Conclusion
==========
In this paper, we investigate the effectiveness of BERT embedding component on the task of End-to-End Aspect-Based Sentiment Analysis (E2E-ABSA). Specifically, we explore to couple the BERT embedding component with various neural models and conduct extensive experiments on two benchmark datasets. The experimental results demonstrate the superiority of BERT-based models on capturing aspect-based sentiment and their robustness to overfitting.
[^1]: Our code is open-source and available at: <https://github.com/lixin4ever/BERT-E2E-ABSA>
[^2]: The work described in this paper is substantially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project Code: 14204418).
[^3]: Due to the limited space, we can not list all of the existing works here, please refer to the survey [@zhou2019deep] for more related papers.
[^4]: In this paper, we generalize the concept of “word embedding” as a mapping between the word and the low-dimensional word representations.
[^5]: Both of ABSA and AOWE assume that the aspects in a sentence are given. Such setting makes them less practical in real-world scenarios since manual annotation of the fine-grained aspect mentions/categories is quite expensive.
[^6]: @hu-etal-2019-open introduce BERT to handle the E2E-ABSA problem but their focus is to design a task-specific architecture rather than exploring the potential of BERT.
[^7]: https://github.com/huggingface/transformers
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A system of nested dichotomies is a method of decomposing a multi-class problem into a collection of binary problems. Such a system recursively applies binary splits to divide the set of classes into two subsets, and trains a binary classifier for each split. Many methods have been proposed to perform this split, each with various advantages and disadvantages. In this paper, we present a simple, general method for improving the predictive performance of nested dichotomies produced by any subset selection techniques that employ randomness to construct the subsets. We provide a theoretical expectation for performance improvements, as well as empirical results showing that our method improves the root mean squared error of nested dichotomies, regardless of whether they are employed as an individual model or in an ensemble setting.'
author:
- |
Tim Leathart, Eibe Frank, Bernhard Pfahringer and Geoffrey Holmes\
Department of Computer Science, University of Waikato, New Zealand
bibliography:
- 'multi\_subset\_nd.bib'
title: Ensembles of Nested Dichotomies with Multiple Subset Evaluation
---
Introduction
============
Multi-class classification problems are commonplace in real world applications. Some models, like neural networks and random forests, are inherently able to operate on multi-class data directly, while other models, such as classic support vector machines, can only be used for binary (two-class) problems. The standard way to bypass this limitation is to convert the multi-class classification problem into a series of binary problems. There exist several methods of performing this decomposition, the most well-known including one-vs-rest [@rifkin2004defense], pairwise classification [@hastie1998classification] and error-correcting output codes [@dietterich1995solving]. Models that are directly capable of working with multi-class problems may also see improved accuracy from such a decomposition [@mayoraz1997decomposition; @furnkranz2002round; @pimenta2005study].
The use of ensembles of nested dichotomies is one such method for decomposing a multi-class problem into several binary problems. It has been shown to outperform one-vs-rest and perform competitively compared to the aforementioned methods [@frank2004ensembles]. In a nested dichotomy [@fox1997applied], the set of classes is recursively split into two subsets in a tree structure. Two examples of nested dichotomies for a four class problem are shown in Figure \[fig:nd\_example\]. At each split node of the tree, a binary classifier is trained to discriminate between the two subsets of classes. Each leaf node of the tree corresponds to a particular class. To obtain probability estimates for a particular class from a nested dichotomy, assuming the base learner can produce probability estimates, one can simply compute the product of the binary probability estimates along the path from the root node to the leaf node corresponding to the class.
For non-trivial multi-class problems, the space of potential nested dichotomies is very large. An ensemble classifier can be formed by choosing suitable decompositions from this space. In the original formulation of ensembles of nested dichotomies, decompositions are sampled with uniform probability [@frank2004ensembles], but several other more sophisticated methods for splitting the set of classes have been proposed [@dong2005ensembles; @duarte2012nested; @leathart2016building]. Superior performance is achieved when ensembles of nested dichotomies are trained using common ensemble learning methods like bagging or boosting [@rodriguez2010forests].
In this paper, we describe a simple method that can improve the predictive performance of nested dichotomies by considering several splits at each internal node. Our technique can be applied to nested dichotomies built with almost any subset selection method, only contributing a constant factor to the training time and no additional cost when obtaining predictions. It has a single hyperparameter $\lambda$ that gives a trade-off between predictive performance and training time, making it easy to tune for a given learning problem. It is also very easy to implement.
The paper is structured as follows. First, we describe existing methods for class subset selection in nested dichotomies. Following this, we describe our method and provide a theoretical expectation of performance improvements. We then discuss related work, before presenting and discussing empirical results for our experiments. Finally, we conclude and discuss future research directions.
Class Subset Selection Methods\[sec:subset\_selection\_methods\]
================================================================
At each internal node $i$ of a nested dichotomy, the set of classes present at the node $\mathcal{C}_i$ is split into two non-empty, non-overlapping subsets, $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$. In this section, we give an overview of existing class subset selection methods for nested dichotomies. Note that other methods than those listed here have been proposed for constructing nested dichotomies—these are not suitable for use with our method and are discussed later in Related Work.
Random Selection
----------------
The most basic form of class subset selection method, originally proposed in [@frank2004ensembles], is to split the set of classes into two subsets such that each member of the space of nested dichotomies has an equal probability of being sampled. This approach has several attractive qualities. It is simple to compute, and does not scale with the size of the dataset, making it suitable for datasets of any size. Furthermore, for an $n$-class problem, the number of possible nested dichotomies is very large, given by the recurrence relation $$\begin{aligned}
T(n) = (2n-3) \times T(n-1)\end{aligned}$$
where $T(1) = 1$. This ensures that, in an ensemble of nested dichotomies, there is a high level of diversity amongst ensemble members. We refer to this function that relates the number of classes to the size of the sample space of nested dichotomies for a given subset selection method as the *growth function*. Growth functions for each method discussed in this section are compared in Figure \[fig:growth\].
Balanced Selection
------------------
An issue with random selection is that it can produce very unbalanced tree structures. While the number of internal nodes (and therefore, binary models) is the same in any nested dichotomy for the same number of classes, an unbalanced tree often implies that the internal binary models are trained on large datasets near the leaves, which has a negative effect on the time taken to train the full model. Deeper subtrees also provide more opportunity for estimation errors to accumulate. Dong *et. al.* mitigate this effect by enforcing $\mathcal{C}_i$ to be split into two subsets $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ such that ${abs}(|\mathcal{C}_{i1}| - |\mathcal{C}_{i2}|) \leq 1$ [@dong2005ensembles]. This has been shown empirically to have little effect on the accuracy in most cases, while reducing the time taken to train nested dichotomies. Balanced selection has greater benefits for problems with many classes.
It is clear that the sample space of random nested dichotomies is larger than that of class balanced nested dichotomies, but it is still large enough to ensure sufficient ensemble diversity. The growth function for class balanced nested dichotomies is given by $$\begin{aligned}
T_{CB}(n) =
\begin{cases}
\frac{1}{2} \binom{n}{n/2} T_{CB}(\frac{n}{2}) T_{CB}(\frac{n}{2}), & \text{if } n \text{ is even} \\
\binom {n}{(n+1)/2} T_{CB}(\frac{n+1}{2}) T_{CB}(\frac{n-1}{2}), & \text{if } n \text{ is odd} \\
\end{cases}\end{aligned}$$
where $T_{CB}(2) = T_{CB}(1) = 1$ [@dong2005ensembles].
Dong *et. al.* also explored a form of balancing where the amount of data in each subset is roughly equal, which gave similar results for datasets with unbalanced classes [@dong2005ensembles].
Random-Pair Selection
---------------------
Random-pair selection provides a non-deterministic method of creating $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ that groups similar classes together [@leathart2016building]. In random-pair selection, the base classifier is used directly to identify similar classes in $\mathcal{C}_i$. First, a random pair of classes $c_1, c_2 \in \mathcal{C}_i$ is selected, and a binary classifier is trained on just these two classes. Then, the remaining classes are classified with this classifier, and its predictions are stored as a confusion matrix $M$. $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ are constructed by $$\begin{aligned}
\mathcal{C}_{i1} &= \{ c \in \mathcal{C}_i \setminus \{c_1, c_2\} : M_{c, c_1} \leq M_{c, c_2} \} \cup \{c_1\} \\
\mathcal{C}_{i2} &= \{ c \in \mathcal{C}_i \setminus \{c_1, c_2\} : M_{c, c_1} > M_{c, c_2} \} \cup \{c_2\}\end{aligned}$$ where $M_{i,j}$ is defined as the number of examples of class $j$ that were classified as class $i$ by the binary classifier. In other words, a class is assigned to $\mathcal{C}_{i1}$ if it is less frequently confused with $c_1$ than with $c_2$, and to $\mathcal{C}_{i2}$ otherwise. Finally, the binary classifier is re-trained on the new meta-classes $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$. This way, each binary split is more easily separable for the base learner than a completely random split, but also exhibits a degree of randomness, which leads to diverse and high-performing ensembles.
Due to the fact that the size of the sample space of nested dichotomies under random-pair selection is dependent on the dataset and base learner (different initial random pairs may lead to the same split), it is not possible to provide an exact expression for the growth function $T_{RP}(n)$; using logistic regression as the base learner [@leathart2016building], it has been empirically estimated to be $$\begin{aligned}
T_{RP}(n) = p(n)T_{RP}(\frac{n}{3})T_{RP}(\frac{2n}{3}) \label{eqn:random_pair_estimation}\end{aligned}$$ where $$\begin{aligned}
p(n) = 0.3812n^2 - 1.4979n + 2.9027\end{aligned}$$ and $T_{RP}(2) = T_{RP}(1) = 1$.
Multiple Subset Evaluation\[sec:multiple\_subset\_selection\]
=============================================================
In existing class subset selection methods, at each internal node $i$, a single class split $(\mathcal{C}_{i1}, \mathcal{C}_{i2})$ of $\mathcal{C}_i$ is considered, produced by some splitting function $S(\mathcal{C}_i) : \mathbb{N}^n \rightarrow \mathbb{N}^a \times \mathbb{N}^b$ where $a+b=n$. Our approach for improving the predictive power of nested dichotomies is a simple extension. We propose to, at each internal node $i$, consider $\lambda$ subsets $\{(\mathcal{C}_{i1}, \mathcal{C}_{i2})_1 \dots (\mathcal{C}_{i1}, \mathcal{C}_{i2})_\lambda\}$ and choose the split for which the corresponding model has the lowest training root mean squared error (RMSE). The RMSE is defined as the square root of the Brier score [@brier1950verification] divided by the number of classes: $$\textrm{RMSE} = \sqrt{\frac{1}{nm}\sum_{i=1}^n \sum_{j=1}^m (\hat{y}_{ij} - y_{ij})^2 }$$ where $n$ is the number of instances, $m$ is the number of classes, $\hat{y}_{ij}$ is the estimated probability that instance $i$ is of class $j$, and $y_{ij}$ is $1$ if instance $i$ actually belongs to class $j$, and $0$ otherwise. RMSE is chosen over other measures such as classification accuracy because it is smoother and a better indicator of generalisation performance. Previously proposed methods with single subset selection can be considered a special case of this method where $\lambda = 1$.
Although conceptually simple, this method has several attractive qualities, which are now discussed.
#### Predictive Performance.
It is clear that by choosing the best of a series of models at each internal node, the overall performance should improve, assuming the size of the sample space of nested dichotomies is not hindered to the point where ensemble diversity begins to suffer.
#### Generality.
Multiple subset evaluation is widely applicable. If a subset selection method $S$ has some level of randomness, then multiple subset evaluation can be used to improve the performance. One nice feature is that advantages pertaining to $S$ are retained. For example, if class-balanced selection is chosen due to a learning problem with a very high number of classes, we can boost the predictive performance of the ensemble while keeping each nested dichotomy in the ensemble balanced. If random-pair selection is chosen because the computational budget for training is high, then we can improve the predictive performance further than single subset selection in conjunction with random-pair selection.
#### Simplicity.
Implementing multiple subset evaluation is very simple. Furthermore, the computational cost for evaluating multiple subsets of classes scales linearly in the size of the tuneable hyperparameter $\lambda$, making the tradeoff between predictive performance and training time easy to navigate. Additionally, multiple subset evaluation has no effect on prediction times.
Higher values of $\lambda$ give diminishing returns on predictive performance, so a value that is suitable for the computational budget should be chosen. When training an ensemble of nested dichotomies, it may be desirable to adopt a *class threshold*, where single subset selection is used if fewer than a certain number of classes is present at an internal node. This reduces the probability that the same subtrees will appear in many ensemble members, and therefore reduce ensemble diversity. In the lower levels of the tree, the number of possible binary problems is relatively low (Fig. \[fig:growthrate\]).
\[sec:growth\_functions\_effect\]Effect on Growth Functions
-----------------------------------------------------------
Performance of an ensemble of nested dichotomies relies on the size of the sample space of nested dichotomies, given an $n$-class problem, to be relatively large. Multiple subset evaluation removes the $\lambda-1$ class splits that correspond to the worst-performing binary models at each internal node $i$ from being able to be used in the tree. The effect of multiple subset evaluation on the growth function is non-deterministic for random selection, as the sizes of $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ affect the values of the growth function for the subtrees that are children of $i$. The upper bound occurs when all worst-performing splits isolate a single class, and the lower bound is given when all worst-performing splits are class-balanced. Class-balanced selection, on the other hand, is affected deterministically as the size of $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$ are the same for the same number of classes.
Growth functions for values of $\lambda \in \{1, 3, 5, 7\}$, for random, class balanced and random-pair selection methods, are plotted in Figure \[fig:growthrate\]. The growth curves for random and class balanced selection were generated using brute-force computational enumeration, while the effect on random-pair selection is estimated.
{width="95.00000%"}
Analysis of error\[sec:theoretical\]
------------------------------------
In this section, we provide a theoretical analysis showing that performance of each internal binary model is likely to be improved by adopting multiple subset evaluation. We also show empirically that the estimates of performance improvements are accurate, even when the assumptions are violated.
Let $E$ be a random variable for the training root mean squared error (RMSE) for some classifier for a given pair of class subsets $\mathcal{C}_{i1}$ and $\mathcal{C}_{i2}$, and assume $E \sim N(\mu, \sigma^2)$ for a given dataset under some class subset selection scheme. For a given set of $\lambda$ selections of subsets $\mathcal{S} = \{(\mathcal{C}_{i1}, \mathcal{C}_{i2})_1, \dots, (\mathcal{C}_{i1}, \mathcal{C}_{i2})_\lambda\}$ and corresponding training s $\mathcal{E} = \{E_1, \dots, E_\lambda\}$, let $\hat{E}_\lambda = min(\mathcal{E})$. There is no closed form expression for the expected value of $\hat{E}_\lambda$, the minimum of a set of normally distributed random variables, but an approximation is given by $$\mathbb{E}[\hat{E}_\lambda] \approx \mu + \sigma \Phi^{-1} \Bigg( \frac{1-\alpha}{\lambda-2\alpha + 1}\Bigg) \label{eqn:expected_order_statistics}$$ where $\Phi^{-1}(x)$ is the inverse normal cumulative distribution function [@royston1982algorithm], and the *compromise value* $\alpha$ is the suggested value for $\lambda$ given by Harter ([-@harter1961expected]).[^1]
Figure \[fig:norm\_drawn\] illustrates how this expected value changes when increasing values of $\lambda$ from $1$ to $5$. The first two rows show the distribution of $E$ and estimated $\mathbb{E}[\hat{E}_\lambda]$ on the UCI dataset `mfeat-fourier`, for a logistic regression model trained on 1,000 random splits of the class set $\mathcal{C}$. These rows show the training and testing RMSE respectively, using 90% of the data for training and the rest for testing. Note that as $\lambda$ increases, the distribution of the train and test error shifts to lower values and the variance decreases.
This reduction in error affects each binary model in the tree structure, so the effects accumulate when constructing a nested dichotomy. The third row shows the distribution of RMSE of 1,000 nested dichotomies trained with multiple subset evaluation on `mfeat-fourier`, using logistic regression as the base learner, considering increasing values of $\lambda$. As expected, a reduction in error with diminishing returns is seen as $\lambda$ increases.
In order to show an example of how the estimate from (\[eqn:expected\_order\_statistics\]) behaves when the error is not normally distributed, the distribution of $E$ for logistic regression trained on the `segment` UCI data is plotted in the bottom row. This assumption is commonly violated in real datasets, as the distribution is often skewed towards zero error. As with the other examples, 1,000 different random choices for $\mathcal{C}_1$ and $\mathcal{C}_2$ were used to generate the histogram. Although the distribution in this case is not very well modelled by a Gaussian, the approximation of $\mathbb{E}[\hat{E}_\lambda]$ from (\[eqn:expected\_order\_statistics\]) still closely matches the empirical mean. This shows that even when the normality assumption is violated, performance gains of the same degree can be expected. This example is not cherry picked; the same behaviour was observed on the entire collection of datasets used in this study.
Related Work\[sec:related\_work\]
=================================
Splitting a multi-class problem into several binary problems in a tree structure is a general technique that has been referred to by different names in the literature. For example, in a multi-class classification context, nested dichotomies in the broadest sense of the term have been examined as filter trees, conditional probability trees, and label trees. @beygelzimer2009conditional proposed algorithms which build balanced trees and demonstrate the performance on datasets with very large numbers of classes. Filter trees, with deterministic splits [@beygelzimer2009error], as well as conditional probability trees, with probabilistic splits [@beygelzimer2009conditional], were explored. @bengio2010label ([-@bengio2010label]) define a tree structure and optimise all internal classifiers simultaneously to minimise the tree loss. They also propose to learn a low-dimensional embedding of the labels to improve performance, especially when a very large number of classes is present. @melnikov2018effectiveness ([-@melnikov2018effectiveness]) also showed that a method called bag-of-$k$ models—simply sampling $k$ random nested dichotomies and choosing the best one based on validation error—gives competitive predictive performance to the splitting heuristics discussed so far for individual nested dichotomies (i.e., not trained in an ensemble). However, it is very expensive at training time, as $k$ independent nested dichotomies must be constructed and tested on a validation set.
A commonality of these techniques is that they attempt to build a single nested dichotomy structure with the best performance. Nested dichotomies that we consider in this paper, while conceptually similar, differ from these methods because they are intended to be trained in an ensemble setting, and as such, each individual nested dichotomy is not built with optimal performance in mind. Instead, a group of nested dichotomies is built to maximise ensemble performance, so diversity amongst the ensemble members is key [@kuncheva2003measures].
Nested dichotomies based on clustering [@duarte2012nested], are deterministic and used in an ensemble by resampling or reweighting the input. They are built by finding the two classes in $\mathcal{C}_i$ for which the class centroids are furthest from each other by some distance metric. The remainder of the classes are grouped based on the distance of their centroids from the initial two centroids.
@wever2018ensembles ([-@wever2018ensembles]) utilise genetic algorithms to build nested dichotomies. In their method, a population of random nested dichotomies is sampled and runs through a genetic algorithm for several generations. The final nested dichotomy is chosen as the best performing model on a held-out validation set. An ensemble of $k$ nested dichotomies is produced by initialising $k$ individual populations, independently evolving each population, and taking the best-performing model from each population.
Experimental Results\[sec:experiments\]
=======================================
All experiments were conducted in WEKA 3.9 [@hall2009weka], and performed with 10 times 10-fold cross validation. We use class-balanced nested dichotomies and nested dichotomies built with random-pair selection and logistic regression as the base learner.
For both splitting methods, we compare values of $\lambda \in \{1,3,5,7\}$ in a single nested dichotomy structure, as well as in ensemble settings with bagging [@breiman1996bagging] and AdaBoost [@freund1996game]. The default settings in WEKA were used for the `Logistic` classifier as well as for the `Bagging` and `AdaBoostM1` meta-classifiers. We evaluate performance on a collection of datasets taken from the UCI repository [@lichman2013uci], as well as the MNIST digit recognition dataset [@lecun1998gradient]. Note that for MNIST, we report results of 10-fold cross-validation over the entire dataset rather than the usual train/test split. Datasets used in our experiments, and their number of classes, instances and features, are listed in Table \[tab:datasets\].
We provide critical difference plots [@demvsar2006statistical] to summarise the results of the experiments. These plots present average ranks of models trained with differing values of $\lambda$. Models producing results that are not significantly different from each other at the 0.05 significance level are connected with a horizontal black bar. Full results tables showing RMSE for each experimental run, including significance tests, are available in the supplementary materials.
Individual Nested Dichotomies
-----------------------------
**Dataset** **Classes** **Instances** **Features**
---------------- ------------- --------------- --------------
audiology 24 226 70
krkopt 18 28056 7
LED24 10 5000 25
letter 26 20000 17
mfeat-factors 10 2000 217
mfeat-fourier 10 2000 77
mfeat-karhunen 10 2000 65
mfeat-morph 10 2000 7
mfeat-pixel 10 2000 241
MNIST 10 70000 784
optdigits 10 5620 65
page-blocks 5 5473 11
pendigits 10 10992 17
segment 7 2310 20
usps 10 9298 257
vowel 11 990 14
yeast 10 1484 9
: \[tab:datasets\]The datasets used in our experiments.
Restricting the sample space of nested dichotomies through multiple subset evaluation is expected to have a greater performance impact on smaller ensembles than larger ones. This is because in a larger ensemble, a poorly performing ensemble member does not have a large impact on the overall performance. On the other hand, in a small ensemble, one poorly performing ensemble member can degrade the ensemble performance significantly. In the extreme case, where a single nested dichotomy is trained, there is no need for ensemble diversity, so a technique for improving the predictive performance of an individual nested dichotomy should be effective. Therefore, we first compare the performance of single nested dichotomies for different values of $\lambda$.
Figure \[fig:cd\_individual\] shows critical difference plots for both subset selection methods. Class balanced selection shows a clear trend that increasing $\lambda$ improves the RMSE, with the average rank for $\lambda=1$ being exactly 4. For random-pair selection, choosing $\lambda=3$ is shown to be statistically equivalent to $\lambda=1$, while higher values of $\lambda$ give superior results on average.
Ensembles of Nested Dichotomies
-------------------------------
Typically, nested dichotomies are utilised in an ensemble setting, so we investigate the predictive performance of ensembles of ten nested dichotomies with multiple subset evaluation, with bagging and AdaBoost employed as the ensemble methods.
### Class Threshold. {#class-threshold. .unnumbered}
[l]{}\
As previously discussed, the number of binary problems is reduced when multiple subset evaluation is applied. This could have negative a effect on ensemble diversity, and therefore potentially reduce predictive performance. To investigate this effect, we built ensembles of nested dichotomies with multiple subset evaluation by introducing a *class threshold*, the number of classes present at a node required to perform multiple subset evaluation, and varying its value from one to seven. We plot the test RMSE, relative to having a class threshold of one, averaged over the datasets from Table \[tab:datasets\], including standard errors, in Figure \[fig:threshold\]. Surprisingly, the RMSE increases monotonically, showing that the potentially reduced ensemble diversity does not have a negative effect on the RMSE for ensembles of this size. Therefore, we use a class threshold of one in our subsequent experiments. However, note that increasing the class threshold has a positive effect on training time, so it may be useful to apply it in practice.
### Number of Subsets. {#number-of-subsets. .unnumbered}
We now investigate the effect of $\lambda$ when using bagging and boosting. Figure \[fig:cd\_bagging\] shows critical difference plots for bagging. Both subset selection methods improve when utilising multiple subset selection. In the case when class-balanced selection is used, as was observed for single nested dichotomies, the average ranks across all datasets closely correspond to the integer values, showing that increasing the number of subsets evaluated consistently improves performance. For random-pair selection, a more constrained subset selection method, each value of $\lambda > 1$ is statistically equivalent and superior to the single subset case.
The critical difference plots in Figure \[fig:cd\_boosting\] (top) show boosted nested dichotomies are significantly improved by increasing the number of subsets sufficiently when class-balanced nested dichotomies are used. Results are less consistent for random-pair selection, with few significant results in either direction. This is reflected in the critical differences plot (Fig. \[fig:cd\_boosting\], bottom), which shows single subset evaluation statistically equivalent to multiple subset selection for all values of $\lambda$, with $\lambda = 7$ performing markedly worse on average. As RMSE is based on probability estimates, this may be in part due to poor probability calibration, which is known to affect boosted ensembles [@niculescu2005predicting] and nested dichotomies [@leathart2018calibration].
\
Conclusion\[sec:conclusion\]
============================
Multiple subset selection in nested dichotomies can improve predictive performance while retaining the particular advantages of the subset selection method employed. We present an analysis of the effect of multiple subset selection on expected RMSE and show empirically in our experiments that adopting our technique can improve predictive performance, at the cost of a constant factor in training time.
The results of our experiments suggest that for class-balanced selection, performance can be consistently improved significantly by utilising multiple subset evaluation. For random-pair selection, $\lambda=3$ yields the best trade-off between predictive performance and training time, but when AdaBoost is used, our experiments show that multiple subset evaluation is not generally beneficial.
Avenues of future research include comparing multiple subset evaluation with base learners other than logistic regression. It is unlikely that training RMSE of the internal models will be a reliable indicator when selecting splits based on more complex models such as decision trees or random forests, so other metrics may be needed. Also, it may be beneficial to choose subsets such that maximum ensemble diversity is achieved, possibly through information theoretic measures such as variation of information [@meilua2003comparing]. Existing meta-heuristic approaches to constructing individual nested dichotomies like genetic algorithms [@lee2003binary; @wever2018ensembles] could also be adapted to optimise ensembles in this way.
Acknowledgements
================
This research was supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.
[^1]: Appropriate values for $\alpha$ for a given $\lambda$ can be found in Table 3 of [@harter1961expected].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The recent results of 13 TeV ATLAS and CMS di-photon searches show an excess at di-photon invariant mass of 750 GeV. We look for possible explanation of this within minimal left right symmetric model (MLRSM). The possible candidate is a neutral Higgs of mass 750 GeV that can decay to di-photon via charged Higgs and right handed gauge boson loop. However, the cross-section is not consistent with the ATLAS and CMS results. We then discuss one possible variation of this model with universal seesaw for fermion masses that can explain this excess.'
author:
- Arnab Dasgupta
- Manimala Mitra
- Debasish Borah
title: 'Minimal Left-Right Symmetry Confronted with the 750 GeV Di-photon Excess at LHC'
---
Introduction
============
The recently reported 13 TeV center of mass energy data of the large hadron collider (LHC) experiment have pointed towards the existence of a resonance of mass about 750 GeV and width around 45 GeV decaying into two photons [@lhcrun2a; @atlasconf; @CMS:2015dxe]. The ATLAS collaboration has reported the presence of this 750 GeV resonance from their $3.2 \; \text{fb}^{-1}$ data with a statistical significance of $3.9\sigma$ ($2.3\sigma$ including look-elsewhere effects) whereas CMS collaboration has reported the same with a significance of $2.6\sigma$ from their $2.6\; \text{fb}^{-1}$ data. Apart from the mass and decay width, these two experiments have also measured the cross section $\sigma(pp\rightarrow \gamma \gamma)$ to be $10\pm3$ fb (ATLAS) and $6\pm3$ fb (CMS). The large decay width as well as the sizeable cross section have made it a challenging task to come up with beyond standard model (BSM) frameworks which can accommodate it.
Although the reported signal could well be a statistical fluctuation, it has drawn significant attention from the particle physics community leading to a large number of interesting possible explanations including two Higgs doublet models, additional scalars coupling to vector like fermions, extra dimensions, dark matter among others as well as the implications of this signal reported in the works [@750pheno16Dec; @750GeVsinglet; @750pheno17Dec; @750pheno18Dec; @750pheno21Dec; @750pheno22Dec; @750pheno23Dec; @750pheno24Dec; @LR750GeV; @750pheno25Dec; @750pheno29Dec]. In this work, we try to scrutinize one of the very popular BSM framework, known as the left right symmetric model [@lrsm; @lrsmpot] in the light of the reported ATLAS and CMS results. Very recently, this model have been analyzed in the context of several other excesses: such as 2.8$\sigma$ excess in the $eejj$ final state reported by CMS [@Khachatryan:2014dka; @Gluza:2015goa], $3.4 \sigma$ diboson excess reported by ATLAS [@lrdiboson; @Dobrescu:2015qna] and [@ATLASdiboson] and the dijet results [@lrdijet] and [@CMSdijet].
The MLRSM model has few additional Higgs states, where few of the scalars can have lower than TeV scale masses. The neutral Higgs state which has 750 GeV mass, can decay to di-photon via charged Higgs and gauge boson loop. We compute its production cross section at LHC and the branching ratio into two photons. We do a parameter scan of the model by varying the different parameters of the potential, with a fixed symmetry breaking scale. The computed cross section $\sigma(pp \to H^0_2 \to \gamma \gamma)$ is way below the observed one with a few fb. We show that the minimal version of this model with the neutral Higgs states as 750 GeV resonance can not explain the observed signal.
We then consider the possible modification to the MLRSM in order to explain the observed signal. The easiest modification is the addition of vector like quarks which can couple directly to the 750 GeV neutral scalar. This will not only enhance the production cross section but also the partial decay width into two photons. This scenario has already been explored within several different models mentioned above. Instead of arbitrarily adding vector like fermions into the MLRSM to explain the observed signal, we try to focus on the possibility of having these exotic fermions to serve another purposes. One possibility could be to realize these fermions within some higher fermion representations of grand unified theories like SO(10). Since TeV scale MLRSM does not give rise to gauge coupling unification at high energy scale, it will be very natural to have these exotic fermions at TeV scale which not only arise naturally within SO(10) framework but also can help to achieve gauge coupling unification. Another interesting motivation for additional vector like fermions is their role in generating masses of the known standard model fermions. This can happen for example, if the standard model Higgs can not directly couple to the left and right handed fermions of the model, but can do so only through additional heavy fermions. There is one such a version of LRSM where such vector like fermions have to be incorporated in order to generate observed fermion masses [@VLQlr; @univSeesawLR]. This model was studied later in the context of cosmology [@gulr] and neutrinoless double beta decay [@lr0nu2beta]. The model gives rise to fermion masses through a universal seesaw framework where the standard model fermion masses arise after integrating out heavy fermions. Leaving the possibility of having new physics source within a grand unified theory framework to a future work, here we focus on the LRSM with universal seesaw for fermion masses. We consider a 750 GeV neutral singlet scalar which can couple to vector like fermions and can give rise to the observed signal.
This paper is organized as follows. In section \[sec1\], we briefly discuss the MLRSM and then discuss the possibility of a 750 GeV neutral scalar in view of the LHC signal in section \[sec2\]. In section \[sec3\], we briefly discuss the LRSM with universal seesaw and in section \[sec4\] we study the possibility of explaining the LHC signal of a 750 GeV resonance decaying into two photons. We finally conclude in section \[sec5\].
Minimal Left-Right Symmetric Model (MSLRM) {#sec1}
==========================================
Left-Right Symmetric Model [@lrsm; @lrsmpot] is one of the very well motivated BSM frameworks where the gauge symmetry of the electroweak theory is extended to $SU(3)_c \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$. The right handed fermions which are singlets under the $SU(2)_L$ of SM, transform as doublets under $SU(2)_R$, making the presence of right handed neutrinos natural in this model. The Higgs doublet of the SM is replaced by a Higgs bidoublet to allow couplings between left and right handed fermions, both of which are doublets under $SU(2)_L$ and $SU(2)_R$ respectively. The enhanced gauge symmetry of the model $SU(2)_R \times U(1)_{B-L}$ is broken down to the $U(1)_Y$ of SM by the vacuum expectation value (vev) of additional Higgs scalar, transforming as triplet under $SU(2)_R$ and having non-zero $U(1)_{B-L}$ charge. This triplet gives rise to the Majorana masses of the right handed neutrinos through symmetry breaking. The heavy right handed neutrinos participate in the seesaw mechanism and generate the Majorana masses of the light neutrinos. On the other hand, the left handed Higgs triplet generates Majorana masses of the light neutrinos through type II.
The fermion content of the minimal LRSM is $$Q_L=
\left(\begin{array}{c}
\ u_L \\
\ d_L
\end{array}\right)
\sim (3,2,1,\frac{1}{3}),\hspace*{0.8cm}
Q_R=
\left(\begin{array}{c}
\ u_R \\
\ d_R
\end{array}\right)
\sim (3^*,1,2,\frac{1}{3}),\nonumber$$ $$\ell_L =
\left(\begin{array}{c}
\ \nu_L \\
\ e_L
\end{array}\right)
\sim (1,2,1,-1), \quad
\ell_R=
\left(\begin{array}{c}
\ \nu_R \\
\ e_R
\end{array}\right)
\sim (1,1,2,-1) \nonumber$$ Similarly, the Higgs content of the minimal LRSM is $$\Phi=
\left(\begin{array}{cc}
\ \phi^0_{11} & \phi^+_{11} \\
\ \phi^-_{12} & \phi^0_{12}
\end{array}\right)
\sim (1,2,2,0)
\nonumber$$ $$\Delta_L =
\left(\begin{array}{cc}
\ \delta^+_L/\surd 2 & \delta^{++}_L \\
\ \delta^0_L & -\delta^+_L/\surd 2
\end{array}\right)
\sim (1,3,1,2), \hspace*{0.2cm}
\Delta_R =
\left(\begin{array}{cc}
\ \delta^+_R/\surd 2 & \delta^{++}_R \\
\ \delta^0_R & -\delta^+_R/\surd 2
\end{array}\right)
\sim (1,1,3,2) \nonumber$$ where the numbers in brackets correspond to the quantum numbers with respect to the gauge group $SU(3)_c\times SU(2)_L\times SU(2)_R \times U(1)_{B-L}$. In the symmetry breaking pattern, the neutral component of the Higgs triplet $\Delta_R$ acquires a vev to break the gauge symmetry of the LRSM into that of the SM and then to the $U(1)$ of electromagnetism by the vev of the neutral component of Higgs bidoublet $\Phi$: $$SU(2)_L \times SU(2)_R \times U(1)_{B-L} \quad \underrightarrow{\langle
\Delta_R \rangle} \quad SU(2)_L\times U(1)_Y \quad \underrightarrow{\langle \Phi \rangle} \quad U(1)_{em}$$ The symmetry breaking of $SU(2)_R \times U(1)_{B-L}$ into the $U(1)_Y$ of standard model can also be achieved at two stages by choosing a non-minimal scalar sector. [[We]{}]{} denote the vev of the two neutral components of the bidoublet as $k_1, k_2$ and that of triplets $\Delta_{L, R}$ as $v_{L, R}$. Considering $g_L=g_R$, $k_2 \sim v_L \approx 0$ and $v_R \gg k_1$, the gauge boson masses after symmetry breaking can be written as $$M^2_{W_L} = \frac{g^2}{4} k^2_1, \;\;\; M^2_{W_R} = \frac{g^2}{2}v^2_R$$ $$M^2_{Z_L} = \frac{g^2 k^2_1}{4\cos^2{\theta_w}} \left ( 1-\frac{\cos^2{2\theta_w}}{2\cos^4{\theta_w}}\frac{k^2_1}{v^2_R} \right), \;\;\; M^2_{Z_R} = \frac{g^2 v^2_R \cos^2{\theta_w}}{\cos{2\theta_w}}$$ where $\theta_w$ is the Weinberg angle. After the symmetry breaking, four neutral scalars emerge, two from the bidoublet $(H^0_0, H^0_1)$, one from right handed triplet $(H^0_2)$ and another from left handed triplet $(H^0_3)$. Similarly there are two neutral pseudoscalars, one from the bidoublet $(A^0_1)$ and another from the left handed triplet $(A^0_2)$. Among the charged scalars, there are two singly charged ones $(H^{\pm}_1, H^{\pm}_2)$ and two doubly charged ones $(H^{\pm \pm}_1, H^{\pm \pm}_2)$.
Under the approximations made above, the scalar masses are given by $$M^2_{H^0_0} = 2 \lambda_1 k^2_1, \, \, \, M^2_{H^0_1} = \frac{1}{2}\alpha_3 v^2_R, \, \, M^2_{H^0_2} = 2\rho_1 v^2_R, \, \, M^2_{H^0_3} = \frac{1}{2}(\rho_3-2\rho_1)v^2_R$$ $$M^2_{A^0_1} = \frac{1}{2}\alpha_3 v^2_R - 2(2\lambda_2-\lambda_3)k^2_1, \, \, M^2_{A^0_2} = \frac{1}{2}v^2_R (\rho_3-2\rho_1), \, \,
M^2_{H^{\pm}_1} = \frac{1}{2} (\rho_3-2\rho_1)v^2_R +\frac{1}{4} \alpha_3 k^2_1$$ $$M^2_{H^{\pm}_2} = \frac{1}{2}\alpha_3 v^2_R + \frac{1}{4} \alpha_3 k^2_1, \, \,
M^2_{H^{\pm \pm}_1} = \frac{1}{2} (\rho_3-2\rho_1)v^2_R+\frac{1}{2}\alpha_3 k^2_1, \, \,
M^2_{H^{\pm \pm}_2} = 2\rho_2 v^2_R +\frac{1}{2} \alpha_3 k^2_1$$ where $\lambda_i, \alpha_i, \rho_i$ are dimensionless couplings of the scalar potential of this model [@lrsmpot]. We take into account the following results and experimental searches that fix the dimensionless parameters.
- In the above, $H^0_0$ can be identified as SM like Higgs of mass 125 GeV, that fixes the coupling $\lambda_1$. Few of the other couplings can be constrained after taking into account the experimental limits on the scalar masses.
- We demand that the Higgs $H^0_2$ has a mass 750 GeV that explain the di-photon bump, that fixes the dimensionless coupling $\rho_1$.
- In order to avoid the flavor changing neutral currents (FCNC) processes, the neutral scalars from bi-doublet $H^0_1, A^0_1$ have to be heavier than 10 TeV [@fcnc], which puts further constraint on $\alpha_3$ for a fixed $v_R$. This also constrain the charged Higgs mass $H^{\pm}_2$ to be heavy.
- The lower bound on the mass of doubly charged scalar $H^{\pm \pm}_1 $ from the multilepton search [@Aad:2014hja] fixes the coupling $\rho_3$. This automatically fixes the mass of singly charged scalar $H^{\pm}_1$ as well.
This leaves only two free parameters $ (2\lambda_2-\lambda_3)$ and $\rho_2$ in the expressions for scalar masses to be varied arbitrarily. The other couplings in the full scalar potential are also free to be varied and are not affected by the chosen spectrum of scalar masses.
750 GeV neutral scalar in MLRSM {#sec2}
===============================
Among the neutral physical scalars in MLRSM, the $H^0_0$ is the standard model like Higgs with mass $125$ GeV. The other two neutral (pseudo) scalars from the bidoublet namely $H^0_1, A^0_1$ are heavier than at least 10 TeV due to tight constraints from FCNC. However, the other three neutral scalars $(H^0_2, H^0_3, A^0_2)$ originating from the scalar triplets can lie at 750 GeV. We consider $H^0_2 \equiv \Delta^0_R$ as a possible candidate for a 750 GeV neutral scalar decaying into two photons.
[ As the heavy Higgs $H^0_2$ does not directly couple with gluons or quarks, hence its production will be governed by the mixing with the SM Higgs. The production cross-section of the $750$ GeV heavy Higgs at 13 TeV LHC is [@physrepdjouadi] $$\sigma (p p \to H^0_2) \sim \theta^2 \times 0.85 \,\rm{pb},$$ ]{} where $\theta \sim \alpha \frac{k_1}{M_{H^0_2}}$ is the mixing between SM Higgs state and the $H^0_2$, and we have considered the dimensionless parameters $\alpha_1=\alpha_2=\alpha$.
As the neutral scalar $H^0_2$ does not couple to a pair of charged fermions at the tree level, the only way it can decay into two photons is through a charged gauge or scalar boson loop. This can happen through a loop containing $W_R$ bosons or one of the charged scalars $H^{\pm}_1, H^{\pm}_2, H^{\pm \pm}_1, H^{\pm \pm}_2$. The total production cross-section of $p p \to H^0_2 \to \gamma \gamma$ is $$\sigma(p p \to H^0_2 \to \gamma \gamma) \sim \theta^2 \times 0.85 \times \rm{Br}(H^0_2 \to \gamma \gamma) \, \rm{pb}.$$
We first consider the following benchmark values of the scalar masses and compute the decay widths. Following this we will provide a full parameter scan. $$m_{H^0_0} = 125 \; \text{GeV}, \;\; m_{H^0_2} = 750 \; \text{GeV}, \;\; m_{H^{\pm}_1} = 380 \; \text{GeV}, \;\; m_{H^0_1} = m_{A^0_1} = m_{H^{\pm}_2} = 10\; \text{TeV}.$$ $$m_{H^{\pm \pm}_1} = 465 \; \text{GeV}, \;\; m_{H^{\pm \pm}_2} = 380 \; \text{GeV}.$$ The values are chosen in such a way to satisfy the current experimental bounds. We show the partial decay width of $H^0_2$ to $\gamma \gamma$ and di-Higgs modes for these illustrative points in the parameter space. Few comments are in order.
- We show the partial decay width to two photons through the $W_R$ loop in Fig. \[fig2\], where the $W_R$ mass has been set to be 3 TeV-in agreement with the collider constraint [@Khachatryan:2014dka]. The partial decay width through gauge boson loop is extremely suppressed and can not explain the diphoton signal.
- We show the partial decay width of $H^0_2 \rightarrow \gamma \gamma$ through charged Higgs loop in Fig. \[fig1\] as a function of the parameter $\alpha$. This also decides the decay mode $H^0_2 \rightarrow H^0_0 H^0_0$. We have set the charged Higgs masses to the lowest possible value, which is consistent with collider searches. The masses of the neutral scalars have been set to 10 TeV and $W_R$ mass as 3 TeV.
- The mixing parameter $\theta$ can be constrained from di-Higgs search [@dihiggscms], which is $\mathcal{O}(0.3)$ for 750 GeV scalar resonance decaying into di-Higgs with 100$\%$ branching ratio.
- From Fig. \[fig1\], it is evident that for the parameter $\alpha > 0.001$, the partial decay width to di-Higgs will be larger than the di-photon. Including both the gauge boson $W_R$ and charged Higgs, this limit goes to $\alpha \sim 0.01$.
- It is straightforward to see from Fig. \[fig2\] and Fig. \[fig1\] that the partial width of $H^0_2 \rightarrow \gamma \gamma$ through all available charged particles in loop falls below 1 GeV for small $\alpha$ and can not explain the large decay width preferred by the ATLAS and CMS data.
- In the above, we have considered large mass > 1 TeV for the heavy neutrinos, which is consistent with the collider constraint [@Khachatryan:2014dka]. Therefore, the decay of $H^0_2 \to N_R N_R$ is absent. However, we have checked that even with lighter heavy neutrino masses, the total decay width of $H^0_2$ is not in agreement with the di-photon results.
- The partial decay width to $\gamma \gamma$ via top loop is extremely suppressed $10^{-7}$ GeV, while its decay to $t-\bar{t}$ and $W-W$ is 2.78 GeV and 11.62 GeV for mixing $\theta \sim 0.3$.
[[ The total cross section $\sigma (pp\rightarrow H^0_2 \rightarrow \gamma \gamma) = \sigma(pp\rightarrow H^0_2) \text{BR}(H^0_2 \rightarrow \gamma \gamma)$ for the above mentioned parameter values is $ < $ 1 fb for smaller branching ratio, and clearly can not take into account the required cross-section $\sim $ 10 fb to fit the ATLAS and CMS data. ]{}]{} Considering the other two neutral scalars $H^0_3$ and $A^0_2$ as potential 750 GeV candidate will not significantly improve the situation. This requires beyond MLRSM physics to explain the recently observed di-photon excess at 750 GeV by ATLAS and CMS.
[ [Following the above discussion with a particular set of parameters, we now provide a full parameter scan where we vary the parameters $ \alpha_1=\alpha_2 , \alpha_3, \rho_3, \rho_2, 2\lambda_2-\lambda_3$ in the following ranges.]{}]{} $$\alpha_1 , \alpha_3, \rho_3, \rho_2, 2\lambda_2-\lambda_3 \equiv 10^{-4} - 4 \pi.$$
In Fig. \[figx\] and Fig. \[figdw\], we show the total cross-section and decay width for these ranges of parameters with $\alpha_1=\alpha_2=\alpha$. The blue region corresponds to the following mass cut on the scalars: $$m_{H^{\pm}_{1}}, m_{H^{\pm \pm}_{2}} \ge 380 \; \text{GeV}, \;\; m_{H^0_1} = m_{A^0_1} = m_{H^{\pm}_2} > 10\; \text{TeV}, \, \, m_{H^{\pm \pm}_1} \ge 465 \; \text{GeV}.
\label{masscut}$$ Note that, with the cut on the decay width of $H^0_2 \le 50 $ GeV, the total cross-section is extremely small and can not explain the di-photon result. In addition to this, although for large value of $\alpha$, the cross-section increases, however the branching ratio to di-Higgs also increases. Hence to avoid any constraint from di-Higgs channel, $\alpha$ should be small.
LRSM with Universal Seesaw (LRSM-US) {#sec3}
====================================
The fermion content of the LRSM with universal seesaw is the extension of the MLRSM fermion content by the following vector like fermions $$U_{L} (3, 1, 1, \frac{4}{3}), \;\; U_{R} (3^*, 1, 1, \frac{4}{3})\;\; D_{L} (3, 1, 1, -\frac{2}{3}), \;\;D_{R} (3^*, 1, 1, -\frac{2}{3})$$ $$E_{L,R} (1,1,1, -2), \;\; N_{L,R} (1,1,1,0)$$ each of which comes in three different copies corresponding to the three fermion generations of the MLRSM or the standard model. The presence of these extra fermions is necessary due to the fact that the usual scalar sector of the MSLRM is replaced by the following scalar fields $$H_L (1, 2, 1, -1), \;\;H_R (1,1,2,-1), \;\; \sigma (1,1,1,0)$$ Due to the absence of the usual bidoublet, the left and right handed fermion doublets of the MSLRM can not directly couple to each other. However, they can couple to the scalar fields $H_{L,R}$ via the additional vector like fermions. $$\begin{aligned}
\mathcal{L} & \supset Y_U (\bar{Q_L} H^{\dagger}_L U_L+\bar{Q_R} H^{\dagger}_R U_R) + Y_D (\bar{Q_L} H_L D_L+\bar{Q_R} H_R D_R) +M_U \bar{U_L} U_R+ M_D \bar{D_L} D_R\nonumber \\
& Y_E (\bar{\ell_L} H_L E_L+\bar{\ell_R} H_R E_R) +M_E \bar{E_L} E_R+ \text{h.c.}\end{aligned}$$ where we have ignored the terms corresponding to neutrino masses. For details of the origin of neutrino masses, one may refer to the discussions in [@gulr]. After integrating out the heavy fermions, the charged fermions of the standard model develop Yukawa couplings to the scalar doublet $H_L$ as follows $$y_u = Y_U \frac{v_R}{M_U} Y^T_U, \;\;y_d = Y_D \frac{v_R}{M_D} Y^T_D, \;\;y_e = Y_E \frac{v_R}{M_E} Y^T_E$$ where $v_R$ is the vev of the neutral component of $H_R$. The apparent seesaw then can explain the observed mass hierarchies among the three generations of fermions. The non-zero vev of the neutral component of $H_R$ also breaks the $SU(2)_R\times U(1)_{B-L}$ symmetry of the model into $U(1)_Y$ of the standard model. The left handed Higgs doublet can acquire a non-zero vev $v_L$ at a lower energy to induce electroweak symmetry breaking. However, the left-right symmetry of the theory forces one to have the same vev for both $H_L$ and $H_R$ that is, $v_L=v_R$ which is unacceptable from phenomenological point of view. To decouple these two symmetry breaking scales, the extra singlet scalar $\sigma$ is introduced into the model. This field is odd under the discrete left-right symmetry and hence couple to the two scalar doublets with a opposite sign. After this singlet acquires a non-zero vev at high scale, this generates a difference between the effective mass squared of $H_L$ and $H_R$ which ultimately decouples the symmetry breaking scales.
750 GeV neutral scalar in LRSM-US
=================================
[^1] \[sec4\] The LRSM with universal seesaw has three neutral scalars: one from $H_L$, one from $H_R$ and one from the singlet $\sigma$. Now the neutral scalar part of $H_L$ is the standard model like Higgs with mass 125 GeV. On the other hand the neutral scalars from $H_R$ is supposed to be heavy in order to avoid dangerous flavor changing neutral currents. The scalar $\sigma$ is naturally heavy as it is responsible for discrete left-right symmetry breaking at a scale above $v_R$. To allow the possibility of a neutral 750 GeV scalar, we add another singlet $\zeta$ into the model which can couple to the vector like quarks and leptons as $Y_{f\zeta} \zeta \bar{f_L}f_R$ where $f$ is the vector like fermion. This essentially boils down to the singlet scalar resonance coupled to additional vector like fermions as an explanation of 750 GeV di-photon excess put forward by [@750GeVsinglet].
The singlet scalar can be produced dominantly in pp collisions by two different ways: (a) through mixing with the standard model Higgs and (b) through gluon gluon fusion via new vector like quarks. The singlet scalar can decay into two photons through the vector like fermions in a loop. Since the mixing with the standard model Higgs is constrained, we assume the corresponding production channel to be negligible. We then consider the production of the singlet scalar $\zeta$ in proton proton collisions dominantly through gluon gluon fusion with the vector like quarks in loop. This singlet scalar can decay either into two photons or two gluons or one photon, one Z boson at one loop level whereas the tree level decay into a pair of standard model Higgs can be neglected assuming small mixing. Using the loop level production cross section and decay width expressions given in [@physrepdjouadi], we calculate for what values of vector like fermion masses $m_f$ and their Yukawa couplings $Y_f$, the desired cross section $\sigma(pp\rightarrow \zeta \rightarrow \gamma \gamma)$ can be obtained. For simplicity we consider all the quark and lepton masses and their Yukawa couplings degenerate. Since the masses of vector like leptons are less constrained than that of vector like quarks, we consider vector like lepton masses to be half of vector like quark masses. It should be noted that vector like quark masses are restricted to be $m_q \geq 750-920$ GeV depending on the particular channel of decay [@VLQconstraint] whereas this bound gets relaxed to $m_q \geq 400$ GeV [@VLQconstraint2] for long lived vector like quarks. Further constraints on vector like quarks can be found in [@vlqhandbook]. The constraints on vector like leptons are much weaker $m_l \geq 114-176$ GeV and allows the possibility of the 750 GeV scalar to decay into them at tree level [@VLLconstraint]. We however, do not allow tree level decay of $\zeta$ into vector like leptons which will reduce the branching ratio $\text{BR} (\zeta \rightarrow \gamma \gamma)$. Considering $m_l = m_q/2$, we then constrain the corresponding Yukawa couplings $Y_q, Y_l$ from the requirement of producing the observed signal. The restricted Yukawa couplings for some benchmark values of quark masses are shown in figure \[figyukawa\].
Further constraints on the model comes in terms of the Yukawa couplings involved in the seesaw relations for fermion masses discussed in the previous section. For standard model fermion mass $m_f$, the Yukawa couplings are constrained as $$\frac{y^2 v_R}{M} = \frac{m_f}{v_L}$$ where $M$ is the heavy vector like fermion mass, $v_R$ is the $SU(2)_R$ breaking scale and $v_L=246$ GeV is the electroweak symmetry breaking scale. If $m_f$ is top quark mass and $v_R \approx 6$ TeV, then for $M=1500$ GeV, the corresponding Yukawa couplings are constrained to be $y \approx 0.42$. However, fitting with all the fermion masses will require non-degenerate heavy vector like fermion masses. Another constraints comes from the mixing of these heavy fermions with the standard model fermions. For the case of vector like quarks, such mixings with standard model quarks are constrained to be small $\theta \leq 0.1$ from precision measurements of electroweak parameters [@vlqhandbook]. Parametrising the heavy-light quark mixing from the seesaw relations as $\sin^2\theta \approx \frac{y^2 v_R v_L}{M^2}$ and using the constraints $\theta \leq 0.1, \frac{y^2 v_R v_L}{M} = m_f$, we get the constraints on the heavy quark masses as $M \geq 100 m_f$. This is possible to achieve for $M=1500$ GeV in case of bottom and lighter quark masses. But for top quark seesaw, the corresponding heavy vector like quark has to be much heavier than 1500 GeV considered in this simple analysis.
It should be noted that there are 6 vector like quarks and 3 vector like charged leptons in the model. However, the singlet scalar can not decay into them at tree level. These exotic fermions only appear at one loop to allow the scalar to decay into $gg, \gamma \gamma$ or $\gamma Z$. However, such loop level decay is not enough to generate the total decay width observed by the LHC. Similar observations were also made by [@750GeVsinglet]. If the LHC confirms the measured decay width in future, this will invite further modification to the left-right model considered in this work.
Conclusion {#sec5}
===========
We have studied the minimal left right symmetric extension of the standard model in view of the latest LHC observations of a 750 GeV neutral resonance decaying into two photons with a cross section of around 10 fb. Since the extra neutral scalars (in addition to the 125 GeV Higgs boson) from the bidoublet of MLRSM are very heavy to be in agreement with flavor constraints, we consider the neutral scalar $H^0_2$ from one of the triplet scalars of the model namely, $\Delta_R$. The discussion will be similar for the neutral component of $\Delta_L$. Since the triplet does not couple to quarks, we consider the production of this scalar only through its mixing with the standard model Higgs. We then consider the possible decay of $H^0_2$ and calculate the total as well as partial decay widths. After incorporating the LHC constraints on neutral as well as charged scalar masses, we find that the total cross section $\sigma(pp\rightarrow H^0_0 \rightarrow \gamma \gamma)$ remain below the observed 10 fb signal, after putting constrain on the decay width of $H^0_2 \le 50$ GeV. The cross section is maximal, close to 1 fb only for very high values of the dimensionless parameter $\alpha = \alpha_1 = \alpha_2$ of the scalar potential. This parameter also decides the size of the $H^0_2$ mixing with the standard model like Higgs $H^0_0$ and hence constrained to be $\alpha \leq 0.3 M_{H^0_2}/k_1$, to avoid collider constraint. Thus, the di-photon cross section will be much smaller than 1 fb after taking the constraint on $H^0_0-H^0_2$ mixing into account. It is observed from Fig. \[figx\], that after taking the experimental lower bounds on neutral and charged scalar masses into account, the total cross section gets shifted to higher side. This is due to the fact that, experimental lower bounds on scalar masses also restricts relevant dimensionless parameters to high values for fixed $v_R$. As the same parameters also appear in the decay widths, they increases the total cross section. We also observe that the neutral scalar $H^0_2$ can have a sizeable total decay width $\approx 50$ GeV as observed by the LHC for allowed parameter space though it can not give rise to the 10 fb di-photon signal simultaneously.
We then briefly mention another possible left right model with universal seesaw for fermion masses. Due to the existence of additional vector like fermions, the production of a neutral scalar and its decay into two photons can he enhanced at the same time. By taking some benchmark values of additional fermion masses, we show how their couplings to a neutral 750 GeV scalar get restricted from the requirement of producing a 10 fb di-photon signal. The neutral scalar in such a scenario however, fails to give rise to the large decay width observed by experiment. Thus, if the di-photon cross section as well as the decay width are both confirmed by future LHC data, then further improvement of the left right symmetric models discussed in this work will be required. We leave such an investigation to future work.
The work of M.M was supported by the DST-INSPIRE Faculty grant. The authors would like to thank Ketan M. Patel, IISER Mohali for very useful discussions.
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[^1]: While preparing this manuscript, we found a similar model proposed by [@LR750GeV] with additional field content.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Gesture recognition is a challenging problem in the field of biometrics. In this paper, we integrate Fisher criterion into Bidirectional Long-Short Term Memory (BLSTM) network and Bidirectional Gated Recurrent Unit (BGRU), thus leading to two new deep models termed as F-BLSTM and F-BGRU. Both Fisher discriminative deep models can effectively classify the gesture based on analyzing the acceleration and angular velocity data of the human gestures. Moreover, we collect a large Mobile Gesture Database (MGD) based on the accelerations and angular velocities containing 5547 sequences of 12 gestures. Extensive experiments are conducted to validate the superior performance of the proposed networks as compared to the state-of-the-art BLSTM and BGRU on MGD database and two benchmark databases (*i.e.* BUAA mobile gesture and SmartWatch gesture).'
address:
- 'Department of Automation, Beihang University, Beijing, China'
- 'Department of Computer Science, China University of Mining and Technology, Beijing, China'
- 'University of Central Florida, Orlando, FL, USA.'
- 'Nortumbria Univesity, Newcastle, UK.'
author:
- Chunyu Xie
- Ce Li
- Baochang Zhang
- Chen Chen
- Jungong Han
bibliography:
- 'mybibfile.bib'
title: Deep Fisher Discriminant Learning for Mobile Hand Gesture Recognition
---
Fisher Discriminant,Hand Gesture Recognition ,Mobile Devices
Introduction {#sec:introduction}
============
Towards natural human-computer interaction, the emergence of smartphones has changed our lives and made our lives more convenient. The interaction between people and mobile phones are through the touch screen, camera and microphone. However, due to environmental constraints, these interaction methods suffer from many uncontrollable problems. For example, the video based methods do not work well in the night time. Nowadays, inertial sensors including accelerometer and gyrometer are built in smartphones [@lane2010a; @choi2005; @mantyla2000hand], which can record the signal of hand movements when their devices are in use. Therefore, gesture action using inertial sensors can be easily achieved and understood [@liu2009uwave]. Different from computer vision based gesture recognition, the requirements for gesture recognition based on inertial sensors (*e.g.* accelerometer and gyrometer) are much simpler[@catal2015on]. However, the mobile gesture recognition inevitably encounters several external variations including signal intensity differences (intense versus weak gestures), temporal variations (slow versus fast movements) and physical differences (users’ physical conditions, etc.). In addition, the noise caused by the hardware device also has a severe impact on the recognition performance. Support vector machine (SVM), hidden Markov model (HMM) and dynamic time warping (DTW) have been introduced to solve the problems mentioned above.
Recently, deep learning methods have significantly push the state-of-the-art in human activity recognition. They avoid feature engineering and are able to learn data representations and classifiers simultaneously. Among those, the recurrent neural network (RNN) and Long Short-Term Memory (LSTM)[@hochreiter1997long] are popular sequential modelling methods. They have been successfully applied to many fields such as language modeling[@mikolov2011extensions; @sundermeyer2012lstm; @mesnil2013investigation], image captioning[@vinyals2015show; @xu2015icml], video analysis[@ng2015cvpr; @alahi2016social; @deng2015structure; @ibrahim2015a] and 3D action recognition. These methods can also be utilized to solve our problem, since the mobile gesture signals are sequential data streams of inertial sensors[@shin2016dynamic; @lefebvre2013ANN].
The publicly available databases are very important to researchers for algorithm development and evaluation. However, most of the previous works in the field of mobile gesture recognition were conducted with self-prepared data. And few databases related to mobile gesture recognition are publicly available. In this paper, we first introduce a mobile based gesture recognition benchmark, which can help researchers to evaluate and compare their algorithms by using the same gesture data. The database was collected based on 32 participants (23 males and 9 females), consisting 12 gestures of 5547 repetitions.
Two Fisher discriminative deep models termed F-BLSTM and a variant F-BGRU are proposed for hand gesture recognition on mobile devices. The Fisher criterion which minimizes the [intra-class variations and maximize inter-class variations]{} is incorporated into softmax loss of LSTM, leading a better classification ability to cope with external variations. The flowchart of the gesture recognition approach is shown in Fig.\[fig:flowchart\].
{width="\linewidth" height="0.4\textheight"}
In summary, we make the following contributions:
1. We collect a large mobile gesture database using an Andriod [Huawei]{} device, which is the largest database in published studies for mobile gesture recognition systems.
2. We integrate Fisher criterion into BLSTM network to improve the traditional softmax loss training function. Extensive experiments on our [MGD, BUAA Mobile Gesture database,]{} and a public database are conducted to verify the superior performance of the proposed networks.
The rest of the paper is organized as follows. Section\[sec:relatedwork\] introduces the related works, and Section\[sec:approach\] describes the details of the proposed method. Experiments and results are presented in Section\[sec:experiments\]. Finally, Section\[sec:conclusion\] concludes the paper.
Related Work {#sec:relatedwork}
============
[Gesture recognition has been extensively investigated in the last two decades, and remarkable advances are achieved using inertial sensors in mobile devices[@rekimoto2001iswc; @jang2003signal; @kallio2003online; @bulling2014a]. For the application of human computer interaction, Rekimoto [*et al.*]{}[@rekimoto2001iswc] detected the movement of arm using a specific wearable device. But it is difficult to get high precision in practice because of large size of the equipment. Fresca [*et al.*]{}[@ferscha2007gestural] studied and recognized human gesticulation and the manipulation of graspable and movable everyday artifacts as a potentially effective means for the interaction with smart things. Parsani [*et al.*]{}[@Parsani2009A] designed an embedded system which could analyze and recognize smartphone gestures involving a combination of straight line motions in three dimensions. Roy [*et al.*]{}[@Roy2014Demo] suggested that the walking direction should be detectable through the accelerometer and get blended into various other motion patterns during the act of walking, including up and down bounce, side-to-side sway, swing of arms or legs, *etc.*. They also analyzed the human walking dynamics to estimate the dominating forces and used this knowledge to find the heading direction of the pedestrian. For the application of biological monitoring, Park [*et al.*]{}[@Park2014Poster] demonstrated a very promising application to classify and monitor heartbeats, while Nandakumar [*et al.*]{}[@Nandakumar2015Contactless] monitored sleep apnea using the sensors in smartphones to develop more convenient conditions for gesture recognition[@hoang2013adaptive]. Besides, there are also some other applications.]{} As reported in[@Agrawal2011Using], Agrawal [*et al.*]{}proposed a system called PhonePoint Pen that uses the built-in accelerometer in mobile phones to recognize the human writing. The system, based on Nokia N95 platform, was evaluated through 10 students and 5 hospital patients. Results showed that English characters can be identified with an average accuracy of 91%. The system presented a promising prospect for mobile based gesture recognition. [Mobile gesture recognition has provided new directions and also delivered compelling performance for the application of machine learning. Hofmann [*et al.*]{}[@hofmann1998velocity] proposed a recognition scheme based on Hidden Markov Models (HMM)[@markov] and used discrete HMM (dHMM) to recognize dynamic gestures. The approach essentially divided the input data into different regions and assigned each of them to a corresponding codebook for classifying them with dHMM. The vector codebook was obtained by a clustering method, serving as an unsupervised learning procedure to model the feature vector distribution in the input data space. The experiments were carried out using 500 training gestures with 10 samples per gesture, yielding an accuracy of 95.6% for 100 test gestures.]{} In[@kallio2003online], gestures were captured with a small wireless sensor-box that produced three dimensional acceleration signal. Kallio [*et al.*]{}trained the dHMM model by using five states and a codebook size of eight. They [measured]{} the recognition accuracy of system using four degrees of complexity. In[@kela2006accelerometer-based], an HMM model was trained with five states, achieving a rate of 96.1% accuracy for classifying 8 gestures. Pylv [*et al.*]{}[@pylvanainen2005accelerometer] proposed a method based on continuous HMM (cHMM), which takes correlated time information into consideration. The experiment achieved reliable results, with 96.67% of correct classification on a database of 20 samples for 10 gestures. In[@Zhang2009Hand], multi-stream HMM consisting of EMG and ACC streams was utilized as decision fusion method to recognize hand gestures. For a data set of 18 gestures, each trained with 10 repetitions, the average recognition accuracy was about 91.7% in real application.
[Besides HMM, a few other popular techniques have been used in gesture recognition. Akl [*et al.*]{} proposed a gesture recognition system based primarily on a single 3 dimensional accelerometer, by employing DTW[@akl2010accelerometer-based].]{} The system defined a dictionary of 18 gestures and a database of 3700 repetitions from 7 users[ and got up to 90% classification accuracy in the experiment.]{} David [*et al.*]{}[@Mace2013Accelerometer] proposed two approaches including Naive Bayes and DTW for recognizing four gesture types from five different subjects in the experiment. The results revealed Bayesian classification is better than DTW. Wu [*et al.*]{}[@wu2009gesture] employed multi-class Support Vector Machine (SVM) for user-independent gesture recognition and demonstrated that SVM significantly outperformed other methods including DTW, Naive Bayes and HMM. In[@Hsu2009Integrating], Wang [*et al.*]{} combined LCS and SVM to perform the classification task and achieved the classification accuracy of 93%. Another line of research focuses on the feature extraction and selection. For example, the principle component analysis was used for feature selection and dimensionality reduction in gesture classification[@Marasovic2011Accelerometer]. [In[@He2011Accelerometer], the hybrid features combined short-time energy with Fast Fourier Transform, denoting the fusion of time-domain features and frequency-domain features, were presented for recognizing seventeen complex gestures on cell phone.]{} An average recognition accuracy of 89.89% was obtained using multi-class SVM.
[Driven by the tremendous success of deep learning, the research paradigm has been shifted from traditional approaches to deep learning methods for mobile gesture recognition[@shin2016dynamic; @lefebvre2013ANN].]{} Shin [*et al.*]{}[@shin2016dynamic] developed a dynamic hand gesture recognition technique using recurrent neural network (RNN) algorithm. The gesture recognition model was trained using the SmartWatch Gestures database[@costante2014eusipco]. Each gesture sequence contains acceleration data from the 3 dimensional accelerometer. An evaluation of the network size was presented, and the best performance was obtained by using the LSTM layer with the size of 128. Lefebvre[@lefebvre2013ANN] carried out gesture recognition experiments on a database consisted of both accelerometer and gyrometer sensors. The sensor data was captured using an Android Nexus S Samsung device. 22 participants, from 20 to 55 years old, contributed to the database of the 14 symbolic gestures. The results showed that gesture recognition utilizing both sensors can achieve better performance than using each individual sensor. [Moreover, the BLSTM based method achieved an accuracy of 95.57% on the database of total 1540 gestures. To the best of our knowledge, the BLSTM based method is currently the state-of-the-art baseline and performs better than previous approaches such as cHMM, DTW, FDSVM and LSTM]{}.
The Proposed Approach {#sec:approach}
=====================
In this section, we first describe the network structures of [Bidirectional Long-Short Term Memory (BLSTM) and its variant – LSTM with Gate Recurrent Unit (BGRU).]{} Then, we propose to incorporate the Fisher criterion to improve the discriminative power of these deep models, dubbed F-BLSTM and F-BGRU.
Bidirectional LSTM {#sec:blstm}
------------------
We briefly describe the LSTM unit which is the basic building block of the proposed F-BLSTM model. The neurons of LSTM contain a constant memory cell name, which has a state $c_t$ at time $t$. [A LSTM neuron unit is presented in detail at the bottom of Fig.\[fig:blstm-architecture\]]{}. Each LSTM unit for reading or modifying is controlled through a sigmoidal input gate $i_t$, a forget gate $f_t$ and an output gate $o_t$. At each time step $t$, [LSTM unit receives inputs from two external sources at each of the three gates.]{} The current frame $x_t$ and previous hidden states $h_{t-1}$ are two sources, and the cell state $c_{t-1}$ in the cell block is an internal source of each gate. The gates are passed through the tanh non-linearity and activated by logistic function. After multiplying the cell state by the forget gate $f_t$, the final output of the LSTM unit $h_t$ is computed by multiplying the activation $o_t$ of the output gates with updated cell state. Denoting all $W_*$ are diagonal matrices, the updates in a layer of LSTM units are summarized as follows: $$\begin{array}{l}
{i_t} = \sigma \left( {{W_{xi}}{x_t} + {W_{hi}}{h_{t - 1}} + {W_{ci}}{c_{t - 1}} + {b_i}} \right),\\
{f_t} = \sigma \left( {{W_{xf}}{x_t} + {W_{hf}}{h_{t - 1}} + {W_{cf}}{c_{t - 1}} + {b_f}} \right),\\
{c_t}={f_t}c_{t-1}+{i_t}tanh\left(W_{xc}x_t+W_{hc}h_{t-1}+b_c\right),\\
o_t=\sigma\left(W_{xo}x_t+W_{ho}h_{t-1}+W_{co}c_t+b_o\right),\\
h_t={o_t}tanh\left(c_t\right).
\end{array}
\label{eq:lstm-unit}$$
{height="0.9\textheight"}
The model of [BLSTM uses Recurrent Neural Nets (RNNs)]{} made of LSTM units, which have shown the great ability to deal with temporal data in many applications [@sundermeyer2012lstm]. We consider the gesture data using 3 dimensional accelerometer and 3 dimensional gyrometer synchronized input vectors through sampling timestep. As shown in Fig.\[fig:blstm-architecture\], the forward and backward LSTM hidden layers are fully connected to the input layer and consist of multiple LSTM neurons each with full recurrent connections. Several experiments have been conducted with different hidden neuron sizes and 128 neurons yield the best results. The output layer has a size equivalent to the number of neuron to classify (*i.e.* $M=128$). $G = \left\{ {G_1,...,G_{T}} \right\}$ is a gesture sequence of $T$ size, $G_t = \left(x_1\left(t\right),...,x_{N}\left(t\right)\right)$ is a vector at timestep $t$ and $N$ denotes the sensor number, and $\left(y_1,...,y_{n}\right)$ is the BLSTM output set with $n$ being the number of gestures to be classified. The softmax activation function is used for this layer to give network response between 0 and 1. Classically, these outputs can be considered as posterior probabilities of the input sequence belonging to a specific gesture class, and the softmax loss function is presented as follows $${\cal L}_s = - \frac{1}{m} \sum\limits_{i = 1}^m \log \frac{{{e^{W_{{y_i}}^T{O_i} + {b_{{y_{i}}}}}}}}{{\sum\nolimits_{j = 1}^n {{e^{W_j^T{O_i} + {b_j}}}} }},
\label{eq:softmaxloss}$$ where $O_i =\left(o_1,...,o_M\right)$ denotes the [$i$th output belonging to the $y_i$th class]{}. $W_j$ denotes the $j$th column of the weights $W$ in the last layer and $b$ is the bias term. $m$ is the size of mini-batch and $n$ is the number of classes.
Deep Fisher Discriminant Learning {#sec:fisherloss}
----------------------------------
To further enhance the performance of BLSTM, we incorporate the Fisher criterion into the softmax loss function, which is shown in Fig.\[fig:blstm-architecture\]. First, the input layer consists of the concatenation of 3 dimensional accelerometer and 3 dimensional gyrometer signals synchronized in time (*i.e.* $N=6$). The sensor data is normalized between 0 and 1 according to the maximum value that sensors can provide. In order to minimize the intra-class variations and maximize the inter-class variations of gesture data, we propose a new Fisher criterion based on Fisher Linear Discrimination as follows: $${\cal L}_f
= \frac{1}{m}{\sum\limits_{i = 1}^m {\left\| {{O_i} - {\mu_{{y_i}}}} \right\|_2^2} - \frac{\delta}{n\left(n-1\right)} \sum\limits_{j = 1,k = 1}^n {\left\| {{\mu_j} - {\mu_k}} \right\|_2^2} }
\label{eq:fisher}$$ where ${\mu_{{y_i}}}$ is the $y_i$th class mean of output vectors, and $\delta$ is the discriminative factor. As updating the mean vector ${\mu_{{y_i}}}$ when learning BLSTM, the Fisher criterion utilizes the whole training set and mean vectors of each class in every iteration. We propose to augment the loss in Eq.(\[eq:softmaxloss\]) with the additional Fisher criterion term in Eq.(\[eq:fisher\]) as follows: $$ {\cal L}
= {\cal L}_s +\theta{\cal L}_f \\
\label{eq:wholeloss}$$ where $\theta$ and $\delta$ are bounded in $[0,1]$, and these two parameters are used for balancing three parts of the loss function. In forward and backward processes, we set output vector ${O_i}$, mean vector ${\mu_j}$, loss parameter $W$, scalar parameters $\theta$, $\delta$ and learning rate $\lambda$, BLSTM parameters $H_f$ and iteration number $e$, respectively. In each iteration, we compute the loss of F-BLSTM by Eq. (\[eq:fisher\]) and Eq. (\[eq:wholeloss\]), and the backpropagation error by $$\frac{{\partial {L^e}}}{{\partial O_i^e}} = \frac{{\partial L_s^e}}{{\partial O_i^e}} + \theta \frac{{\partial L_f^e}}{{\partial O_i^e}}.
\label{eq:backerror}$$ Then, we update the parameter $W$, mean vector $\mu_{j}$ and BLSTM parameter $H_f$ in the $e+1$ iteration by the following formulas until the converge stopping criterion. $$\begin{array}{l}
{W^{e + 1}} = {W^e} - {\lambda ^e} \cdot \frac{{\partial L_f^e}}{{\partial {W^e}}},\\
\mu _j^{e + 1} = \mu _j^e - \alpha \cdot \Delta \mu _j^e,\\
H_f^{e + 1} = H_f^e - {\lambda ^e}\sum\nolimits_i^m {\frac{{\partial {L^e}}}{{\partial O_i^e}}} \cdot \frac{{\partial O_i^e}}{{\partial H_f^e}}.
\end{array}
\label{eq:parmean}$$ With proper scalar parameters $\theta$, $\delta$ and $\alpha$, the discriminative power of F-LSTM can be significantly enhanced for hand gesture recognition. This network is learned using classical online backpropagation through time with momentum. For classifying a new gesture sequence, we use a majority voting rule over the outputs along the sequence (*i.e.* keeping only the most probable class $argmax _{i \in \left[ {1,n} \right]}{O_i}$) to determine the final gesture class. A detailed parameter analysis of $\theta$, $\delta$ and $\alpha$ is presented in Section\[sec:parameter\].
Bidirectional GRU and F-BGRU {#sec:bgru}
----------------------------
To further enhance the performance of network, a variant of BLSTM termed Bidirectional Gated Recurrent Unit (BGRU) was proposed in[@cho2014arxiv; @chung2014eprint] to make each recurrent unit to adaptively capture dependence of different time scales. Similarly to the BLSTM unit, the BGRU has the activation $h_t$, candidate activation $\tilde{h}_t$, update gate $z_t$ and reset gate $r_t$ units to modulate the flow of information in unit without some separate memory cells, as shown in Fig.\[fig:bgru-unit\]. The flows in BGRU are summarized as follows:
$$\begin{array}{l}
{z_t} = \sigma \left( {{W_z}{x_t} + {W_{zf}}{h_t} + {b_z}} \right),\\
{r_t} = \sigma \left( {{W_r}{x_t} + {W_{rf}}{h_t} + {b_r}} \right),\\
{{\tilde h}_t} = tanh\left( {W{x_t} + U\left( {{r_t} \odot {h_{t - 1}}} \right) + {b_h}} \right),\\
{h_t} = \left( {1 - {z_t}} \right){h_{t - 1}} + {z_t}{{\tilde h}_t}
\end{array}.
\label{eq:bgru-unit}$$
where the activation $h_t$ at time $t$ is a linear interpolation between previous activation $h_{t-1}$ and the candidate activation $\tilde{h}_t$, the candidate activation $\tilde{h}_t$ is computed same as traditional recurrent unit. The update gate $z_t$ decides the number of units to update its activation, and so as the reset gate $r_t$.
It is easy to notice that the BGRU unit also controls the flow of information like the BLSTM unit, but without having to use a memory unit. Similar to F-BLSTM, we also apply the Fisher discriminative function for BGRU and learn a new variant named F-BGRU model to recognize hand gestures.
{width="0.65\linewidth"}
Experiments {#sec:experiments}
===========
Hardware Device {#sec:database}
---------------
Our mobile hand gesture database is collected using a Huawei device with Android system, which has a 3 dimensional accelerometer and a 3 dimensional gyrometer. According to[@lefebvre2013ANN], we collect the data of both accelerometer and gyrometer, [and record each gesture by pressing, holding and releasing the “Sensor" button on the touch screen.]{}
### Gesture Dictionary
As shown in Fig.\[fig:gesture-dic\], the gesture dictionary consists of two categories including Arabic numerals (1, 2, 3, 4, 5, 6) and English capital letters (A, B, C, D, E, F). Furthermore, the stroke order of gestures is set in advance to ensure the consistency of gestures drawed by left or right hand of each participant. We directly collect all accelerometer and gyrometer data, then transfer the data from the cellphone memory to the computer for gesture recognition.
{width="\linewidth"}
### Database Collection
The database named MGD consists of 12 gestures performed by 32 participants (23 males and 9 females) with about fifteen times per gesture. Therefore, there are a total of 5547 gesture sequences. The sampling time of accelerometer and gyrometer sensors is 5ms corresponding to a frequency of 200Hz. To the best of our knowledge, it is the largest database so far for mobile based gesture recognition, [which is of benefit to the research community]{}.
Implementation Details {#sec:preprocess}
----------------------
In this section, we present the details of the implementation of our experiments. We use Tensorflow toolbox as the deep learning platform and an NVIDIA GTX 1070 GPU to run the experiments. In order to validate the effectiveness of our proposed Fisher criterion in LSTM for modeling temporal sequences, we compare our methods, F-BLSTM and F-BGRU, with the state-of-the-art baselines (BLSTM and BGRU[@chung2014eprint]) on three benchmarks including our proposed database (MGD), and two previous databases: the BUAA Mobile Gesture database[@xie2016ccbr] and the SmartWatch Gestures database[@chung2014eprint]. We comprehensively evaluate the performance of the proposed model under different parameter settings of $\delta$, $\alpha$ and $\theta$ in Sec.\[sec:parameter\], and provide extensive experimental comparison in Sec.\[sec:comparison\].
**Data preprocessing.** The main objective for data preprocessing is to facilitate gesture recognition. In real world applications, the sensor data often contain a lot of noise due to complex environmental conditions and hardware limitations. Therefore, we first carry out a filtering process to suppress noise (i.e. data smoothing). We experimented with Average Filter, Median Filter and Butterworth Filter, and selected the Average Filter in terms of its good performance and computational efficiency. Fig.\[fig:filtered-data\] shows the original accelerometer and gyrometer data and the processed data using the Average Filter.
\
\
The gesture execution speed of different participants may vary considerably, which leads to different signal lengths due to a fixed sampling frequency (200HZ) of accelerometer and gyrometer in the mobile phone. For example, gestures completed relatively faster will have fewer sampling points. Also, the signal strength of gesture sequences may vary. To cope with signal strength and speed variations, we apply amplitude and sequence normalization to the original signal sequences. Specifically, we first normalize a signal $x_i^n\left(t\right)$ by $$x_i^n\left( t \right) = \frac{{{x_i}\left( t \right) - \min _{t = 1}^T{x_i}\left( t \right)}}{{\max _{t = 1}^T{x_i}\left( t \right) - \min _{t = 1}^T{x_i}\left( t \right)}},\;\;\;
\forall i \in \left\{ {1,...,6} \right\}.
\label{eq:normalize}$$ Then, we use cubic spline interpolation to normalize the length of a sequence to a fixed size (we set this size as 1000 in our experiments). Fig.\[fig:preprocessed-data\] shows the preprocessed accelerometer and gyrometer data, where the sequence has been filtered and normalized.
\
Study of Fisher Criterion Parameters {#sec:parameter}
------------------------------------
In our model, the hyperparameter $\theta$ impacts the Fisher criterion, $\alpha$ controls the update rate of mean $\mu$ in F-BLSTM, and $\delta$ adjusts the relationship of intra-class distance and inter-class distance between features. These parameters would affect the performance of gesture recognition. In order to configure an optimal parameter setting, we conduct parameter tuning experiments for the F-BLSTM model as follows.
1. We fix $\alpha$ to 0.5, $\delta$ to 0.01 and vary $\theta$ from 0 to 1 to investigate the effect of parameter $\theta$. Fig.\[fig:parameter-theta\] shows the classification accuracy on testing set. The result shows that the model trained with only softmax loss has poor performance, which certificates the necessity of introducing Fisher criterion.
2. We fix $\alpha$ to 0.5, $\theta$ to 0.1 and vary $\delta$ from 1e-5 to 0.1 to verify that the inter-class distance’s joining promote the classification ability. As shown in Fig.\[fig:parameter-delta\], $\delta$ balances the relationship of intra-class distance and inter-class distance. We can set $\delta$ to an appropriate value to make the classification better according to different circumstances.
3. We fix $\theta$ to 0.1, $\delta$ to 0.01 and vary $\alpha$ from 0 to 1 to train different models. The classification accuracy of these models on our gesture database are illustrated in Fig.\[fig:parameter-alpha\]. We find that the classification performance of our model remains relatively stable across a wide range of $\alpha$, but a moderate value of $\alpha$ has a better performance.
{width="0.5\linewidth"}
\
{width="0.5\linewidth"}
{width="0.5\linewidth"}
\
\
Analysis of Model Effect {#sec:discussion}
------------------------
As shown in the analysis mentioned above, the F-BLSTM and FBGRU achieves a better discriminant ability than the baseline BLSTM and BGRU. This section further discusses the better feature disctribution has been achieved. By the study in last section, we set $\theta$ to 0.1, $\delta$ to 0.01 and $\alpha$ to 0.5 in the F-BLSTM model, and set the parameters to 0.3, 0.01, 0.5 in the F-BGRU model, respectively.
Fig. \[fig:featurevisualization\] shows the feature visualizations on MGD database. In Fig. \[fig:featurevisualization\](a) and Fig. \[fig:featurevisualization\](b), the BLSTM and BGRU features of 12 classes are visualized by t-SNE [@Maaten2009Science] with the parameters of initial dimension 100 and perplexity dimension 30, while the F-BLSTM and F-BGRU features are illustrated in Fig. \[fig:featurevisualization\](c) and Fig. \[fig:featurevisualization\](d), respectively. Clearly, the fisher discriminant learning features are more discriminative than the original baseline features, especially the F-BGRU feature in Fig. \[fig:featurevisualization\](d) can be better discriminated than the BLSTM feature in Fig. \[fig:featurevisualization\](a). As another verification, the quantitative evaluation is performed based on three databases in the next section.
Comparison with State-of-the-Arts {#sec:comparison}
---------------------------------
**Experiment on [MGD]{} Database.** For the proposed database, we select 3500 sequences for training our model and 2047 sequences for testing. After preprocessing, the length of each data sequence is set to 1000, thus each input sample (3-axis accelerometer and gyrometer signals) is a matrix of [$1000 \times 6$]{}. Here, we train the network using adaptive moment estimation, with the learning rate as 0.002 and the batch size as 200. For the F-BLSTM model, we set $\theta$ to 0.1, $\delta$ to 0.01 and $\alpha$ to 0.5. We complete the training of BLSTM and F-BLSTM models at 1.5K iterations. The parameters of F-BGRU model are set to 0.3, 0.01, 0.5 respectively. The training of BGRU and F-BGRU is completed at 1.2K iterations. We report the performance of different methods on the testing set based on the average over 5 runs. The class-wise classification accuracy comparison of different mothods is presented in Table.\[table:ourdatabase\]. It is clear that by incorporating the Fisher criterion to the base models (BLSTM and BGRU), the recognition performance can be improved. Fig.\[fig:curve-proposed\] also shows the behaviors of F-BLSTM and F-BGRU models. Dotted lines denote training errors, and solid lines denote testing errors for different methods. As can be observed from the figure, our proposed softmax function with Fisher criterion effectively speeds up the convergence of training and achieves a smaller error rate.
BLSTM F-BLSTM BGRU F-BGRU
--------- ------- ----------- ------- -----------
A 97.41 97.85 97.09 98.09
B 94.17 96.50 97.24 98.78
C 98.95 99.40 99.85 100.00
D 96.88 99.04 98.02 98.87
E 96.88 97.40 98.48 98.61
F 96.86 98.59 97.62 99.54
1 93.80 95.33.82 96.62 98.53
2 98.60 98.82 99.03 99.35
3 96.69 97.56 98.29 99.42
4 98.77 98.97 99.28 99.29
5 96.55 98.16 99.77 100.00
6 99.10 99.32 99.77 99.61
Overall 97.05 **98.04** 98.38 **99.15**
{width="\linewidth"}
**Experiment on BUAA Mobile Gesture Database[@xie2016ccbr].** This database has 1120 samples for gestures A, B, C, D, 1, 2, 3, 4. Each sample includes three-dimensional acceleration and angular velocity of the mobile phone. The training and testing sets are divided randomly into 70% and 30%, respectively. We conduct the experiments using the same setting for F-BLSTM and F-BGRU. We set $\theta$ to 0.1, $\delta$ to 0.03 and $\alpha$ to 0.5. Model training is finished at 400 iterations. Table.\[table:buaadatabase\] shows that LSTMs with Fisher criterion still have better results than baselines on a smaller dataset. Similarly, the models converge faster and yield lower classification error rates with the Fisher criterion as shown in Fig.\[fig:curve-buaa\].
BLSTM F-BLSTM BGRU F-BGRU
--------- -------- ----------- -------- -----------
A 100.00 99.17 98.34 99.58
B 97.29 98.92 97.84 98.37
C 100.00 100.00 100.00 100.00
D 99.26 97.42 96.77 99.35
1 97.87 99.57 100.00 100.00
2 100.00 100.00 100.00 100.00
3 97.06 100.00 100.00 100.00
4 95.83 97.50 97.08 97.08
Overall 98.44 **99.06** 98.75 **99.25**
{width="\linewidth"}
**Experiment on SmartWatch Gesture Database[@costante2014eusipco].**
{width="0.6\linewidth"}\
The database has been used to evaluate gesture recognition algorithms for interacting with mobile applications using gestures. Eight different users performed twenty repetitions of twenty different gestures for a total of 3200 sequences. The gestures are depicted in Fig.\[fig:example-smartwatch\]. Different from the 6-dimensional sequences of previous two databases, each sequence in this dataset only contains acceleration data from the 3-axis accelerometer of a first generation Sony SmartWatch. Furthermore, due to the lower sampling frequency, we set the length of each gesture sequence preprocessed to 50. We randomly select 2400 sequences as training set and the rest 800 sequences as testing set. The parameters of Fisher criterion follow the same setting in previous experiment. Adaptive moment estimation is used to train the network, and the initial learning rate $\lambda$ is set to 0.0001. The batch size is 1000. Training for BLSTM and F-BLSTM is stopped at 1.4K iterations, and BGRU and F-BGRU at 2K iterations. [Fig.\[fig:curve-smart\] shows the training error and testing error during the training process. Like Fig.\[fig:curve-proposed\] and Fig.\[fig:curve-buaa\], dotted lines denote training errors, and solid lines denote testing errors. Table.\[table:smartwatch-database\] lists the classification results for different gestures. Notice that our proposed models perform considerably better than the baselines across the 20 gestures.]{}
BLSTM F-BLSTM BGRU F-BGRU
--------- ------- ----------- -------- -----------
1 94.58 97.91 97.08 97.50
2 95.00 97.22 95.56 95.56
3 86.90 87.59 93.10 93.10
4 95.91 97.27 97.27 97.73
5 96.88 98.13 96.88 98.13
6 93.33 94.07 96.30 100.00
7 96.44 96.89 98.22 99.56
8 97.62 98.57 100.00 100.00
9 93.49 96.74 96.74 97.67
10 94.84 98.06 100.00 100.00
11 89.76 94.15 94.15 95.12
12 92.89 92.44 96.00 97.33
13 90.42 95.00 94.17 95.42
14 94.88 96.30 96.30 97.21
15 95.14 95.14 100.00 97.84
16 92.20 89.27 93.17 93.17
17 96.52 95.65 99.13 100.00
18 96.22 97.30 96.76 95.68
19 94.29 94.76 94.76 96.67
20 97.21 98.60 100.00 100.00
Overall 94.30 **95.65** 96.80 **97.40**
{width="\linewidth"}
Based on the experimental evaluations, we can observe that LSTM models with Fisher criterion consistently outperform the standard LSTMs on three different databases, validating the advantage of the proposed Fisher criterion. Furthermore, with even less data, the proposed Fisher criterion helps the LSTMs obtain a better result. This proves that it has a significant effect on a small database, showing the Fisher criterion has a widespread application scope.
Conclusion {#sec:conclusion}
==========
[In this paper, we have collected a large gesture database, namely MGD, for the application of mobile based gesture recognition. We proposed a novel Fisher criterion for F-BLSTM network to effectively classify the mobile hand gestures. Based on F-BLSTM, we also extended the F-BLSTM to a variant F-BGRU. By conducting numerous experiments, we proved that the mobile gesture recognition based on F-BLSTM networks which is supervised by the Fisher criterion has an outstanding performance. An appropriate values of Fisher criterion parameters contribute to the better results.]{}
Acknowledgement {#acknowledgement .unnumbered}
===============
The work was supported in part by the Natural Science Foundation of China under Contract 61672079, 61473086, 61601466. The work of B. Zhang was supported in part by the Program for New Century Excellent Talents University within the Ministry of Education, China, and in part by the Beijing Municipal Science and Technology Commission under Grant Z161100001616005. Baochang Zhang is the correspondence.
References {#references .unnumbered}
==========
Biography {#biography .unnumbered}
=========
**Chunyu Xie** received the B.S. degree and is a master in automation from Beihang University. His current research interests include signal and image processing, pattern recognition and computer vision.\
\
**Ce Li** received the B.E. degree in Computer Science from Tianjin University, Tianjin, China, in 2008, the M.S. and Ph.D. degrees in Computer Science from the School of Electronic, Electrical and Communication Engineering at the University of Chinese Academy of Sciences, Beijing, China, in 2012 and 2015, respectively. She is currently a research assistant with China University of Mining & Technology, Beijing, China. Her current interests include computer vision, video analysis and machine learning. She was supported by the Natural Science Foundation of China for Youth.\
\
**Baochang Zhang** received the B.S., M.S. and Ph.D. degrees in Computer Science from Harbin Institue of the Technology, Harbin, China, in 1999, 2001, and 2006, respectively. From 2006 to 2008, he was a research fellow with the Chinese University of Hong Kong, Hong Kong, and with Griffith University, Brisban, Australia. Currently, he is an associate professor with the Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing, China. He was supported by the Program for New Century Excellent Talents in University of Ministry of Education of China. His current research interests include pattern recognition, machine learning, face recognition, and wavelets.\
\
**Chen Chen** received the B.E. degree in automation from Beijing Forestry University, Beijing, China, in 2009, the M.S. degree in electrical engineering from Mississippi State University, Starkville, MS, USA, in 2012, and the Ph.D. degree from the University of Texas at Dallas, Richardson, TX, USA, in 2016. He is currently a Postdoctoral Fellow with the Center for Research in Computer Vision, University of Central Florida, Orlando, FL, USA. His current research interests include compressed sensing, signal and image processing, pattern recognition, and computer vision. He has published over 40 papers in refereed journals and conferences in the above areas.\
\
**Jungong Han** is currently a Senior Lecturer with the Department of Computer Science and Digital Technologies at Northumbria University, Newcastle, UK. Previously, he was a Senior Scientist (2012-2015) with Civolution Technology (a combining synergy of Philips Content Identification and Thomson STS), a Research Staff (2010-2012) with the Centre for Mathematics and Computer Science (CWI), and a Senior Researcher (2005-2010) with the Technical University of Eindhoven (TU/e) in Netherlands. Dr. Han’s research interests include Multimedia Content Identification, Multi-Sensor Data Fusion, Computer Vision and Multimedia Security. He is an Associate Editor of Elsevier Neurocomputing (IF 2.4) and an Editorial Board Member of Springer Multimedia Tools and Applications (IF 1.4). He has been (lead) Guest Editor for five international journals, such as IEEE-T-SMCB, IEEE-T-NNLS. Dr. Han is the recipient of the UK Mobility Award Grant from the UK Royal Society in 2016.\
\
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Summarizing Likert scale ratings from human annotators is an important step for collecting human judgments. In this project we study a novel, graph theoretic method for this purpose. We also analyze a few interesting properties for this approach using real annotation datasets.'
author:
- |
M. Iftekhar Tanveer\
Electrical and Computer Engineering\
University of Rochester\
Rochester, NY 14627\
`[email protected]`
bibliography:
- 'project.bib'
title: Use of Laplacian Projection Technique for Summarizing Likert Scale Annotations
---
Introduction {#sec:intro}
============
Likert scale is a popular method for quantifying and gathering human opinion. In disciplines like Behavioral Science, Psychology, or Human-Computer-Interaction, scientists use Likert scale to measure subjective opinions from human annotators. As human annotations are inherently noisy, it is often customary to collect the data from more than one annotators (Fig. \[Fig:annotation\]). Traditionally, average of these ratings are computed to get a summary.
{width="0.5\linewidth"}
\[Fig:annotation\]
However, human annotators usually have their own bias on the ratings. For example, some annotators are biased towards high ratings, some towards low ratings; some annotators rate on a wide range, some others rate on a narrow range. As a result, computing averages without addressing these variations might lead to erroneous results.
In classical literature on label denoising, researchers attempt to learn the underlying distribution of annotator ratings to rescale and compute the average. On the other hand, in this project we are interested to capture the neighborhood information from the datapoints. The rationale for this approach is as follows: Although humans have personal bias in assigning the exact values in the Likert scale, the idea of relative positions of the datapoints is universal. We employ a graph based technique to capture this neighborhood relationship within the datapoints.
Literature Review
=================
In recent years, acquiring ground truth via accumulation of several unreliable crowd annotators is considered to be an important problem to address. An increasing body of literature is addressing this problem from various perspectives. For example, there are works involving complicated generative models [@Whitehill2009][@Raykar2010] for denoising and aggregating crowd opinion. As these models involve latent variables, the solution approach typically involve Expectation Maximization (EM) [@Dempster1977] framework. These approaches are often criticized[@Karger2011] as EM is sensitive to initialization and can stick to local optima. Liu et al. [@Liu2012] approached this problem as an inference problem in probabilistic models. They used variational methods such as Belief Propagation and Mean Fields.
However, none of these approaches consider the relative neighborhood of the vertices from graph perspective. As the annotators assign ratings to the subjects in comparison to one another; it should be reasonable to assume that the ground truth annotations captured in the relative distances among the datapoints.
Problem Formulation {#sec:probfor}
===================
![The structure of the Annotation Matrix[]{data-label="Fig:dataStruct"}](./figures/data.pdf)
Let us consider the annotation data is stored in an $m\times n$ matrix as shown in Fig. \[Fig:dataStruct\]. Each row of the matrix represents a unique annotator and each column represents a unique datapoint. The ratings are given in a K-point Likert scale. The goal of this problem is to formulate a linear embedding that projects the $m$ dimensional data points on to a one dimensional line while preserving the relative neighborhood. This idea is inspired from the concept of Locality Preserving Projection proposed by Niyogi et al. [@Niyogi2004]. If $\mathbf{a}$ be the intended embedding, then the projection of datapoint $X_{:j}$ is $y_j$, which can be written as in Eq. . $$y_j = \mathbf{a}^T X_{:j} \implies \mathbf{y}^T = \mathbf{a}^T X
\label{eq:1dEmbedding}$$
In order to capture the neighborhood structure among the datapoints, we formulate a graph for each individual annotator. In the graph each node corresponds to a datapoint. In order to avoid encoding the subjective biases, we take the following simplest rule for forming the edges — two nodes will be connected by an edge if and only if the two datapoints receive identical nonzero rating. That is, $$w_{ij}^{(m)} =
\begin{cases}
1 & \text{if node i and j has nonzero identical score by annotator m} \\
0 & \text{otherwise}.
\end{cases}
\label{eq:graphweight}$$
This weight assignment captures the proximity of the datapoints in a higher dimensional space. The more annotators agree with this proximity information, the closer the datapoints are considered to be. To capture this structure, we formulate the final neighborhood graph by averaging the proximity weights $w_{ij}^{(m)}$ between nodes $i$ and $j$ for all the annotators. Now we project the datapoints on a 1D space so that the distances among the projections preserve the neighborhood structure composed by all the $w$’s. Mathematically we want to minimize the following. $$\begin{split}
& \frac{1}{M}\sum_{m=1}^{M}\sum_{(i,j)\in E_m} w_{ij}^{(m)}(y_i - y_j)^2 \\
= & \frac{1}{M}\sum_{m=1}^{M}\mathbf{y}^T L_m \mathbf{y}\\
= & \mathbf{y}^T \left( \frac{1}{M}\sum_{m=1}^{M} L_m \right) \mathbf{y}\\
= & \mathbf{a}^T X L_{\text{avg}} X^T \mathbf{a}
\label{eq:ObjFun}
\end{split}$$ where, $L_m := D_m - A_m$ is called the graph Laplacian for $m^{\text{th}}$ annotator graph. $D_m$ and $A_m$ are the degree matrix and adjacency matrix respectively for the $m^{\text{th}}$ annotator graph. As the graphs are undirected, so $A_m$ is symmetric. $D_m$ is obtained by performing row or column-wise sum and then placing the sums in the diagonal. As the degree of the nodes capture a natural measure of node importance, we impose a constraint $\mathbf{a}^TX D_{\text{avg}} X^T \mathbf{a} = 1$. Here $D_{\text{avg}}$ is the average degree matrix for all the annotators. Therefore, the optimization problem becomes: $$\begin{split}
\operatorname*{\arg\!\min}_{\mathbf{a}} \quad & \mathbf{a}^T X L_{\text{avg}} X^T \mathbf{a}\\
s.t. \quad & \mathbf{a}^TX D_{\text{avg}} X^T \mathbf{a} = 1
\end{split}
\label{eq:Optimization}$$
Optimization {#sec:opt}
============
We use Lagrange multiplier to construct the objective function as follows: $$\mathcal{L}(\mathbf{a}) = \mathbf{a}^T X L_{\text{avg}} X^T \mathbf{a} + \lambda (1 - \mathbf{a}^TX D_{\text{avg}} X^T \mathbf{a}).
\label{eq:opjFun2}$$ As both the Laplacian matrix and Degree matrix are positive semi-definite, $\mathcal{L}$ is a convex function of $\mathbf{a}$. Therefore, differentiating $\mathcal{L}$ with respect to $\mathbf{a}$ and setting to zero we get, $$\frac{\partial \mathcal{L}}{\partial \mathbf{a}} = 2X L_{\text{avg}} X^T \mathbf{a} - 2\lambda X D_{\text{avg}} X^T \mathbf{a} = 0 \implies X L_{\text{avg}} X^T \mathbf{a} = \lambda X D_{\text{avg}} X^T \mathbf{a}.
\label{eq:opjFun3}$$ Therefore $\mathbf{a}$ can be obtaining by solving the generalized Eigenvalue problem as shown in Eq. . In order to get the solution that minimizes $\mathcal{L}$, we take the Eigenvector with smallest corresponding Eigenvalue.
Algorithm
=========
From the discussions in Section \[sec:probfor\] and Section \[sec:opt\], we can formulate an algorithm for computing the intended 1D embedding. We show it in Algorithm \[algo:denoise\].
**Initialize:** $A_{\text{avg}} \leftarrow \frac{1}{M}\sum_{m=1}^{M} A_m$ Construct a diagonal matrix, $D_{\text{avg}}$, with entries $\sum_{\text{rows}}A_{\text{avg}}$ $L:= D_{\text{avg}} - A_{\text{avg}}$ Calculate Generalized Eigenvalue solution for Eq. Return normalized Eigenvector with smallest corresponding Eigenvalue
Data
====
We apply the algorithm on the following two datasets:
- Job interview dataset [@Hoque2013], and
- Public speaking dataset [@Tanveer2015].
We used the annotators’ response on the overall performance of the study participants. The data matrix, $X$, for both datasets are shown in Fig. \[Fig:datasets\]. There are 4 annotators and 138 datapoints (participants) in the job interview dataset. It is particularly evident from the picture of job interview dataset that the annotator 1 and 4 have a tendency to give higher rating than annotator 2 and 3. In the public speaking dataset, there are 15 annotators and 51 datapoints. However, there are a number of missing values (rating is zero) in the public speaking dataset.
![Job Interview Dataset (Left) and Public speaking Dataset (Right)[]{data-label="Fig:datasets"}](./figures/jint_data.pdf "fig:"){width="0.45\linewidth"} ![Job Interview Dataset (Left) and Public speaking Dataset (Right)[]{data-label="Fig:datasets"}](./figures/pspeak_data.pdf "fig:"){width="0.45\linewidth"}
Results
=======
The results for the job interview dataset are shown in Fig. \[Fig:jint\_result\]. The topmost row shows the original data matrix ($X$) on the left and the average of annotators’ scores on the right.
![Results of denoising for Job Interview Dataset[]{data-label="Fig:jint_result"}](./figures/jint_result1.pdf){width="\linewidth"}
In the second row, we sort the datapoints in ascending order of the mean ratings for representational convenience. The columns of the data matrix is also sorted accordingly. We also plot the denoised values of the ratings (i.e. the projected values obtained by the proposed algorithm) in red markers. It is interesting to notice that the projected values follow a flipped sequence than the mean ratings. This is due to the fact that our proposed optimization algorithm selects an embedding by only preserving the relative neighborhood among the datapoints. It does not preserve the absolute values. Consequently, the datapoints might arbitrarily take positive or negatively correlation with the mean values. However, in practice, it is not of a big concern as we can always flip the sequence by subtracting all the values in the sequence from the maximum allowable rating.
The third row represents the data matrix and the accumulated plots which are sorted based on the denoised ratings. If we compare the scatter plots (right hand side plot) of the second row with the third one, it is evident that the denoised ratings give finer discrimination among the datapoints than the mean values. As there are only four annotators, the means cannot discriminate among the datapoints less than $\frac{1}{4}^{\text{th}}$ of the ratings. As a result, significantly more datapoints receive same rating (notice the blue line on the right hand plot of the second row). However, as the denoised ratings consider the neighborhoods of the datapoints, it can discriminate with finer detail (notice the red line on the right hand plot of third row).
In the fourth row, the denoised ratings are flipped by subtracting them from the maximum rating, 7. This made the denoised ratings to be positively correlated with the mean values of the ratings. The datapoints were resorted in ascending order of the flipped denoised ratings.
In Fig. \[Fig:pspeak\_result\], we show similar results for public speaking dataset. While calculating the mean, we totally omitted the missing values so that they do not bias the mean ratings. In this dataset, the mean values are capable to show finer discrimination than the job interview dataset. This is due to the higher number of annotators. An interesting phenomenon is that the denoised ratings particularly enhanced the poor quality of three datapoints (notice the three leftmost points in the red line on 2nd row) while the mean ratings “smooth out” the differences.
![Results of denoising for Public Speaking Dataset[]{data-label="Fig:pspeak_result"}](./figures/pspeak_result1.pdf){width="\linewidth"}
Future Work and Conclusion
==========================
In this project, we proposed a novel technique for summarizing the annotators opinion. The technique employs graph structure which captures the relative neighborhood among the datapoints. We applied the techniques on two datasets and compared with simple mean ratings.
In future, we will try to apply this technique with ground truth information of subjective data. Having the ground truth will enable us to better quantify the quality of this metric.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Information transfer which reveals the state variation of variables usually plays a vital role in big data analytics and processing. In fact, the measures for information transfer could reflect the system change by use of the variable distributions, similar to KL divergence and Renyi divergence. Furthermore, in terms of the information transfer in big data, small probability events usually dominate the importance of the total message to some degree. Therefore, it is significant to design an information transfer measure based on the message importance which emphasizes the small probability events. In this paper, we propose a message importance transfer measure (MITM) and investigate its characteristics and applications on three aspects. First, the message importance transfer capacity based on MITM is presented to offer an upper bound for the information transfer process with disturbance. Then, we extend the MITM to the continuous case and discuss the robustness by using it to measuring information distance. Finally, we utilize the MITM to guide the queue length selection in the caching operation of mobile edge computing.'
author:
-
-
-
title: |
State Variation Mining: On Information Divergence with Message Importance in Big Data\
[^1]
---
information transfer measure, message importance measure, big data analysis, mobile edge computing (MEC), queue theory
Introduction
============
Recently, the amount of data is exploding rapidly and the computing complexity for data processing is also increasing. To some degree, this phenomenon is resulted from more and more mobile devices as well as the growing service of clouds. In the literature, it is favored to process the collected data to dig out the hidden important information. On one hand, it is necessary to improve the computation platforms for big data processing, such as cloud computing, fog computing and mobile edge computing (MEC). On the other hand, a series of algorithmic technologies for big data analysis and mining are required, such as neural networks and machine learning, as well as distributed parallel computing, etc.
In many scenarios of big data, the small probability events attract more attention than the large probability ones. That is, the rarity of small probability events has higher value in use. For instance, on anti-terrorist activities, there are only a few illegal people and hazardous agent that should be supervised especially [@Counterterrorism-systems]. Moreover, in terms of the synthetic ID detection, it just focuses on a small number of artificial identities for financial frauds [@A-comprehensive-survey]. Actually, how to mine and characterize small probability events becomes more challenging and more significant in modern life.
From the perspective of information theory, small probability events detection can be regarded as a kind of clustering problem. In particular, a graph-based rare category detection was presented based on the global similarity matrix [@Graph-based-rare]. Furthermore, a time-flexible rare category detection was also designed by resorting to the time-evolving of graphs [@Rare-category-detection]. In spite of these efficient algorithms for some special applications, it is worth noting that they were designed by traditional information measures and theory, which originate from the viewpoint of typical events, namely the large probability events.
Review of Message Importance Measure
------------------------------------
As two fundamental measures in information theory, Shannon entropy and Renyi entropy play a crucial role in many applications including communication engineering, estimation theory, hypothesis testing and pattern recognition. However, they are not suitable enough for small probability events mining in the big data scenarios. To do this, the message importance measure (MIM), a new information measure, is proposed to reflect the significance of small probability events. Thus, let us review the definition of MIM briefly first [@message-importance-measure-and-its-application-to-minority-subset-detection-in-big-data].
In a finite alphabet, for a given probability distribution $P=\{ p(x_1), p(x_2),..., p(x_n)\}$, the MIM with importance coefficient $\varpi \geq 0$ is defined as $$L(P,\varpi) = \log\big\{ \sum\limits_{x_i} p(x_i)e^{\varpi\left(1-p(x_i)\right)} \big\},$$ which measures the information importance of the distribution. Then, by setting the parameter $\varpi=1$ and simplifying the form of MIM, it is easy to obtain its fundamental definition as follows.
\[defn:MIM\] For the discrete probability $P$=$\{p(x_1)$, $p(x_2)$, ...,$ p(x_n)\}$, the MIM can be given by $$\label{MIM_discrete1}
\begin{aligned}
L(P)
& = \sum\limits_{x_i} p(x_i) e^{-p(x_i)}.\\
\end{aligned}$$
Comparing with Shannon entropy and Renyi entropy, the MIM replaces the corresponding logarithm operator or polynomial operator with the exponential form so that the weight factors of small probability elements can be amplified much more. This can help to reflect the significance of small probability events from the viewpoint of information measure.
In addition, as a kind of evaluation criteria, Fadeev’s postulates are commonly used to describe the information measures including Shannon entropy and Renyi entropy [@On-measures-of-entropy-and-information]. In this case, for two independent random distributions $P$ and $Q$, Renyi entropy has a weaker postulate than Shannon entropy, that is $$H(PQ) = H(P) + H(Q),$$ where the function $H(\cdot)$ denotes the corresponding information measure. Similarly, the MIM has a much weaker postulate than Renyi entropy, as follows $$H(PQ) \le H(P) + H(Q).$$ Therefore, in the sense of generalized Fadeev’s postulates, the MIM can be reasonably viewed as a kind of information measure similar to Shannon entropy and Renyi entropy.
Message Importance Transfer Measure
-----------------------------------
For an information transfer process, we consider such a model that all the $P$ and $Q$ satisfies the Lipschitz condition as follows, $$\label{eq.Lipschitz}
|H(P)- H(Q)| \le \lambda\|P-Q\|_{1},$$ where $P$ and $Q$ denote the original probability distribution and the final one respectively in the information transfer process; $\lambda>0$ is the Lipschitz constant; $H(\cdot)$ denotes a kind of information measure function; $\| \cdot \|_{1}$ denotes the $l_1$-norm measure.
Here, we shall investigate and measure information transfer process by use of the message importance. Actually, how to characterize the message importance variation in the processing of big data is a critical and interesting problem. On account of Definition \[defn:MIM\], it is available to regard the MIM as an element to measure the message importance variation for a dynamic system. Then, a new information transfer measure based on the MIM is defined as follows.
\[defn:MID\] For two discrete probability $Q=\{q(x_1), q(x_2),$ $... , q(x_n)\}$ and $P=\{p(x_1), p(x_2), ... , p(x_n)\}$ satisfying the constraint in Eq. (\[eq.Lipschitz\]), the message importance transfer measure (MITM) is defined as $$\label{MID_discrete}
\begin{aligned}
& D_{I}(Q||P)
= \sum\limits_{x_i} \{ q(x_i) e^{-q(x_i)} -p(x_i) e^{-p(x_i)} \}.\\
\end{aligned}$$
Note that the Definition \[defn:MID\] characterizes the information transfer from the statistics. That is, we can make use of MITM to measure the change of message importance focusing on small probability events in an information transfer process. Actually, there exist a variety of different information measures handling the problem of information transfer process. Shannon entropy and Renyi entropy are applicable to intrinsic dimension estimation [@On-local-intrinsic-dimension-estimation]. As well, the NMIM can be used in anomaly detection [@Non-parametric-Message-Important-Measure]. Moreover, the directed information and Schreiber’s transfer entropy [@Measuring-information-transfer] are commonly applied to inferring the causality structure and characterizing the information transfer process. In addition, referring to the idea from dynamical system theory, new information transfer measures are proposed to explore and exploit the causality between states in the system control [@Causality-preserving-information-transfer-measure].
However, in spite of numerous kinds of information measures, few works focus on how to characterize the information transfer from the perspective of message importance in big data. To this end, the MITM different from the above information measures is introduced.
Organization
------------
We organize the rest of this paper as follows. In Section II, we introduce the message importance transfer capacity measured by the MITM to describe the information transfer with disturbance. In Section III, we extend the MITM to the continuous case to investigate the variation of message importance in the information transfer process. In Section IV, the MITM is used to discuss the queue length selection for the data caching in MEC from the viewpoint of queue theory. Moreover, some simulations are presented to validate our theoretical results. Finally, we conclude it in Section VI.
Message Importance Transfer Capacity Based on Message Importance Transfer Measure
=================================================================================
In this section, we will introduce the MITM to characterize the information transfer process shown in Fig. \[fig\_transfer\_system\]. To do so, we define the message importance transfer capacity measured by the MITM as follows.
![Information transfer system model.[]{data-label="fig_transfer_system"}](transfer_system.eps){width="3.6in"}
\[defn:C\_D\] Assume that there exists an information transfer process (from the variable $X$ to $Y$) as, $$\label{eq.relation_1}
\begin{aligned}
\big\{ X, p(y-\delta_0|x), Y-\delta_0 \big| \delta_0 \in \{ \delta | \delta \sim p(\delta)\} \big\},
\end{aligned}$$ where $\delta$ denotes a disturbance following distribution $p(\delta)$ and $\delta_0$ is a certain element from the support set of $\delta$. In brief, Eq. (\[eq.relation\_1\]) can also be written as $$\label{eq.relation}
\begin{aligned}
\big\{ X, p(\tilde y|x), \tilde Y \big\},
\end{aligned}$$ where $\tilde y=y-\delta_0$ and $\tilde Y=Y-\delta_0$. Furthermore, $p(\tilde y|x)$ denotes a probability distribution matrix describing the information transfer from the variable $X$ following the distribution $p(x)$ to $\tilde Y$ following the distribution $p(\tilde y)$. We define the message importance transfer capacity as $$\begin{aligned}\label{eq.D_channel_average}
& C = \sum\limits_{\delta_0 \in \{\delta | \delta \sim p(\delta) \}} p(\delta_0) \tilde C(\delta_0), \\
\end{aligned}$$ in which $$\begin{aligned}\label{eq.D_channel}
& \tilde C(\delta_0)= \max\limits_{p(x)} \{ L(\tilde Y) - L(\tilde Y|X)\}, \\
\end{aligned}$$ where $p(\tilde y_j) = \sum\limits_{x_i} p(x_i)p(\tilde y_j|x_i)$, $L(\tilde Y)=\sum\limits_{\tilde y_j} p(\tilde y_j)e^{-p(\tilde y_j)}$, $L(\tilde Y|X) = \sum_{\tilde y_j}\sum_{x_i} p(x_i,\tilde y_j)e^{-p(\tilde y_j|x_i)}$ with the constraint $|L(\tilde Y)- L(\tilde Y|X)| \le \lambda\|p(\tilde y)-p(\tilde y|x)\|_{1}$.
In order to have an insight into the applications of message importance transfer capacity, some specific information transfer scenarios are discussed as follows.
Binary symmetric information transfer
-------------------------------------
\[prop.symmetric\] Assume that there exists an information transfer process as same as that mentioned in Eq. (\[eq.relation\_1\]) and Eq. (\[eq.relation\]), where the disturbance $\delta$ follows a binary uniform distribution (namely $p$($\delta$)= (1/2, 1/2)), and the information transfer matrix is $$\begin{aligned}
p(\tilde y|x) = \left [
\begin{matrix}
1-\beta & \beta \\
\beta & 1-\beta
\end{matrix}
\right ],
\end{aligned}$$ which indicates that variables $X$ and $\tilde Y$ both obey the binary distributions. In this case, the message importance transfer capacity is $$\begin{aligned}
C(\beta) = e^{ -\frac{1}{2} } - L(\beta),
\end{aligned}$$ where $L(\beta)= \beta e^{-\beta} + (1-\beta)e^{-(1-\beta)} $ ($0<\beta<1 $) and $|C(\beta)| \le \lambda\|p(\tilde y)-p(\tilde y|x)\|_{1}$ ($\lambda \ge \frac{e^{ -\frac{1}{2} } - \beta e^{-\beta} + (1-\beta)e^{-(1-\beta)}}{|1-2\beta|} $).
Considering a variable $X$ following the binary distribution $(p, 1-p)$, it is not difficult to see that $$\begin{aligned}
L(\tilde Y|X)
& = \beta e^{-\beta} + (1-\beta)e^{-(1-\beta)}.
\end{aligned}$$ Moreover, according to Eq. (\[eq.D\_channel\_average\]) and Eq. (\[eq.D\_channel\]), we have message importance transfer capacity as $$\begin{aligned}
& C(p, \beta) = \max\limits_{p} \Big\{ [p+ \beta(1-2p)]e^{-[p+ \beta(1-2p)]} \\
& + [(1-p)+ \beta(2p-1)]e^{-[(1-p)+ \beta(2p-1)]} \Big\} - L(\beta).\\
\end{aligned}$$ Then, it is readily seen that $$\begin{aligned}
& \frac{\partial C(p, \beta)}{\partial p} = (1-2\beta)\Big\{ [1-p-\beta(1-2p)]e^{-[p+ \beta(1-2p)]} \\
& - [1-(1-p)-\varepsilon(2p-1)]e^{-[(1-p)+ \beta(2p-1)]} \Big\}. \\
\end{aligned}$$
In the light of the monotonically decreasing of $\frac{\partial C(p, \beta)}{\partial p}$ for $p \in [0,1]$, it is apparent that $p=1/2$ is the only solution for $\frac{\partial C(p, \beta)}{\partial p} =0 $. Therefore, the proposition can be testified.
According to Proposition \[prop.symmetric\], on one hand, when $\beta=1/2$, that is, the information transfer process is just random, we will gain the lower bound of $C(\beta)$, namely $C(\beta) =0$. On the other hand, when $\beta=0$, we will have the maximum message importance transfer capacity.
Strongly symmetric information transfer
---------------------------------------
Assume that the information transfer process described by Eq. (\[eq.relation\_1\]) and Eq. (\[eq.relation\]), has a strongly symmetric information transfer matrix $$\begin{aligned}
p(\tilde y|x) = \left [
\begin{matrix}
1-\beta & \frac{\beta}{K-1} &...& \frac{\beta}{K-1} \\
\frac{\beta}{K-1} & 1-\beta & ...& \frac{\beta}{K-1} \\
...& ... & ... & ...\\
\frac{\beta}{K-1} &...& \frac{\beta}{K-1} & 1-\beta
\end{matrix}
\right ],
\end{aligned}$$ and its disturbance $\delta$ follows an uniform distribution (namely $p$($\delta$)= (1/K,... 1/K)), which indicates that variables $X$ and $\tilde Y$ both follow $K$-ary distributions. Then, we have the message importance transfer capacity as $$\begin{aligned}
C(\beta) = e^{-\frac{1}{K}}- \{ (1-\beta)e^{- (1-\beta) } + \beta e^{-\frac{\beta}{K-1} } \},
\end{aligned}$$ where the parameter $\beta \in (0,1)$ and $|C(\beta)| \le \lambda\|p(\tilde y)-p(\tilde y|x)\|_{1}$ ($\lambda \ge \frac{ e^{-{1}/{K}}- (1-\beta)e^{- (1-\beta) } - \beta e^{-{\beta}/{K-1} }}{2|1-\beta-1/K|} $).
This Corollary is an extension of Proposition \[prop.symmetric\]. First, on account of the information transfer matrix and the Eq. (\[MIM\_discrete1\]), we have $$\begin{aligned}
L(\tilde Y|X)
& = \beta e^{-\frac{\beta}{K-1}} + (1-\beta)e^{-(1-\beta)}.
\end{aligned}$$
Then, similar to the proof of Proposition \[prop.symmetric\], we can also use Lagrange multiplier method to obtain the message information transfer capacity. In this case, the distribution of $\tilde Y$ should satisfy $p(\tilde y_1)=p(\tilde y_2)=...=p(\tilde y_K)=1/K$.
In addition, consider that the probability distribution of variable $X$ is $\{p(x_1),p(x_2),...,p(x_K) \}$. In the strongly symmetric transfer matrix, if the variable $X$ follows uniform distribution, namely $p(x_1)=p(x_2)=...=p(x_K)=1/K$, we will have $$\begin{aligned}
p(\tilde y_j)
& = \sum\limits_{i=1}^{K} p(x_i, \tilde y_j)= \sum\limits_{i=1}^{K}p(x_i) p( \tilde y_j|x_i)\\
& = \frac{1}{K} \sum\limits_{i=1}^{K} p(\tilde y_j|x_i)= \frac{1}{K},
\end{aligned}$$ which indicates that $\tilde Y$ also follows the uniform distribution.
Therefore, it is testified that when the variable $X$ follows an uniform distribution which leads to the uniform distribution for variable $\tilde Y$, we will obtain the message importance transfer capacity $C(\beta)$.
Message Importance Transfer Measure in Continuous Cases
=======================================================
Similar to the definition \[defn:MIM\] and \[MIM\_discrete1\], we can extend the two definition to the case with continuous distributions as follows $$\label{MIM_continue1}
\begin{aligned}
L(f(x))
& = \int_{S_x} f(x) e^{-f(x)}dx, \quad \quad x \in S_x,\\
\end{aligned}$$ $$\label{MID_continue}
\begin{aligned}
D_{I}(g(x)||f(x))
& = L(g(x))-L(f(x)) \\
& = \int_{S_x} { g(x) e^{-g(x)}- f(x) e^{-f(x)} }dx , x \in S_x,\\
\end{aligned}$$ where $g(x)$ and $f(x)$ are two probability distributions with respect to the variable $X$ in a given interval $S_x$. Moreover, $L(f(x))$ and $D_{I}(g(x)||f(x))$ can be regarded as the continuous MIM and MITM. Then, we investigate the variation of message importance by using the continuous MITM, which can also reflect the robustness of continuous MITM. Consider the observation model, $\mathcal{P}_{g_0|f_0}$: $f_0(x) \to g_0(x)$, that denotes an information transfer map for the variable $X$ from the probability distribution $f_0(x)$ to $g_0(x)$. By using the similar way in [@An-information-theoretic-approach], the relationship between $f_0(x)$ and $g_0(x)$ can be described as $$\label{gx0_fx0}
g_0(x)= f_0(x) + \epsilon f_0^{\alpha}(x)u(x),$$ and the constraint condition satisfies $$\label{condition.gx0_fx0}
\int_{S_x}\epsilon f_0^{\alpha}(x)u(x) dx=0,$$ where $\epsilon$ and $\alpha$ are adjustable coefficients. $u(x)$ is a perturbation function of the variable $X$ in the interval $S_x$.
Then, by using the above model, the end-to-end information distance measured by continuous MITM is given as follows.
\[prop.disturbance\] For two probability distributions $g_0(x)$ and $f_0(x)$ whose relationship satisfies the conditions Eq. (\[gx0\_fx0\]) and Eq. (\[condition.gx0\_fx0\]), the information distance measured by continuous MITM is given by $$\begin{aligned}
& D_{I}(g_0(x)||f_0(x))\\
& = \int_{S_x} \left\{ g_0(x) e^{-g_0(x)}- f_0(x) e^{-f_0(x)} \right\} dx \\
& = \epsilon \sum\limits_{i=1}^{\infty}\frac{(-1)^i(i+1)}{i!} \int_{S_x} f_0^{i+\alpha}(x)u(x) dx\\
& + \frac{\epsilon^2}{2} \sum\limits_{i=1}^{\infty}\frac{(-1)^i(i+1)}{(i-1)!} \int_{S_x} f_0^{i-1+2\alpha}(x)u^2(x) dx + o(\epsilon^2),
\end{aligned}$$ where $\epsilon$ and $\alpha$ denote parameters, $u(x)$ is a function of the variable $X$ in the interval $S_x$, $|D_{I}(g_0(x)||f_0(x))| \le \int_{S_x}|\epsilon f_0^{\alpha}(x)u(x)| dx$ which satisfies the constraint Eq. (\[eq.Lipschitz\]).
In fact, Proposition \[prop.disturbance\] describes the perturbation between $f_0(x)$ and $g_0(x)$. Furthermore, we can obtain the continuous MITM between two distributions $g_1^{(u)}$ and $g_2^{(u)}$ based on the same reference distribution $f_0(x)$, which is given by $$\begin{aligned}
& D_{I}(g_1^{(u)}(x)||g_2^{(u)}(x))\\
& = [L(g_1^{(u)}(x))-L(f_0(x))]-[L(g_2^{(u)}(x))-L(f_0(x))]\\
& = \epsilon \sum\limits_{i=1}^{\infty}\frac{(-1)^i(i+1)}{i!} \int_{S_x} f_0^{i+\alpha}(x)[u_1(x) - u_2(x)]dx\\
& + \frac{\epsilon^2}{2} \sum\limits_{i=1}^{\infty}\frac{(-1)^i(i+1)}{(i-1)!} \int_{S_x} f_0^{i-1+2\alpha}(x) [u_1^2(x)-u_2^2(x)] dx\\
& + o(\epsilon^2),
\end{aligned}$$ where the $\epsilon$ and $\alpha$ are parameters, $u_1(x)$ and $u_2(x)$ are functions of the variable $X$, and $$g_1^{(u)}(x)= f_0(x) + \epsilon f_0^{\alpha}(x)u_1(x), \quad \forall x\in S_x,$$ $$g_2^{(u)}(x)= f_0(x) + \epsilon f_0^{\alpha}(x)u_2(x), \quad \forall x\in S_x,$$ with the constraint $|D_{I}(g_1(x)||g_2(x))| \le \int_{S_x}|\epsilon f_0^{\alpha}(x)\{u_{1}(x)-u_{2}(x)\}| dx$.
It is apparent that when the parameter $\epsilon$ is small enough, the continuous MITM is convergent with the order of $O(\epsilon)$. Actually, this provides a way to apply the continuous MITM to measure the variantion of message importance, if the system does not have relatively large change.
Application in Mobile Edge Computing with the M/M/s/k queue
============================================================
Consider the MEC system that consists of numerous mobile users, an edge server, and a central cloud. The queue model on the edge server can be considered as the M/M/s/k queue, where the first and the second $M$ denote the request interarrival time of mobile users and service request time in the edge server respectively, and both of them follow exponential distribution; $s$ is the parallel processing core number; $k$ denotes the queuing buffer size [@multi-objective-optimization-for-computation-offloading].
In order to save resources of system, we now consider a more complicated M/M/s/k model which has the request lose depending on the queue length, namely the real arrival rate satisfies $\tilde \lambda_j=\tilde \lambda \cdot h_j$ ($\tilde \lambda$ is the original arrival rate and the parameter $h_j= \frac{1}{1+j}$ depends on the queue length $j$) [@An-explicit-solution; @On-multiserver-feedback]. In fact, the state probability of this queue model is derived from the stationary process, namely a dynamic equilibrium based on birth and death process. In this case, we can obtain the steady queue state probability $p_{k,j}$ ($j=0,..., s+k$) as follows $$\label{eq.p_0}
p_{k,0}= \Big[ \sum\limits_{j=0}^{s-1} \frac{a^j}{j!j!} + \frac{a^s}{s!} \cdot \sum\limits_{j=s}^{s+k} \frac{\rho^{j-s} }{j!} \Big]^{-1}, \ $$ $$\label{eq.p_k_j}
p_{k,j} = \frac{a^j}{j!j!}p_{k,0}, \quad (0<j<s),\qquad \quad \ $$ $$\label{eq.p_k_s}
p_{k,j} = \frac{a^s}{s!j!} \rho^{j-s} p_{k,0}, \quad ( s \le j \le s+k),$$ where $s$ is the number of servers, $k$ is the buffer or caching size, the traffic intensity $\rho= a /s$ as well as $a= \tilde \lambda / \tilde \mu$ ($\tilde \lambda$ and $\tilde \mu$ are the original arrival rate and service rate respectively).
As for the MITM, it can be used to distinguish the state probability distributions in the above M/M/s model. By use of Taylor series expansion, the approximate MIM is given by $$\label{eq.MIM_MMsk}
\begin{aligned}
& \sum\limits_{j=0}^{s+k} p_{k,j} e^{-p_{k,j}}
= \sum\limits_{j=0}^{s+k} p_{k,j}[1-p_{k,j} + O(p_{k,j}^2)]\\
& \doteq 1 - p_{k,0}^2 \bigg\{ \sum\limits_{j=0}^{s-1} {(\frac{a^j}{j!j!})^2}+ (\frac{a^s}{s!})^2 \sum\limits_{j=s}^{s+k} (\frac{\rho^{j-s}}{j!})^2 \bigg\}. \\
\end{aligned}$$ Then, referring to Eq. (\[eq.MIM\_MMsk\]), we can use MITM to characterize the message importance gap for the M/M/s model as follows.
\[pro.MID\_queue\] As for the M/M/s model mentioned in Eq. (\[eq.p\_0\])-(\[eq.p\_k\_s\]), the information difference between two queue state probability distributions $P_{k}= \{{p}_{k,0}, {p}_{k,1}, ..., {p}_{k,s+k}, 0, 0,..., 0\}$ and $P_{k+1} = \{{p}_{k+1,0},$ ${p}_{k+1,1}, ..., {p}_{k+1,s+k+1}, 0,..., 0\}$ with buffer size $k$ and $k+1$ respectively, can be measured by MITM as $$\label{MID_queueing}
\begin{aligned}
& D_I(P_{k+1}||P_{k})\\
& = \sum\limits_{j=0}^{s+k+1} {p}_{k+1,j} e^{-{p}_{k+1,j}} - \sum\limits_{j=0}^{s+k}{p}_{k,j} e^{-{p}_{k,j}} \\
& \doteq \Big\{ \frac{1}{(\varphi_1 + \varphi_2 \sum\limits_{j=s}^{s+k}\frac{\rho^{j-s}}{j!})^2}
- \frac{1}{(\varphi_1 + \varphi_2 \sum\limits_{j=s}^{s+k+1}\frac{\rho^{j-s}}{j!})^2} \Big\} \\
& \quad \cdot \Big\{ \sum\limits_{j=0}^{s-1} {(\frac{a^j}{j!j!})^2}
+ \varphi_2^2 \sum\limits_{j=s}^{s+k} (\frac{\rho^{j-s}}{j!})^2 \Big\} \\
& \quad - \frac{ \varphi_2^2 \rho^{2k+2} }{ [(s+k+1)!]^2
\big(\varphi_1^2+ \varphi_2^2\sum\limits_{j=s}^{s+k} \frac{\rho^{j-s}}{j!}\big) },
\end{aligned}$$ where $p_{k,j}$ and $ p_{k+1,j}$ are queue state probability in the M/M/s/[k]{} and M/M/s/[k+1]{} models with the constraint $|D_I(P_{k+1}||P_{k})| \le \lambda\|P_{k+1}-P_{k}\|_{1}$, as well as the parameter $\varphi_1$ and $\varphi_2$ are given by $ \varphi_1= \sum_{j=0}^{s-1} {a^j}/{(j!j!)}$ and $\varphi_2= {a^s}/{s!}$.
Similarly, it is not difficult to derive the MITM between the queue state probability distributions $P_{\infty} = \{ {p}_{\infty,0}, {p}_{\infty,1}, ...,$ ${p}_{\infty,\infty} \}$ and $P_{k}= \{{p}_{k,0}, {p}_{k,1}, ..., {p}_{k,s+k}, 0, 0,..., 0\}$ with buffer size $\infty$ and $k$, which is given by $$\label{MID_queueing_infty}
\begin{aligned}
& D_I(P_{\infty}|| P_{k}) \\
& \doteq \Big\{ \frac{1}{(\varphi_1 + \varphi_2 \sum\limits_{j=s}^{s+k}\frac{\rho^{j-s}}{j!})^2}
- \frac{1}{\big[ \varphi_1 + \varphi_2 (\frac{e^{\rho}}{\rho^s}
- \sum\limits_{j=0}^{s-1}\frac{\rho^{j-s}}{j!}) \big]^2} \Big\}\\
& \quad \cdot \Big\{ \sum\limits_{j=0}^{s-1} {(\frac{a^j}{j!j!})^2}
+ \varphi_2^2 \sum\limits_{j=s}^{s+k} (\frac{\rho^{j-s}}{j!})^2 \Big\}\\
& \quad - \frac{ \varphi_2^2 \Big(\frac{e^{\rho}}{\rho^s}
- \sum\limits_{j=0}^{s+k}\frac{\rho^{j-s}}{j!}\Big) }{ \Big[ \varphi_1 + \varphi_2 (\frac{e^{\rho}}{\rho^s}
- \sum\limits_{j=0}^{s-1}\frac{\rho^{j-s}}{j!}) \Big]^2}.
\end{aligned}$$
Moreover, for the queue length selection, it is required that the distinction between two distribution $P_{\infty}$ and $P_{k}$ should be small enough, namely, $|D_{I}(P_{\infty}|| P_{k})| \le \epsilon $ ($\epsilon$ is a small parameter). Since that the lower bound of buffer size is complicated, we have a looser lower bound as follows $$\begin{aligned}
& k \ge
\frac{ \ln \Big\{ 1-\frac{1-\rho}{\varphi_2} \big[ ({\varphi}/{\sum\limits_{j=0}^{s-1} (\frac{a^j}{j!j!})^2})^{-\frac{1}{2}} -\varphi_1 \big] \Big\} }{\ln \rho} -1,
\end{aligned}$$ where the parameter $\varphi$ is given by $$\begin{aligned}
\varphi= \epsilon +{\sum\limits_{j=0}^{s-1}(\frac{a^j}{j!j!})^2+\frac{\varphi_2^2 e^{\rho}}{\rho^s} }{\Big[ \varphi_1 + \varphi_2 (\frac{e^{\rho}}{\rho^s}
- \sum\limits_{j=0}^{s-1}\frac{\rho^{j-s}}{j!}) \Big]^{-2} }.
\end{aligned}$$ It is easy to see that $\epsilon$ plays a key role in the caching size selection when using finite size caching to imitate the infinite caching working mode.
Similar to MITM, the KL divergence between the queue state probability distributions with buffer size $k+1$ and $k$ is given by $$\begin{aligned}
& D(P_{k} || P_{k+1})\\
& = \sum\limits_{j} p_{k,j} \log \frac{1}{p_{k+1,j}} - \sum\limits_{j} p_{k,j} \log \frac{1}{p_{k,j}}\\
& = \log \Big\{ 1+ \frac{ \rho^{k+1}}{(s+k+1)!\big(\varphi_1+\varphi_2 \sum\limits_{j=s}^{s+k} \frac{\rho^{j-s}}{j!}\big)} \Big\},
\end{aligned}$$ where the parameters $p_{k,j}$, $p_{k+1,j}$, $\varphi_1$ and $\varphi_2$ are the same as them in Proposition \[pro.MID\_queue\].
Likewise, we can derive the KL divergence between the queue state distributions with buffer size $\infty$ and $k$ as $$\begin{aligned}
D(P_{k}||P_{\infty})
& = \log \frac{ \varphi_1 + \varphi_2 (\frac{e^{\rho}}{\rho^s}- \sum_{j=0}^{s-1} \frac{\rho^{j-s}}{j!})}{ \varphi_1+ \varphi_2 \sum_{j=s}^{s+k}\frac{\rho^{j-s}}{j!} }.
\end{aligned}$$
For the queue length selection with KL divergence, we have a looser lower bound of buffer size as follows $$\begin{aligned}
k \ge \frac{\ln \Big\{ 1- \frac{(1-\rho)}{ 2^{\epsilon} \varphi_2}\big[ \varphi_1(1-2^{\epsilon})
+ \varphi_2(\frac{e^{\rho}}{\rho^s} - \sum\limits_{j=0}^{s-1} \frac{\rho^{j-s}}{j!})
\big] \Big\}}{\ln \rho} -1.
\end{aligned}$$ To validate our derived results in theory, some simulations are presented. The events arrivals are listed in Table \[table\_arrival\]. It is readily seen that they have the same average interarrival time as $1/\tilde \lambda_{j,0}$. Besides, the traffic intensity is selected as $\rho = 0.9$ in all discussed cases.
[|c|c|c|c|]{}
--------------
Type of
Distribution
--------------
: The Interarrival Time Distributions of Events’ Arrivals[]{data-label="table_arrival"}
&
--------------
Exponential
Distribution
--------------
: The Interarrival Time Distributions of Events’ Arrivals[]{data-label="table_arrival"}
&
--------------
Uniform
Distribution
--------------
: The Interarrival Time Distributions of Events’ Arrivals[]{data-label="table_arrival"}
&
--------------
Normal
Distribution
--------------
: The Interarrival Time Distributions of Events’ Arrivals[]{data-label="table_arrival"}
\
$P(X)$ & $X \sim E({\tilde \lambda_{j,0}})$ & $X \sim U(0,2/{\tilde \lambda_{j,0}})$ & $X \sim N(\frac{1}{\tilde \lambda_{j,0}}, \frac{1}{{\tilde \lambda_{j,0}}^2} )$\
In Fig. \[fig\_kk\_performance\] and \[fig\_inftyk\_performance\], the legends $D_I$-$Sim$, $D_I$-$Ana$ and $D$-$Sim$, $D$-$Ana$ denote the simulation results and the analytical results for MITM and KL divergence, respectively. It is illustrated that the convergence of MITM is faster than that of KL divergence, which indicates that MITM may provide a reasonable lower bound to select the caching size for MEC. In addition, we can see that the Poisson distribution corresponds the worst case for the arrival process among the three discussed cases with respect to the convergence of both MITM and KL divergence.
Conclusion
==========
In this paper, we investigated the information transfer problem in big data and proposed an information measure, i.e., MITM. Furthermore, this information measure has its own dramatic characteristics on paying more attention to the message importance hidden in big data. This makes the information measure as a promising tool for information transfer measure in big data. We presented the message importance transfer capacity measured by the MITM which can give an upper bound for the information transfer with disturbance. Furthermore, the MITM was extended to the continuous case to investigate the variation of message importance in the information transfer process. In addition, we employed the MITM to discuss the caching size selection in the MEC.
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[^1]: We indeed appreciate the support of the National Natural Science Foundation of China (NSFC) No. 61771283.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Modern approaches for multi-person pose estimation in video require large amounts of dense annotations. However, labeling every frame in a video is costly and labor intensive. To reduce the need for dense annotations, we propose a PoseWarper network that leverages training videos with sparse annotations (every $k$ frames) to learn to perform dense temporal pose propagation and estimation. Given a pair of video frames—a labeled Frame A and an unlabeled Frame B—we train our model to predict human pose in Frame A using the features from Frame B by means of deformable convolutions to implicitly learn the pose warping between A and B. We demonstrate that we can leverage our trained PoseWarper for several applications. First, at inference time we can reverse the application direction of our network in order to propagate pose information from manually annotated frames to unlabeled frames. This makes it possible to generate pose annotations for the entire video given only a few manually-labeled frames. Compared to modern label propagation methods based on optical flow, our warping mechanism is much more compact ($6$M vs $39$M parameters), and also more accurate ($88.7\%$ mAP vs $83.8\%$ mAP). We also show that we can improve the accuracy of a pose estimator by training it on an augmented dataset obtained by adding our propagated poses to the original manual labels. Lastly, we can use our PoseWarper to aggregate temporal pose information from neighboring frames during inference. This allows us to obtain state-of-the-art pose detection results on PoseTrack2017 and PoseTrack2018 datasets. Code has been made available at: <https://github.com/facebookresearch/PoseWarper>.'
author:
- |
Gedas Bertasius$^{1,2}$, Christoph Feichtenhofer$^{1}$, Du Tran$^{1}$, Jianbo Shi$^{2}$, Lorenzo Torresani$^{1}$\
$^{1}$Facebook AI, $^{2}$University of Pennsylvania
bibliography:
- 'gb\_bibliography.bib'
title: |
Learning Temporal Pose Estimation\
from Sparsely-Labeled Videos
---
Introduction
============
In recent years, visual understanding methods [@gberta_2015_CVPR; @xie2016groups; @DBLP:journals/corr/ToshevS13; @gberta_2017_CVPR; @SPP; @he2017maskrcnn; @lin2017focal; @ren2015faster; @girshick15fastrcnn; @girshick2014rcnn; @guptaECCV14; @44872; @dai16rfcn; @DBLP:journals/corr/RedmonDGF15; @DBLP:journals/corr/RedmonF16] have made tremendous progress, partly because of advances in deep learning [@NIPS2012_4824; @43022; @Simonyan14c; @He2016DeepRL], and partly due to the introduction of large-scale annotated datasets [@imagenet_cvpr09; @coco_paper]. In this paper we consider the problem of pose estimation, which has greatly benefitted from the recent creation of large-scale datasets [@Iqbal_CVPR2017; @xiao2018simple]. Most of the recent advances in this area, though, have concentrated on the task of pose estimation in still-images [@DBLP:journals/corr/ToshevS13; @xiao2018simple; @cao2017realtime; @DBLP:conf/eccv/NewellYD16; @DBLP:conf/cvpr/WeiRKS16; @sun2019deep]. However, directly applying these image-level models to video is challenging due to nuisance factors such as motion blur, video defocus, and frequent pose occlusions. Additionally, the process of collecting annotated pose data in multi-person videos is costly and time consuming. A video typically contains hundreds of frames that need to be densely-labeled by human annotators. As a result, datasets for video pose estimation [@Iqbal_CVPR2017] are typically smaller and less diverse compared to their image counterparts [@coco_paper]. This is problematic because modern deep models require large amounts of labeled data to achieve good performance. At the same time, videos have high informational redundancy as the content changes little from frame to frame. This raises the question of whether every single frame in a training video needs to be labeled in order to achieve good pose estimation accuracy.
To reduce the reliance on densely annotated video pose data, in this work, we introduce the PoseWarper network, which operates on sparsely annotated videos, i.e., videos where pose annotations are given only every $k$ frames. Given a pair of frames from the same video—a labeled Frame A and an unlabeled Frame B—we train our model to detect pose in Frame A, using the features from Frame B. To achieve this goal, our model leverages deformable convolutions [@8237351] across space and time. Through this mechanism, our model learns to sample features from an unlabeled Frame B to maximize pose detection accuracy in a labeled Frame A.
Our trained PoseWarper can then be used for several applications. First, we can leverage PoseWarper to propagate pose information from a few manually-labeled frames across the entire video. Compared to modern optical flow propagation methods such as FlowNet2 [@DBLP:journals/corr/IlgMSKDB16], our PoseWarper produces more accurate pose annotations ($88.7\%$ mAP vs $83.8\%$ mAP), while also employing a much more compact warping mechanism ($6$M vs $39$M parameters). Furthermore, we show that our propagated poses can serve as effective pseudo labels for training a more accurate pose detector. Finally, our PoseWarper can be used to aggregate temporal pose information from neighboring frames during inference. This naturally renders the approach more robust to occlusion or motion blur in individual frames, and leads to state-of-the-art pose detection results on the PoseTrack2017 and PoseTrack2018 datasets [@Iqbal_CVPR2017].
Related Work
============
**Multi-Person Pose Detection in Images.** The traditional approaches for pose estimation leverage pictorial structures model [@conf/cvpr/AndrilukaRS09; @5540156; @BMVC.24.12:abbreviated; @DBLP:conf/iccv/PishchulinAGS13; @YiYang:2011:APE:2191740.2192012], which represents human body as a tree-structured graph with pairwise potentials between the connected body parts. These approaches have been highly successful in the past, but they tend to fail if some of body parts are occluded. These issues have been partially addressed by the models that assume a non-tree graph structure [@DGLG13; @1541292; @Sigal:CVPR:2006; @10.1007/978-3-540-88690-7_53]. However, most modern approaches for single image pose estimation are based on convolutional neural networks [@DBLP:journals/corr/ToshevS13; @he2017maskrcnn; @xiao2018simple; @cao2017realtime; @DBLP:conf/eccv/NewellYD16; @DBLP:conf/cvpr/WeiRKS16; @sun2019deep; @7780881; @NIPS2014_5291; @6909696; @insafutdinov2016deepercut; @pishchulin16cvpr; @NIPS2014_5573; @DBLP:conf/cvpr/PapandreouZKTTB17]. The method in [@DBLP:journals/corr/ToshevS13] regresses $(x,y)$ joint coordinates directly from the images. More recent work [@DBLP:conf/eccv/NewellYD16] instead predicts pose heatmaps, which leads to an easier optimization problem. Several approaches [@cao2017realtime; @DBLP:conf/cvpr/WeiRKS16; @7780881; @Belagiannis17] propose an iterative pose estimation pipeline where the predictions are refined at different stages inside a CNN or via a recurrent network. The methods in [@he2017maskrcnn; @xiao2018simple; @DBLP:conf/cvpr/PapandreouZKTTB17] tackle pose estimation problem in a top-down fashion, first detecting bounding boxes of people, and then predicting the pose heatmaps from the cropped images. The work in [@cao2017realtime] proposes part affinity fields module that captures pairwise relationships between different body parts. The approaches in [@insafutdinov2016deepercut; @pishchulin16cvpr] leverage a bottom-up pipeline first predicting the keypoints, and then assembling them into instances. Lastly, a recent work in [@sun2019deep], proposes an architecture that preserves high resolution feature maps, which is shown to be highly beneficial for the multi-person pose estimation task.
**Multi-Person Pose Detection in Video.** Due to a limited number of large scale benchmarks for video pose detection, there has been significantly fewer methods in the video domain. Several prior methods [@Iqbal_CVPR2017; @insafutdinov2017; @girdhar2018detecttrack] tackle a video pose estimation task as a two-stage problem, first detecting the keypoints in individual frames, and then applying temporal smoothing techniques. The method in [@SongWVH17] proposes a spatiotemporal CRF, which is jointly optimized for the pose prediction in video. The work in [@Charles16] proposes a personalized video pose estimation framework, which is accomplished by finetuning the model on a few frames with high confidence keypoints in each video. The approaches in [@Pfister15a; @Zhang_2018_CVPR] leverage flow based representations for aligning features temporally across multiple frames, and then using such aligned features for pose detection in individual frames.
In contrast to these prior methods, our primary objective is to learn an effective video pose detector from sparsely labeled videos. Our approach has similarities to the methods in [@Pfister15a; @Zhang_2018_CVPR], which use flow representations for feature alignment. However, unlike our model, the methods in [@Pfister15a; @Zhang_2018_CVPR] do not optimize their flow representations end-to-end with respect to the pose detection task. As we will show in our experiments, this is important for strong performance.
![A high level overview of our approach for using sparsely labeled videos for pose detection. Faces in the figure are artificially masked for privacy reasons. In each training video, pose annotations are available only every $k$ frames. During training, our system considers a pair of frames–a labeled Frame A, and an unlabeled Frame B, and aims to detect pose in Frame A, using the features from Frame B. Our training procedure is designed to achieve two goals: 1) our model must be able to extract motion offsets relating these two frames. 2) Using these motion offsets our model must then be able to rewarp the detected pose heatmap extracted from an unlabeled Frame B in order to optimize the accuracy of pose detection in a labeled Frame A. After training, we can apply our model in reverse order to propagate pose information across the entire video from ground truth poses given for only a few frames.[]{data-label="main_fig"}](./paper_figures/main/main_approach6_noface.pdf){width="0.82\linewidth"}
The PoseWarper Network
======================
**Overview.** Our goal is to design a model that learns to detect pose from sparsely labeled videos. Specifically, we assume that pose annotations in training videos are available every $k$ frames. Inspired by a recent self-supervised approach for learning facial attribute embeddings [@Wiles18a], we formulate the following task. Given two video frames—a labeled Frame A and an unlabeled Frame B—our model is allowed to compare Frame A to Frame B but it must predict Pose A (i.e., the pose in Frame A) using the features from Frame B, as illustrated in Figure \[main\_fig\] (top).
At first glance, this task may look overly challenging: how can we predict Pose A by merely using features from Frame B? However, suppose that we had body joint correspondences between Frame A and Frame B. In such a scenario, this task would become trivial, as we would simply need to spatially “warp” the feature maps computed from frame B according to the set of correspondences relating frame B to frame A. Based on this intuition, we design a learning scheme that achieves two goals: 1) By comparing Frame A and Frame B, our model must be able to extract motion offsets relating these two frames. 2) Using these motion offsets our model must be able to rewarp the pose extracted from an unlabeled Frame B in order to optimize pose detection accuracy in a labeled Frame A. To achieve these goals, we first feed both frames through a backbone CNN that predicts pose heatmaps for each of the frames. Then, the resulting heatmaps from both frames are used to determine which points from Frame B should be sampled for detection in Frame A. Finally, the resampled pose heatmap from Frame B is used to maximize accuracy of Pose A.
![An illustration of our PoseWarper architecture. Given a labeled Frame A and an unlabeled Frame B, which are separated by $\delta$ steps in time, our goal is to detect pose in a labeled Frame A using the features from an unlabeled Frame B. First, we predict pose heatmaps for both frames. Then, we compute the difference between pose heatmaps in Frame A and Frame B and feed it through a stack of $3 \times 3$ residual blocks. Afterwards, we attach five $3 \times 3$ convolutional layers with dilation rates $d \in \{3, 6, 12, 18, 24\}$ and predict five sets of offsets $o^{(d)}(p_n)$ for each pixel location $p_n$. The predicted offsets are used to rewarp pose heatmap B. All five rewarped heatmaps are then summed and the resulting tensor is used to predict pose in Frame A.[]{data-label="arch_fig"}](./paper_figures/arch/arch5_noface.pdf){width="1\linewidth"}
**Backbone Network.** Due to its high efficiency and accuracy, we use the state-of-the-art High Resolution Network (HRNet-W48) [@sun2019deep] as our backbone CNN. However, we note that our system can easily integrate other architectures as well. Thus, we envision that future improvements in still-image pose estimation will further improve the effectiveness of our approach.
**Deformable Warping.** Initially, we feed Frame A and Frame B through our backbone CNN, which outputs pose heatmaps and . Then, we compute the difference . The resulting feature tensor is provided as input to a stack of $3 \times 3$ simple residual blocks (as in standard ResNet-18 or ResNet-34 models), which output a feature tensor . The feature tensor is then fed into five $3 \times 3$ convolutional layers each using a different dilation rate $d \in\{3, 6, 12, 18, 24\}$ to predict five sets of offsets $o^{(d)} (p_n)$ at all pixel locations $p_n$. The motivation for using different dilation rates at the offset prediction stage comes from the need to consider motion cues at different spatial scales. When the body motion is small, a smaller dilation rate may be more useful as it captures subtle motion cues. Conversely, if the body motion is large, using large dilation rate allows us to incorporate relevant information further away. Next, the predicted offsets are used to spatially rewarp the pose heatmap $f_{B}$. We do this for each of the five sets of offsets $o^{(d)}$, and then sum up all five rewarped pose heatmaps to obtain a final output , which is used to predict pose in Frame A.
We implement the warping mechanism via a deformable convolution [@8237351], which takes 1) the offsets $o^{(d)}(p_n)$, and 2) the pose heatmap $f_{B}$ as its inputs, and then outputs a newly sampled pose heatmap . The subscript $(A,B)$ is used to indicate that even though was resampled from tensor , the offsets for rewarping were computed using , which contains information from both frames. An illustration of our architecture is presented in Figure \[arch\_fig\].
**Loss Function.** As in [@sun2019deep], we use a standard pose estimation loss function which computes a mean squared error between the predicted, and the ground-truth heatmaps. The ground-truth heatmap is generated by applying a 2D Gaussian around the location of each joint.
**Pose Annotation Propagation.** During training, we force our model to warp pose heatmap from an unlabeled frame B such that it would match the ground-truth pose heatmap in a labeled Frame A. Afterwards, we can reverse the application direction of our network. This then, allows us to propagate pose information from manually annotated frames to unlabeled frames (i.e. from a labeled Frame A to an unlabeled Frame B). Specifically, given a pose annotation in Frame A, we can generate its respective ground-truth heatmap by applying a 2D Gaussian around the location of each joint (identically to how it was done in [@xiao2018simple; @sun2019deep]. Then, we can predict the offsets for warping ground-truth heatmap to an unlabeled Frame B, from the feature difference . Lastly, we use our deformable warping scheme to warp the ground-truth pose heatmap to Frame B, thus, propagating pose annotations to unlabeled frames in the same video. See Figure \[main\_fig\] (bottom) for a high-level illustration of this scheme.
**Temporal Pose Aggregation at Inference Time.** Instead of using our model to propagate pose annotations on training videos, we can also use our deformable warping mechanism to aggregate pose information from nearby frames during inference in order to improve the accuracy of pose detection. For every frame at time $t$, we want to aggregate information from all frames at times $t+\delta$ where $\delta \in \{-3,-2,-1,0,1,2,3\}$. Such a pose aggregation procedure renders pose estimation more robust to occlusions, motion blur, and video defocus.
Consider a pair of frames, $I_t$ and $I_{t+\delta}$. In this case, we want to use pose information from frame $I_{t+\delta}$ to improve pose detection in frame $I_t$. To do this, we first feed both frames through our trained PoseWarper model, and obtain a spatially rewarped (resampled) pose heatmap , which is aligned with respect to frame $I_t$ using the features from frame $I_{t+\delta}$. We can repeat this procedure for every $\delta$ value, and then aggregate pose information from multiple frames via a summation as $\sum_{\delta}g_{t,t+\delta}$. **Implementation Details.** Following the framework in [@sun2019deep], for training, we crop a $384 \times 288$ bounding box around each person and use it as input to our model. During training, we use ground truth person bounding boxes. We also employ random rotations, scaling, and horizontal flipping to augment the data. To learn the network, we use the Adam optimizer [@DBLP:journals/corr/KingmaB14] with a base learning rate of $10^{-4}$, which is reduced to $10^{-5}$ and $10^{-6}$ after $10$, and $15$ epochs, respectively. The training is performed using $4$ Tesla M40 GPUs, and is terminated after $20$ epochs. We initialize our model with a HRNet-W48 [@sun2019deep] pretrained for a COCO keypoint estimation task. To train the deformable warping module, we select Frame B, with a random time-gap $\delta \in [-3,3]$ relative to Frame A. To compute features relating the two frames, we use twenty $3 \times 3$ residual blocks each with $128$ channels. Even though this seems like many convolutional layers, due to a small number of channels in each layer, this amounts to only $5.8$M parameters (compared to $39$M required to compute optical flow in [@DBLP:journals/corr/IlgMSKDB16]). To compute the offsets , we use five $3 \times 3$ convolutional layers, each using a different dilation rate ($d=3,6,12,18,24$). To resample the pose heatmap , we employ five $3 \times 3$ deformable convolutional layers, each applied to one of the five predicted offset maps . The five deformable convolution layers too employ different dilation rates of $3,6,12,18,24$. During testing, we follow the same two-stage framework used in [@sun2019deep; @xiao2018simple]: first, we detect the bounding boxes for each person in the image using the detector in [@girdhar2018detecttrack], and then feed the cropped images to our pose estimation model.
![The results of a video pose propagation task by our PoseWarper and FlowNet2 [@DBLP:journals/corr/IlgMSKDB16]. The first frame in each $3$-frame sequence illustrates a [*labeled*]{} reference frame at time t. For simplicity, we show only the “right ankle” body joint for one person, denoted by a **[ pink]{}** circle in each of the frames (please zoom in for a clearer view). The second frame depicts our propagated “right ankle” detection from the labeled frame in time t to the unlabeled frame in time t+1. The third frame shows the propagated detection in frame t+1 produced by the FlowNet2 baseline. In contrast to our method, FlowNet2 fails to propagate poses when there is large motion, blurriness or occlusions.[]{data-label="flownet2_comparison_fig"}](./paper_figures/flownet2_comparison/flownet2_comparison5_v2_noface.pdf){width="0.92\linewidth"}
Experiments
===========
In this section, we present our results on the PoseTrack [@Iqbal_CVPR2017] dataset. We demonstrate the effectiveness of our approach on three applications: 1) video pose propagation, 2) training a network on annotations augmented with propagated pose pseudo-labels, 3) temporal pose aggregation during inference.
Video Pose Propagation
----------------------
**Quantitative Results.** To verify that our model learns to capture pose correspondences, we apply it to the task of video pose propagation, i.e., propagating poses across time from a few labeled frames. Initially, we train our PoseWarper in a sparsely labeled video setting according to the procedure described above. In this setting, every $7^{th}$ frame of a training video is labeled, i.e. there are $6$ unlabeled frames between each pair of manually labeled frames. Since each video contains on average 30 frames, we have approximately $5$ annotated frames uniformly spaced out in each video. Our goal then, is to use our learned PoseWarper to propagate pose annotations from manually-labeled frames to all unlabeled frames in the same video. Specifically, for each labeled frame in a video, we propagate its pose information to the three preceding and three subsequent frames. We train our PoseWarper on sparsely labeled videos from the training set of PoseTrack2017 [@Iqbal_CVPR2017] and then perform our evaluations on the validation set.
To evaluate the effectiveness of our approach, we compare our model to several relevant baselines. As our weakest baseline, we use our trained HRNet [@sun2019deep] model that simply predicts pose for every single frame in a video. Furthermore, we also include a few propagation baselines based on warping annotations using optical flow. The first of these uses a standard Farneback optical flow [@Farneback:2003:TME:1763974.1764031] to warp the manually-labeled pose in each labeled frame to its three preceding and three subsequent frames. We also include a more advanced optical flow propagation baseline that uses FlowNet2 optical flow [@DBLP:journals/corr/IlgMSKDB16]. Finally, we evaluate our PoseWarper model.
In Table \[pose\_propagation\_table\], we present our quantitative results for video pose propagation. The evaluation is done using an mAP metric as in [@insafutdinov2016deepercut]. Our best model achieves a $88.7\%$ mAP, while the optical flow propagation baseline using FlowNet2 [@DBLP:journals/corr/IlgMSKDB16] yields an accuracy of $83.8\%$ mAP. We also note that compared to the FlowNet2 [@DBLP:journals/corr/IlgMSKDB16] propagation baseline, our PoseWarper warping mechanism is not only more accurate, but also significantly more compact ($6$M vs $39$M parameters).
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Method Head Shoulder Elbow Wrist Hip Knee Ankle Mean
---------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------- ---------- ------- ------- ------ ------ ------- ------
Pseudo-labeling w/ HRNet [@sun2019deep] 79.1 86.5 81.4 74.7 81.4 79.4 72.3 79.3
Optical Flow Propagation (Farneback [@Farneback:2003:TME:1763974.1764031]) 76.5 82.3 74.3 69.2 80.8 74.8 70.1 75.5
Optical Flow Propagation (FlowNet2 [@DBLP:journals/corr/IlgMSKDB16]) 82.7 91.0 83.8 78.4 89.7 83.6 78.1 83.8
PoseWarper (no dilated convs) 86.1 91.7 88.0 83.5 90.2 87.3 84.6 87.2
PoseWarper (1 dilated conv) 85.0 91.6 88.0 83.7 89.6 87.3 84.7 87.0
PoseWarper (2 dilated convs) 85.8 92.4 88.8 84.9 91.0 88.4 86.0 88.0
PoseWarper (3 dilated convs) 86.1 92.6 89.2 85.5 91.3 88.8 86.3 88.4
PoseWarper (4 dilated convs) **86.3 & 92.6 & **89.5 & 85.9 & **91.9 & 88.8 & 86.4 & 88.6\
PoseWarper (5 dilated convs) & 86.0 & **92.7 & **89.5 & **86.0 & 91.5 & **89.1 & **86.6 & **88.7\
******************
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: The results of video pose propagation on the PoseTrack2017 [@Iqbal_CVPR2017] validation set (measured in mAP). We propagate pose information across the entire video from the manual annotations provided in few frames. To study the effect of different levels of dilated convolutions in our PoseWarper architecture, we also include several ablation baselines (see the bottom half of the table).[]{data-label="pose_propagation_table"}
**Ablation Studies on Dilated Convolution.** In Table \[pose\_propagation\_table\], we also present the results investigating the effect of different levels of dilated convolutions in our PoseWarper architecture. We evaluate all these variants on the task of video pose propagation. First, we report that removing dilated convolution blocks from the original architecture reduces the accuracy from $88.7$ mAP to $87.2$ mAP. We also note that a network with a single dilated convolution (using a dilation rate of $3$) yields $87.0$ mAP. Adding a second dilated convolution level (using dilation rates of $3,6$) improves the accuracy to $88.0$. Three dilation levels (with dilation rates of $3,6,12$) yield a mAP of $88.4$ and four levels (dilation rates of $3,6,12,18$) give a mAP of $88.6$. A network with $5$ dilated convolution levels yields $88.7$ mAP. Adding more dilated convolutions does not improve the performance further. Additionally, we also experimented with two networks that use dilation rates of $1,2,3,4,5$, and $4,8,16,24,32$, and report that such models yield mAPs of $88.6$ and $88.5$, respectively, which are slightly lower.
**Qualitative Comparison to FlowNet2.** In Figure \[flownet2\_comparison\_fig\], we include an illustration of the motion encoded by PoseWarper, and compare it to the optical flow computed by FlowNet2 for the video pose propagation task. The first frame in each $3$-frame sequence illustrates a [*labeled*]{} reference frame at time t. For a cleaner visualization, we show only the “right ankle” body joint for one person, which is marked with a **[ pink]{}** circle in each of the frames. The second frame depicts our propagated “right ankle” detection from the labeled frame in time t to the unlabeled frame in time t+1. The third frame shows the propagated detection in frame t+1 produced by the FlowNet2 baseline. These results suggest that FlowNet2 struggles to accurately warp poses if 1) there is large motion, 2) occlusions, or 3) blurriness. In contrast, our PoseWarper handles these cases robustly, which is also indicated by our results in Table \[pose\_propagation\_table\] (i.e., $88.7$ vs $83.8$ mAP w.r.t. FlowNet2).
Data Augmentation with PoseWarper
---------------------------------
Here we consider the task of propagating poses on sparsely labeled training videos using PoseWarper, and then using them as pseudo-ground truth labels (in addition to the original manual labels) to train a standard HRNet-W48 [@sun2019deep]. For this experiment, we study the pose detection accuracy as a function of two variables: 1) the total number of sparsely-labeled videos, and 2) the number of manually-annotated frames per video. We aim to study how much we can reduce manual labeling through our mechanism of pose propagation, while maintaining strong pose accuracy. Note, that we first train our PoseWarper on sparsely labeled videos from the training set of PoseTrack2017 [@Iqbal_CVPR2017]. Then, we propagate pose annotations on the same set of training videos. Afterwards, we retrain the model on the joint training set comprised of sparse manual pose annotations and our propagated poses. Lastly, we evaluate this trained model on the validation set.
![A figure illustrating the value of a) training a standard HRNet [@sun2019deep] using our propagated pose pseudo labels (left), and b) our temporal pose aggregation scheme during inference. In both settings, we study pose detection performance as a function of 1) number of sparsely-labeled training videos (with $1$ manually-labeled frame per video), and 2) number of labeled frames per video (with $50$ sparsely-labeled videos in total). All baselines are based on retraining the standard HRNet [@sun2019deep] model on the different training sets. The “GT (1x)” baseline is trained in a standard way on sparsely labeled video data. The “GT (7x)” baseline uses $7$x more manually annotated data relative to the “GT (1x)” baseline. Our approach on the left subfigure (“GT (1x) + pGT (6x)”), augments the original sparsely labeled video data with our propagated pose pseudo labels ($6$ nearby frames for every manually-labeled frame). Lastly, in b) “GT (1x) + T-Agg” denotes the use of PoseWarper to fuse pose information from multiple neighboring frames during inference (training is done as in “GT (1x)” baseline). From the results, we observe that both application modalities of PoseWarper provide an effective way to achieve strong pose accuracy while reducing the number of manual annotations.[]{data-label="results_fig"}](./paper_figures/results/results6.pdf){width="1\linewidth"}
All results are based on a standard HRNet [@sun2019deep] model trained on different forms of training data. “GT (1x)” refers to a model trained on sparsely labeled videos using ground-truth annotations only. “GT (7x)” baseline employs $7$x more manually-annotated poses relative to “GT (1x)” (the annotations are part of the PoseTrack2017 training set). In comparison, our approach (“GT (1x) + pGT (6x)”), is trained on a joint training set consisting of sparse manual pose annotations (same as “GT (1x)” baseline) and our propagated poses (on the training set of PoseTrack2017), which we use as pseudo ground truth data (pGT). As before, for every labeled frame we propagate the ground truth pose to the $3$ previous and the $3$ subsequent frames, which allows us to expand the training set by $7$ times.
Based on the results in the left subfigure of Figure \[results\_fig\], we can draw several conclusions. First, we note that when there are very few labeled videos (i.e., $5$), all three baselines perform poorly (leftmost figure). This indicates that in this setting there is not enough data to learn an effective pose detection model. Second, we observe that when the number of labeled videos is somewhat reasonable (e.g., $50-100$), our approach significantly outperforms the “GT (1x)” baseline, and is only slightly worse relative to the “GT (7x)” baseline. As we increase the number of labeled videos, the gaps among the three methods shrink, suggesting that the model becomes saturated.
As we vary the number of labeled frames per video (second leftmost figure), we notice several interesting patterns. First, we note that for a small number of labeled frames per video (i.e., $1-2$) our approach outperforms the “GT (1x)” baseline by a large margin. Second, we note that the performance of our approach and the “GT (7x)” becomes very similar as we add $2$ or more labeled frames per video. These findings further strengthen our previous observation that PoseWarper allows us to reduce the annotation cost without a significant loss in performance.
Improved Pose Estimation via Temporal Pose Aggregation
------------------------------------------------------
In this subsection we assess the ability of PoseWarper to improve the accuracy of pose estimation at test time by using our deformable warping mechanism to aggregate pose information from nearby frames. We visualize our results in Figure \[results\_fig\] b), where we evaluate the effectiveness of our temporal pose aggregation during inference for models trained a) with a different number of labeled videos (second rightmost figure), and b) with a different number of manually-labeled frames per video (rightmost figure). We compare our approach (“GT (1x) + T-Agg.”) to the same “GT (7x)” and “GT (1x)” baselines defined in the previous subsection. Note that our method in this case is trained exactly as “GT (1x)” baseline, the only difference comes from the inference procedure.
When the number of training videos and/or manually labeled frames is small, our approach provides a significant accuracy boost with respect to the “GT (1x)” baseline. However, once, we increase the number of labeled videos/frames, the gap between all three baselines shrinks, and the model becomes more saturated. Thus, our temporal pose aggregation scheme during inference is another effective way to maintain strong performance in a sparsely-labeled video setting.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Dataset Method Head Shoulder Elbow Wrist Hip Knee Ankle Mean
--------- ------------------------------------------------------------------------------------------------------- ------ ---------- ------- ------- ------ ------ ------- ------
Girdhar et al. [@girdhar2018detecttrack] 72.8 75.6 65.3 54.3 63.5 60.9 51.8 64.1
Xiu et al. [@xiu2018poseflow] 66.7 73.3 68.3 61.1 67.5 67.0 61.3 66.5
Bin et al [@xiao2018simple] 81.7 83.4 80.0 72.4 75.3 74.8 67.1 76.7
HRNet [@sun2019deep] 82.1 83.6 80.4 73.3 75.5 75.3 68.5 77.3
MDPN [@DBLP:conf/eccv/GuoTLCLW18] 85.2 88.5 83.9 77.5 79.0 77.0 71.4 80.7
**PoseWarper & 81.4 & 88.3 & 83.9 & 78.0 & 82.4 & 80.5 & 73.6 & **81.2\
& Girdhar et al. [@girdhar2018detecttrack] & - & - & - & - & - & - & - & 59.6\
& Xiu et al. [@xiu2018poseflow] & 64.9 & 67.5 & 65.0 & 59.0 & 62.5 & 62.8 & 57.9 & 63.0\
& Bin et al [@xiao2018simple] & 80.1 & 80.2 & 76.9 & 71.5 & 72.5 & 72.4 & 65.7 & 74.6\
& HRNet [@sun2019deep] & 80.1 & 80.2 & 76.9 & 72.0 & 73.4 & 72.5 & 67.0 & 74.9\
& **PoseWarper & 79.5 & 84.3 & 80.1 & 75.8 & 77.6 & 76.8 & 70.8 & **77.9\
& AlphaPose [@Fang_2017_ICCV] & 63.9 & 78.7 & 77.4 & 71.0 & 73.7 & 73.0 & 69.7 & 71.9\
& MDPN [@DBLP:conf/eccv/GuoTLCLW18] & 75.4 & 81.2 & 79.0 & 74.1 & 72.4 & 73.0 & 69.9 & 75.0\
& **PoseWarper & 79.9 & 86.3 & 82.4 & 77.5 & 79.8 & 78.8 & 73.2 & **79.7\
& AlphaPose++ [@DBLP:conf/eccv/GuoTLCLW18; @Fang_2017_ICCV] & - & - & - & 66.2 & - & - & 65.0 & 67.6\
& MDPN [@DBLP:conf/eccv/GuoTLCLW18] & - & - & - & 74.5 & - & - & 69.0 & 76.4\
& **PoseWarper & 78.9 & 84.4 & 80.9 & 76.8 & 75.6 & 77.5 & 71.8 & **78.0\
****************
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Multi-person pose estimation results on the validation and test sets of PoseTrack2017 and PoseTrack2018 datasets. Even though our model is designed to improve pose detection in scenarios involving sparsely-labeled videos, here we show that our temporal pose aggregation scheme during inference is also useful for models trained on densely labeled videos. We improve upon the state-of-the-art single-frame baselines [@xiao2018simple; @sun2019deep; @DBLP:conf/eccv/GuoTLCLW18].[]{data-label="sota_table"}
Comparison to State-of-the-Art
------------------------------
We also test the effectiveness of our temporal pose aggregation scheme, when the model is trained on the full PoseTrack [@Iqbal_CVPR2017] dataset. Table \[sota\_table\] compares our method to the most recent approaches in this area [@girdhar2018detecttrack; @xiu2018poseflow; @xiao2018simple; @sun2019deep]. These results suggest that although we designed our method to improve pose estimation when training videos are sparsely-labeled, our temporal pose aggregation scheme applied at inference is also useful for models trained on densely-labeled videos. Our PoseWarper obtains $81.2$ mAP and $77.9$ mAP on PoseTrack2017 validation and test sets respectively, and $79.7$ mAP and $78.0$ mAP on PoseTrack2018 validation and test sets respectively, thus outperforming prior single frame baselines [@girdhar2018detecttrack; @xiu2018poseflow; @xiao2018simple; @sun2019deep].
Interpreting Learned Offsets
----------------------------
Understanding what information is encoded in our learned offsets is nearly as difficult as analyzing any other CNN features [@journals/corr/ZeilerF13; @journals/corr/YosinskiCNFL15]. The main challenge comes from the high dimensionality of offsets: we are predicting $c \times k_h \times k_w$ $(x,y)$ displacements for every pixel for each of the five dilation rates $d$, where $c$ is the number of channels, and $k_h, k_w$ are the convolutional kernel height and width respectively.
In columns $3,4$ of Figure \[offsets\_fig\], we visualize two randomly-selected offset channels as a motion field. Based on this figure, it appears that different offset maps encode different motions rather than all predicting the same solution (say, the optical flow between the two frames). This makes sense, as the network may decide to ignore motions of uninformative regions, and instead capture the motion of different human body parts in different offset maps (say, a hand as opposed to the head). We also note that the magnitudes of our learned offsets encode salient human motion (see Column 5 of Figure \[offsets\_fig\]).
Lastly, to verify that our learned offsets encode human motion, for each point $p_n$ denoting a body joint, we extract our predicted offsets and train a [*linear*]{} classifier to regress the ground truth $(x,y)$ motion displacement of this body joint. In Column 7 of Figure \[offsets\_fig\], we visualize our predicted motion outputs for every pixel. We show Farneback’s optical flow in Column 6. Note that in regions containing people, our predicted human motion matches Farneback optical flow. Furthermore, we point out that compared to the standard Farneback optical flow, our motion fields look less noisy.
Conclusions
===========
In this work, we introduced PoseWarper, a novel architecture for pose detection in sparsely labeled videos. Our PoseWarper can be effectively used for multiple applications, including video pose propagation, and temporal pose aggregation. In these settings, we demonstrated that our approach reduces the need for densely labeled video data, while producing strong pose detection performance. Furthermore, our state-of-the-art results on PoseTrack2017 and PoseTrack2018 datasets demonstrate that our PoseWarper is useful even when the training videos are densely-labeled. Our future work involves improving our model ability to propagate labels and aggregate temporal information when the input frames are far away from each other. We are also interested in exploring self-supervised learning objectives for our task, which may further reduce the need of pose annotations in video. We will release our source code and our trained models upon publication of the paper.
![In the first two columns, we show a pair of video frames used as input for our model. The $3^{rd}$ and $4^{th}$ columns depict $2$ randomly selected offset channels visualized as a motion field. Different channels appear to capture the motion of different body parts. In the $5^{th}$ column, we display the offset magnitudes, which highlight salient human motion. Finally, the last two columns illustrate the standard Farneback flow, and the human motion predicted from our learned offsets. To predict human motion we train a **linear** classifier to regress the ground-truth $(x,y)$ displacement of each joint from the offset maps. The color wheel, at the bottom right corner encodes motion direction.[]{data-label="offsets_fig"}](./paper_figures/interpreting_offsets/interpreting_offsets2_noface.pdf){width="1\linewidth"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A system of three particles undergoing inelastic collisions in arbitrary spatial dimensions is studied with the aim of establishing the domain of “inelastic collapse”—an infinite number of collisions which take place in a finite time. Analytic and simulation results show that for a sufficiently small restitution coefficient, $0\leq r<7-4\sqrt{3}\approx 0.072$, collapse can occur. In one dimension, such a collapse is stable against small perturbations within this entire range. In higher dimensions, the collapse can be stable against small variations of initial conditions, within a smaller $r$ range, $0\leq r<9-4\sqrt{5}\approx 0.056$.
PACS: 47.50.+d, 05.20.Dd
author:
- |
Tong Zhou[^1] and Leo P. Kadanoff[^2]\
The James Franck Institute\
The University of Chicago\
Chicago, IL 60637
title: Inelastic Collapse of Three Particles
---
[2]{}
Introduction
============
A system of particles interacting only through inelastic collisions is a useful idealization of granular materials, and has been much investigated recently[@1; @2; @3; @4; @5; @6; @7; @8; @9; @10]. Inelasticity can make such a system evolve into a collapse state, in which several of the particles collide an infinite number of times in a finite time interval.
Inelastic collapse in one dimension is well understood[@1; @2; @3; @4]. In two dimensions, McNamara and Young carried out numerical investigations and found some evidence for inelastic collapses of three particles[@5]. To understand the collapse mechanism in higher dimensions, we study the behavior of three particles in a particular model. Our model involves collisions which preserve the total momentum and the components of the momentum perpendicular to the line of centers. The component of the relative velocity along the line of centers is reversed (as in an elastic collision) and reduced by the restitution coefficient, $0\le r \le 1$. We look at a situation in which one particle (labeled zero) takes part in all collisions. The other two particles (labeled one and two) are alternatively a collider and a spectator. We assume that all particles have the same mass, and that particles one and two have identical radii.
There are two possible reasons that previous numerical studies might have shown collapse. One scenario is that the collapsed state is represented by one or many attractive fixed points, so that the collapsing orbit can be stable against small variations in initial data. The other scenario is that each orbit is unstable but that the infinity of collapsing orbits produces an observable collapse probability. For the specific example of three particles, we find an attractive fixed point for all dimensions and sufficiently small coefficient of restitution, $r$. Thus, we establish the possibility of the first scenario. The second scenario is still possible, but we have seen no evidence for it. For a larger $r$, there is an interval in which the fixed point is unstable against changes in the initial conditions.
For dimensions greater than one, after the collapse has occurred, the particles can separate from one another. Thus beyond $d=1$, inelastic collapse is an event, not an end-point, in the “lives” of the particles.
To study the collapse, we use the methods of dynamical systems theory. Specifically, we examine the situation in which particles one and two are very close to particle zero and aimed so that the system is very close to the collapsing fixed point. Figure 1 shows a typical configuration. Particle $0$ keeps colliding with particle $1$ and particle $2$ repeatedly, and an inelastic collapse may occur. After many collisions, the distances between the particles become small, and the remaining collisions take place so rapidly that the relative motion of the particles is small. Therefore, in the inelastic collapse, the angle $\theta$ approaches a limiting value as the number of collision goes to infinity. We identify the constant-$\theta$ fixed point and then investigate its stability. We find that a fixed point of an inelastic collapse exists only when the final $\theta$ obeys: $$\cos\theta \geq \frac{4\sqrt{r}}{1+r}.
\label{eq:A}$$ A collapse state will occur whenever this criterion holds and also steric effect do not block off the required collisions. (For example such a blockage will always occur at $\theta = \pi$.) The stability analysis implies that for the collapse to be stable against small perturbations in the initial velocities, a stronger condition is required, namely $$\cos\theta >\frac{2\root 3 \of r (1+\root 3 \of r)}{1+r}.
\label{eq:B}$$
The collision model
===================
We use the standard model of inelastic collision: due to a collision the component of the relative velocity of the colliders along the line of centers, changes by a factor of $-r$. We denote by $\vec{u}_j$ and $\vec{x}_j$ the velocity and the position of the $j$th particle at the instant before a collision occurs. Let us consider a collision between particles $1$ and $0$. In the course of the collision, the velocities of the particles change to: $$\begin{aligned}
\vec{u}_1' & = & \vec{u}_1 - \vec{\Delta}, \nonumber \\
\vec{u}_0' & = & \vec{u}_0 + \vec{\Delta}, \nonumber \\
\vec{u}_2' & = & \vec{u}_2.
\label{eq:1}\end{aligned}$$ Here, the momentum transfer is $\vec{\Delta}$. It must point in the direction of the line of centers. In terms of the coefficient of restitution, $r$, this transfer is given by the expression $$\vec{\Delta} = \frac{1+r}{2} (\vec{x}_1-\vec{x}_0) [(\vec{x}_1-\vec{x}_0)
\cdot
(\vec{u}_1-\vec{u}_0)].$$ Here we have assumed that the radii of the colliding particles sum to unity so that, at the point of collision: $$(\vec{x}_1-\vec{x}_0)^2 = 1.
\label{eq:3}$$
To do dynamical systems theory, we wish to look at the very same process repeatedly. Therefore, we introduce the superscripts, $c$, denoting the collider and $s$ denoting the spectator particle, as well as a subscript, $n$, to denote the instant before the $n$th collision occurs (Figure 1). For simplicity, we take the velocity and the position of particle $0$ to be zero. In order to make sure that the velocity of particle $0$ continues to vanish after the collision, we view the post-collision system from a frame moving with velocity $\vec{u}_0'=
\vec{\Delta}$. Then equations (\[eq:1\]) - (\[eq:3\]) read $$\begin{aligned}
\vec{u}_{n+1}^s & = & \vec{u}_{n}^c - 2 \vec{\Delta}, \nonumber \\
\vec{u}_{n+1}^c & = & \vec{u}_{n}^s-\vec{\Delta}.
\label{eq:3a}\end{aligned}$$ These equations are supplemented by the conditions: $$\vec{\Delta} = \frac{1+r}{2} (\vec{x}_{n}^c) (\vec{x}_{n}^c \cdot
\vec{u}_{n}^c),$$ $$(\vec{x}_{n}^c)^2 = 1.$$ Additionally, both the velocity and the position of particle $0$ vanish. Notice that a collider becomes a spectator immediately following a collision. The above equations are complemented by the equations corresponding to the positions of the particles at the next collision, $$\begin{aligned}
\vec{x}_{n+1}^s & = & \vec{x}_{n}^c + t_n \vec{u}_{n+1}^s, \nonumber \\
\vec{x}_{n+1}^c & = & \vec{x}_{n}^s + t_n \vec{u}_{n+1}^c.
\label{eq:5}\end{aligned}$$ The time interval between the $n$th and the $(n+1)$th collisions, $t_n$, is such that the magnitude of $\vec{x}_{n+1}^c$ is unity.
Flat Surface Approximation
==========================
We now seek fixed points in these equations. We assume that the time between collisions is sufficiently small so that the $t_n$ terms in equations (\[eq:5\]) are negligible and consequently $$\begin{aligned}
\vec{x}_{n+1}^s & = & \vec{x}_{n}^c, \nonumber\\
\vec{x}_{n+1}^c & = & \vec{x}_{n}^s,
\label{eq:apr}\end{aligned}$$ during the approach to the fixed point.
We wish to find a fixed point in the components of the velocity in the direction of the lines of centers. Specifically, we would like to investigate how this component decreases in each iteration. We can define: $$\vec{x}_{n+1}^c \cdot \vec{u}_{n+1}^c \equiv k_n \vec{x}_{n}^c
\cdot \vec{u}_{n}^c.
\label{eq:8}$$
Taking the dot product of equations (\[eq:3a\]) respectively into $\vec{x}_{n}^s$ and $\vec{x}_{n}^c$, and using equations (\[eq:apr\]) gives $$\vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s = -r \vec{x}_{n}^c \cdot \vec{u}_{n}^c,
\label{eq:9a}$$ $$\vec{x}_{n+2}^c \cdot \vec{u}_{n+2}^c = \vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s+\frac{1+r}{2}
\vec{x}_{n+1}^c \cdot \vec{u}_{n+1}^c\cos\theta.
\label{eq:9}$$ Equations (\[eq:8\])-(\[eq:9\]) thus imply the recursion satisfied by the scaling factor $k_n$ $$k_{n+1}=-\frac{r}{k_n}+\frac{1+r}{2}\cos\theta.
\label{eq:rok}$$ Fixed points can be found by setting $k_{n+1}=k_n$ in (\[eq:rok\]), $$k^2 - k \frac{1+r}{2}\cos\theta +r=0.$$ As a result, the fixed point of the scaling factor has two possible values $$k_\pm =\frac{1+r}{4}\cos\theta \pm
\sqrt{\left(\frac{1+r}{4}\cos\theta\right)^2-r}.
\label{eq:11}$$ Equation (\[eq:11\]) is one of the major results of our study.
In every collision, the colliding particle must approach particle $0$. Hence, $\vec{x}_c \cdot \vec{u}_c$ must be negative in every iteration. This is possible only if $k$ is a positive real number. One kind of failure arises when $k$ is complex. Then the real part of the dot product will change sign infinitely often and no fixed point can be reached. Thus, for inelastic collapse to occur, the quantity under the square root in equation (\[eq:11\]) must be positive. This positivity still permits both signs of $\cos\theta$. However, if the roots are real and the cosine is negative, both roots will be negative. Hence neither is a possible solution for inelastic collapse. The only remaining possibility is that inelastic collapse may occur under the condition on the cosine given by equation (\[eq:A\]). According to that statement, when $r\rightarrow 0$, $\theta$ can have a value between $0$ and $\frac{\pi}{2}$. On the other hand, when $\theta =0$, the well-known one-dimensional result is recovered [@2], [*i.e.*]{}, inelastic collapse is possible for $0\le r< 7-4\sqrt{3}$. Regions (a) and (b) in figure 2 are the region of $r$ and $\theta$ for which we may have inelastic collapse.
We now consider the stability of the above fixed points. Stability will imply that a small change in the initial conditions will leave the system in a collapse state, or in other words, changes will still permit an infinite number of collisions. There are two collapse fixed points distinguished by the values of the two multipliers $k=k_+$ and $k=k_-$. Subtracting $k_\pm$ from both sides of (\[eq:rok\]) yields $$\frac{k_{n+1}-k_\pm }{k_n-k_\pm }=\frac{k_\mp}{k_n}.$$ It is seen that $k_-$ corresponds to an unstable fixed point, while $k_+$ corresponds to a stable one. Henceforth, we use $k$ to denote the stable fixed point $k\equiv k_+$.
Next, we investigate the time interval between successive collisions. Assume that relative motion of the two colliding particles between each pair of collisions covers a distance which is very small in comparison to their radii. Then, we can think of the surface of the particles as flat. After the $n$th collision, the colliding particle moves away from the surface and covers a distance $t_n \vec{x}_{n+1}^s
\cdot \vec{u}_{n+1}^s$. In the next step, this particle moves back over the same distance, which is given by -$t_{n+1} \vec{x}_{n+2}^c
\cdot \vec{u}_{n+2}^c $ and reaches the surface. Thus, the ratio of times is $$\frac{t_{n+1}}{t_n}= \frac{-\vec{x}_{n+1}^s \cdot \vec{u}_{n+1}^s}
{\vec{x}_{n+2}^c \cdot \vec{u}_{n+2}^c}.$$ This result may be simplified with the aid of equations (\[eq:8\])-(\[eq:9a\]) to give $$\frac{t_{n+1}}{t_n}=\frac{r}{k^2}.
\label{eq:14}$$ Let $d$ denote the shortest distance between two particles. The distance ratio equals the product of the time ratio and the velocity ratio, $k$, $$\frac{d_{n+1}}{d_n}=\frac{r}{k}.
\label{eq:15}$$
We performed numerical simulations of the collision process by considering three inelastic particles moving in two-dimension with random initial conditions. When collapse happens, we compared the ratios calculated from simulations with the predictions of equations (\[eq:8\]), (\[eq:14\]), (\[eq:15\]). We found excellent agreement (Figure 3), indicating that the fixed points are attractive and indeed correspond to collapse
Validation of the Flat Surface Approximation
============================================
As observed in the last section, making the approximation (\[eq:apr\]) simplifies dramatically the original system and it can be described by a single ratio $k_n$. It is as if the particles have flat surfaces, so that the effect of the tangential components of the velocities of particles $1$ and $2$ can be ignored. This is true only if the time intervals are negligible, so that (\[eq:apr\]) can be obtained from (\[eq:5\]). We will see when such a simplification is valid, and we will set a criterion for the range of validity of the approximation.
In our way to equations (\[eq:9a\])-(\[eq:9\]), we neglected the terms like $t_n (u_{n}^c)^2$ in comparison to $\vec{x}_{n}^c \cdot
\vec{u}_{n}^c$, by using the approximation (\[eq:apr\]). As noted above, the former terms decrease as $(\frac{r}{k^2})^n$, while the latter terms decrease as $k^n$. Thus, the flat surface approximation is reliable only when $r<k^3$ so that terms proportional to $t_n$ can be safely ignored. This condition can be explicitly written as, $$r<\left(\frac{1+r}{4}\cos\theta
+\sqrt{\left(\frac{1+r}{4}\cos\theta\right)^2-r}\right)^3,$$ which can then be simplified into the form of condition (\[eq:B\]). The region of stability determined by this condition is region (a) in Figure 2. The maximum possible value of $r$ for stable behavior is $r_c=9-4\sqrt{5}$.
To this point, our calculation did not rely on the circular geometry of the particles. The name “flat surface” suggests that when criterion (\[eq:B\]) is satisfied, particles do not experience the curvature of their surfaces, and collide as if they are flat. This calculation is valid for arbitrary particle shapes when (\[eq:B\]) is satisfied.
We also observe that when criterion (\[eq:B\]) is satisfied, the time interval $t_n$ decreases faster than the radial component of the velocities. In such a situation, collapse happens so quickly that all other effects, external or internal, have no essential influences to the process. One can further consider arbitrary interactions between the particles as well as arbitrary external fields, as long as all the interactions depends only on the relative positions. Since the particles’ relative positions only change very little during the process of collapse, all the effects of the interactions on, say, particle $1$ can be replaced by a constant total force acting on it which induces an constant acceleration. This acceleration has very little effect in the tangential direction since time interval is too small for it to change the tangential component of the velocity. When the flat surface approximation is valid, the time interval is even too small to change the radial velocity component. We conclude that the the previously obtained fixed points are unchanged.
Circular Geometry
=================
In the previous sections, the calculation were performed by neglecting the $t_n$ terms completely out of equations (\[eq:5\]) and the fixed points for inelastic collapse were found when the final state satisfied condition (\[eq:B\]). After understanding the characteristic behavior of the collapse, we can do a more rigorous calculation to investigate how the system behaves outside the region satisfying (\[eq:B\]).
We now see that during the collapse process, the radial velocity components of particles $1$ and $2$ monotonically decrease till they vanish at the moment of singularity, while their tangential components approach limiting values as the number of collision diverges. Hence, we take those tangential components as constants, and concentrate on the radial components in the calculation of fixed points.
For simplicity we study in detail one collision in the situation where particle $1$ has a zero radius while particle $0$ has a unit radius. Since the theory depends only upon the sum of the two radii, this case subsumes all others. Here we denote the initial instant with a subscript $i$, and the instant before the collision with $f$. We drop the superscripts since only particle $1$ is considered. We also assume the particles are very close, $d\ll 1$, and $-\vec{u}\cdot \vec{x}_i
\ll u_t$, where $u_t$ is the magnitude of the tangential velocity component of particle $1$. The collision time is $$t=\frac{1}{u_t}\left(\frac{-\vec{u}\cdot \vec{x}_i
}{u_t}-\sqrt{\left(\frac{\vec{u}\cdot
\vec{x}_i}{u_t}\right)^2-2d}\right).
\label{eq:19}$$ Immediately before the collision, the radial velocity component of particle $1$ equals $$-\vec{u}\cdot \vec{x}_f =\sqrt{(\vec{u}\cdot \vec{x}_i )^2-2du_t^2}.
\label{eq:20}$$
Equation (\[eq:19\]) gives the quantity needed to complete the equation set (\[eq:3a\])-(\[eq:5\]). In order to abbreviate this calculation, we introduce an effective centrifugal acceleration. In the above calculation, if we view the situation in a frame rotating with an angular velocity of $u_t$, around an axis perpendicular to both $\vec{u}$ and the line of centers, passing through the center of particle $0$, then particle $1$ has zero tangential velocity, and the effect of the tangential velocity can be represented by a centrifugal acceleration $a_1=u_t^2$. This substitution is justified by noticing that we can get exactly the same expressions as (\[eq:19\]) and (\[eq:20\]) by using this acceleration. We do not need really use such a rotating frame. We use $a_1$ as an effective centrifugal acceleration to replace the tangential component of the velocity with the same effects.
After these preparations, we are in a much clearer position. Particle $1$ and particle $2$ respectively have centrifugal accelerations $a_1$ and $a_2$, which are all in the radial directions. Equation (\[eq:apr\]) is again a good approximation. Particle $1$ is moving on a line, and so is particle $2$. We can further drop the vector notation. In the following, we use $u$ to denote the radial component of the velocity of particles $1$ and $2$ immediately before a collision, with its positive direction pointing towards the center of particle $0$. Consequently, $a_1=-u_{1t}^2$ and $a_2=-u_{2t}^2$.
The equation set (\[eq:3a\])-(\[eq:5\]) reduces to $$\begin{aligned}
u_{n+1}^s & = & -ru_n^c+a_{n+1}^st_n \nonumber \\
u_{n+1}^c & = & u_n^s+\frac{1+r}{2}\cos\theta\cdot u_n^c +a_{n+1}^ct_n \nonumber \\
d_n & = & u_{n+1}^c\cdot t_n-\frac{1}{2}a_{n+1}^ct_n^2 \nonumber \\
-d_{n+1} & = & -ru_n^c \cdot t_n+\frac{1}{2}a_{n+1}^st_n^2 \nonumber\\
a_{n+1}^c & = & a_n^s \nonumber \\
a_{n+1}^s & = & a_n^c
\label{eq:set}\end{aligned}$$ where $d_n$ is the distance between the spectator and particle $0$ at the instant of the $n$th collision. Recall that $t_n$ is the time interval between the $n$th and $(n+1)$th collisions.
One simple case can be fully carried through, the case $a_1=a_2\equiv
a$. In this case, particle $1$ and particle $2$ are in a symmetrical situation so that the recursion relation of the system can be obtained after a single collision.
As we did before, we use two nondimensional numbers $k_n$ and $\alpha_n$ to describe the evolution of the system: $$k_n\equiv\frac{u_{n+1}^c}{u_n^c}\quad\mbox{and}\quad\alpha_n\equiv\frac{-a\cdot t_n}{u_n^c}.$$
Starting from (\[eq:set\]), after some straightforward calculation, we get following recursion relation: $$k_{n+1}=\sqrt{\left(\frac{1+r}{2}\cos\theta
-\frac{r}{k_n}\right)^2-(1+r)\cos\theta \frac{\alpha_n}{k_n}},
\label{eq:27}$$ and $$\begin{aligned}
\alpha_{n+1} & = & \frac{1+r}{2}\cos\theta -\frac{r}{k_n}
-\frac{\alpha_n}{k_n}\nonumber\\ & - &
\sqrt{\left(\frac{1+r}{2}\cos\theta
-\frac{r}{k_n}\right)^2-(1+r)\cos\theta \frac{\alpha_n}{k_n}}.
\label{eq:28}\end{aligned}$$
Suppose the fixed point is $(\alpha,k)$, then from (\[eq:28\]), we have $$\alpha =\frac{-k^2+\frac{1+r}{2}\cos\theta\cdot k-r}{k+1}.
\label{eq:29}$$ Since $\alpha\ge 0$ from its definition, the right hand side of the above equality must be nonegative. Thus, there must exist two real solutions of $k$ satisfying r.h.s.$=0$. And we readily recover the condition (\[eq:A\]).
Substituting (\[eq:29\]) into (\[eq:27\]), we have $$\begin{aligned}
k^4 & = & \left(\frac{1+r}{2}\cos\theta\cdot k-r\right)^2\nonumber\\ &
& -(1+r)\cos\theta\cdot k\frac{-k^2+\frac{1+r}{2}\cos\theta\cdot
k-r}{k+1}.
\label{eq:31}\end{aligned}$$ Of course, $k_\pm$, which appeared before, are solutions of (\[eq:31\]). From (\[eq:29\]) we learn that $(0,k_\pm )$ are fixed points. Let us look at other solutions of equation (\[eq:31\]) which satisfy $$\begin{aligned}
k^3+\left(1+\frac{1+r}{2}\cos\theta\right)k^2-\left(r+\frac{1+r}{2}\cos\theta\right)k\nonumber \\
-r=0.\end{aligned}$$ This equation has one and only one solution of $k$ in the interval $[0:1]$. We denote the corresponding fixed point as $(\alpha_0,k_0)$. Of the three relevant fixed points, $(0,k_-)$ is unstable, and we concentrate on the stability condition for the other two fixed points.
Suppose that $(\alpha_n,k_n)$ has a small deviation from the fixed point $(\alpha, k)$. then $$\left( \begin{array}{c} \delta k_{n+1}\\ \delta\alpha_{n+1}\end{array} \right) =\left( \begin{array}{cc} A_{11} & A_{12}\\A_{21} & A_{22} \end{array} \right) \left( \begin{array}{c} \delta k_n\\ \delta\alpha_n\end{array} \right)$$ where $$\begin{aligned}
A_{11} & = & \frac{1}{2k^3}\left[ 2\left(\frac{1+r}{2}\cos\theta -\frac{r}{k}\right)r+(1+r)\cos\theta\cdot\alpha\right] \nonumber \\
A_{12} & = & -\frac{(1+r)\cos\theta}{2k^2} \nonumber \\
A_{21} & = & \frac{\alpha +r}{k^2}-A_{11} \nonumber \\
A_{22} & = & \frac{(1+r)\cos\theta-2k}{2k^2} \nonumber\end{aligned}$$ Let us denote by $\lambda$ the eigenvalue of matrix $A$, hence we have $$\lambda^2-b\lambda+c=0
\label{eq:39}$$ where $$\begin{aligned}
b & = & \frac{1}{2}-\frac{1}{k}+\left[(1+r)\cos\theta-\left(\frac{1+r}{2}\cos\theta\right)^2\right]\frac{1}{2k^2}\nonumber\\
& & +r(1+r)\cos\theta\cdot\frac{1}{k^3}-\frac{3r^2}{2k^4},\nonumber\\
c & = & \frac{r^2}{k^5}.\nonumber\end{aligned}$$ For the fixed point $(0,k_+)$, $$b=\frac{r}{k_+^3}(1+k_+)\quad\mbox{and}\quad c=\frac{r^2}{k_+^5}.$$ So $$\lambda_1=\frac{r}{k_+^2}\quad\mbox{and}\quad \lambda_2=\frac{r}{k_+^3}.
\label{eq:eig}$$ The point $(0,k_+)$ is an attractive fixed point if and only if $r<k_+^3$. Hence we recover the condition of (\[eq:B\]). If we only require $\lambda_1<1$, we can recover condition (\[eq:A\]).
For the fixed point $(\alpha_0,k_0)$, we find that $\alpha_0>0$ if and only if the condition of (\[eq:B\]) is satisfied, but that condition also decides the range of $(r,\theta)$ inside which we will have $$1-b+c<0$$ From equation (\[eq:39\]) we know that one eigenvalue of matrix $A$ is larger than $1$, implying that the corresponding fixed point is unstable. We conclude that there are no additional stable fixed points beyond those which satisfy condition (\[eq:B\]), and there are no stable collapses outside that range in the case $a_1=a_2$.
We believe this is true for the general situation of $a_1\ne a_2$. When condition (\[eq:B\]) is violated, our simulations show that even though the particles can be very close, they will get apart before having collided an infinite number of times.
From the above calculation, and specifically equation (\[eq:eig\]), we can see the parameter space $(r,\theta)$ can be divided into three regions (Figure 2):
a\) When condition (\[eq:B\]) is satisfied, both the eigenvalues of matrix $A$ are smaller than unity. The fixed point is stable in all directions in space $(\alpha, k)$. It is the collapse region.
b\) When condition (\[eq:B\]) is violated but the condition (\[eq:A\]) is satisfied, the eigenvalue $\lambda_2>1$, while $\lambda_1<1$. In this region, particles can have any number of collisions before they might eventually separate.
c\) When the condition (\[eq:A\]) is violated, both eigenvalues of matrix $A$ are larger than unity. Collapse does not occur.
The above calculation is independent of the sources of the accelerations, which could be the interactions between particles. The calculation confirms our previous argument that interactions are irrelevant in the process of collapse[^3].
We also observe that in the above calculation, the circular geometry of the particles is not essential. The calculation is also valid for arbitrary shape of the particles, with the corresponding radius of curvature replacing the radii used. Even though the radius of curvature is not relevant in deciding the behavior of collapse process—the radii do not show up in the expression of the stable fixed point or condition (\[eq:B\]), it does have some effects. The centrifugal acceleration, which obviously is important in deciding the probability of collapse, is inversely proportional to the radius. So chances are larger for collapse to happen when the colliding point is at a position on the surface with a larger radius of curvature.
Conclusion
==========
We demonstrated analytically the existence of inelastic collapse for three particles in all dimensions. At the last moment of collapse, the three particles have a cyclic behavior, which is characterized by a fixed point. We have established the range of the parameters for which the fixed point exists and the range for which it is stable.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to express our thanks to Y.Du, S.Esipov, H.M.Jaeger, M.Mungan, S.R.Nagel, N.Schörghofer and W.R.Young for very helpful discussions and especially to Eli Ben-Naim for continued advice and discussions. This work was supported in part by NSF-DMR and in part by DOE.
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[^1]: Email: [email protected]
[^2]: Email: [email protected]
[^3]: But when there is a sufficiently strong attraction, e.g. gravity, between particles so that the directions of the accelerations are reversed, we want fixed points with $\alpha\le 0$. Then the fixed point $(\alpha_0,k_0)$ becomes stable when condition (\[eq:B\]) is violated. Inelastic collapse can happen in a much larger region of $(r,\theta)$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This paper addresses the radiation back reaction problem for cosmological branes. A general framework is provided in which results are given for the radiation reaction with massles and massive scalar fields with flat extra dimensions and massless conformal fields in anti-de Sitter extra dimensions. For massless scalar field radiation the back reaction terms in the equation of motion are non-analytic. The interpretation of the radiation reaction terms is discussed and the equations of motion solved in simple cases. Nucleosynthesis bounds on dark radiation give a lower bound on the string vacuum energy scale of $\sqrt{A_T}\,m_p$, where $A_T$ is the tensor perturbation amplitude in the cosmic microwave background.'
author:
- 'Ian G. Moss'
- James P Norman
bibliography:
- 'radiation.bib'
title: Radiation back reaction on moving branes
---
introduction
============
Attempts to construct a unified theory of gravity and the other fundamental forces using superstrings have provided a new impetus for the study of higher dimensions and branes. In certain low energy limits, superstring theory reduces to a brane universe, or universes, where ordinary matter is confined to surfaces embedded in higher dimensions [@horava96-2]. The cosmological evolution of these models can be different from the standard cosmological scenario [@cline99; @binetruy99; @shiromizu99] and their study is therefore worthwhile.
The matter fields on the brane interact with higher dimensional fields, which include the graviton and any particles associated with the graviton by supersymmetry. Cosmological equations, which describe the evolution from the point of view of the brane, can be obtained by decomposing the higher dimensional Einstein equations and applying boundary or junction conditions for the brane [@shiromizu99]. These equations reduce to the standard four dimensional equations for energies small compared to the vacuum energy scale of the brane.
Our interest here is mainly in the effects of radiation generated by the movement of the brane. We shall focus our attention on the radiation back reaction force on the brane. The radiation is closely related to the radiation due to a moving mirror [@davies76]. The mirror, like the brane, disturbs the vacuum fluctuations of the radiation field through the effect of the boundary condition at the surface of the mirror. The vacuum fluctuations react back on the mirror though the pressure component in the stress energy tensor.
It may be useful to recall some of the old results on moving mirrors with massless scalar field radiation. The pressure force is proportional to the 3’rd time derivative of the mirror’s position in two dimensions [@davies76] and the 5’th time derivative in four dimensions[@ford82]. There is no radiation reaction force on a uniformly accelerating mirror. A rather perplexing result is that the pressure component of the stress energy tensor diverges as the distance from the mirror shrinks to zero . It is believed that the correct pressure on the mirror is obtained by dropping the divergent terms. Some support for this view is provided by the fact that the total work done by the divergent terms is zero, at least when the mirror motion has constant velocity asymptotically.
The moving mirror might also be placed in a heat bath with temperature $T$. In this case, for a mirror moving with a speed $v$, the thermal contribution to the pressure force is proportional to $T^2v$ in two dimensions [@jaekel92] and $T^4v$ in four dimensions [@machado02]. These agree with the forces which would be expected according to elementary kinetic theory applied to a mirror and a gas of photons.
The equation of motion for the mirror can be modified to take these radiation reaction forces into account. The modified equation of motion is analogous to the Abraham-Lorentz equation of motion for an electron with radiation damping [@landau71], which offers a guide to the interpretation of the radiation reaction effects for the moving mirror and the moving brane. The Abraham-Lorentz equation for the electron’s momentum is $${dp\over dt}=F+\tau{d^2p\over dt^2}$$ where $F$ is an external force and $\tau=2e^2/3mc^3$. A feature of this equation is the existence of ‘runaway’ solutions $p\propto \exp(t/\tau)$, which are rejected on the grounds that the radiation reaction term should be only a small correction to the equation of motion. The runaway solutions can be eliminated by replacing the equation by an integrodifferential equation, but this introduces another feature, ‘preacceleration’, where the electron begins to respond a short time before the external force is applied. We shall see that these features can be found in the brane system.
Recent work related to radiating branes includes [@setare04], which examined the radiation due to a uniformly accelerated brane. The authors did not consider the radiation reaction, but it would be expected to vanish for this type of motion. The brane can also radiate though interactions of the type $X+X\to Y$, where the $X$’s are particles of matter and $Y$ is a bulk graviton. The effect of these gravitons on the brane persists, and resembles ordinary radiation. In some very specific models of brane cosmology, the gravitons can make a small contribution to the expansion rate during the nucleosynthesis era. This has been investigated by a number of authors [@hebecker01; @langlois02; @langlois03]. A similar type of reaction, where $X$ is the inflaton and $Y$ a bulk scalar, could also change the nature of reheating after a period of inflation [@enqvist04] .
The next section contains some general results for the radiation from moving branes and mirrors. We take some care with the perturbation theory because we would like to ensure that the calculations give the vacuum expectation value $\langle \hbox{in}|T_{ab}|\hbox{in}\rangle$ of the stress energy tensor rather than $\langle \hbox{out}|T_{ab}|\hbox{in}\rangle$. It is also important to separate the radiation reaction calculation from effective action and casimir energy calculations. From the point of view of the brane and its cosmological evolution, the radiation back reaction effect is a dissipative phenomenon and cannot be derived from a reduced effective action which only depends on the degrees of freedom attached to the brane. However, we shall see that the Schwinger-Keldeysh formalism gives a reduced effective action which can be used to analyse both dissipative and non-dissipative effects.
Radiation back reaction
=======================
Backreaction equations
----------------------
Our approach to the back reaction on a brane moving in one extra dimension is closely based on the back reaction problem for a moving mirror [@ford82]. In both cases the radiation is produced because the radiation satisfies boundary conditions on a moving boundary. In moving mirror problems the back reaction force is determined by the pressure components of the expectation value of the stress energy tensor.
The back reaction of this radiation on the motion of a brane in five dimensions can be obtained by reducing the five dimensional Einstein equations in the manner described by Shiromizu, Maeda and Sasaki [@shiromizu99]. This reduction assumes a reflection symmetry between to two sides of the brane. For the $5$-dimensional Einstein equations we have $$G_{ab}=\kappa_5^2T_{ab}$$ For simplicity, we shall consider just a 5-dimensional cosmological constant $\Lambda_5$, a brane vacuum energy $\lambda$ and 5-dimensional radiation, $$T_{ab}=-\kappa_5^{-2}\Lambda_5 g_{ab}-\lambda h_{ab}\delta(\Sigma)+
\langle\hbox{in}|T^r_{ab}|\hbox{in}\rangle$$ where $h_{ab}$ is the metric induced on the brane by the 5-dimensional metric $g_{ab}$. Using the Gauss-Codacci equations, Shiromizu, Maeda and Sasaki show that the Einstein tensor ${}^{(4)}G_{\mu\nu}$ satisfies $${}^{(4)}G_{\mu\nu}+\Lambda h_{\mu\nu}={2\kappa_5^2\over 3}
\langle\hbox{in} |T^r_{\mu\nu}
+(T^r_{nn}-\frac14 T^r)h_{\mu\nu}|\hbox{in}\rangle
-E_{\mu\nu}\label{smk}$$ where $E_{\mu\nu}$ is a projection of the Weyl tensor, $C_{n\mu n\nu}$ and therefore trace-free. The 4-dimensional cosmological constant term and Newton’s constant are related to their 5-dimensional counterparts by $$\begin{aligned}
\Lambda&=&{1\over 12}
\left( 6\Lambda_5+\kappa_5^4\lambda^2\right)\\
G&=&{\kappa_5^4\lambda\over 48\pi}\label{bigG}\end{aligned}$$ In order to recover the correct low energy limit for the single brane we require $\lambda>0$ and, for a small or vanishing cosmological constant, $\Lambda_5<0$.
The $E_{\mu\nu}$ term can be eliminated by taking the trace of (\[smk\]), giving an equation for the Ricci scalar of the brane, $${}^{(4)}R=4\Lambda-2\kappa_5^2\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle
\label{rbr}$$ In a cosmological context, this is the equation of motion of the scale factor. The expectation value on the right includes the radiation damping effect on the expansion rate of the universe. In general, the expectation value depends on both the five dimensional metric and the brane motion. However, in this paper we shall assume that the back reaction of the radiation on the five dimensional metric can be neglected in the quantum calculation.
For a spatially flat, homogeneous universe with scale factor $a(t)$ and expansion rate $H(t)$, the cosmological evolution equation (\[rbr\]) becomes $$6\dot H+12H^2=4\Lambda
-2\kappa_5^2\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle.\label{Heom}$$ The initial condition as $t\to-\infty$ is provided by the Friedman equation $3H^2=\Lambda$. The Friedman equation is modified at all other times by the Weyl term, $$E_{00}={2\kappa_5^2\over 3}
\langle\hbox{in}|(T^r_{00}-T^r_{nn})|\hbox{in}\rangle
+\Lambda-3H^2,$$ which can be interpreted as energy lost by the brane. Note, however, that the quantum term can be negative.
Typically, the radiation reaction $\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle$ will only be significant for very early times, but it leaves behind a residual ‘dark radiation’ term [@binetruy99b] obtained by integrating (\[Heom\]), $$E_{00}\sim{2\kappa_5^2\over a^4}\int_{-\infty}^\infty
H(t')a(t')^4\langle\hbox{in}|T^r_{nn}(t')|\hbox{in}\rangle\,dt'\label{dr}$$ In principle, this term behaves like a contribution to the radiation energy density, but we have not, so far, considered the effects of ordinary matter. If, for example, matter is generated by reheating after a period of inflation, then the dark radiation produced during or prior to inflation would be inflated away [@flanagan99].
Quantum stress energy tensor
----------------------------
We shall obtain some general results for the expectation value of the stress energy tensor using perturbation theory. The classical action of the radiation field $\phi$ is $$S_r=\frac12\int_{\cal M}\phi\Delta\phi\,dv\label{raction}$$ where $\Delta$ is a second order operator and $dv$ is the volume measure. The unperturbed system is chosen to be in vacuum state where the green functions of the radiation with the prescribed boundary conditions are known explicitly. For scalar fields, the boundary conditions can be Dirichlet, where the field vanishes on the boundary, or Robin, where the normal derivatives of the field are prescribed. Our notation for Green functions follows ref. [@birrell82].
The relation between the perturbed and unperturbed boundaries can be defined by a diffeomorphism $f$ as shown in figure \[fig1\]. If the perturbation is small, then the vacuum state should be invariant. Expectation values calculated for the brane $\Sigma$ and the metric $g$ should be equivalent to the expectation values calculated with the unperturbed brane $f^{-1}(\Sigma)$ and the metric $f^* g$. By this means we can replace a perturbation of the brane by a perturbation of the metric.
The number of dimensions $n$ is left arbitrary. Local coordinates can be adapted to the unperturbed brane so that $n-1$ coordinates $x^\mu$ describe the position on the brane and a coordinate $z=-x^n$ describes the distance to the perturbed brane. The perturbed brane is then described by the function $z(x^\mu)$.
We begin with the Wightman functions $G^{\pm}$ of the scalar field $\phi$, defined by $$\begin{aligned}
G^+(x.x')&=&\langle\hbox{in}|\phi(x)\phi(x')|\hbox{in}\rangle\label{wight}\\
G^-(x.x')&=&\langle\hbox{in}|\phi(x')\phi(x)|\hbox{in}\rangle.\end{aligned}$$ The Wightman functions satisfy the homogeneous wave equation $$\Delta G^{\pm}=0$$ Their sum defines the Hadamard function, $$G^{(1)}=G^++G^-$$ The change in Hadamard’s function when the operator is perturbed by the moving brane is $$\delta G^{(1)}=-G_R(\delta\Delta)G^{(1)}-G^{(1)}(\delta\Delta)G_A$$ where $G_R$ and $G_A$ are the retarded and advanced propagators, $$G_R(x,x')=G_A(x',x)=-i\langle\hbox{in}|[\phi(x)\phi(x')]|\hbox{in}\rangle
\theta(t-t')$$ The perturbed form of the Hadamard function is real, symmetric and causal.
The first order change in the Hadamard function can be cast into a form which is more convenient for calculations, $$\delta G^{(1)}=-4{\rm Im}(G_0^>(\delta\Delta)G_0^<)$$ where the subscript denotes the use of the unperturbed green function, and $$G_0^>(x,x')=G_0^<(x',x)=iG_0^+(x,x')\theta(t-t')\label{gbig}$$ The subscripts will subsequently be dropped.
The next step is to find an expression for $\delta\Delta$. Consider the general second order operator $$\Delta=-\nabla^2+\xi R+m^2\label{operator}$$ The change in the operator is induced by the coordinate transformation $\xi^a=-z(x^\mu)n^a$ as shown in figure \[fig1\]. The diffeomorphism invariance of the classical action of the radiation (\[raction\]) implies that $$\int_{\cal M}\left(\phi(\delta\Delta)\phi
+(\xi^a\nabla_a\phi)\Delta\phi+\phi\Delta(\xi^a\nabla_a\phi)
\right)dv+\int_{\Sigma}z\phi\Delta\phi\,dv=0$$ After integration by parts we deduce that $$\delta\Delta=\nabla_a\xi^a\Delta-\Delta\xi^a\nabla_a\label{ddelta}$$ The required combination of green functions can now be found explicitly by a covariant volume integral, $$\delta G^{(1)}=
4\,{\rm Im}\,\int_{\cal M}
G^>(x,x')(\Delta\xi^{a'}\nabla_{a'}-\nabla_{a'}\xi^{a'}\Delta)
G^>(y,x')dv'$$ The imaginary part of the integrand consists of a total divergence which reduces by the divergence theorem to $$\delta G^{(1)}=
4\,{\rm Im}\,\int_{\Sigma}\left( \nabla_{n'}G^>(x,x')\nabla_{n'}G^>(y,x')
-G^>(x,x')\nabla_{n'}^2G^>(y,x')\right)z(x')dv'\label{deltag1}$$
The reason for obtaining this expression for the Hadamard function is that it can be used to find the expectation value of the stress energy tensor. The expectation value of the stress energy tensor contains terms such as $$\langle\hbox{in}|(\nabla_a\phi\nabla_b\phi
+\nabla_b\phi\nabla_a\phi)|\hbox{in}\rangle,$$ which can be evaluated by applying an operator to the hadamard function and taking a coincidence limit [@birrell82], $$[\nabla_a\nabla_{b'}G^{(1)}].$$ The brackets denote the coincidence limit $x=x'$. The full stress energy tensor is given by $$\langle\hbox{in}|T^r_{ab}|\hbox{in}\rangle=
\frac12[D_{a'b}G^{(1)}]\label{tdg}$$ where the distribution valued operator $D_{ab}$ is given by $$D_{ab}=\frac12{\delta^2\over \delta\phi^2} T_{ab}=
{\delta\Delta\over\delta g^{ab}}.\label{dab}$$ In the minimal case, for example, $$D_{ab}(x,x')=
\left(\nabla_{a'}\nabla_b-\frac12g_{ab}g^{cd}\nabla_{c'}\nabla_d\right)
\delta(y,x)\delta(y,x')$$ For Dirichlet boundary conditions, the change in the normal component of the stress energy tensor at the brane position is given by (\[deltag1\]), $$\delta\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle=
{\rm Im}\int_{\Sigma}
\left(\nabla_n\nabla_{n'}G^>(x,x')\right)^2z(x')\,dv'\label{tnn}$$ which is equivalent to an old result due to Ford and Vilenkin [@ford82]. For a moving mirror, this stress energy component generates the pressure force on the mirror. Since the unperturbed value of the stress energy does not affect the motion we shall drop the $\delta$ and only keep the perturbed value from now on.
Schwinger-Keldeysh version
--------------------------
Before proceeding, it is interesting to see how the same results can be obtained more directly from an effective action using Schwinger-Keldeysh methods [@schwinger61; @keldysh64; @calzetta87]. In this approach, the operator $\Delta$ is a $2\times2$ matrix which depends on two copies of the background fields, $$\Delta_{SK}=\pmatrix{\Delta[g_1]&0\cr
0&\Delta[g_2]\cr}$$ The radiation reaction forces can be obtained by variation of the one loop correction to the Schwinger-Keldeysh effective action, $$W_{SK}=-\frac{i}2\log\det(iG_{SK})\label{actionsk}$$ where $G_{SK}$ is the green function in the Schwinger-Keldeysh approach. Let $S[g_1]$ be the classical gravitational action, then $\delta_1 S+\delta_1 W_{SK}=0$, where $\delta_1$ denotes variation of $g_1$, and $g=g_1=g_2$ after the variation.
The Schwinger-Keldeysh green function is a $2\times2$ matrix $$G_{SK}=\pmatrix{G_T&iG^-\cr-iG^+&-G_{\bar T}\cr}\label{skp}$$ where $$\begin{aligned}
G_T(x,x')&=&iG^+\theta(t-t')+iG^-\theta(t'-t)\\
G_{\bar T}(x.x')&=&iG^-\theta(t-t')+iG^+\theta(t'-t).\end{aligned}$$ (Note that $G_T$ is identical to the Feynman Green function if the in and out vacua are the same state, but not necessarily identical otherwise). The Schwinger-Keldeysh Green function changes under perturbations of the operator by $$\delta G_{SK}=-G_{SK}(\delta\Delta)_{SK}G_{SK}$$ Multiplying out the matrices gives, for example, $$\delta G_T=-G_T(\delta\Delta)G_T+G^-(\delta\Delta)G^+$$ Under the same restrictions as before, this is equivalent to $$\delta G_T=-2\,{\rm Im}(G^>(\delta\Delta)G^<)\label{pertsk}$$ The variation of the Schwinger-Keldeysh effective action to first order in perturbation theory is therefore $$\delta_1W_{SK}={\rm Im}\,{\rm tr}((\delta_1\Delta)G^>(\delta\Delta)G^<).$$ By (\[tdg\]) and (\[dab\]), this is equivalent to the previous result for the stress energy tensor of the radiation.
Alternatively, we we can regard the action as a function of the brane position and obtain a reduced action formulation, often called the moduli space approximation. Let $$W[z,g]=\frac12{\rm Im}\,{\rm tr}\,\left(\delta\Delta \,G^>\,\delta\Delta\,
G^<\right).$$ Variation with respect to $z$ can be related to a metric variation $\delta_1W$ by diffeomorphism invariance, as in figure \[fig1\]. The effective equations of motion are then $${\delta S\over\delta z}+{\delta W\over\delta z}=0$$
The Schwinger-Keldeysh formalism is usually applied to non-equilibrium thermal field theory, using finite temperature version of the propagator (\[skp\]). Our results are therefore equally applicable to finite temperatures (for a time-independent background) if we replace the function $G^>(x,x')$ by the finite temperature version.
Branes moving in flat space
===========================
We shall examine the radiation back reaction force on the brane to leading order in the displacement from a flat hyperplane in flat space. Consider the massless field with Dirichlet boundary conditions to begin with. The unperturbed green function (\[gbig\]) can be expressed in terms of basis functions which vanish at $z=0$, $$G^>(x,x')=\int_0^\infty{dq\over 2\pi}\int{d^{n-2}k\over (2\pi)^{n-2}}
{i\over 2\omega}4\sin qz\,\sin qz'\,
e^{i{\bf k}({\bf x}-{\bf x}')-i\omega(t-t')}\theta(t-t')$$ where $\omega=(k^2+q^2)^{1/2}$. Inserting this into equation (\[tnn\]) for the stress energy tensor gives $$\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle=
{\rm Im}\,{\rm reg}\int_{-\infty}^\infty{dq\over 2\pi}{dq'\over 2\pi}
\int{d^{n-2}k\over (2\pi)^{n-2}}{q^2q^{\prime 2}\over \omega\omega'}
I[z]$$ where ‘reg’ indicates some form of regularisation has been performed to make the integral finite and $$I[z]=\int_{-\infty}^te^{-i(\omega+\omega')(t-t')}z(t')dt'
=-i\left((\omega+\omega')-i\partial_t\right)^{-1}z$$ Note the importance of taking the imaginary part of the expression [*after*]{} regularisation. It is useful to introduce $$F_n(x)={\rm reg}\,\int{dq\over 2\pi}{dq'\over 2\pi}
{d^{n-2}k\over (2\pi)^{n-2}}{q^2q^{\prime 2}\over \omega\omega'}
{-i\over (\omega+\omega')-ix}$$ and then $$\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle=
{\rm Im}\, F_n(\partial_t)z$$
The integral diverges for all values of the dimension $n$ which rules out dimensional regularisation. However, an analytic regularisation scheme can be used where we define $$F_{ns}(x)={i\over \Gamma(s+1)}
\int_0^\infty d\lambda \lambda^s\int{dq\over 2\pi}{dq'\over 2\pi}
{d^{n-2}k\over (2\pi)^{n-2}}{-q^2q^{\prime 2}\over
\omega\omega'}e^{-\lambda(\omega+\omega')+i\lambda x}\label{fns}$$ and take the value at $s=0$, removing pole terms if necessary. After integrating over $q$, we have $$F_{ns}(x)={-4i\over \Gamma(s+1)}
\int_0^\infty d\lambda \lambda^s\int
{d^{n-2}k\over (2\pi)^{n-2}} (\lambda k)^{-2}K_1(\lambda k)^2\,e^{i\lambda x}$$ where $K_1$ is a Bessel function of the second kind. The remaining integrals give $$F_{ns}(x)={1\over\Gamma(s+1)}{x^{n-s+1}\over 2\pi^{n/2}}
{\Gamma(n/2)^3\over (n+1)\Gamma(n)^2}{i^{n-s}\over \sin\pi(n-s)}$$ After removing the pole at $s=0$, the regularised expression is $$F_n(x)=-{(-ix)^{n+1}\over 4\pi^{n/2}}
\left(1-i\ln\left({x\over\mu}\right)\right)
{\Gamma(n/2)^3\over(n+1)\Gamma(n)^2}$$ where $\mu$ is a renormalisation constant.
Some examples are $$\begin{aligned}
{\rm Im}\,F_2(x)&=&-{x^3\over 12\pi}\\
{\rm Im}\,F_4(x)&=&{x^5\over 720\pi^2}\\
{\rm Im}\,F_5(x)&=&-{x^6\over16384\pi^2}\ln\left({x\over\mu}\right)\end{aligned}$$ The results for two and four dimensions agree with those found by Ford and Vilenkin for the moving mirror problem[@ford82].
The dependence on $\mu$ in odd dimensions can be traced to need for new counterterms in the classical action [@moss03] which arise from the $n$’th heat kernel coefficient of the operator $\Delta$ [@gibbons03]. In five dimensions, the heat kernel coefficient contains boundary terms of the form $$S_{5s}={1\over s}\int_\Sigma\,\nabla_\mu K_{\nu\rho}
\nabla^\mu K^{\nu\rho}\,dv,$$ and other permutations of the indices, where $K$ is the extrinsic curvature of the brane. These counterterms produce a ${\partial_t}^6z$ in the equation of motion (since $K=-{\partial_t}^2z$ to leading order) which cancels the divergence in the radiation reaction and leaves the logarithmic dependence on $\mu$. How the logarithmic derivatives are interpreted, and how to solve the equations of motion for the brane will be covered in section 4.
The radiation back reaction from a massive scalar field with mass $m$ can be found by redefining $\omega=(k^2+q^2+m^2)^{1/2}$ in eq. (\[fns\]). In even dimensions, the integral results in $${\rm Im}\,F_n(x)={1\over 8(4\pi)^{n/2}}m^{n-1}x^2
\Gamma\left(\frac{1-n}2\right)\Gamma\left(\frac12\right)
{}_2F_1\left(\frac{1-n}2,\frac12;2;-\frac{x^2}{4m^2}\right)$$ where ${}_2F_1$ is a hypergeometric function. In odd dimensions, we discard the pole term and retain only the finite part of the same expression.
For small values of $x$, $F_n$ is of order $x^2$. If the position of the brane is oscillating with a frequency $\omega<m$, the radiation reaction force is proportional to $\omega^2$, which is similar to a known result for a domain wall in four dimensions [@vachaspati84]. For large values of $x$, the hypergeometric functions have branch cuts in the region $|x|>2m$ and logarithmic terms appear in the large $x$ limits for both even and odd dimensions. The leading terms agree with the massless results given above.
Branes moving in anti-De Sitter space
=====================================
Branes in anti-de Sitter space are interesting from a cosmological point of view. The intrinsic geometry of a homogeneous brane moving in anti-de Sitter space is similar to a cosmological model [@kraus99; @kehagias99]. The effects of the higher dimensional cosmological constant and the vacuum energy on the brane can be fine-tuned to give a relatively small effective cosmological constant on the brane [@randall99a; @randall99b].
As before, we will neglect the back reaction of the motion of the brane on the bulk metric when calculating the quantum stress energy tensor. We shall consider small perturbations of a single flat brane whose penrose diagram is shown in figure \[fig2\]. We take the vacuum state related to the timelike translation symmetry along the brane, which we call the brane vacuum.
The brane vacuum has the disadvantage that there are horizons where the timelike killing field vanishes. The horizons can be avoided, for example by considering a two brane system and taking the limit where the separation between the branes becomes large. If, instead, we use a vacuum state which covers the whole of anti-de Sitter space, then we have to consider the effects of Bogolubov coefficients [@birrell82]. In the case of the de Sitter brane, the Euclidean vacuum state can be used [@moss03]. The Euclidean vacuum becomes a thermal state with respect to the brane, creating additional thermal effects in the radiation reaction force. We shall not consider this further here.
One of the many ways to express the Anti-de Sitter metric in $n$ dimensions is the conformally flat form, $$\sigma^2z^{-2}(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu)$$ with Minkowski metric $\eta_{\mu\nu}$. The Anti-de Sitter radius $\sigma$ is related to the cosmological constant in $n$ dimensions by $$\Lambda_n=-{(n-1)(n-2)\over2\sigma^2}.$$ The unperturbed brane will be placed at $z=\sigma$ and the manifold extends in the $z>\sigma$ direction only. The covering space can be filled in by reflection symmetry about the brane.
The Green functions can be obtained from the normalised modes $u_{kq}$ of the scalar wave equation. These are $$u_{kq}={1\over (2\omega)^{1/2}}
\left({z\over\sigma}\right)^{(n-1)/2}
H_\nu(q,z)e^{i({\bf k}\cdot{\bf x}-\omega t)}$$ with $\omega=(k^2+q^2)^{1/2}$ and $H_\nu$ a combination of Hankel functions of order $$\nu^2=\frac14+\sigma^2\left((\xi-\xi(n))R+m^2\right).$$ Conformal curvature coupling corresponds to the case $\xi=\xi(n)$ and $\nu=1/2$.
For modes which vanish on the brane, $$H_\nu(q,z)={i(\pi q\sigma)^{1/2}\over 2|H^{(1)}_\nu(q\sigma)|}
\left(H_\nu^{(1)}(qz)H_\nu^{(2)}(q\sigma)
-H_\nu^{(2)}(qz)H_\nu^{(1)}(q\sigma)\right)$$ Note that the normal derivatives of the mode functions when evaluated on the brane $z=\sigma$ are then given by $$\nabla_n u_{kq}={1\over (2\omega)^{1/2}}
\left({2q\sigma\over \pi}\right)^{1/2}{2^{1/2}q\over|H^{(1)}_\nu(q\sigma)|}
\,e^{i({\bf k}\cdot{\bf x}-\omega t)}$$ In the conformal case $\nu=1/2$, this reduces to $$\nabla_n u_{kq}={1\over (2\omega)^{1/2}}
\,2^{1/2}q\, e^{i({\bf k}\cdot{\bf x}-\omega t)}$$ which is also the value obtained in flat space.
The normal derivatives of the Wightman function are needed to obtain the stress energy tensor. The Wightman function can be obtained from a mode sum, $$G^+(x,x')=\int_0^\infty{dq\over2\pi}\int{d^{n-2} k\over (2\pi)^{n-2}}
\,u_{kq}(x)u^*_{kq}(x')$$ The normal derivatives at the brane can be expressed as $$\nabla_n\nabla_{n'}G^+(x,x')=\int_0^\infty{dq\over2\pi}
\int{d^{n-2} k\over (2\pi)^{n-2}}
{2q^2\over 2\omega} \left|{H^{(1)}_{1/2}(q\sigma)\over
H^{(1)}_\nu(q\sigma)}\right|^2
e^{i({\bf k}\cdot({\bf x}-{\bf x'})-\omega (t-t'))}$$ As in the flat space example, the expectation value of the normal components of the stress energy tensor are given in terms of an integral by $$\langle\hbox{in}|T^r_{nn}|\hbox{in}\rangle=
{\rm Im}\, F_n(\partial_t)z$$ where the anti-De Sitter space version of $F$ is $$F_n(x)={\rm reg}\,\int{dq\over 2\pi}{dq'\over 2\pi}
{d^{n-2}k\over (2\pi)^{n-2}}{q^2q^{\prime 2}\over \omega\omega'}
\left|{H^{(1)}_{1/2}(q\sigma)H^{(1)}_{1/2}(q'\sigma)\over
H^{(1)}_\nu(q\sigma)H^{(1)}_\nu(q'\sigma)}\right|^2
{-i\over (\omega+\omega')-ix}\label{newf}$$
The first striking feature of eq. (\[newf\]) is that, in the conformal case $\nu=1/2$, the result for radiative back reaction is identical to the flat space result. In the non-conformal case, we can obtain the large $\sigma x$ limit from the large argument expansion of the Hankel functions, for example $${\rm Im}\,F_5(x)\sim -{x^6\over16384\pi^2}\ln\left({x\over\mu}\right)
\label{F5}$$ in five dimensions. The small $\sigma x$ limit can correspondingly be obtained from the small argument expansion of the Hankel functions, which leads to $${\rm Im}\,F_5(x)\sim C x^6(\sigma x)^{4\nu-2}$$ when $\sigma x\ll 1$, where $C$ is a constant.
Equations of motion with logarithmic terms
==========================================
We can now construct the equation of motion for the brane. For a small perturbation $z$ of a flat brane, the scale factor $a\approx1$ and $\partial_t z\approx-\sigma H$. For conformal scalar field radiation, eq. (\[F5\]) suggests that eq. (\[Heom\]) would become $${dH\over dt}+2H^2-2H_0^2=
-A\kappa_5^2\sigma\ln\left({1\over\mu}{d\over dt}\right){d^5H\over dt^5}.
\label{neweom}$$ where $H_0^2=\Lambda/3$ and $A$ is a numerical coefficient. Strictly speaking, this equation is incomplete because there may be additional radiation damping terms of order $H^2$.
The radiation damping term calculated here is small in recent cosmological eras. Returning to eq (\[bigG\]), we see that the combination $\kappa_5^2\sigma\sim G\sigma^2$ in order of magnitude. The experimental lower bound on $(G\sigma^2)^{-1/4}$ is around $1TeV$, but there is no reason for the the value to be much smaller than the usual Planck scale. When $\dot H\sim H^2$, the damping term is only important when $H^4\sim
(G\sigma^2)^{-1}$.
There is a small residual dark radiation effect given by eq. (\[dr\]). If we take into account a period of inflation, ending at time $t_I$, then the ratio of dark radiation to ordinary radiation is $$6A\kappa_5^2\sigma\int_{t_I}^\infty{H\over \rho}
\ln\left({1\over\mu}{d\over dt}\right){d^5H\over dt^5}dt\label{dri}$$ which is of order $G\sigma^2H(t_I)^4$. In principle, $H(t_I)$ should be close to the value of $H$ determined by the tensor perturbation amplitude $A_T$ in the cosmic microwave background, $A_T^2=8\pi GH^2$ [@liddle93]. To be within nucleosynthesis constraints, the amount of dark radiation must be small [@flanagan99; @hebecker01], and therefore there is a lower bound on the string vacuum energy scale of approximately $$(G\sigma^2)^{-1/4}>A_T^{-1/2}m_p$$ where $m_p$ is the Planck mass.
We shall analyse a slightly more general type of equation of motion, $${d p\over dt}-F(p)=-\eta
\ln\left({1\over\mu}{d\over dt}\right){d^np\over dt^n}\label{motion}$$ A reasonable definition of the logarithmic term should be linear and causal, to ensure that the back reaction depends on the history of the source. We can begin be defining the action of the logarithmic derivative on exponentials, $$\ln\left({1\over\mu}{d\over dt}\right)\,e^{\alpha t}
=\ln\left({\alpha\over\mu}\right)\,e^{\alpha t}$$ where we take $\alpha>0$. This leads to the following definition, $$\ln\left({1\over\mu}{d\over dt}\right)f
=-\int_{-\infty}^t\ln\left(e^\gamma\mu(t-t')\right){df\over dt'}dt'$$ where $\gamma$ is Euler’s constant. The definition is linear, causal and correctly reproduces the action on exponentials. The integral can be evaluated for a broad class of functions, including discontinuous functions, for example $$\ln\left({1\over\mu}{d\over dt}\right)\theta(t)e^{\alpha t}=
\left({\rm E}_1(\alpha t)+\ln\left({\alpha\over\mu}\right)
\right)\theta(t)e^{\alpha t}$$ where ${\rm E}_1$ is the exponential integral. If $f$ is integrable, then the logarithmic derivative of $f$ is $O(t^{-1})$ as $t\to\infty$.
For a linear equation of motion (\[motion\]), with $F=-\lambda p$, we have $${d p\over dt}+\lambda p=-\eta
\ln\left({1\over\mu}{d\over dt}\right){d^5p\over dt^5}$$ The exponential solutions $\exp(\mu zt)$ satisfy $$z+\alpha=-\beta z^n\ln(z)$$ where $\alpha=\mu^{-1}\lambda$ and $\beta=\eta\mu^{n-1}$. The roots for a sample case are plotted in figure \[fig3\]. When there is no reaction term there is only a decaying mode corresponding to $z=-\alpha$. The reaction term shifts this solution and introduces an oscillation. There are two other decaying solutions and two growing modes. These modes have $|z|>1$ and damping or growth rates larger than the renormalisation scale $\mu$. We tentatively identify these as unphysical runaway solutions.
Runaway solutions can be excluded by replacing the equation of motion by an integrodifferential equation. First, we introduce a Green function which decays exponentially as the time $t\to\pm\infty$ and satisfies the following equation, $$G+\eta
\ln\left({1\over\mu}{d\over dt}\right){d^{n-1}G\over dt^{n-1}}=\delta(t-t')$$ The equation of motion (\[motion\]) can then be rewritten in an alternative form $${dp\over dt}=\int_{-\infty}^\infty G(t-t')F(t').$$ The runaway solutions are excluded at the expense of a preacceleration term, as we can see from the expression for the Green function $$G(t-t')=
\sum_{{\rm Re}z_i<0}c_ie^{z_i (t-t')}\theta(t-t')
-\sum_{{\rm Re}z_i>0}c_ie^{z_i (t-t')}\theta(t'-t),$$ where $z_i$ are the roots of $\eta z^{n-1}\ln(z/\mu)=-1$ and $c_i$ are constants fixed by continuity relations at $t=t'$. The acausality is represented by the final term, but it is a relatively minor effect if ${\rm Re}z_i\gg\mu$, because then the acausality only occurs on timescales small compared to the renormalisation timescale.
An interesting variant of the equation of motion (\[motion\]) occurs for the case $n=1$ and $F=0$. There are two real solutions, $p=0$ and $p=\exp(\exp(-\mu\eta^{-1})t)$. Because of the small size of the exponent it would not be appropriate to regard the exponential solution as a runaway solution. In a cosmological context, a flat brane would spontaneously begin to accelerate with a tiny, positive cosmological constant. We have not been able to find a model with this particular radiation reaction so far.
Conclusion
==========
This paper has been devoted to calculating the radiation back reaction forces on a moving brane. We have found that massless, conformally invariant scalar radiation into the bulk results in logarithmic terms in the equations of motion. The radiation reaction leads to both physical and runaway solutions. The runaway solutions can be excluded by introducing an integrodifferential formulation of the equation of motion. This form of the equation could be used if we were to attempt to solve the equations of motion numerically. The question of whether the brane’s initial trajectory might be destabilised by radiation reaction forces remains an open one.
The radiation from the moving brane also leaves a residual effect in the form of dark radiation. The amount of this radiation depends on the brane trajectory (\[dri\]). An order of magnitude estimate combined with a crude nucleosynthesis constaint gives a lower bound on the string vacuum energy scale of $\sqrt{A_T}\,m_p$, where $A_T$ is the tensor perturbation amplitude in the cosmic microwave background.
The radiation reaction problem has been set up in a way which can be generalised to non-scalar fields. One issue which would be of interest is the possibility of supersymetric cancellations amongst some of the terms. The regularisation methods used in this paper have enabled us to neglect the non-logarithmic divergences, but it would be better to see direct cancellation of these divergent terms. For broken supersymmetry, we expect that the renormalisation scale, which appears in our results, would be replaced by a supersymmetry breaking scale.
The results for the radiation reaction of branes moving in anti-de Sitter space generalises in a very staightforward way to any background where the modes of the radiation fields are known explicitly. An example of this is the low energy limit of the heterotic string [@horava96-2; @lukas98; @lukas98-2], where the modes of the graviton multiplet are known and can be found in [@moss04].
Another aspect of the radiation reaction which is worth further study is the effect of different higher dimensional vacuum states. These states can have thermal properties and the reaction forces would have similarities to those acting on moving mirrors at finite temperatures.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'This is a survey of the historical development of the Spectral Standard Model and beyond, starting with the ground breaking paper of Alain Connes in 1988 where he observed that there is a link between Higgs fields and finite noncommutative spaces. We present the important contributions that helped in the search and identification of the noncommutative space that characterizes the fine structure of space-time. The nature and properties of the noncommutative space are arrived at by independent routes and show the uniqueness of the Spectral Standard Model at low energies and the Pati–Salam unification model at high energies.'
title: |
A survey of spectral models of gravity\
coupled to matter
---
Dedicated to Alain Connes
Introduction
============
In 1988, at the height of the string revolution, there appeared an alternative way to think about the structure of space-time, based on the breathtaking progress in the new field of noncommutative geometry. Despite the success of string theory in incorporating gravity, consistency of the theory depended on the existence of supersymmetry as well as six or seven extra dimensions. Enormous amount of research was carried out to obtain the Standard Model from string compactification, which even up to day did not materialize. Most compactifications start in ten dimensions with the Yang–Mills gauge group $E_{8}\times E_{8}$ requiring a very large number of fields to become massive at high energies. In a remarkable paper, Alain Connes laid down the blue print of a new innovative approach to uncover the origin of the Standard Model and its symmetries [@C90]. The foundation of this approach was based on noncommutative geometry, a field he founded few years before [@C85] (see also [@C94]). Alain realized that by making space slightly noncommutative by tensoring the four dimensional space with a space of two points, one gets a parallel universe where the distance between the two sheets is of the order of $10^{-16}$ cm, with the unexpected bonus of having the Higgs scalar field mediating between them. Although this looked similar to the idea of Kaluza–Klein, there were essential differences, mainly in avoiding the huge number of the massive tower of states as well as obtaining the Higgs field in a representation which is not the adjoint. Soon after this work inspired similar approaches also based on extending the four-dimensional space to become noncommutative [@Dub88; @DKM89; @DKM89b; @DKM90; @CFF92].
In this survey we will review the key developments that allowed noncommutative geometry to deepen our understanding of the structure of space-time and explain from first principles why and how nature dictates the existence of the elementary particles and their fundamental interactions. In Section 2 we will start by reviewing the pioneering work of Alain Connes [@C90] introducing the basic mathematical definitions and structures needed to define a noncommutative space. We summarize the characteristic ingredients in the construction of the Connes–Lott model and later generalizations by others. We then consider how to develop the analogue of Riemannian geometry for non-commutative spaces, and to incorporate the gravitational field in the Connes–Lott model. In Section 3 we present a breakthrough in the development of noncommutative geometry with the introduction of the reality operator which led to the definition of KO dimension of a noncommutative space. With this it became possible to present the reconstruction theorem of Riemannian geometry from noncommutative geometry. Section 4 covers the formulation and applications of the spectral action principle where the spectrum of the Dirac operator plays a dominant role in the study of noncommutative spaces. This key development allowed to obtain the dynamics of the Standard Model coupled to gravity in a non-ambiguous way, and to study geometric invariants of noncommutative spaces. We then show that incorporating right-handed neutrinos with the fundamental fermions forces a change in the algebra of the noncommutative space and the use of real structures to impose simultaneously the reality and chirality conditions on fundamental states, singling out the KO dimension to be 6. We show in detail how the few requirements about KO dimension, Majorana masses for right-handed neutrinos and the first order condition on the Dirac operator, singles out the geometry of the Standard Model. In Section 5 we present a classification of finite noncommutative spaces of KO dimension 6 showing the almost uniqueness of the Standard spectral model. In Section 6 we give a prescription of constructing spectral models from first principles and show that the spectral Standard Model agrees with the available experimental limits, provided that the scale giving mass to the right-handed neutrinos is promoted to a singlet scalar field. We then show that there exists a more general case where the first order condition on the Dirac operator is removed, the singlet scalar fields become part of a larger representation of the Pati–Salam model. The Standard Model becomes a special point in the spontaneous breaking of the Pati–Salam symmetries. In Section 6 we show that a different starting point where a Heisenberg like quantization condition in terms of the Dirac operator considered as momenta and two possible Clifford-algebra valued maps from the four-dimensional manifold to two four-spheres $S^{4}$ result in noncommutative spaces with quantized volumes. The Pati–Salam model and its various truncations are uniquely determined as the symmetries of the spaces solving the constraint. Section 7 contains the conclusions and a discussion of possible directions of future research.
### Acknowledgements {#acknowledgements .unnumbered}
The work of A. H. C. is supported in part by the National Science Foundation Grant No. Phys-1518371. He also thanks the Radboud Excellence Initiative for hosting him at Radboud University where this research was carried out. We would like to thank Alain Connes for sharing with us his insights and ideas.
Early days of the spectral Standard Model
=========================================
The first serious attempt to utilize the ideas of noncommutative geometry in particle physic was made by Alain Connes in 1988 in his paper “Essay on physics and noncommutative geometry” [@C90]. He observed that it is possible to change the structure of the (Euclidean) space-time so that the action functional gives the Weinberg-Salam model. The main emphasis was on the conceptual understanding of the Higgs field, which he calls, the black box of the standard model. The qualitative picture was taken to be of a two-sheeted Euclidean space-time separated by a distance of the order of $10^{-16}$ cm. In order to simplify the presentation, and to easily follow the historical development, we will use a uniform notation, representing old results in a new format. It is therefore more efficient to start with the basic definitions.
Noncommutative spaces and differential calculus
-----------------------------------------------
A noncommutative space is determined from the spectral data $\left(
\mathcal{A},\mathcal{H},D,\gamma,J\right) $ where $\mathcal{A}$ is an associative algebra with unit element $1$ and involution \*, $\mathcal{H}$ a Hilbert space carrying a faithful representation $\pi$ of the algebra, $D$ is a self-adjoint operator on $\mathcal{H}$ with $\left( D^{2}+1\right) ^{-1}$ compact, $\gamma$ is the unitary chirality operator and $J$ an anti-unitary operator on $\mathcal{H}$, the reality structure. The operator $J$ was introduced later in 1994 [@C95].
In the model proposed in 1988, there were ambiguities in defining the algebra and the action on the Hilbert space. These were rectified in the 1990 paper [@CL91] with John Lott, in what became known as the Connes–Lott model. In order to appreciate the enormous progress made over the years, we will summarize this model in a simplified presentation. A more detailed account can be found in the early reviews [@VG93; @MGV98; @Kas93; @Kas96; @KasS96; @KasS97]. Note that at around the same time a derivation based differential calculus was introduced by others in [@Dub88; @DKM89; @DKM89b; @DKM90] with many similarities to the model proposed by Connes in 1988.
We first need to first introduce new ingredients. Given a unital involutive algebra $\mathcal{A}$, the universal differential algebra over $\mathcal{A}$ is defined as $$\Omega^{\ast}\left( \mathcal{A}\right) ={\displaystyle\bigoplus\limits_{n=0}^{\infty}}
\Omega^{n}\left( \mathcal{A}\right)$$ where we set $\Omega^{0}\left( \mathcal{A}\right) =\mathcal{A}$, and take$$\Omega^{n}\left( \mathcal{A}\right) =\left\{
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}da_{2}^{i}\cdots da_{n}^{i},\qquad a_{j}^{i}\in
\mathcal{A},\quad\forall i,j\right\} ,\quad n=1,2,\cdots$$ Here $da$ denotes an equivalence class of $\mathcal{A}$, modulo the following relations $$d\left( a\cdot b\right) =da\cdot b+a\cdot db,\qquad d1=0,\qquad d^{2}=0$$ An element of $\Omega^{n}\left( \mathcal{A}\right) $ is called a form of degree $n.$ One forms can be considered as connections on a line bundle whose space of sections is given by the algebra $\mathcal{A}$. A one form $\rho
\in\Omega^{1}\left( \mathcal{A}\right) $ is expressed in the form$$\rho={\displaystyle\sum\limits_{i}}
a^{i}db^{i},\qquad a^{i},b^{i}\in\mathcal{A}$$ and since $d1=0,$ we may impose the condition ${\displaystyle\sum\limits_{i}}
a^{i}b^{i}=1,$ without any loss in generality. We say that $\left(
\mathcal{H},D\right) $ is a Dirac K-cycle for $\mathcal{A}$ if and only if there exists an involutive representation $\pi$ of $\mathcal{A}$ on $\mathcal{H}$ satisfying $\pi\left( a\right) ^{\ast}=\pi\left( a^{\ast}\right) $ with the properties that $\pi\left( a\right) $ and $\left[ D,\pi\left(
a\right) \right] $ are bounded operators on $\mathcal{H}$ for all $a\in\mathcal{A}$. The K-cycle is called even if there exists a chirality operator $\gamma$ such that $\gamma D=-D\gamma,$ $\gamma=\gamma^{-1}=\gamma^{\ast}$ and $\left[ \gamma,\pi\left( a\right) \right] =0,$ otherwise it is odd. The action of $\pi$ on $\Omega^{\ast}\left(
\mathcal{A}\right) $ is defined as $$\pi\left(
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\right) ={\displaystyle\sum\limits_{i}}
\pi\left( a_{0}^{i}\right) [D,\pi\left( a_{1}^{i}\right) ]\cdots\lbrack
D,\pi\left( a_{n}^{i}\right) ]$$ The space of auxiliary fields is defined by $$\mathrm{Aux}=\ker\pi+d\,\ker\pi$$ where $$\ker\pi={\displaystyle\bigoplus\limits_{n=0}^{\infty}}
\left\{
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\,:\pi\left(
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\right) =0\right\}$$ and $$d\ker\pi={\displaystyle\bigoplus\limits_{n=0}^{\infty}}
\left\{
{\displaystyle\sum\limits_{i}}
da_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\,:\pi\left(
{\displaystyle\sum\limits_{i}}
a_{0}^{i}da_{1}^{i}\cdots da_{n}^{i}\right) =0\right\}$$ The integral of a form $\alpha\in\Omega^{\ast}\left( \mathcal{A}\right) $ over a noncommutative space of metric dimension $d$ is defined by setting$${\displaystyle\int}
\alpha=\mathrm{Tr}_{w}\left( \pi\left( \alpha\right) D^{-d}\right)$$ where $\mathrm{Tr}_{w}$ is the Dixmier trace.
Two-sheeted spacetime
---------------------
A simple extension of space-time is taken as a product of continuous four-dimensional manifold times a discrete set of two points. The algebra is $\mathcal{A}=\mathcal{A}_{1}\otimes\mathcal{A}_{2}$ acting on the Hilbert space $\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ where $\mathcal{A}_{1}=C^{\infty}\left( M\right) $ and $\mathcal{A}_{2}=M_{2}\left(
\mathbb{C}\right) \oplus M_{1}\left( \mathbb{C}\right) ,$ the algebra of $2\times2$ and $1\times1$ matrices. The Hilbert space is that of spinors of the form $$L=\left(
\begin{array}
[c]{c}l\\
e
\end{array}
\right)$$ where $l$ is a doublet and $e$ is a singlet. The spinor $L$ satisfies the chirality condition $\gamma_{5}\otimes\Gamma_{1}L=L$ where $\Gamma
_{1}=\mathrm{diag}\left( 1_{2},-1\right) $ is a grading operator. From this we deduce that $l$ is a left-handed spinor and $e$ is right handed, and we thus write $l=\left(
\begin{array}
[c]{c}\nu_{L}\\
e_{L}\end{array}
\right) $ and $e=e_{R}.$ The Dirac operator is given by $D=D_{1}\otimes1+\gamma_{5}\otimes D_{2}$ where $D_{1}=\gamma^{\mu}\partial_{\mu}$ and $D_{2}$ is the Dirac operator on $\mathcal{A}_{2}$ such that $$D_{l}=\left(
\begin{array}
[c]{cc}\gamma^{\mu}\partial_{\mu}\otimes1_{2}\otimes1_{3} & \gamma_{5}\otimes
M_{12}\otimes k\\
\gamma_{5}\otimes M_{21}\otimes k^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes1\otimes1_{3}\end{array}
\right)$$ where $M_{21}=M_{12}^{\ast}$ and $k$ is a $3\times3$ family mixing matrix representing Yukawa couplings for the leptons. The $1\times2$ matrix $M_{12}$ turns out to be the vev of the Higgs field and is taken as $M_{12}=\mu\left(
\begin{array}
[c]{c}0\\
1
\end{array}
\right) =H_{0}.$ The elements $a\in\mathcal{A}$ have the representation $a=\left(
\begin{array}
[c]{cc}a_{1} & 0\\
0 & a_{2}\end{array}
\right) $ where $a_{1},$ $a_{2}$ are $2\times2$ and $1\times1$ unitary valued functions. A quick calculation shows that the self-adjoint one-form $\rho$ has the representation$$\pi_{1}\left( \rho\right) =\left(
\begin{array}
[c]{cc}A_{1}\otimes1_{3} & \gamma_{5}\otimes H\otimes k\\
\gamma_{5}\otimes H^{\ast}\otimes k^{\ast} & A_{2}\otimes1_{3}\end{array}
\right)$$ where $$\begin{aligned}
A_{1} & =\gamma^{\mu}{\displaystyle\sum\limits_{i}}
a_{1}^{i}\partial_{\mu}b_{1}^{i},\qquad A_{2}=\gamma^{\mu}{\displaystyle\sum\limits_{i}}
a_{2}^{i}\partial_{\mu}b_{2}^{i},\\
H & =H_{0}+{\displaystyle\sum\limits_{i}}
a_{1}^{i}H_{0}b_{2}^{i}.\end{aligned}$$ The quarks are introduced by taking for the finite space a bimodule structure relating two algebras $\mathcal{A}$ and $\mathcal{B}$ where the algebra $\mathcal{B}$ is taken to be $M_{1}\left( \mathbb{C}\right) \oplus
M_{3}\left( \mathbb{C}\right) $ commuting with the action of $\mathcal{A}.$ In addition, the mass matrices in the Dirac operator are taken to be zero when acting on elements of $\mathcal{B}.$ The one-form $\eta\in\Omega^{1}\left(
\mathcal{B}\right) $ has the simple form $B_{1}\mathrm{diag}\left(
1_{2},1\right) $ where $B_{1}$ is a gauge field associated with $M_{1}\left(
\mathbb{C}\right) .$ The Hilbert space for the quarks is $$Q=\left(
\begin{array}
[c]{c}q_{L}\\
u_{R}\\
d_{R}\end{array}
\right) ,\qquad q_{L}=\left(
\begin{array}
[c]{c}u_{L}\\
d_{L}\end{array}
\right)$$ The representation of $a\in\mathcal{A}$ is $a\rightarrow\left( a_{1},a_{2},\overline{a}_{2}\right) $ where $a_{1}$ and $a_{2}$ are a $2\times2$ and $1\times1$ complex valued functions. The Dirac operator acting on the quark Hilbert space is $$D_{q}=\left(
\begin{array}
[c]{ccc}\gamma^{\mu}\left( \partial_{\mu}+\cdots\right) \otimes1_{2}\otimes1_{3} &
\gamma_{5}\otimes M_{12}\otimes k^{\prime} & \gamma_{5}\otimes\widetilde{M}_{12}\otimes k^{^{\prime\prime}}\\
\gamma_{5}\otimes M_{12}^{\ast}\otimes k^{\prime\ast} & \gamma^{\mu}\left(
\partial_{\mu}+\cdots\right) \otimes1_{3} & 0\\
\gamma_{5}\otimes\widetilde{M}_{12}^{\ast}\otimes k^{^{\prime\prime}\ast} &
0 & \gamma^{\mu}\left( \partial_{\mu}+\cdots\right) \otimes1_{3}\end{array}
\right)$$ where $k^{\prime}$ and $k^{\prime\prime}$ are $3\times3$ family mixing matrices and $\widetilde{M}_{12}=\mu\left(
\begin{array}
[c]{c}1\\
0
\end{array}
\right) .$ The one form in $\Omega^{1}\left( \mathcal{A}\right) $ has then the representation $$\pi_{q}\left( \rho\right) =\left(
\begin{array}
[c]{ccc}A_{1}\otimes1_{3} & \gamma_{5}\otimes H\otimes k^{\prime} & \gamma_{5}\otimes\widetilde{H}\otimes k^{^{\prime\prime}}\\
\gamma_{5}\otimes H^{\ast}\otimes k^{\prime\ast} & A_{2}\otimes1_{3} & 0\\
\gamma_{5}\otimes\widetilde{H}^{\ast}\otimes k^{^{\prime\prime}\ast} & 0 &
\overline{A}_{2}\otimes1_{3}\end{array}
\right)$$ where $\widetilde{H}_{a}=\epsilon_{ab}H^{b}.$ When acting on the algebra $\mathcal{B}$ the Dirac operator has zero mass matrices and the one-form $\eta$ in $\Omega^{1}\left( \mathcal{B}\right) $ has the representation $\pi_{q}\left( \eta\right) =B_{2}\mathrm{diag}\left( 1_{2},1\right) $ where $B_{2}$ is the gauge field associated with $M_{3}\left( \mathbb{C}\right) .$ Imposing the unimodularity condition on the algebras $\mathcal{A}$ and $\mathcal{B}$ would then relate the $U\left( 1\right) $ factors in both algebras so that $\mathrm{tr}\left( A_{1}\right) =0,$ $A_{2}=B_{1}=-\mathrm{tr}\left( B_{2}\right) \equiv\frac{i}{2}g_{1}B$. With these we can then write $$\begin{aligned}
A_{1} & =-\frac{i}{2}g_{2}\sigma^{a}A_{a}\\
B_{2} & =-\frac{i}{6}g_{1}B-\frac{i}{2}g_{3}V^{i}\lambda_{i}$$ where $g_{1},$ $g_{2}$ and $g_{3}$ are the $U\left( 1\right) ,$ $SU\left(
2\right) $ and $SU\left( 3\right) $ gauge coupling constants, $\sigma^{a}$ and $\lambda^{i}$ are the Pauli and Gell-Mann matrices respectively. The fermionic actions for the leptons and quarks are then given by $$\begin{aligned}
\left\langle L,\left( D+\rho+\eta\right) L\right\rangle & ={\displaystyle\int}
d^{4}x\sqrt{g}\left( \overline{L}\left( D_{l}+\pi_{l}\left( \rho\right)
+\pi_{l}\left( \eta\right) \right) L\right) \\
\left\langle Q,\left( D+\rho+\eta\right) Q\right\rangle & ={\displaystyle\int}
d^{4}x\sqrt{g}\left( \overline{Q}\left( D_{q}+\pi_{q}\left( \rho\right)
+\pi_{q}\left( \eta\right) \right) Q\right)\end{aligned}$$ These terms can be easily checked to reproduced all the fermionic terms of the Standard Model.
The bosonic action is the sum of the square of curvatures in both the lepton and quark sectors. These are given by $$\begin{aligned}
I_{l} & =\mathrm{Tr}\left( C_{l}\left( \theta_{\rho}+\theta_{\eta}\right)
^{2}D_{l}^{-4}\right) \\
I_{q} & =\mathrm{Tr}\left( C_{q}\left( \theta_{\rho}+\theta_{\eta}\right)
^{2}D_{q}^{-4}\right)\end{aligned}$$ where $$\theta_{\rho}\equiv d\rho+\rho^{2}$$ is the curvature of $\rho,$ and $C_{l}$ and $C_{q}$ are constant elements of the algebra. Since the representation $\pi$ has a kernel, the auxiliary fields must be projected out. This step mainly affects the potential. After some algebra one can show that the bosonic action given above reproduces all the bosonic interactions of the Standard Model with the same number of parameters. If one assumes that $C_{l}$ and $C_{q}$ belong to the center of the algebra, then one can get fixed values for the top quark mass and Higgs mass. The main advantage of the noncommutative construction of the Standard Model is that one gets a geometrical understanding of the origin of the Higgs field and a unification of the gauge and Higgs sectors. One sees that the Higgs fields are the components of the one form along discrete directions.
Constructions beyond the Standard Model
---------------------------------------
The early constructions of the Standard Model provided encouragements to look further into noncommutative spaces. The construction was also complicated with some ambiguities such as the independence of the lepton and quark sectors, the construction of the Higgs potential and projecting out the auxiliary fields. It was then natural to ask whether it is possible to go beyond the Standard Model. In particle physics the route taken was to consider larger groups such as $SU\left( 5\right) $ or $SO(10)$ which contains $U\left( 1\right)
\times SU\left( 2\right) \times SU\left( 3\right) $ as a subgroup. The main advantage of GUT is that the fermionic fields are unified in one or two representations, the most attractive possibility being $SO(10)$ where the spinor representation $16_{s}$ contains all the known fermions in addition to the right-handed neutrino. The simplicity in the fermionic sector did not make the theory more predictive because of the arbitrariness of the Higgs sector. There are many possible Higgs representations that can break the symmetry spontaneously from $SO(10)$ to $SU\left( 3\right) \times U\left( 1\right)
.$ In the noncommutative construction the Higgs sector is more constrained which was taken as an encouragement to explore the possibility of considering larger matrix algebras. As an example if one arranges the leptons in the form $L=\left(
\begin{array}
[c]{c}l_{L}\\
l_{R}\end{array}
\right) $ where $l=\left(
\begin{array}
[c]{c}\nu\\
e
\end{array}
\right) $ then the corresponding algebra will be $M_{2}\left( \mathbb{C}\right) \oplus M_{2}\left( \mathbb{C}\right) .$ A natural possibility is then to consider a discrete space of four points and where the fermions are arranged in the format $\psi=\left(
\begin{array}
[c]{c}l_{L}\\
l_{R}\\
l_{L}^{c}\\
l_{R}^{c}\end{array}
\right) $ and the representation $\pi$ acting on $\mathcal{A}$ is given by $\pi\left( a\right) =\mathrm{diag}\left( a_{1},a_{2},\overline{a}_{1},\overline{a}_{2}\right) $ where $a_{1},$ $a_{2}$ are $2\times2$ complex matrices. The resulting model has $SU\left( 2\right) _{L}\times SU\left(
2\right) _{R}\times U\left( 1\right) _{B-L}$ with the Higgs fields in the representations $\left( 2,2\right) ,$ $\left( 3,1\right) +\left(
1,3\right) $ of $SU\left( 2\right) _{L}\times SU\left( 2\right) _{R}.$ We can summarize the steps needed to construct noncommutative particle physics models. First we specify the fermion representations then we choose the number of discrete points and the symmetry between them. From this we deduce the appropriate algebra and the map $\pi$ acting on the Hilbert space of spinors. Finally we write down the Dirac operator acting on elements of the algebra and choose the mass matrices to correspond to the desired vacuum of the Higgs fields.
To illustrate these steps consider the chiral space-time spinors $P_{+}\psi$ to be in the $16_{s}$ representation of $SO(10),$ where $P_{+}$ is the $SO(10)$ chirality operator, and the number of discrete points to be four. The Hilbert space is taken to be $\Psi=\left(
\begin{array}
[c]{c}P_{+}\psi\\
P_{+}\psi\\
P_{-}\psi^{c}\\
P_{-}\psi^{c}\end{array}
\right) $ where $\psi^{c}=BC\overline{\psi}^{T},$ $C$ being the charge conjugation matrix while $B$ is the $SO\left( 10\right) $ conjugation matrix. The finite algebra is taken to be $\mathcal{A}_{2}=P_{+}\left(
\mathsf{Cliff\,SO}\left( 10\right) \right) P_{+},$ and the finite Hilbert space $\mathcal{H}_{2}=\mathbb{C}^{32}.$ Let $\pi_{0}$ denote the representation of the algebra $\mathcal{A}$ on the Hilbert space $\mathcal{H}$ and let $\overline{\pi}_{0}$ denote the anti representation defined by $\overline{\pi}_{0}\left( a\right) =B\overline{\pi_{0}\left( a\right)
}B^{-1}.$ We then define $\pi\left( a\right) =\pi_{0}\left( a\right)
\oplus\pi_{0}\left( a\right) \oplus\overline{\pi}_{0}\left( a\right)
\oplus\overline{\pi}_{0}\left( a\right) .$ The Dirac operator is taken to be $$\left(
\begin{array}
[c]{cccc}\gamma^{\mu}\partial_{\mu}\otimes1_{32}\otimes1_{3} & \gamma_{5}\otimes
M_{12}\otimes K_{12} & \gamma_{5}\otimes M_{13}\otimes K_{13} & \gamma
_{5}\otimes M_{14}\otimes K_{14}\\
\gamma_{5}\otimes M_{12}^{\ast}\otimes K_{12}^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes1_{32}\otimes1_{3} & \gamma_{5}\otimes M_{23}\otimes
K_{23} & \gamma_{5}\otimes M_{24}\otimes K_{24}\\
\gamma_{5}\otimes M_{13}^{\ast}\otimes K_{13}^{\ast} & \gamma_{5}\otimes
M_{23}^{\ast}\otimes K_{23}^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes
1_{32}\otimes1_{3} & \gamma_{5}\otimes M_{34}\otimes K_{34}\\
\gamma_{5}\otimes M_{14}^{\ast}\otimes K_{14}^{\ast} & \gamma_{5}\otimes
M_{24}^{\ast}\otimes K_{24}^{\ast} & \gamma_{5}\otimes M_{34}^{\ast}\otimes
K_{34}^{\ast} & \gamma^{\mu}\partial_{\mu}\otimes1_{32}\otimes1_{3}\end{array}
\right)$$ where the $K_{mn}$ are $3\times3$ family mixing matrices commuting with $\pi\left( a\right) .$ We may impose the exchange symmetries $1\leftrightarrow2$ and $3\leftrightarrow4$ so that $M_{12}=M_{12}^{\ast
}=\mathcal{M}_{0},$ $M_{13}=M_{14}=M_{23}=M_{24}=\mathcal{N}_{0},$ $M_{34}=M_{34}^{\ast}=B\overline{\mathcal{M}}_{0}B^{-1}.$ Computing $\pi\left( \rho\right) $ we get$$\pi\left( \rho\right) =\left(
\begin{array}
[c]{cccc}A & \gamma_{5}\mathcal{M}K_{12} & \gamma_{5}\mathcal{N}K_{13} & \gamma
_{5}\mathcal{N}K_{14}\\
\gamma_{5}\mathcal{M}K_{12}^{\ast} & A & \gamma_{5}\mathcal{N}K_{23} &
\gamma_{5}\mathcal{N}K_{24}\\
\gamma_{5}\mathcal{N}^{\ast}K_{13}^{\ast} & \gamma_{5}\mathcal{N}^{\ast}K_{23}^{\ast} & B\overline{A}B^{-1} & \gamma_{5}B\overline{\mathcal{M}}B^{-1}K_{34}\\
\gamma_{5}\mathcal{N}^{\ast}K_{14}^{\ast} & \gamma_{5}\mathcal{N}^{\ast}K_{24}^{\ast} & \gamma_{5}B\overline{\mathcal{M}}B^{-1}K_{34}^{\ast} &
B\overline{A}B^{-1}\end{array}
\right)$$ where $$\begin{aligned}
A & =P_{+}{\displaystyle\sum\limits_{i}}
a^{i}\gamma^{\mu}\partial_{\mu}b^{i}P_{+}\\
\mathcal{M}+\mathcal{M}_{0} & =P_{+}{\displaystyle\sum\limits_{i}}
a^{i}\mathcal{M}_{0}b^{i}P_{+}\\
\mathcal{N}+\mathcal{N}_{0} & =P_{+}{\displaystyle\sum\limits_{i}}
a^{i}\mathcal{N}_{0}B\overline{b}^{i}B^{-1}P_{-}$$ One sees immediately that the Higgs fields $\mathcal{M}$ and $\mathcal{N}$ are in the $16_{s}\times16_{s}$ and $16_{s}\times\overline{16}_{s}$ representations. Equating the action of $A$ on $\psi$ and $\psi^{c}$ will reduce it to an $SO\left( 10\right) $ gauge field. Specifying $\mathcal{M}_{0}$ and $\mathcal{N}_{0}$ determines the breaking pattern of $SO\left(
10\right) .$ One can then proceed to construct the bosonic sector and project out the auxiliary fields to determine the potential. There are very limited number of models one can construct. These models, however, will suffer the same problems encountered in the GUT construction, mainly that of low unification scale of $10^{14}$ Gev implying fast rate of proton decay which is ruled out experimentally.
Coupling matter to gravity
--------------------------
The dynamics of the gravitational force is based on Riemannian geometry. It is therefore natural to study the nature of the gravitational field in noncommutative geometry. The original attempt [@CFF93; @CFG95] was based on generalizing the basic notions of Riemannian geometry, notably the theory of linear connections on differential forms. (Note that an alternative route that takes vector fields as a starting point ends with a derivation based differential calculus as in [@Dub88] ([*cf.*]{} [@Mad95]). In line with the Connes–Lott model, we will instead take differential forms as our starting point. For more details we also refer to the exposition in [@Lnd97 Sect. 10.3]).
First one defines the metric as an inner product on a cotangent space. Then one shows that every cycle over $\mathcal{A}$ yields a notion of cotangent bundle associated with $\mathcal{A}$ and a Riemannian metric on the cotangent bundle $\Omega_{D}^{1}\left( \mathcal{A}\right) .$ With the connection $\nabla$ the Riemann curvature of $\nabla$ on $\Omega_{D}^{1}\left( \mathcal{A}\right) $ is defined by $R\left( \nabla\right) :=-\nabla^{2}$ and the torsion by $T=d-m\circ\nabla$ where $m$ is the tensor product. Requiring $\nabla$ to be unitary and the torsion to vanish we obtain the Levi–Civita connection. If $\Omega_{D}^{1}\left( \mathcal{A}\right) $ is a finitely generated module, then it admits a basis $e^{A},$ $A=1,2,\cdots,N,$ and the connection $\omega_{B}^{A}\in\Omega_{D}^{1}\left( \mathcal{A}\right) $ is defined by $\nabla e^{A}=-\omega_{B}^{A}\otimes e^{B}.$ The components of the torsion $T\left( \nabla\right) $ are defined by $T^{A}=T\left( \nabla\right)
e^{A}$ then $T^{A}\in\Omega_{D}^{2}\left( \mathcal{A}\right) $ is given by $$T^{A}=de^{A}+\omega_{B}^{A}e^{B}$$ Similarly, components of the curvature $R_{B}^{A}\in\Omega_{D}^{2}\left(
\mathcal{A}\right) $ satisfy the defining property that $R\left( \nabla\right) e^{A}=R_{B}^{A}\otimes e^{B}$ so that $$R_{B}^{A}=d\omega_{B}^{A}+\omega_{C}^{A}\omega_{B}^{C}.$$ The analogue of the Einstein–Hilbert action is then $$I\left( \nabla\right) :=\kappa^{-2}\left\langle R_{B}^{A}e^{B},e_{A}\right\rangle$$ where $\kappa^{-1}$ is the Planck scale. Computing this action for the product space $M_{4}\times Z_{2}$ one finds that $$I\left( \nabla\right) =2{\displaystyle\int\limits_{M}}
d^{4}x\sqrt{g}\left( \kappa^{-2}r-2\partial_{\mu}\sigma\partial^{\mu}\sigma\right)$$ where $r$ is the scalar curvature of the Levi–Civita connection of the Riemannian manifold $M_{4}$ coupled to a scalar field $\sigma.$ Applying this construction to the Connes–Lott model is rather involved because the two sheets are not treated symmetrically, being associated with two different algebras. The complication arise because the projective module is not free and the basis $e^{A}$ is constrained. The Einstein–Hilbert action in this case is given by $$I\left( \nabla\right) =2{\displaystyle\int\limits_{M}}
d^{4}x\sqrt{g}\left( \kappa^{-2}\frac{3}{2}r-2\left( 3+\lambda\right)
\partial_{\mu}\sigma\partial^{\mu}\sigma+c\left( \lambda\right) e^{-2\sigma
}\right)$$ where $\lambda=\mathrm{Tr}\left( kk^{\ast}\right) ^{2}-1.$ To understand the significance of the field $\sigma$, we note that by examining the Dirac operator one finds that the field $\phi=e^{-\kappa\sigma}$ now replaces the weak scale. Thus quantum corrections to the classical potential will depend on $\sigma,$ thus the vev of $\sigma$ could be determined from the minimization equations.
The spectral action principle
=============================
Despite the success of the Connes–Lott model and the generalizations that followed in giving a geometrical meaning to the Higgs field and unifying it with the gauge fields, it was felt that the construction is not satisfactory. The first unpleasant feature was the use of the bimodule structure to introduce the $SU\left( 3\right) $ symmetry and the second is the use of unimodularity condition to get the correct hypercharge assignments to the particles. Another major problem was the existence of mirror fermions as a consequence of the fact that the conjugation operator on fermions gives independent fields. In addition, there was arbitrariness in the construction of the potential in the bosonic sector associated with the step of eliminating the auxiliary fields.
Real structures on spectral triples {#sect:st}
-----------------------------------
The first breakthrough came in 1995 with the publication of Alain Connes’ paper “Noncommutative geometry and reality” [@C95]. In this paper, the notion of real structure is introduced, motivated by Atiyah’s KR theory and Tomita’s involution operator $J.$ A hint for the necessity of the reality operator can be taken from physics. We have seen that space-time spinors, which are elements of the Hilbert space satisfy a chirality condition. The charge conjugation operator, when acting on these spinors, produces a conjugate element, which in general is independent. It is possible to replace the chirality condition, with a reality one, known as the Majorana condition which equates the two. Imposing both conditions, chirality and reality, simultaneously can only occur in certain dimensions. The action of the anti-linear isometry $J$ on the algebra $\mathcal{A}$ satisfies the commutation relation $\left[ a,b^{\mathrm{o}}\right] =0,$ $\forall
a,b\in\mathcal{A}$ where$$b^{\mathrm{o}}=Jb^{\ast}J^{-1},\qquad\forall b\in\mathcal{A}$$ so that $b^{\mathrm{o}}\in$ $\mathcal{A}^{\mathrm{o}}.$ This gives a bimodule, using the representation of $\mathcal{A}\otimes\mathcal{A}^{\mathrm{o}}$, given by $$a\otimes b^{\mathrm{o}}\rightarrow aJb^{\ast}J^{-1},\qquad\forall
a,b\in\mathcal{A}$$ We define the fundamental class $\mu$ of the noncommutative space as a class in the $KR$-homology of the algebra $\mathcal{A}\otimes\mathcal{A}^{\mathrm{o}}$ having the involution $$\tau\left( a\otimes b^{\mathrm{o}}\right) =b^{\ast}\otimes\left( a^{\ast
}\right) ^{\mathrm{o}},\qquad\forall a,b\in\mathcal{A}$$ The $KR$-homology cycle implements the involution $\tau$ given by $$\tau\left( w\right) =JwJ^{-1},\qquad\forall w\in\mathcal{A}\otimes
\mathcal{A}^{\mathrm{o}}$$ These imply that the $KR$-homology is periodic with period $8$ and the dimension $n$ modulo $8$ is determined from the commutation rules $$J^{2}=\varepsilon,\qquad JD=\varepsilon^{\prime}DJ,\qquad J\gamma
=\varepsilon^{\prime\prime}\gamma J$$ where $\varepsilon,$ $\varepsilon^{\prime},$ $\varepsilon^{\prime\prime}\in\left\{ -1,1\right\} $ are given as function of $n$ modulo $8$ according to the table$$\begin{tabular}
[c]{l|llllllll}$n$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$\\
\hline
$\varepsilon$ & $1$ & $1$ & $-1$ & $-1$ & $-1$ & $-1$ & $1$ & $1$\\
$\varepsilon^{\prime}$ & $1$ & $-1$ & $1$ & $1$ & $1$ & $-1$ & $1$ & $1$\\
$\varepsilon^{\prime\prime}$ & $1$ & & $-1$ & & $1$ & & $-1$ &
\end{tabular}$$ It is not surprising that this table agrees with the one obtained by classifying in which dimensions a spinor obey the Majorana and Weyl conditions. The intersection form $K_{\ast}\left( \mathcal{A}\right) \times
K_{\ast}\left( \mathcal{A}\right) \rightarrow\mathbb{Z}$ is obtained from the Fredholm index of $D$ in $K_{\ast}\left( \mathcal{A}\otimes
\mathcal{A}^{\mathrm{o}}\right) .$ Using the Kasparov intersection product, Poincare duality is formulated in terms of the invertibility of $\mu$ and that $D$ is an operator of order one implies the condition$$\left[ \left[ D,a\right] ,b^{\mathrm{o}}\right] =0,\qquad\forall
a,b\in\mathcal{A}$$ Next we consider automorphisms of the algebra $\mathcal{A}$ denoted by $\mathrm{Aut}\left( \mathcal{A}\right) .$ This comprises both of inner and outer automorphisms. Inner automorphisms $\mathrm{Int}\left( \mathcal{A}\right) $ is a normal subgroup of $\mathrm{Aut}\left( \mathcal{A}\right) $ defined by $$\alpha\left( f\right) =ufu^{\ast},\qquad\forall f\in\mathcal{A},\qquad
u\,u^{\ast}=u^{\ast}u=1$$ The group $\mathrm{Aut}^{+}\left( \mathcal{A}\right) $ of automorphisms of the involutive algebra $\mathcal{A}$ are implemented by a unitary operator $U$ in $\mathcal{H}$ commuting with $J$ satisfying $$\alpha\left( x\right) =UxU^{-1}\,\,\qquad\forall x\in\mathcal{A}$$ For Riemannian manifolds $M$, this plays the role of the group of diffeomorphisms $\mathrm{Diff}^{+}\left( M\right) ,$ which preserves the $K$-homology fundamental class of $M.$ Let $\mathcal{E}$ be a finite projective, hermitian right $\mathcal{A}$-module, and define the algebra $\mathcal{B}=\mathrm{End}\left( \mathcal{A}\right) $ as the Morita equivalence of the algebra $\mathcal{A}$ with a hermitian connection $\nabla$ on $\mathcal{E}$ defined as the linear map $\nabla:\mathcal{E\rightarrow
E\otimes}_{\mathcal{A}}\Omega_{D}^{1}$ satisfying $$\begin{aligned}
\nabla\left( \zeta a\right) & =\left( \nabla\zeta\right) a+\zeta\otimes
da,\qquad\forall\zeta\in\mathcal{E},\,a\in\mathcal{A}\\
d\left( \zeta,\eta\right) & =\left( \zeta,\nabla\eta\right) -\left(
\nabla\zeta,\eta\right) ,\qquad\forall\zeta,\,\eta\in\mathcal{E}$$ where $da=\left[ D,a\right] $ and $\Omega_{D}^{1}$ is the bimodule of operators of the form $$A={\displaystyle\sum\limits_{i}}
a_{i}\left[ D,b_{i}\right] ,\qquad a_{i},b_{i}\in\mathcal{A}$$ Since any algebra is Morita equivalent to itself with $\mathcal{E}=\mathcal{A},$ applying the construction given above yields the inner deformation of the spectral geometry. The unitary equivalence is implemented by the representation $u\rightarrow\widetilde{U}=u\left( Ju\,J^{-1}\right)
=u\left( u^{\mathrm{o}}\right) ^{\ast}$ so that the Dirac operator that includes inner fluctuations$$D_{A}=D+A+JAJ^{-1}$$ where $A=A^{\ast}$ transforms as $D_{A}\rightarrow\widetilde{U}D_{A}\widetilde{U}^{-1}$ provided that $$A\rightarrow u\,Au^{\ast}+u\left[ D,u^{\ast}\right]$$ This will ensure that the inner product $$\left( \Psi,D_{A}\Psi\right)$$ is invariant under the transformation $\Psi\rightarrow\widetilde{U}\Psi.$ This expression will then take care of all fermionic interactions which, as will be seen in the next section, removes the arbitrariness in specifying the action of the connection on the Hilbert space.
The spectral action principle
-----------------------------
The next breakthrough came a year later in 1996 in the work of Chamseddine and Connes entitled “The spectral action principle” [@CC96]. The basic observation is that for a noncommutative space defined by spectral data, the emphasis is shifted from the coordinates $x$ of a geometric space to the spectrum $\Sigma
\sqsubset\mathbb{R}$ of the operator $D$. We postulate the following hypothesis$$\text{The physical action depends only on }\Sigma$$ The existence of Riemannian manifolds which are isospectral but not isometric shows that the spectral action principle is stronger than the usual diffeomorphism invariance. In the usual Riemannian case the group $\mathrm{Diff}\left( M\right) $ of diffeomorphisms of $M$ is canonically isomorphic to the group $\mathrm{Aut}\left( \mathcal{A}\right) $ of automorphisms of the algebra $\mathcal{A}=C^{\infty}\left( M\right) .$ To each $\varphi\in\mathrm{Diff}\left( M\right) $ one associates the algebra preserving map $\alpha_{\varphi}:\mathcal{A}\rightarrow\mathcal{A}$ given by $$\alpha_{\varphi}\left( f\right) =f\circ\varphi^{-1}\qquad\forall f\in
C^{\infty}\left( M\right) =\mathcal{A}$$ The prescription to determine the bosonic action with some cutoff energy scale $\Lambda$ is to first replace the Hilbert space $\mathcal{H}$ by the subspace $\mathcal{H}_{\Lambda}$ defined by $$\mathcal{H}_{\Lambda}=\mathrm{range\,}\chi\left( \frac{D}{\Lambda}\right)$$ where $\chi$ is a suitable smooth positive function, restricting both $D$ and $\mathcal{A}$ to this subspace maintaining the commutation relations for the algebra. This procedure is superior to the lattice approximation because it does respect the geometric symmetry group. The [*spectal action functional*]{} is then given by the $$\operatorname{Tr}\chi\left( \frac D \Lambda \right).$$ For a noncommutative space which is a tensor product of a continuous manifold times a discrete space, the functional $\operatorname{Tr}\chi\left( \frac{D}{\Lambda}\right) $ can be expanded in an asymptotic series in $\Lambda$, rendering the computation amenable to a heat kernel expansion. This procedure will be illustrated in the next section. More general methods to analyze the spectral action have also been developed, see [@FGLV98] for an early result and also the recent book [@EI18]. An interpretation of the spectral action as the von Neumann entropy of a second-quantized spectral triple has been found recently in [@CCS18] ([*cf.*]{} [@DK19]).
To summarize, the breakthroughs carried out in the short period 1995-1996, defining the reality operator $J$ and developing the spectral action principle will allow to remove the ambiguities encountered before in the construction of the noncommutative spectral Standard Model.
The spectral Standard Model
===========================
At the time that the spectral action was formulated, it was clear that this principle forms a unifying framework for gravity and particle physics of the Standard Model. As said, this led to much activity ([*cf.*]{} [@SUW02]) in the years that followed. Also shortcomings of the approach were pointed out quite quickly, such as the notorious fermion-doubling problem [@LMMS97; @GIS98]. This doubling —or actually, quadrupling— was due to the incorporation of left-right, particle-anti-particle degrees of freedom both in the continuum spinor space and in the finite noncommutative space. At the technical level this was a crucial starting point, allowing for a product geometry to describe gravity coupled to the Standard Model.
Nevertheless, it was a somewhat disturbing feature which, together with the apparent arbitrariness of the choice of a finite geometry and the abscence of neutrino mixing in the model, led Connes to eventually resolve these problems in [@C06]. At the same time John Barrett [@Bar06] arrived at the same conclusion (see also the recent uniqueness result [@Bes19]), even though his motivation came from the desire to have noncommutative geometry with a Lorentzian signature.
The crucial insight in both of these works is that one should allow for a KO-dimension for the finite space $F$ which is different from the metric dimension (which is zero). More specifically, the KO-dimension of the finite space should be 6 (modulo 8), so that the product of the continuum $M$ with $F$ is 10 modulo 8. The precise structure of the spectral Standard Model (see Section \[sect:spectr-SM\]) is then best understood using the classification of all irreducible finite noncommutative geometries of KO-dimension 6 which we now briefly recall.
Classification of irreducible geometries {#sect:irr}
----------------------------------------
In [@CC07b] Chamseddine and Connes classified [*irreducible*]{} finite real spectral triples of KO-dimension 6. This lead to a remarkably concise list of spectral triples, based on the matrix algebras $M_N(\C) \oplus M_N(\C)$ for some $N$. We remark that earlier classification results were obtained [@Kra97; @PS98] which were also exploited in a search Beyond the Standard Model (see Remark \[rem:beyond-sm\] below).
A finite real spectral triple $(A,H,D;J, \gamma)$ is called [*irreducible*]{} if the triple $(A,H,J)$ is irreducible. More precisely, we demand that:
1. The representations of $A$ and $J$ in $H$ are irreducible;
2. The action of $A$ on $H$ has a separating vector.
We will prove the main result of [@CC07b] using an alternative approach which is based on [@Sui14 Sect. 3.4].
\[thm:irr-geom\] Let $(A,H,D;J,\gamma)$ be an irreducible finite real spectral triple of KO-dimension 6. Then there exists a positive integer $N$ such that $A \simeq M_N(\C) \oplus M_N(\C)$.
Let $(A,H,D;J,\gamma)$ be an arbitrary finite real spectral triple. We may then decompose $$A= \bigoplus_{i=1}^N M_{n_i}(\C), \qquad H = \bigoplus_{i,j=1}^N \C^{n_i} \otimes( \C^{n_j})^\circ \otimes V_{ij},$$ with $V_{ij}$ corresponding to the multiplicities as before. Now each $\C^{n_i} \otimes \C^{n_j}$ is an irreducible representation of $A$, but in order for $H$ to support a real structure $J:H \to H$ we need both $\C^{n_i} \otimes (\C^{n_j})^\circ$ and $\C^{n_j} \otimes (\C^{n_i})^\circ$ to be present in $H$. Moreover, an old result of Wigner [@Wig60] for an anti-unitary operator with $J^2 =1$ assures that already with multiplicities $\dim V_{ij}=1$ there exists such a real structure. Hence, the irreducibility condition (1) above yields $$H = \C^{n_i} \otimes (\C^{n_j})^\circ \oplus \C^{n_j} \otimes (\C^{n_i})^\circ,$$ for some $i,j \in \{ 1,\ldots, N\}$. Then, let us consider condition (2) on the existence of a separating vector. Note first that the representation of $A$ in $H$ is faithful only if $A= M_{n_i}(\C) \oplus M_{n_j}(\C)$. Second, the stronger condition of a separating vector $\xi$ then implies $n_i = n_j$, as it is equivalent to $A' \xi = H$ for the commutant $A'$ of $A$ in $H$. Namely, since $A' = M_{n_j}(\C) \oplus M_{n_i}(\C)$ with $\dim A' = n_i^2 + n_j^2$, and $\dim H = 2n_i n_j$ we find the desired equality $n_i=n_j$.
With the complex finite-dimensional algebras $A$ given as a direct sum $M_N(\C) \oplus M_N(\C)$,[^1] the additional demand that $H$ carries a symplectic structure $I^2=-1$ yields real algebras of which $A$ is the complexification. We see that this requires $N=2k$ so that one naturally considers triples $(A,H,J)$ for which $$\label{eq:classif}
A= M_{k}(\bH) \oplus M_{2k}(\C); \qquad H= \C^{2(2k)^2}.$$
Noncommutative geometry of the Standard Model {#sect:spectr-SM}
---------------------------------------------
The above classification of irreducible finite geometries of KO-dimension 6 forms the starting point for the derivation of the Standard Model from a noncommutative manifold [@CCM07]. Hence, it is based on the matrix algebra $M_N(\C) \oplus M_N(\C)$ for $N \geq 1 $. Let us make the following two additional requirements on the irreducible finite geometry $(A,H_F,D_F;J_F,\gamma_F)$:
1. The finite-dimensional Hilbert space $H_F$ carries a symplectic structure $I^2 = -1$;
2. the grading $\gamma_F$ induces a non-trivial grading on $A$, by mapping $$a \mapsto \gamma_F a \gamma_F,$$ and selects an even subalgebra $A^\ev \subset A$ consisting of elements that commute with $\gamma_F$.
But the first demand sets $A=M_k(\bH) \oplus M_{2k}(\C)$, represented on the Hilbert space $\C^{2(2k)^2}$. The second requirement sets $k \geq 2$; we will take the simplest $k=2$ so that $H_F = \C^{32}$. [^2] Indeed, this allows for a $\gamma_F$ such that $$\begin{aligned}
A^\ev &= \bH_R \oplus \bH_L \oplus M_4(\C), \nn
\intertext{where $\bH_R$ and $\bH_L$ are two copies (referred to as {\em right} and {\em left}) of the quaternions; they are the diagonal of $M_2(\bH) \subset A$. The Hilbert space can then be decomposed according to the defining representations of $A^\ev$, }
\label{eq:HF-PS}
H_F &= (\C^2_R \oplus \C^2_L) \otimes (\C^4)^{\circ} \oplus \C^4 \otimes ( (\C^2_R)^{\circ} \oplus (\C^2_L)^{\circ}).
\intertext{According to this direct sum decomposition, we write }
\label{eq:dirac-sm}
D_F&= \begin{pmatrix} {S}&{T^*}\\ {T}&{\bar S} \end{pmatrix}
$$ Moreover, $J_F$ is the anti-unitary operator that flips the two 16-dimensional components in Equation .
The key result is that if we assume that $T$ is non-trivial, then the first-order condition selects the maximal subalgebra of the Standard Model, that is to say, $A_F= \C \oplus \bH \oplus M_3(\C)$.
[[@CCM07 Prop. 2.11]]{} \[prop:subalg-sm\] Up to $*$-automorphisms of $A^\ev$, there is a unique $*$-subalgebra $A_F\subset A^\ev$ of maximal dimension that allows $T \neq 0$ in . It is given by $$A_F = \left\{ \left( q_\lambda, q , \begin{pmatrix} q &0 \\ 0&m \end{pmatrix} \right): \lambda \in \C, q \in \bH_L, m \in M_3(\C) \right\} \subset \bH_R \oplus \bH_L \oplus M_4(\C),$$ where $\lambda \mapsto q_\lambda$ is the embedding of $\C$ into $\bH$, with $$q_\lambda = \begin{pmatrix} \lambda & 0 \\ 0 & \bar\lambda \end{pmatrix}.$$ Consequently, $A_F \simeq \C \oplus \bH \oplus M_3(\C)$.
The restriction of the representation of $A$ on $H_F$ to the subalgebra $A_F$ gives a decomposition of $H_F$ into irreducible (left and right) representations of $\C$, $\H_L$ and $M_3(\C)$. For instance, $$\label{eq:decomp-subalg}
(\C^2_R \oplus \C^2_L) \otimes (\C^4)^{\circ} \leadsto (\C \oplus \overline \C \oplus \C^2_L) \otimes \left ((\C)^\circ \oplus (\C^3)^\circ \right).$$ and similarly for $\C^4 \otimes ( (\C^2_R)^{\circ} \oplus (\C^2_L)^{\circ})$. In order to connect to the physics of the Standard Model, let us introduce an orthonormal basis for $H_F$ that can be recognized as the fermionic particle content of the Standard Model, and subsequently write the representation of $A_F$ in terms of this basis.
We let the subspace of $H_F$ displayed in Equation be represented by basis vectors $\{ \nu_R ,e_R, (\nu_L,e_L)\}$ of the so-called [*lepton space*]{} $H_l$ and basis vectors $\{ u_R,d_R,(u_L,d_L)\}$ of the [*quark space*]{} $H_q$. Their reflections with respect to $J_F$ are the [*anti-lepton space*]{} $H_{\bar l}$ and the [*anti-quark space*]{} $H_{\bar q}$, spanned by $\{ \bar{\nu_R} ,\bar{e_R}, (\bar{\nu_L},\bar{e_L})\}$ and $\{\bar{u_R}, \bar{d_R},( \bar{u_L},\bar{d_L})\}$, respectively. The three colors of the quarks are given by a tensor factor $\C^3$ and when we take into account [*three generations*]{} of fermions and anti-fermions by tripling the above finite-dimensional Hilbert space we obtain $$\begin{aligned}
H_F := \left( H_l \oplus H_{\bar l} \oplus H_q \oplus H_{\bar q} \right)^{\oplus3} .\end{aligned}$$ Note that $H_l = \C^4$, $H_q=\C^4 \otimes \C^3$, $H_{\bar l} = \C^4$, and $H_{\bar q} = \C^4 \otimes \C^3$.
An element $a=(\lambda,q,m)\in A_F$ acts on the space of leptons $H_l$ as $q_\lambda \oplus q$, and acts on the space of quarks $H_q$ as $(q_\lambda \oplus q) \otimes 1_3$. For the action of $a$ on an anti-lepton $\bar l\in H_{\bar l}$ we have $a\bar l = \lambda 1_4\bar l$, and on an anti-quark $\bar q\in H_{\bar q}$ we have $a\bar q = (1_4 \otimes m) \bar q$.
The $\Z_2$-grading $\gamma_F$ is such that left-handed particles have eigenvalue $+1$ and right-handed particles have eigenvalue $-1$. The anti-linear operator $J_F$ interchanges particles with their anti-particles, so $J_F f = \bar f$ and $J_F \bar f = f$, with $f$ a lepton or quark.
The first indication that the subalgebra $A_F$ is relevant for the Standard Model —to say the least— comes from the fact that the Standard Model gauge group can be derived from the unitaries in $A_F$. We restrict to the [*unimodular gauge group*]{}, $$\mathrm{SU}(A_F) = \left\{ u \in A_F: u^* u = uu^* = 1, \det(u) = 1\right\}$$ where $\det$ is the determinant of the action of $u$ in $H_F$. It then follows that, up to a finite abelian group we have $$\mathrm{SU}(A_F) \sim U(1) \times SU(2) \times SU(3)$$ and the hypercharges are derived from the unimodularity condition to be the usual ones: $$\begin{aligned}
\begin{array}{l|cccccccc}
\text{Particle} & \nu_R & e_R & \nu_L & e_L & u_R & d_R & u_L & d_L \\
\hline
\text{Hypercharge} & 0 & -2 & -1 & -1 & \frac43 & -\frac23 & \frac13 & \frac13 \\
\end{array}\end{aligned}$$
Let us now turn to the form of the finite Dirac operator, and see what we can say about the components of the matrix $D_F$ as displayed in . Recall that we are looking for a self-adjoint operator $D_F$ in $H_F$ that commutes with $J_F$, anti-commutes with $\gamma_F$, and fulfills the first-order conditions with resepct to $A_F$: $$[[D,a],J b J^{-1}]=0; \qquad (a,b \in A_F).$$ We also require that $D_F$ commutes with the subalgebra $\C_F = \{ (\lambda,\lambda,0) \} \subset A_F$ which physically speaking corresponds to the fact that the photon remains massless. Then it turns out [@C06 Theorem 1] (see also [@CCM07 Theorem 2.21]\[page:moduli-dirac\]) that any $D_F$ that satisfies these assumptions is of the following form: in terms of the decomposition of $H_F$ in particle ($H_l^{\oplus 3} \oplus H_q^{\oplus 3}$) and anti-particles ($H_{\bar l}^{\oplus 3} \oplus H_{\bar q}^{\oplus 3}$) the operator $S$ is $$\begin{aligned}
S_l &:= \left.S\right|_{H_l^{\oplus 3}} = \begin{pmatrix} 0&0&Y_\nu^*&0 \\0&0&0&Y_e^*\\ Y_\nu&0&0&0\\ 0&Y_e&0&0\end{pmatrix} ,
\label{eq:yukawa-l} \\
S_q \otimes 1_3 &:= \left.S\right|_{H_q^{\oplus 3}} =\begin{pmatrix} 0&0&Y_u^*&0 \\0&0&0&Y_d^* \\Y_u&0&0&0 \\0&Y_d&0&0\end{pmatrix} \otimes1_3 ,\label{eq:yukawa-q}\end{aligned}$$ where $Y_\nu$, $Y_e$, $Y_u$ and $Y_d$ are some $3\times3$ matrices acting on the three generations, and $1_3$ acting on the three colors of the quarks. The symmetric operator $T$ only acts on the right-handed (anti)neutrinos, so it is given by $T\nu_R = Y_R\bar{\nu_R}$, for a certain $3\times3$ symmetric matrix $Y_R$, and $Tf=0$ for all other fermions $f\neq\nu_R$. Note that $\nu_R$ here stands for a vector with $3$ components for the number of generations.
The above classification result shows that the Dirac operators $D_F$ give all the required features, such as mixing matrices for quarks and leptons, unbroken color and the see-saw mechanism for right-handed neutrinos. Let us illustrate the latter in some more detail. The mass matrix restricted to the subspace of $H_F$ with basis $\{\nu_L, \nu_R, \bar{\nu_L}, \bar{\nu_R}\}$ is given by $$\begin{aligned}
\begin{pmatrix} 0&Y_\nu^*&Y_R^*&0 \\ Y_\nu&0&0&0\\ Y_R&0&0&\bar Y_\nu^* \\ 0&0&\bar Y_\nu&0\end{pmatrix} .\end{aligned}$$ Suppose we consider only one generation, so that $Y_\mu = m_\nu$ and $Y_R = m_R$ are just scalars. The eigenvalues of the above mass matrix are then given by $$\begin{aligned}
\pm \frac12 m_R \pm \frac12 \sqrt{{m_R}^2 + 4{m_\nu}^2} .\end{aligned}$$ If we assume that $m_\nu \ll m_R$, then these eigenvalues are approximated by $\pm m_R$ and $\pm\frac{{m_\nu}^2}{m_R}$. This means that there is a heavy neutrino, for which the Dirac mass $m_\nu$ may be neglected, so that its mass is given by the Majorana mass $m_R$. However, there is also a light neutrino, for which the Dirac and Majorana terms conspire to yield a mass $\frac{{m_\nu}^2}{m_R}$, which is in fact much smaller than the Dirac mass $m_\nu$. This is called the *seesaw mechanism*. Thus, even though the observed masses for these neutrinos may be very small, they might still have large Dirac masses (or Yukawa couplings).
Of course, in the physical applications one chooses $Y_\nu, Y_e$ to be the [*Yukawa mass matrices*]{} and $Y_R$ is the [*Majorana mass matrix*]{}. There has been searches for additional conditions to be satisfied by the spectral triple $(A_F,H_F,D_F)$ to further constrain the form of $D_F$, see for instance [@BBB15; @BF18; @KL18; @DAS18; @DS18].
The gauge and scalar fields as inner fluctuations
-------------------------------------------------
We here derive the precise form of internal fluctuations $A_\mu$ for the above spectral triple of the Standard Model (following [@CCM07 Sect. 3.5] or [@Sui14 Sect. 11.5]).
Take two elements $a=(\lambda,q,m)$ and $b=(\lambda',q',m')$ of the algebra $\A = C^\infty(\C\oplus\bH \oplus M_3(\C))$. According to the representation of $A_F$ on $H_F$, the inner fluctuations $A_\mu = -ia\partial_\mu b$ decompose as $$\begin{aligned}
\Lambda_\mu &:= -i\lambda\partial_\mu\lambda'; \qquad
\Lambda_\mu' := -i\bar\lambda\partial_\mu\bar\lambda'
\intertext{on $\nu_R$ and $e_R$, respectively, and as }
Q_\mu &:= -iq\partial_\mu q';\qquad
V_\mu' := -im\partial_\mu m'\end{aligned}$$ acting on $(\nu_l,e_L)$ and $H_{\bar q}$, respectively. On all other components of $H_F$ the gauge field $A_\mu$ acts as zero. Imposing the hermiticity $\Lambda_\mu=\Lambda_\mu^*$ implies $\Lambda_\mu\in\R$, and also automatically yields $\Lambda_\mu' = -\Lambda_\mu$. Furthermore, $Q_\mu = Q_\mu^*$ implies that $Q_\mu$ is a real-linear combination of the Pauli matrices, which span $i\,su(2)$. Finally, the condition that $V_\mu'$ be hermitian yields $V_\mu' \in i\,u(3)$, so $V_\mu'$ is a $U(3)$ gauge field. As mentioned above, we need to impose the unimodularity condition to obtain an $SU(3)$ gauge field. Hence, we require that the trace of the gauge field $A_\mu$ over $H_F$ vanishes, and we obtain $$\begin{aligned}
\left.\operatorname{Tr}\right|_{H_{\bar l}}\big( \Lambda_\mu 1_4 \big) + \left.\operatorname{Tr}\right|_{H_{\bar q}}\big( 1_4\otimes V_\mu' \big) = 0 \quad\Longrightarrow\quad \operatorname{Tr}(V_\mu') = - \Lambda_\mu .\end{aligned}$$ Therefore, we can define a traceless $SU(3)$ gauge field $V_\mu$ by $\bar V_\mu := - V_\mu' - \frac13 \Lambda_\mu$. The action of the gauge field $B_\mu = A_\mu - J_FA_\mu J_F^{-1}$ on the fermions is then given by $$\begin{aligned}
\left.B_\mu\right|_{H_l} &= \begin{pmatrix} 0&0& \\ 0&-2\Lambda_\mu& \\ &&Q_\mu-\Lambda_\mu1_2\end{pmatrix} , \notag\\
\label{eq:Gauge_field_SM}
\left.B_\mu\right|_{H_q} &= \begin{pmatrix} \frac43\Lambda_\mu1_3+V_\mu&0& \\ 0&-\frac23\Lambda_\mu1_3+V_\mu& \\ &&(Q_\mu+\frac13\Lambda_\mu1_2)\otimes1_3+1_2\otimes V_\mu\end{pmatrix} .\end{aligned}$$ for some $U(1)$ gauge field $\Lambda_\mu$, an $SU(2)$ gauge field $Q_\mu$ and an $SU(3)$ gauge field $V_\mu$.
Note that the coefficients in front of $\Lambda_\mu$ in the above formulas are precisely the aforementioned (and correct!) hypercharges of the corresponding particles.
Next, let us turn to the scalar field $\phi$, which is given by $$\begin{aligned}
\label{eq:higgs_field_SM}
\left.\phi\right|_{H_l} &= {
\left(\!\!\!\begin{array}{c@{~}c}0&Y^*\\#3&0\\\end{array}\!\!\!\right)
} , & \left.\phi\right|_{H_q} &= {
\left(\!\!\!\begin{array}{c@{~}c}0&X^*\\#3&0\\\end{array}\!\!\!\right)
} \otimes1_3 , & \left.\phi\right|_{H_{\bar l}} &= 0 , & \left.\phi\right|_{H_{\bar q}} &= 0 ,\end{aligned}$$ where we now have, for complex fields $\phi_1,\phi_2$, $$\begin{aligned}
Y &= {
\left(\!\!\!\begin{array}{c@{~}c}Y_\nu\phi_1&-Y_e\bar\phi_2\\#3&Y_e\bar\phi_1\\\end{array}\!\!\!\right)
} , & X &= {
\left(\!\!\!\begin{array}{c@{~}c}Y_u\phi_1&-Y_d\bar\phi_2\\#3&Y_d\bar\phi_1\\\end{array}\!\!\!\right)
} . \end{aligned}$$ The scalar field $\Phi$ is then given by $$\begin{aligned}
\label{eq:Higgs_field_SM}
\Phi = D_F + {
\left(\!\!\!\begin{array}{c@{~}c}\phi&0\\#3&0\\\end{array}\!\!\!\right)
} + J_F{
\left(\!\!\!\begin{array}{c@{~}c}\phi&0\\#3&0\\\end{array}\!\!\!\right)
}J_F^* = {
\left(\!\!\!\begin{array}{c@{~}c}S+\phi&T^*\\#3&\bar{(S+\phi)}\\\end{array}\!\!\!\right)
} .\end{aligned}$$ Finally, one can compute that the action of the gauge group $\mathrm{SU}(A_F)$ by conjugation on the fluctuated Dirac operator $$\begin{aligned}
D_\omega = \dirac\otimes 1 + \gamma^\mu\otimes B_\mu + \gamma_M\otimes\Phi\end{aligned}$$ is implemented by $$\begin{gathered}
\Lambda_\mu \mapsto \Lambda_\mu - i \lambda\partial_\mu\bar\lambda , \quad
Q_\mu \mapsto qQ_\mu q^* - iq\partial_\mu q^* , \quad
\bar V_\mu \mapsto m\bar V_\mu m^* - im\partial_\mu m^* , \\
H \mapsto \bar\lambda\,q H ,\end{gathered}$$ for $\lambda\in C^\infty\big(M,U(1)\big)$, $q\in C^\infty\big(M,SU(2)\big)$ and $m\in C^\infty\big(M,SU(3)\big)$ and we have written the [*Higgs doublet*]{} as $$H:= {
\left(\!\!\!\begin{array}{c}\phi_1+1\\#2\\\end{array}\!\!\!\right)
}$$ For the detailed computation we refer to [@CCM07 Sect. 3.5] or [@Sui14 Prop. 11.5].
Summarizing, the gauge fields derived take values in the Lie algebra $u(1) \oplus su(2) \oplus su(3)$ and transform according to the usual Standard Model gauge transformations. The scalar field $\phi$ transforms as the Standard Model Higgs field in the defining representation of $SU(2)$, with hypercharge $-1$.
Spectral action
---------------
The spectral action for the above spectral Standard Model has been computed in full detail in [@CCM07 Section 4.2] and confirmed in [*e.g.*]{} [@Sui14 Theorem 11.10]. Since it would lie beyond the scope of the present review, we refrain from repeating this computation. Instead, we summarize the main result, which is that the Lagrangian derived from the spectral action is $$\begin{aligned}
S_B= \int &\Bigg( \frac{48\chi_4\Lambda^4}{\pi^2} - \frac{c\chi_2\Lambda^2}{\pi^2} + \frac{d\chi(0)}{4\pi^2} + \left(\frac{c\chi(0)}{24\pi^2} - \frac{4\chi_2\Lambda^2}{\pi^2} \right) s - \frac{3\chi(0)}{10\pi^2} (C_{\mu\nu\rho\sigma})^2 \notag\\
&\quad+ \frac14 Y_{\mu\nu} Y^{\mu\nu} + \frac14 W_{\mu\nu}^a W^{\mu\nu,a} + \frac14 G_{\mu\nu}^i G^{\mu\nu,i} + \frac{b\pi^2}{2a^2\chi(0)} |H|^4 \notag\\
&\quad- \frac{2a\chi_2\Lambda^2 - e\chi(0)}{a\chi(0)} |H|^2 + \frac{1}{12} s |H|^2 + \frac12 |D_\mu H|^2 \Bigg) \sqrt{g} d^4x ,\end{aligned}$$ where $\chi_j = \int_0^\infty \chi(v) v^{j-1} dv$ are the moments of the function $\chi$, $j>0$, $s=-R$ is the scalar curvature, $Y_{\mu\nu}, W_{\mu\nu}$ and $G_{\mu\nu}$ are the field strengths of $Y_\mu, Q_\mu$ and $V_\mu$, respectively and the covariant derivative $D_\mu H$ is given by $$\begin{aligned}
\label{eq:Higgs_kin_gauge}
D_\mu H = \partial_\mu H + \frac12 i g_2 W_\mu^a \sigma^a H - \frac12 i g_1 Y_\mu H .\end{aligned}$$ Moreover, we have defined the following constants $$\begin{aligned}
\label{eq:abcde_SM}
a &= \operatorname{Tr}\big(Y_\nu^*Y_\nu + Y_e^*Y_e + 3Y_u^*Y_u + 3Y_d^*Y_d\big) , \notag\\
b &= \operatorname{Tr}\big((Y_\nu^*Y_\nu)^2 + (Y_e^*Y_e)^2 + 3(Y_u^*Y_u)^2 + 3(Y_d^*Y_d)^2\big) , \notag\\
c &= \operatorname{Tr}\big(Y_R^*Y_R\big) , \\
d &= \operatorname{Tr}\big((Y_R^*Y_R)^2\big) , \notag\\
e &= \operatorname{Tr}\big(Y_R^*Y_R Y_\nu^*Y_\nu\big) . \notag\end{aligned}$$ The normalization of the kinetic terms imposes a relation between the coupling constants $g_1,g_2,g_3$ and the coefficients $\chi_0$, of the form $$\begin{aligned}
\label{eq:couplings_norm}
\frac{\chi(0)}{2\pi^2} {g_3}^2 = \frac{\chi(0)}{2\pi^2} {g_2}^2 = \frac{5\chi(0)}{6\pi^2} {g_1}^2 = \frac14 .\end{aligned}$$ The coupling constants are then related by $$\begin{aligned}
{g_3}^2 = {g_2}^2 = \frac53 {g_1}^2 ,\end{aligned}$$ which is precisely the relation between the coupling constants at unification, common to grand unified theories (GUT). We shall further discuss this in Section \[sect:pheno\].
Fermionic action in KO-dimension 6
----------------------------------
As already announced above, the shift to KO-dimension 6 for the finite space solved the fermion doubling problem of [@LMMS97]. Let us briefly explain how this works, following [@C06].
The crucial observation is that in KO-dimension $2 \equiv 4+6 \mod 8$ the following pairing $$( \psi, \psi') \mapsto (J \psi, D_\omega \psi')$$ is a skew-symmetric form on the $+1$-eigenspace of $\gamma$ in $\H$. This skew-symmetry is in concordance with the Grassmann nature of fermionic fields $\psi$, guaranteeing that the following action functional is in fact non-zero: $$S_F = \frac 12 \langle J \xi , D_A \xi \rangle$$ for $\xi$ a Grassmann variable in the $+1$-eigenspace of $\gamma$.
This then solves the fermion doubling, or actually quadrupling as follows. First, the restriction to the chiral subspace of $\gamma$ takes care of a factor of two. Then, the functional integral involving anti-commuting Grassman variables delivers a Pfaffian, which takes care of a square root. That this indeed works has been worked out in full detail for the case of the Standard Model in [@CCM07 Section 4.4.1] or [@Sui14 Section 11.4].
Phenomenological consequences {#sect:pheno}
-----------------------------
The first phenomenological consequence one can derive from the spectral Standard Model is an upper bound on the mass of the top quark. In fact, the appearance of the constant $a$ in both the fermionic and the bosonic action allows to derive $$\begin{aligned}
\label{eq:masses_ferm_W}
\operatorname{Tr}\big(m_\nu^*m_\nu + m_e^*m_e + 3m_u^*m_u + 3m_d^*m_d\big) = 2{g_2}^2{v}^2 = 8 {M_W}^2 .\end{aligned}$$ It is natural to assume that the mass $m_{\text{top}}$ of the top quark is much larger than all other fermion masses, except possibly a Dirac mass that arises from the seesaw mechanism as was described above. If we write $m_\nu = \rho m_{\text{top}}$ then the above relation would yield the constraint $$\begin{aligned}
\label{eq:top_mass}
m_{\text{top}} \lesssim \sqrt\frac8{3+\rho^2} M_W .\end{aligned}$$ The relations between the coupling constants and $\chi(0)$ suggests that we have grand unification of the coupling constants. Moreover, from the action functional we see that the quartic Higgs coupling constant $\lambda$ is related to $\chi(0)$ as well via $$\lambda = 24 \frac{b}{a^2} g_2^2.$$ Thus, the spectral Standard Model imposes relations between the coupling constants and bounds on the fermion masses. These relations were used in [@CCM07] as input at (or around) grand unification scale $\Lambda_\GUT$, and then run down using one-loop renormalization group equations to ’low energies’ where falsifiable predictions were obtained.
![Observed and expected exclusion limits for a Standard Model Higgs boson at the 95-percent confidence level for the combined CDF and DZero analyses. (Fermilab) []{data-label="fig:fermilab"}](fermilab.jpg)
In fact, the mass of the top quark can indeed be found to get an acceptable value, however, for the Higgs mass it was found that $$167 \operatorname{GeV}\leq m_h \leq 176 \operatorname{GeV}.$$ Given that there were not much models in particle physics around that could produce falsifiable predictions, it is somewhat ironical that the first exclusion results on the mass of the Higgs that appeared in 2009 from Fermilab hit exactly this region. See Figure \[fig:fermilab\]. And, of course, with the discovery of the Higgs at $m_h \approx 125.5 \operatorname{GeV}$ in [@ATLAS12; @CMS12] one could say that the spectral Standard Model was not in a particularly good shape at that time.
Beyond the Standard Model with noncommutative geometry
======================================================
Even though the incompatibility between the spectral Standard Model and the experimental discovery of the Higgs with a relatively low mass was not an easy stroke at the time, it also led to a period of reflection and reconsideration of the premises of the noncommutative geometric approach. In fact, it was the beginning of yet another exciting chapter in our story on the spectral model of gravity coupled with matter. As we will see in this and the next chapter, once again the input from experiment is taken as a guiding principle in our search for the spectral model that goes Beyond the Standard Model.
\[rem:beyond-sm\] We do not pretend to give a complete overview of the literature here, but only indicate some of the highlights and actively ongoing research areas.
Other searches beyond the Standard Model with noncommutative geometry include [@ISS04; @Ste06; @Ste07; @Ste09; @Ste09b; @Ste13], adopting a slightly different approach to almost-commutative manifolds as we do.
There is another aspect that was studied is the connection between supersymmetry and almost-commutative manifolds. It turned out to be very hard —if not impossible— to combine the two. A first approach is [@Cha94] and more recently the intersection was studied in [@BroS10; @BroS11; @BBS16].
Resilience of the spectral Standard Model
-----------------------------------------
In 2012 it was realized how a small correction of the spectral Standard Model gives an intriguing possibility to go beyond the Standard Model, solving at the same time a problem with the stability of the Higgs vacuum given the measured low mass $m_h$. This is based on [@CC12], but for which some of the crucial ingredients surprisingly enough were already present in the 2010 paper [@CC10].
Namely, in the definition of the finite Dirac operator $D_F$ of Equation \[eq:dirac-sm\], we can replace $Y_R$ by $Y_R \sigma$, where $\sigma$ is a real scalar field on $M$. Strictly speaking, this brings us out of the class of almost-commutative manifolds $M \times F$, since part of $D_F$ now varies over $M$ and this was the main reason why it was disregarded before. However, since from a physical viewpoint there was no reason to assume $Y_R$ to be constant, it was treated as a scalar field already in [@CC10]. This was only fully justified in subsequent papers (as we will see in the next subsections) where the scalar field $\sigma$ arises as the relic of a spontaneous symmetry breaking mechanism, similar to the Higgs field $h$ in the electroweak sector of the Standard Model. We will discuss a few of the existing approaches in the literature in the next few sections. For now, let us simply focus on the phenomenological consequences of this extra scalar field.
Thus we replace $Y_R$ by $Y_R \sigma$ and analyze the additional terms in the spectral action. The scalar sector becomes $$\begin{gathered}
S_H' := \int_M \bigg( \frac{bf(0)}{2\pi^2} |H|^4 - \frac{2af_2\Lambda^2}{\pi^2} |H|^2 +\frac{ef(0)}{\pi^2} \sigma^2|H|^2\\
-\frac{cf_2\Lambda^2}{\pi^2} \sigma^2 + \frac{df(0)}{4\pi^2} \sigma^4 + \frac{af(0)}{2\pi^2} |D_\mu H|^2 + \frac{1}{4 \pi^2} f(0) c ( \partial_\mu \sigma)^2 \bigg) \sqrt{g} dx,\end{gathered}$$ where we ignored the coupling to the scalar curvature.
We exploit the approximation that $m{_{\scriptscriptstyletop}}$, $m_\nu$ and $m_R$ are the dominant mass terms. Moreover, as before we write $m_\nu = \rho m{_{\scriptscriptstyletop}}$. That is, the expressions for $a,b,c,d$ and $e$ in now become $$\begin{aligned}
a &\approx m{_{\scriptscriptstyletop}}^2 (\rho^2 +3),\\
b &\approx m{_{\scriptscriptstyletop}}^4 (\rho^4 +3),\\
c & \approx m_R^2,\\
d&\approx m_R^4,\\
e&\approx \rho^2 m_R^2 m{_{\scriptscriptstyletop}}^2. \end{aligned}$$ In a unitary gauge, where $H = \begin{pmatrix} h \\ 0 \end{pmatrix}$, we arrive at the following potential: $$\L{_{\scriptscriptstylepot}}(h,\sigma) = \frac{1}{24} \lambda_h h^4 + \frac12 \lambda_{h \sigma} h^2 \sigma^2 + \frac14 \lambda_\sigma \sigma^4 - \frac{4 g_2^2}{\pi^2} f_2 \Lambda^2 (h^2 + \sigma^2),$$ where we have defined coupling constants $$\begin{aligned}
\label{eq:scalar-couplings}
\lambda_h &= 24 \frac{\rho^4 + 3}{(\rho^2+3)^2} g_2^2,&
\lambda_{h \sigma} &= \frac{8 \rho^2}{\rho^2 +3} g_2^2,&
\lambda_\sigma &= 8 g_2^2.\end{aligned}$$ This potential can be minimized, and if we replace $h$ by $v+h$ and $\sigma$ by $w+ \sigma$, respectively, expanding around a minimum for the terms quadratic in the fields, we obtain: $$\begin{aligned}
\L{_{\scriptscriptstylepot}}(v+h,w+\sigma)|{_{\scriptscriptstyle\text{quadratic}}} &= \frac16 v^2 \lambda_h v^2 + 2 vw \lambda_{h \sigma} \sigma h + w^2 \lambda_\sigma \sigma^2 \\
&= \frac12 \begin{pmatrix} h & \sigma \end{pmatrix} M^2 \begin{pmatrix} h \\ \sigma \end{pmatrix},\end{aligned}$$ where we have defined the mass matrix $M$ by $$M^2 = 2 \begin{pmatrix} \frac16 \lambda_h v^2 & \lambda_{h \sigma} vw \\ \lambda_{h \sigma} vw & \lambda_\sigma w^2 \end{pmatrix}.$$ This mass matrix can be easily diagonalized, and if we make the natural assumption that $w$ is of the order of $m_R$, while $v$ is of the order of $M_W$, so that $v \ll w$, we find that the two eigenvalues are $$\begin{aligned}
m_+^2 &\sim 2 \lambda_\sigma w^2 + 2 \frac{\lambda_{h \sigma}^2}{\lambda_\sigma} v^2,\\
m_-^2 &\sim 2 \lambda_h v^2 \left( \frac16 - \frac{\lambda_{h \sigma}^2}{\lambda_h \lambda_\sigma}\right).\end{aligned}$$ We can now determine the value of these two masses by running the scalar coupling constants $\lambda_h, \lambda_{h \sigma}$ and $\lambda_\sigma$ down to ordinary energy scalar using the renormalization group equations for these couplings that were derived in [@GLPR10], referring to [@CC12; @Sui14] for full details.
![A contour plot of the Higgs mass $m_h$ as a function of $\rho^2$ and $t = \log (\Lambda{_{\scriptscriptstyleGUT}}/M_Z)$. The red line corresponds to $m_h = 125.5~\operatorname{GeV}$.[]{data-label="fig:higgsmass"}](higgsmass.png)
The result varies with the chosen value for $\Lambda_\GUT$ and the parameter $\rho$. The mass of $\sigma$ is essentially given by the largest eigenvalue $m_+$ which is of the order $10^{12}~ \operatorname{GeV}$ for all values of $\Lambda_\GUT$ and the parameter $\rho$. The allowed mass range for the Higgs, [*i.e.*]{} for $m_-$, is depicted in Figure \[fig:higgsmass\]. The expected value $m_h=125.5 ~\operatorname{GeV}$ is therefore compatible with the above noncommutative model. Moreover, without the $\sigma$ the $\lambda_h$ turns negative at energies around $10^{12} \operatorname{GeV}$. Furthermore, this calculation implies that there is a relation (given by the red line in the Figure) between the ratio $m_\nu/m{_{\scriptscriptstyletop}}$ and the unification scale $\Lambda_\GUT$.
Pati–Salam unification and first-order condition {#sect:patisalam}
------------------------------------------------
In order to see how we one can use the noncommutative geometric approach to go beyond the Standard Model it is important to trace our steps that led to the spectral Standard Model in the previous Section. The route started with the classification of the algebras of the finite space ([*cf.*]{} Equation ). The results show that the only algebras which solve the fermion doubling problem are of the form $M_{2a}(\mathbb{C})\oplus M_{2a}(\mathbb{C})$ where $a$ is an even integer. An arbitrary symplectic constraint is imposed on the first algebra restricting it from $M_{2a}(\mathbb{C})$ to $M_{a}(\mathbb{H}).$ The first non-trivial algebra one can consider is for $a=2$ with the algebra $$M_{2}(\mathbb{H})\oplus M_{4}(\mathbb{C}).$$ Coincidentally, and as explained in the introduction, the above algebra comes out as a solution of the two-sided Heisenberg quantization relation between the Dirac operator $D$ and the two maps from the four spin-manifold and the two four spheres $S^{4}\times S^{4}$ [@CCM14; @CCM15]. This removes the arbitrary symplectic constraint and replaces it with a relation that quantize the four-volume in terms of two quanta of geometry and have far reaching consequences on the structure of space-time. We will come back to this in the last Section.
The existence of the chirality operator $\gamma$ that commutes with the algebra breaks the quaternionic matrices $M_{2}(\mathbb{H})$ to the diagonal subalgebra and leads us to consider the finite algebra $$\mathcal{A}_{F}=\mathbb{H}_{R}\oplus\mathbb{H}_{L}\oplus M_{4}(\mathbb{C}).$$ This algebras is the simplest candidate to search for new physics beyond the Standard Model. In fact, the inner automorphism group of $\mathcal{A=C}^{\infty}\left(
M\right) \otimes\mathcal{A}_{F}$ is recognized as the Pati–Salam gauge group $SU(2)_{R}\times SU(2)_{L}\times SU(4)$, and the corresponding gauge bosons appear as inner perturbations of the (spacetime) Dirac operator [@CCS13b]. Thus, we are considering a spectral Pati–Salam model as a candidate beyond the Standard Model. Let us further analyze this model and its phenomenological consequences.
An element of the Hilbert space $\Psi\in\mathcal{H}$ is represented by $$\Psi_{M}=\left(
\begin{array}
[c]{c}\psi_{A}\\
\psi_{A^{^{\prime}}}\end{array}
\right) ,\quad\psi_{A^{\prime}}=\psi_{A}^{c}$$ where $\psi_{A}^{c}$ is the conjugate spinor to $\psi_{A}.$ Thus all primed indices $A^{\prime}$ correspond to the Hilbert space of conjugate spinors. It is acted on by both the left algebra $M_{2}\left( \mathbb{H}\right) $ and the right algebra $M_{4}\left( \mathbb{C}\right) $. Therefore the index $A$ can take $16$ values and is represented by $$A=\alpha I$$ where the index $\alpha$ is acted on by quaternionic matrices and the index $I$ by $M_{4}\left( \mathbb{C}\right) $ matrices. Moreover, when the grading breaks $M_{2}\left( \mathbb{H}\right) $ into $\mathbb{H}_{R}\oplus\mathbb{H}_{L}$ the index $\alpha$ is decomposed to $\alpha
=\overset{.}{a},a$ where $\overset{.}{a}=\overset{.}{1},\overset{.}{2}$ (dotted index) is acted on by the first quaternionic algebra $\ \mathbb{H}_{R}$ and $a=1,2$ is acted on by the second quaternionic algebra $\ \mathbb{H}_{L}$. When $M_{4}\left( \mathbb{C}\right) $ breaks into $\mathbb{C}\oplus M_{3}\left( \mathbb{C}\right) $ (due to symmetry breaking or through the use of the order one condition as in [@CC07b]) the index $I$ is decomposed into $I=1,i$ and thus distinguishing leptons and quarks, where the $1$ is acted on by the $\mathbb{C}$ and the $i$ by $M_{3}\left(
\mathbb{C}\right) .$ Therefore the various components of the spinor $\psi
_{A}$ are $$\begin{aligned}
\psi_{\alpha I} & =\left(
\begin{array}
[c]{cccc}\nu_{R} & u_{iR} & \nu_{L} & u_{iL}\\
e_{R} & d_{iR} & e_{L} & d_{iL}\end{array}
\right) ,\qquad i=1,2,3\\
& =\left( \psi_{\overset{.}{a}1},\psi_{\overset{.}{a}i},\psi_{a1},\psi
_{ai}\right) ,\qquad a=1,2,\quad\overset{.}{a}=\overset{.}{1},\overset{.}{2}\nonumber\end{aligned}$$ This is a general prediction of the spectral construction that there is $16$ fundamental Weyl fermions per family, $4$ leptons and $12$ quarks.
The (finite) Dirac operator can be written in matrix form$$D_{F}=\left(
\begin{array}
[c]{cc}D_{A}^{B} & D_{A}^{B^{^{\prime}}}\\
D_{A^{^{\prime}}}^{B} & D_{A^{^{\prime}}}^{B^{^{\prime}}}\end{array}
\right) ,\label{eq:dirac}$$ and must satisfy the properties $$\gamma_{F}D_{F}=-D_{F}\gamma_{F}\qquad J_{F}D_{F}=D_{F}J_{F}$$ where $J_{F}^{2}=1.$ A matrix realization of $\gamma_{F}$ and $J_{F}$ are given by $$\gamma_{F}=\left(
\begin{array}
[c]{cc}G_{F} & 0\\
0 & -\overline{G}_{F}\end{array}
\right) ,\qquad G_{F}=\left(
\begin{array}
[c]{cc}1_{2} & 0\\
0 & -1_{2}\end{array}
\right) ,\qquad J_{F}=\left(
\begin{array}
[c]{cc}0_{4} & 1_{4}\\
1_{4} & 0_{4}\end{array}
\right) \circ\mathrm{cc}$$ where $\mathrm{cc}$ stands for complex conjugation. These relations, together with the hermiticity of $D$ imply the relations $$\left( D_{F}\right) _{A^{^{\prime}}}^{B^{^{\prime}}}=\left( \overline
{D}_{F}\right) _{A}^{B}\,\qquad\left( D_{F}\right) _{A^{^{\prime}}}^{B}=\left( \overline{D}_{F}\right) _{B}^{A^{\prime}}$$ and have the following zero components [@CC10] $$\begin{aligned}
\left( D_{F}\right) _{aI}^{bJ} & =0=\left( D_{F}\right) _{\overset{.}{a}I}^{\overset{.}{b}J}\\
\left( D_{F}\right) _{aI}^{\overset{.}{b}^{\prime}J^{\prime}} & =0=\left(
D_{F}\right) _{\overset{.}{a}I}^{b^{\prime}J\prime}$$ leaving the components $\left( D_{F}\right) _{aI}^{\overset{.}{b}J}$, $\left( D_{F}\right) _{aI}^{b^{\prime}J^{\prime}}$ and $\left(
D_{F}\right) _{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}$ arbitrary. These restrictions lead to important constraints on the structure of the connection that appears in the inner fluctuations of the Dirac operator. In particular the operator $D$ of the full noncommutative space given by $$D=D_{M}\otimes1+\gamma_{5}\otimes D_{F}$$ gets modified to $$D_{A}=D+A_{\left( 1\right) }+JA_{\left( 1\right) }J^{-1}+A_{\left(
2\right) }$$ where $$A_{\left( 1\right) }={\displaystyle\sum}
a\left[ D,b\right] ,\,\qquad A_{2}={\displaystyle\sum}
\widehat{a}\left[ A_{\left( 1\right) },\widehat{b}\right] ,\qquad
\widehat{a}=JaJ^{-1}$$
We have shown in [@CCS13b] that components of the connection $A$ which are tensored with the Clifford gamma matrices $\gamma^{\mu}$ are the gauge fields of the Pati–Salam model with the symmetry of $SU\left( 2\right) _{R}\times
SU\left( 2\right) _{L}\times SU\left( 4\right) .$ On the other hand, the non-vanishing components of the connection which are tensored with the gamma matrix $\gamma_{5}$ are given by $$\left( A\right) _{aI}^{\overset{.}{b}J}\equiv\gamma_{5}
\Sigma _{aI}^{\overset{.}{b}J},\qquad\left( A\right) _{aI}^{b^{\prime}J^{\prime}}=\gamma_{5}H_{aIbJ},\qquad\left( A\right)
_{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}\equiv\gamma
_{5}H_{\overset{.}{a}I\overset{.}{b}J}$$ where $H_{aIbJ}=H_{bJaI}$ and $H_{\overset{.}{a}I\overset{.}{b}J}=H_{\overset{.}{b}J\overset{.}{a}I}$, which is the most general Higgs structure possible. These correspond to the representations with respect to $SU\left( 2\right) _{R}\times SU\left( 2\right) _{L}\times SU\left(
4\right) :$$$\begin{aligned}
\Sigma_{aI}^{\overset{.}{b}J} & =\left( 2_{R},2_{L},1\right) +\left(
2_{R},2_{L},15\right) \\
H_{aIbJ} & =\left( 1_{R},1_{L},6\right) +\left( 1_{R},3_{L},10\right) \\
H_{\overset{.}{a}I\overset{.}{b}J} & =\left( 1_{R},1_{L},6\right) +\left(
3_{R},1_{L},10\right)\end{aligned}$$ We note, however, that the inner fluctuations form a semi-group and if a component $\left( D_{F}\right) _{aI}^{\overset{.}{b}J}$ or $\left(
D_{F}\right) _{aI}^{b^{\prime}J^{\prime}}$ or $\left( D_{F}\right)
_{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}$ vanish, then the corresponding $A$ field will also vanish. We can distinguish three cases: 1) Left-right symmetric Pati–Salam model with fundamental Higgs fields $\Sigma_{aI}^{\overset{.}{b}J},$ $H_{aIbJ}$ and $H_{\overset{.}{a}I\overset{.}{b}J}.$ In this model the field $H_{aIbJ}$ should have a zero vev. 2) A Pati–Salam model where the Higgs field $H_{aIbJ}$ that couples to the left sector is set to zero which is desirable because there is no symmetry between the left and right sectors at low energies. 3) If one starts with $\left( D_{F}\right) _{aI}^{\overset{.}{b}J}$ or $\left( D_{F}\right)
_{aI}^{b^{\prime}J^{\prime}}$ or $\left( D_{F}\right) _{\overset{.}{a}I}^{\overset{.}{b}^{\prime}J^{\prime}}$ whose values are given by those that were derived for the Standard Model, then the Higgs fields $\Sigma
_{aI}^{\overset{.}{b}J},$ $H_{aIbJ}$ and $H_{\overset{.}{a}I\overset{.}{b}J}$ will become composite and expressible in terms of more fundamental fields $\Sigma_{I}^{J},$ $\Delta_{\overset{.}{a}J}$ and $\phi_{\overset{.}{a}}^{b}$ . We refer to this as the composite model. It has the scalar field $\sigma$ discussed in the previous section as a remnant after spontaneous symmetry breaking [@CCS13b]. In fact, contrary to some claims in the literature it is possible to perform the potential analysis in this case in unitarity gauge and arrive at the conclusion that the field content contains the scalar field $\sigma$ ([*cf.*]{} Appendix \[app:potential\]).
Depending on the precise particle content we may determine the renormalization group equations of the Pati–Salam gauge couplings $g_{R},g_{L},g$. In [@CCS15] we have run them to look for unification of the coupling $g_{R}=g_{L}=g$. The boundary conditions are taken at the intermediate mass scale $\mu=m_{R}$ to be the usual (e.g. [@Moh86 Eq. (5.8.3)]) $$\frac{1}{g_{1}^{2}}=\frac{2}{3}\frac{1}{g^{2}}+\frac{1}{g_{R}^{2}},\qquad
\frac{1}{g_{2}^{2}}=\frac{1}{g_{L}^{2}},\qquad\frac{1}{g_{3}^{2}}=\frac
{1}{g^{2}},\label{eq:couplings-relations}$$ in terms of the Standard Model gauge couplings $g_{1},g_{2},g_{3}$. At the mass scale $m_{R}$ the Pati–Salam symmetry is broken to that of the Standard Model, and we take it to be the same scale that is present in the see-saw mechanism. It should thus be of the order $10^{11}-10^{13}$GeV. What we have found in [@CCS15] (and this was confirmed by others in [@AMST15]) is that in all three cases it is possible to achieve grand unification of the couplings, while connecting to Standard Model physics in the broken, low-energy phase. An example of a running of the gauge coupling is illustrated in Figure \[fig:ps-running\].
![Running of the gauge couplings of the Standard Model gauge couplings (below scale $m_R \approx 10^{11} \operatorname{GeV}$) and the Pati–Salam gauge coupling (above scale $m_R$) in case 2.[]{data-label="fig:ps-running"}](PSrunningNoOrder1.png)
Grand symmetry and twisted spectral triples
-------------------------------------------
In [@DLM14] the next-to-next case[^3] in the list of irreducible geometries in Equation was considered: $k=4$. Thus, one considers $$A_G = M_4(\bH) \oplus M_8(\C); \qquad H_F := \C^{128}.$$ where $128$ is exactly the number of spinor and internal degrees of freedom combined (including the aforementioned fermion quadruplication). The geometry is then $$\left( C^\infty(M,A_G), L^2(M) \otimes H_F, D_M + \gamma_M D_F \right)$$ where one has to assume that the spinor bundle on $M$ has been trivialized to gather the spinor and internal fermionic degrees of freedom in a single Hilbert space $H_F$.
Note that the above geometry is not a direct product of the continuum with a discrete space. In fact, both the algebra and the Dirac operator $D_M$ contain spinor indices. As a consequence the commutator $[D_M, a]$ can become unbounded, thus challenging one of the basic axioms of spectral triples. Instead, it is possible to guarantee that [*twisted*]{} commutators are bounded so that this example fits in the general framework of twisted spectral triples developed in [@CM08]. In [@DM14] the authors identify an inner automorphism $\rho = R (\cdot )R$ of $A_G$ such that $$[D,a]_\rho = D a - \rho (a) D$$ is bounded.
An interesting question that arises at this point is how to generate inner fluctuations of twisted spectral triples. This was analyzed in full detail from a mathematical viewpoint in [@LM16; @LM17]. One of the intriguing aspects is the self-adjointness of the Dirac operator under fluctuations (even gauge transformations): for this to be respected one has to impose a compatibility between the twist and the fluctuation.
An alternative route was suggested in [@DFLM17]. Namely, one may drop the above condition of self-adjointness and instead look for operators that are Krein-self-adjoint, using the Krein structure on the Hilbert space that is induced by the operator $R$ (defining the twist $\rho$). This will have an intriguing appearance of the Lorentzian structure (given by the Krein inner product) from a purely algebraic and Euclidean starting point. Here we also refer to the nice overview given in [@Liz18].
Algebraic constraints on the finite geometry
--------------------------------------------
An interesting question to consider —in particular in light of theories that go Beyond the Standard Model— is whether one can [*derive*]{} the restricted form of the Dirac operator $D_F$ in . We highlight a few approaches to this question that are present in the literature.
First of all, as mentioned already on page , the form of the $D_F$ in terms of the matrices $Y_\nu, Y_e, Y_u, Y_d$ and $Y_R$ as in Equations and appears naturally in the study of moduli of finite Dirac operators. The only constraint (in addition to the usual conditions layed out in Section \[sect:st\]) there was that the photon remained massless.
An attempt was made to make the latter condition less [*ad hoc*]{} is [@BF13; @BF14; @BF18]. They proposed to generalize noncommutative geometry to non-associative noncommutative geometry, thus allowing for non-associative algebras. The crucial idea —which goes back to Eilenberg— is to combine the (differential) algebra and (Hilbert space) bimodule into a single algebra, and understand the conditions such as commutant property and first-order conditions as consequences of associativity of the pertinent algebra $B$. However, this associativity is a strong constraint and accordingly further restrict the geometry described by $D_F$. Note that non-associative algebras have also been used in the context of noncommutative geometry and particle physics to predict the number of families (to be three) [@TD18]
Another approach to analyzing the form of the Dirac operator $D_F$ by imposing algebraic conditions is taken by [@Dab17; @DAS18]. Here the authors adopt the principle that, similar to differential forms in the continuum, the finite Hilbert space should be a Morita equivalence between $A$ and the Clifford algebra generated by $A_F$ and $D_F$. One finds that the aforementioned form of $D_F$ does not satisfy this condition but additional entries in $D_F$ should be non-zero. This gives rise to a model Beyond the Standard Model: an analysis of the phenomenological consequences is performed in [@KL18; @DS18]. In [@Ayd19] it was then found that this model does not exhibit grand unification of the Standard Model couplings.
Volume quantization and uniqueness of SM
========================================
In the classification of finite noncommutative spaces we arrived at the result that the algebra $\mathcal{A}_{F}=\left( \mathbb{H}_{R}\mathbb{\oplus H}_{L}\right) \oplus M_{4}\left( \mathbb{C}\right) $ was the first possibility out of many of the form $\mathcal{A}_{F}=\left( M_{n}\left(
\mathbb{H}\right) _{R}\mathbb{\oplus}M_{n}\left( \mathbb{H}\right)
_{L}\right) \oplus M_{4n}\left( \mathbb{C}\right) $. in addition we made an assumption, that seemed arbitrary, of the existence of antilinear isometry that reduced the algebra $M_{4n}\left( \mathbb{C}\right) $ to $\left(
M_{n}\left( \mathbb{H}\right) _{R}\mathbb{\oplus}M_{n}\left( \mathbb{H}\right) _{L}\right) $. It is necessary to have a stronger evidence of the uniqueness of our conclusions that helps us to avoid making the above mentioned assumptions. Surprisingly, the new evidence came in the process of solving a seemingly completely independent problem, encoding low dimensional geometries, and in particular dimension four.
Higher form of Heisenberg’s commutation relations
-------------------------------------------------
Starting with the simple example of one dimensional geometries, consider the equation $$U^{\ast}\left[ D,U\right] =1,\qquad U^{\ast}U=1$$ where $D$ is self-adjoint operator. Assuming that the one dimensional space is a closed curve parameterized by coordinate $x$ and the Dirac operator to be $D=-i\frac{d}{dx}+\alpha$ the above equation simplifies to $$-iU^{\ast}dU=dx$$ Writing $U=e^{in\theta}$ we obtain $dx=nd\theta.$ Integrating both sides implies that the length of the one dimensional curve is an integer multiple of $2\pi$, the length of $S^1$$${\displaystyle\oint\limits_{C}}
dx=n\left( 2\pi\right)$$ To adopt this construction to higher dimensions, we note that we can characterize the circle $S^{1}$ by the equation $Y^{A}Y^{A}=1,$ $A=1,2$, $Y^{A\ast}=Y^{A}.$ Assembling the two coordinates $Y^{1},$ $Y^{2}$ in one matrix, define $Y=Y^{A}\Gamma_{A},$ where $\Gamma_{A},$ $A=1,2$ are taken to be $2\times2.$ In addition we identify $\Gamma_{1}=\sigma_{1},$ $\Gamma
_{2}=\sigma_{2},$ the Pauli matrices, and define $\Gamma=-i\Gamma_{1}\Gamma_{2}=\sigma_{3}$ so that $\Gamma_{+}=\frac{1}{2}\left( 1+\Gamma\right)
$ is a projection operator. We notice that we can write $$Y=\left(
\begin{array}
[c]{cc}0 & Y^{1}-iY^{2}\\
Y^{1}+iY^{2} & 0
\end{array}
\right) =\left(
\begin{array}
[c]{cc}0 & U^{\ast}\\
U & 0
\end{array}
\right)$$ where $U=Y^{1}-iY^{2}$ and $U^{\ast}U=1.$ The expression $$\left\langle \Gamma_{+}Y\left[ D,Y\right] \right\rangle =1\label{oned}$$ where $\left\langle {}\right\rangle $ is defined to be the trace over the Clifford algebra defined by $\Gamma_{A},$ gives back the equation $U^{\ast
}\left[ D,U\right] =1.$
For higher dimensional geometries we consider a Riemannian manifold with dimension $n$ and where the algebra $\mathcal{A}$ is taken to be $C^{\infty
}\left( M\right) ,$ the algebra of continuously differentiable functions, while the operator $D$ is identified with the Dirac operator given by $$D_{M}=\gamma^{\mu}\left( \frac{\partial}{\partial x^{\mu}}+\omega_{\mu
}\right) ,$$ where $\gamma^{\mu}=e_{a}^{\mu}\gamma^{a}$ and $\omega_{\mu}=\frac{1}{4}\omega_{\mu bc}\gamma^{bc}$ is the $SO(n)$ Lie-algebra valued spin-connection with the (inverse) vielbein $e_{a}^{\mu}$ being the square root of the (inverse) metric $g^{\mu\nu}=e_{a}^{\mu}\delta^{ab}e_{b}^{\nu}.$ The gamma matrices $\gamma^{a}$ are anti-hermitian $\left( \gamma^{a}\right) ^{\ast
}=-\gamma^{a}$ that define the Clifford algebra $\left\{ \gamma^{a},\gamma
^{b}\right\} =-2\delta^{ab}.$ The Hilbert space $\mathcal{H}$ is the space of square integrable spinors $L^{2}\left( M,S\right) .$ The chirality operator $\gamma$ in even dimensions is then given by $$\gamma=\left( i\right) ^{\frac{n}{2}}\gamma^{1}\gamma^{2}\cdots\gamma^{n}$$ Starting with manifolds of dimension $2$ we first define the two sphere by the equation $Y^{A}Y^{A}=1,$ $A=1,2,3$, $Y^{A\ast}=Y^{A}.$ Assembling the three coordinates $Y^{1},$ $Y^{2},$ $Y^{3}$ in one matrix, defining $Y=Y^{A}\Gamma_{A},$ where $\Gamma_{A},$ $A=1,2,3$ are taken to be $2\times2$ Pauli matrices. Notice that in this case $\Gamma\equiv-i\Gamma_{1}\Gamma_{2}\Gamma_{3}=1$ and to generalize equation (\[oned\]) to two dimensions the factor $\Gamma$ can be dropped, and we write instead$$\frac{1}{2!}\left\langle Y\left[ D,Y\right] ^{2}\right\rangle =\gamma
\label{heisenberg2}$$ The reason we have to include the chirality operator $\gamma$ on the two dimensional manifold $M$ is that the Dirac operator $D$ appears twice yielding a product of the form $\gamma_{1}\gamma_{2}=-i\gamma.$ A simple calculation shows that the above equation in component form is given by $$\frac{1}{2!}\epsilon^{\mu\nu}\epsilon_{ABC}Y^{A}\partial_{\mu}Y^{B}\partial_{\nu}Y^{C}=\det\left( e_{\mu}^{a}\right)$$ which is a constraint on the volume form of $M_{2}.$ This implies that the volume of $M_{2}$ will be an integer multiple of the area of the unit $2$-sphere$$\begin{aligned}
{\displaystyle\int\limits_{M_{2}}}
d^{2}x\sqrt{g} & ={\displaystyle\int}
\epsilon_{ABC}Y^{A}dY^{B}dY^{C}\\
& =n(4\pi)\end{aligned}$$ where $n$ is the winding number. An example of a map $Y$ with winding number $n$ is $$Y\equiv Y^{1}+iY^{2}=\frac{2z^{n}}{\left\vert z\right\vert ^{2n}+1},\qquad
Y^{3}=\frac{\left\vert z\right\vert ^{2n}-1}{\left\vert z\right\vert ^{2n}+1},\qquad z=x^{1}+ix^{2}$$ From this we deduce that the pullback $Y^{\ast}\left( w_{n}\right) $ is a differential form that does not vanish anywhere. This in turn implies that the Jacobian of the map $Y$ does not vanish anywhere, and that $Y$ is a covering of the sphere. The sphere is simply connected, and on each connected component $M_{j}\subset M_{n}$, the restriction of the map $Y$ to $M_{j}$ is a diffeomorphism, implying that the manifold must be disconnected, with each piece having the topology of a sphere. To allow for two dimensional manifolds with arbitrary topology, our first observation is that condition (\[heisenberg2\]) involves the commutator of the Dirac operator $D$ and the coordinates $Y.$ In momentum space $D$ is the Feynman-slashed $\gamma^{\mu
}p_{\mu}$ momentum and $Y$ are the Feynman-slashed coordinates. This suggests that the quantization condition is a higher form of Heisenberg commutation relation quantizing the phase space formed by coordinates and momenta. We first notice that although the quantization condition is given in terms of the noncommutative data, the operator $J$ is the only one missing. We therefore modify the condition to take $J$ into account. The operator $J$ transforms $Y$ into its commutant $Y^{\prime}=iJYJ^{-1}$ so that $\left[ Y,Y^{\prime
}\right] =0$. Thus let $Y=Y^{A}\Gamma_{A}$ and $Y^{\prime}=iJYJ^{-1}$ and $\Gamma_{A}^{\prime}=iJ\Gamma_{A}J^{-1}$ so that we can write $$Y=Y^{A}\Gamma_{A},\qquad Y^{\prime}=Y^{\prime A}\Gamma_{A}^{\prime},$$ satisfying $Y^{2}=1$ and $Y^{\prime2}=1$ with the Clifford algebras $C_{\pm}$$$\begin{aligned}
\left\{ \Gamma_{A},\Gamma_{B}\right\} & =2\,\delta_{AB},\ \quad(\Gamma
_{A})^{\ast}=\Gamma_{A}\label{Cplus}\\
\left\{ \Gamma_{A}^{\prime},\Gamma_{B}^{\prime}\right\} & =-2\,\delta
_{AB},\ \quad(\Gamma_{A}^{\prime})^{\ast}=-\Gamma_{A}^{\prime}\label{Cminus}$$ We immediately see that the Clifford algebra $C_{+}=M_{2}\left(
\mathbb{C}\right) $ and $C_{-}=\mathbb{H}.$ We then define the projection operator $e=\frac{1}{2}\left( 1+Y\right) $ satisfying $e^{2}=e$ and similarly $e^{\prime}=\frac{1}{2}\left( 1+Y^{\prime}\right) $ satisfying $e^{\prime2}=e^{\prime}.$ From the tensor product of $E=ee^{\prime}$ satisfying $E^{2}=E,$ we construct $Z=2E-1$ satisfying $Z^{2}=1$ and allowing us to write $$\frac{1}{2}\left\langle Z\left[ D,Z\right] ^{2}\ \right\rangle =\gamma$$ A straightforward calculation reveals that this relation splits as the sum of two non-interfering parts$$\frac{1}{2}\left\langle Y\left[ D,Y\right] ^{2}\right\rangle +\frac{1}{2}\left\langle Y^{\prime}\left[ D,Y^{\prime}\right] ^{2}\right\rangle
=\gamma$$ which in component form reads$$\frac{1}{2!}\epsilon^{\mu\nu}\epsilon_{ABC}\left( Y^{A}\partial_{\mu}Y^{B}\partial_{\nu}Y^{C}+Y^{^{\prime}A}\partial_{\mu}Y^{^{\prime}B}\partial_{\nu}Y^{^{\prime}C}\right) =\det\left( e_{\mu}^{a}\right)$$ We will show later, when considering the four dimensional case that this modification allows to reconstruct two dimensional manifolds of arbitrary topology from the pullbacks of the maps $Y,$ $Y$’.
For three dimensional manifolds $\gamma=1$ and in analogy with the one-dimensional case we write $$\frac{1}{3!}\left\langle \Gamma_{+}Y\left[ D,Y\right] ^{3}\right\rangle
=1\label{universal}$$ where $Y=Y^{A}\Gamma_{A},$ $A=1,\ldots4,$ $Y^{2}=1,$ $Y=Y^{\ast},$ $\Gamma
_{A}$ are $4\times4$ Clifford algebra matrices $C_{+}$ where $\left\{
\Gamma_{A},\Gamma_{B}\right\} =2\,\delta_{AB}$. In this representation of the $\Gamma$ matrices we have $\Gamma=\Gamma_{5}=\Gamma_{1}\Gamma_{2}\Gamma
_{3}\Gamma_{4}=\left(
\begin{array}
[c]{cc}1_{2} & 0\\
0 & -1_{2}\end{array}
\right) $ so that $\Gamma_{+}=\frac{1}{2}\left( 1+\Gamma\right) $ is a projection operator. In $d=3,$ we can write $$Y=Y^{A}\Gamma_{A}=\left(
\begin{array}
[c]{cc}0 & U^{\ast}\\
U & 0
\end{array}
\right)$$ where $U$ is a unitary $2\times2$ matrix such that it could be written in the form $U=\exp\left( i\left( \alpha_{0}1+\alpha_{a}\sigma^{a}\right) \right)
$ so that $U^{\ast}U=1$. It is easy to check that $\left\langle Y\left[
D,Y\right] ^{3}\right\rangle =0$ and that the component form of the above relation is $$\det\left( e_{\mu}^{a}\right) =\frac{1}{3!}\epsilon^{\mu\nu\rho}\mathrm{Tr}\left( U^{\ast}\partial_{\mu}UU^{\ast}\partial_{\nu}UU^{\ast
}\partial_{\rho}U\right)$$ whose integral is the winding number of the $SU(2)$ group manifold. Again, using the reality operator $J$ we act on the Clifford algebra $Y^{\prime}=iJYJ^{-1}$ so that $\left[ Y,Y^{\prime}\right] =0$, then $\Gamma_{A}^{\prime}=iJ\Gamma_{A}J^{-1}$ satisfies $\left\{ \Gamma_{A}^{\prime
},\Gamma_{B}^{\prime}\right\} =-2\,\delta_{AB},$ $(\Gamma_{A}^{\prime})^{\ast}=-\Gamma_{A}^{\prime}$. Forming the projection operators $e=\frac
{1}{2}\left( 1+Y\right) ,$ $e^{\prime}=\frac{1}{2}\left( 1+Y^{\prime
}\right) $, we form the tensor product $E=ee^{\prime}$ we define the field $Z=2E-1,$ and thus the two sided relation becomes $$\frac{1}{3!}\left\langle \Gamma_{+}\Gamma_{+}^{\prime}Z\left[ D,Z\right]
^{3}\right\rangle =1$$ A lengthy calculation shows that the component form of this relation separates into two parts without interference terms$$\begin{aligned}
\det\left ( e_{\mu}^{a}\right) & =\frac{1}{3!}\epsilon^{\mu\nu\rho}\bigg(
\mathrm{Tr}\left( U^{\ast}\partial_{\mu}UU^{\ast}\partial_{\nu}UU^{\ast
}\partial_{\rho}U\right)\\
&\qquad \qquad +\mathrm{Tr}\left( U^{^{\prime}\ast}\partial_{\mu
}U^{\prime}U^{^{\prime}\ast}\partial_{\nu}U^{\prime}U^{^{\prime}\ast}\partial_{\rho}U^{\prime}\right) \bigg)\end{aligned}$$ Finally, for four dimensional manifolds the Clifford algebras $C_{+}$ and $C_{-}$ defined as in (\[Cplus\]) (\[Cminus\]) with $\Gamma_{A},$ $\Gamma_{A}^{\prime}$, $A=1,\cdots,5$ are known to be given by $C_{+}=M_{2}\left( \mathbb{H}\right) $ and $C_{-}=M_{4}\left(
\mathbb{C}\right) .$ The quantization condition takes the same form as the two dimensional case$$\frac{1}{4!}\left\langle Z\left[ D,Z\right] ^{4}\ \right\rangle
=\gamma\label{Heisenberg}$$ This relation separates into two non-interfering terms $$\frac{1}{4!}\left\langle Y\left[ D,Y\right] ^{4}\ \right\rangle +\frac
{1}{4!}\left\langle Y^{\prime}\left[ D,Y^{\prime}\right] ^{4}\ \right\rangle
=\gamma$$ the component form of which is given by $$\begin{aligned}
\det\left( e_{\mu}^{a}\right) &=\frac{1}{4!}\epsilon^{\mu\nu\kappa\lambda
}\epsilon_{ABCDE}\bigg( Y^{A}\partial_{\mu}Y^{B}\partial_{\nu}Y^{C}\partial_{\kappa}Y^{D}\partial_{\lambda}Y^{E}\\
&\qquad \qquad \qquad +Y^{^{\prime}A}\partial_{\mu
}Y^{^{\prime}B}\partial_{\nu}Y^{^{\prime}C}\partial_{\kappa}Y^{^{\prime}D}\partial_{\lambda}Y^{^{\prime}E}\bigg)\end{aligned}$$ One can verify that similar considerations fail when the dimension of the manifold $n>4$ as there are interference terms between the $Y$ and $Y^{\prime
}.$ Integrating both sides imply$${\displaystyle\int\limits_{M_{4}}}
d^{4}x\sqrt{g}=\frac{8}{3}\pi^{2}\left( N+N^{\prime}\right)$$ where $N$, $N^{\prime}$ are the winding numbers of the two maps $Y,$ $Y^{\prime}.$ An example of a map $Y$ with winding number $n$ is given by $$\begin{aligned}
Y & \equiv Y^{4}1+Y^{i}e_{i}=\frac{2x^{n}}{x^{n}\overline{x}^{n}+1},\\
Y^{5} & =\frac{x^{n}\overline{x}^{n}-1}{x^{n}\overline{x}^{n}+1},\end{aligned}$$ where $x=x^{4}1+x^{i}e_{i}$ and $e_{i},$ $i=1,2,3$ are the quaternionic complex structures $e_{i}^{2}=-1,$ $e_{i}e_{j}=\epsilon_{ijk}e_{k},$ $i\neq
j.$
Volume quantization
-------------------
Consider the smooth maps $\phi_{\pm}:M_{n}\rightarrow S^{n}$ then their pullbacks $\phi_{\pm}^{\ast}$ would satisfy $$\phi_{+}^{\ast}\left( \alpha\right) +\phi_{-}^{\ast}\left( \alpha\right)
=\omega, \label{integer}$$ where $\alpha$ is the volume form on the unit sphere $S^{n}$ and $\omega\left( x\right) $ is an $n-$form that does not vanish anywhere on $M_{n}.$ We have shown that for a compact connected smooth oriented manifold with $n<4$ one can find two maps $\phi_{+}^{\ast}\left( \alpha\right) $ and $\phi_{-}^{\ast}\left( \alpha\right) $ whose sum does not vanish anywhere, satisfying equation (\[integer\]) such that $\int \omega\in\mathbb{Z}.$ The proof for $n=4$ is more difficult and there is an obstruction unless the second Stieffel–Whitney class $w_{2}$ vanishes, which is satisfied if $M$ is required to be a spin-manifold and the volume to be larger than or equal to five units. The key idea in the proof is to note that the kernel of the Jacobian of the map $Y$ is a hypersurface $\Sigma$ of co-dimension $2$ and therefore $$\dim\Sigma=n-2.$$ We can then construct a map $Y^{\prime}=Y\circ\psi$ where $\psi$ is a diffeomorphism on $M$ such that the sum of the pullbacks of $Y$ and $Y^{\prime}$ does not vanish anywhere. The coordinates $Y$ are defined over a Clifford algebra $C_{+}$ spanned by $\left\{ \Gamma_{A},\Gamma_{B}\right\}
=2\delta_{AB}.$ For $n=2$, $C_{+}=M_{2}\left( \mathbb{C}\right) $ while for $n=4$, $C_{+}=M_{2}\left( \mathbb{H}\right) \oplus M_{2}\left(
\mathbb{H}\right) $ where $\mathbb{H}$ is the field of quaternions. However, for $n=4,$ since we will be dealing with irreducible representations we take $C_{+}=M_{2}\left( \mathbb{H}\right) .$ Similarly the coordinates $Y^{\prime}$ are defined over the Clifford algebra $C_{-}$ spanned by $\left\{ \Gamma_{A}^{\prime},\Gamma_{B}^{\prime}\right\} =-2\delta_{AB}$ and for $n=2$, $C_{-}=\mathbb{H\oplus H}$ and for $n=4$, $C_{-}=M_{4}\left(
\mathbb{C}\right) .$ The operator $J$ acts on the two algebras $C_{+}\oplus
C_{-}$ in the form $J\left( x,y\right) =\left( y^{\ast},x^{\ast}\right) $ (i.e. it exchanges the two algebras and takes the Hermitian conjugate). The coordinates $Z=\frac{1}{2}\left( Y+1\right) \left( Y^{\prime}+1\right)
-1,$ then define the matrix algebras [@CCM14] $$\begin{aligned}
\mathcal{A}_{F} & =M_{2}\left( \mathbb{C}\right) \oplus\mathbb{H},\qquad
n=2\\
\mathcal{A}_{F} & =M_{2}\left( \mathbb{H}\right) \oplus M_{4}\left(
\mathbb{C}\right) ,\qquad n=4.\end{aligned}$$ One, however, must remember that the maps $Y$ and $Y^{\prime}$ are functions of the coordinates of the manifold $M$ and therefore the algebra associated with this space must be $$\begin{aligned}
\mathcal{A} & =C^{\infty}\left( M,\mathcal{A}_{F}\right) \\
& =C^{\infty}\left( M\right) \otimes\mathcal{A}_{F}.\end{aligned}$$ To see this consider, for simplicity, the $n=2$ case with only the map $Y.$ The Clifford algebra $C_{-}=\mathbb{H}$ is spanned by the set $\left\{
1,\Gamma^{A}\right\} ,$ $A=1,2,3,$ where $\left\{ \Gamma^{A},\Gamma
^{B}\right\} =-2\delta^{AB}.$ We then consider functions which are made out of words of the variable $Y$ formed with the use of constant elements of the algebra [@C00] $${\displaystyle\sum\limits_{i=1}^{\infty}}
a_{1}Ya_{2}Y\cdots a_{i}Y,\qquad a_{i}\in\mathbb{H},$$ which will generate arbitrary functions over the manifold which is the most general form since $Y^{2}=1$. One can easily see that these combinations generate all the spherical harmonics. This result could be easily generalized by considering functions of the fields $$Z=\frac{1}{2}\left( Y+1\right) \left( Y^{\prime}+1\right) -1,\qquad
Y\in\mathbb{H},\quad Y^{\prime}\in M_{2}\left( \mathbb{C}\right) ,$$ showing that the noncommutative algebra generated by the constant matrices and the Feynman slash coordinates $Z$ is given by [@C00] $$\mathcal{A}=C^{\infty}\left( M_{2}\right) \otimes\left( \mathbb{H+}M_{2}\left( \mathbb{C}\right) \right) .$$ We now restrict ourselves to the physical case of $n=4.$ Here the algebra is given by $$\mathcal{A}=C^{\infty}\left( M_{4}\right) \otimes\left( M_{2}(\mathbb{H})\mathbb{+}M_{4}\left( \mathbb{C}\right) \right) .$$ The associated Hilbert space is $$\mathcal{H}=L^{2}\left( M_{4},S\right) \otimes\mathcal{H}_{F}.$$ The Dirac operator mixes the finite space and the continuous manifold non-trivially$$D=D_{M}\otimes1+\gamma_{5}\otimes D_{F},$$ where $D_{F\text{ }}$ is a self adjoint operator in the finite space. The chirality operator is $$\gamma=\gamma_{5}\otimes\gamma_{F},$$ and the anti-unitary operator $J$ is given by $$J=J_{M}\gamma_{5}\otimes J_{F},$$ where $J_{M}$ is the charge-conjugation operator $C$ on $M$ and $J_{F}$ the anti-unitary operator for the finite space. Thus an element $\Psi
\in\mathcal{H}$ is of the form $\Psi=\left(
\begin{array}
[c]{c}\psi_{A}\\
\psi_{A^{\prime}}\end{array}
\right) $ where $\psi_{A}$ is a $16$ component $L^{2}\left( M,S\right) $ spinor in the fundamental representation of $\mathcal{A}_{F}$ of the form $\psi_{A}=\psi_{\alpha I}$ where $\alpha=1,\cdots,4$ with respect to $M_{2}\left( \mathbb{H}\right) $ and $I=1,\cdots,4$ with respect to $M_{4}\left( \mathbb{C}\right) $ and where $\psi_{A^{\prime}}=C\psi
_{A}^{\ast}$ is the charge conjugate spinor to $\psi_{A}$ [@CC10]. The chirality operator $\gamma$ must commute with elements of $\mathcal{A}$ which implies that $\gamma_{F}$ must commute with elements in $\mathcal{A}_{F}.$ Commutativity of the chirality operator $\gamma_{F}$ with the algebra $\mathcal{A}_{F}$ and that this $\mathbb{Z}/2$ grading acts non-trivially reduces the algebra $M_{2}\left( \mathbb{H}\right) $ to $\mathbb{H}_{R}\oplus\mathbb{H}_{L}$ [@CCM14]. Thus the $\gamma_{F}$ is identified with $\gamma_{F}=\Gamma^{5}=\Gamma^{1}\Gamma^{2}\Gamma^{3}\Gamma^{4}$ and the finite space algebra reduces to $$\mathcal{A}_{F}=\mathbb{H}_{R}\oplus\mathbb{H}_{L}\oplus M_{4}\left(
\mathbb{C}\right) .$$ This can be easily seen by noting that an element of $M_{2}\left(
\mathbb{H}\right) $ takes the form $\left(
\begin{array}
[c]{cc}q_{1} & q_{2}\\
q_{3} & q_{4}\end{array}
\right) $ where each $q_{i},$ $i=1,\cdots,4,$ is a $2\times2$ matrix representing a quaternion. Taking the representation of $\Gamma^{5}=\left(
\begin{array}
[c]{cc}1_{2} & 0\\
0 & -1_{2}\end{array}
\right) $ to commute with $M_{2}\left( \mathbb{H}\right) $ implies that $q_{2}=0=q_{3},$ thus reducing the algebra to $\mathbb{H}_{R}\oplus
\mathbb{H}_{L}.$ Therefore the index $\alpha=1,\cdots,4$ splits into two parts, $\overset{.}{a}=\overset{.}{1},\overset{.}{2}$ which is a doublet under $\mathbb{H}_{R}$ and $a=1,2$ which is a doublet under $\mathbb{H}_{L}$. The spinor $\Psi$ further satisfies the chirality condition $\gamma\Psi=\Psi$ which implies that the spinors $\psi_{\overset{.}{a}I}$ are in the $\left(
2_{R},1_{L},4\right) $ with respect to the algebra $\mathbb{H}_{R}\mathbb{\oplus H}_{L}\oplus M_{4}\left( \mathbb{C}\right) $ while $\psi
_{aI}$ are in the $\left( 1_{R},2_{L},4\right) $ representation. The finite space Dirac operator $D_{F}$ is then a $32\times32$ Hermitian matrix acting on the $32$ component spinors $\Psi.$ In addition we take three copies of each spinor to account for the three families, but will omit writing an index for the families. At present we have no explanation for why the number of generations should be three. The Dirac operator for the finite space is then a $96\times96$ Hermitian matrix. The Dirac action is then given by [@CCM07] $$\left( J\Psi,D\Psi\right) .$$ We note that we are considering compact spaces with Euclidean signature and thus the condition $J\Psi=\Psi$ could not be imposed. It could, however, be imposed if the four dimensional space is Lorentzian [@Bar06].The reason is that the $KO$ dimension of the finite space is $6$ because the operators $D_{F},$ $\gamma_{F}$ and $J_{F}$ satisfy$$J_{F}^{2}=1,\qquad J_{F}D_{F}=D_{F}J_{F},\qquad J_{F}\gamma_{F}=-\gamma
_{F}J_{F}.$$ The operators $D_{M},$ $\gamma_{M}=\gamma_{5},$ and $J_{M}=C$ for a compact manifold of dimension $4$ satisfy $$J_{M}^{2}=-1,\qquad J_{M}D_{M}=D_{M}J_{M},\qquad J_{M}\gamma_{5}=\gamma
_{5}J_{M}. \label{Euclidean}$$ Thus the $KO$ dimension of the full noncommutative space $\left(
\mathcal{A},\mathcal{H},D\right) $ with the decorations $J$ and $\gamma$ included is $10$ and satisfies $$J^{2}=-1,\qquad JD=DJ,\qquad J\gamma=-\gamma J.$$ We have shown in [@CCM07] that the path integral of the Dirac action, thanks to the relations $J^{2}=-1$ and $J\gamma=-\gamma J$, yields a Pfaffian of the operator $D$ instead of its determinant and thus eliminates half the degrees of freedom of $\Psi$ and have the same effect as imposing the condition $J\Psi=\Psi.$
We have also seen that the operator $J$ sends the algebra $\mathcal{A}$ to its commutant, and thus the full algebra acting on the Hilbert space $\mathcal{H}$ is $\mathcal{A\otimes A}^{o}.$ Under automorphisms of the algebra $$\Psi\rightarrow U\Psi,$$ where $U=u\widehat{u}$ with $u\in\mathcal{A},$ $\widehat{u}\in\mathcal{A}^{o}$ with $\left[ u,\widehat{u}\right] =0$, it is clear that Dirac action is not invariant.
At this point it is clear that we have retrieved all our conclusions we have before arriving at a unique possibility, which is to have a noncommutative space corresponding to the Pati–Salam Model we considered before, and in the special case where the Dirac operator and algebra satisfy the order one condition, the result is the noncommutative space of the Standard Model. We have thus succeeded in obtaining the Pati–Salam Model and Standard Model as unique possibilities starting with the two sided Heisenberg like equation (\[Heisenberg\]) thus eliminating all other possibilities obtained in classifying finite noncommutative spaces of KO dimension $6.$ There is no need to assume the existence of an isometry that reduces the first algebra from $M_{4}\left( \mathbb{C}\right) $ to $M_{2}\left( \mathbb{H}\right) $, and no need to assume that the KO dimension of the finite space to be $6.$ These results are very satisfactory and serve to enhance our confidence of the fine structure of space time as given by the above derived noncommutative space.
Outlook: towards quantization
=============================
Starting with the simple observation that the Higgs field could be interpreted as the link between two parallel sheets separated by a distance of the order of $10^{-16}$ cm it took enormous effort to identify a noncommutative space where the spectrum of the Standard Model could fit. Small deviations from the model, such as the need for a real structure and a KO dimension $6$, were taken as input to fine tune and determine precisely the noncommutative space. The spectral action principle proved to be very efficient way in evaluating the bosonic sector of the theory. Having identified the noncommutative space, the next target was to understand why nature would chose the Standard Model and not any other possibility. A classification of finite spaces revealed the special nature of the the finite part of the noncommutative space identified. Work on encoding manifolds with dimensions equal to four satisfying a higher form of Heisenberg type equation showed that the most general solution of this equation is that of a noncommutative space which is a product of a four-dimensional Riemannian spin-manifold times the finite space corresponding to a Pati–Salam unification model. The Standard Model is a special case of this space where a first order differential condition is satisfied. After a long journey the reasons why nature chose the Standard Model is now reduced to determining solutions of a higher form of Heisenberg equation. With such little input, it is quite satisfying to learn that it is possible to answer many of the questions which puzzled theorists for a long time. We now know why there are 16 fermions per generation, why the gauge group is $SU\left(
3\right) \times SU\left( 2\right) \times U\left( 1\right) ,$ an explanation of the Higgs field and origin of spontaneous symmetry breaking. The Spectral model also predicts a Majorana mass for the right-handed neutrinos and explains the see-saw mechanism. We thus understand unification of all fundamental forces as a geometrical theory based on the spectral action principle of a noncommutative space.
Naturally, there are many questions that are still unanswered, and this motivates the need for further research to address these problems using noncommutative geometry considerations. To conclude, we mention few of the possible directions of future research. One important aspect to consider is the renormalizability properties of the spectral model. Another problem is to study the quantum properties of the Dirac operator and whether it could be related to the pullbacks of the maps used in determining the quanta of geometry. The future of noncommutative geometry in the program of unification of all fundamental interactions looks now to be very promising.
Pati–Salam model: potential analysis {#app:potential}
====================================
We here include the scalar potential analysis for the composite Pati–Salam model, as described in Section \[sect:patisalam\] above.
If there is unification of lepton and quark couplings, then $\rho=1$ so that the $\Sigma^I_J$-field decouples. In that case we have $$\begin{gathered}
\L_{pot} (\phi_{\overset{.}{a}}^{b},\Delta_{\overset{.}{a}I})=
-\mu^2 \phi_{\overset{.}{a}}^{c} \phi_{c}^{\overset{.}{a}}
- \nu^2 \left( \Delta_{\overset{.}{a}K}\overline{\Delta}^{\overset{.}{a}K}\right) ^{2}+ \lambda_{\Sigma}
\phi_{a}^{\overset{.}{c}}
\phi_{\overset{.}{c}}^{b}
\phi_{b}^{\overset{.}{d}}
\phi_{\overset{.}{d}}^{a}
\\
+ \lambda_H \left(\Delta_{\overset{.}{a}K}\overline{\Delta}^{\overset{.}{a}L}\Delta_{\overset{.}{b}L}\overline{\Delta}^{\overset{.}{b}K} \right)^2
+\lambda_{H \Sigma}
\left( \Delta_{\overset{.}{a}J}\overline{\Delta}^{\overset{.}{a}J}\Delta_{\overset{.}{c}I}\overline{\Delta
}^{\overset{.}{d}I}\right) \phi_{b}^{\overset{.}{c}} \phi_{\overset{.}{d}}^{b}\end{gathered}$$ where we have absorbed some constant factors by redefining the couplings $\lambda_H, \lambda_{H \Sigma}$ and $\lambda_\Sigma$.
We choose unitarity gauge for the $\Delta$ and $\phi$-fields, in the following precise sense.
\[lma:unitary-gauge1\] For each value of the fields $\{ \phi_{\dot a}^b, \Delta_{\dot a I}\}$ there is an element $(u_R,u_L,u) \in SU(2)_R \times SU(2)_L \times SU(4)$ such that $$\begin{aligned}
u_R \begin{pmatrix} \phi_{\dot 1}^1 & \phi_{\dot 1}^2 \\[1mm] \phi_{\dot 2}^1 & \phi_{\dot 2}^2
\end{pmatrix} u_L^* & = \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix}
\intertext{and}
u_R \begin{pmatrix} \Delta_{\dot 1 1}& \Delta_{\dot 1 2}& \Delta_{\dot 1 3}& \Delta_{\dot 1 4}\\ \Delta_{\dot 2 1}& \Delta_{\dot 2 2}& \Delta_{\dot 2 3}& \Delta_{\dot 2 4}\end{pmatrix} u^t &= \begin{pmatrix} 1+\delta_0 & 0& 0 & 0 \\ \delta_1 & \eta_1& 0 &0 \end{pmatrix}\end{aligned}$$ where $h ,\delta_0, \delta_1, \eta_1$ are real fields and $\chi$ is a complex field.
Consider the singular value decomposition of the $2 \times 2$ matrix $(\phi_{\dot a}^b)$: $$\begin{pmatrix} \phi_{\dot 1}^1 & \phi_{\dot 1}^2 \\[1mm] \phi_{\dot 2}^1 & \phi_{\dot 2}^2
\end{pmatrix} = U \begin{pmatrix} h & 0 \\ 0 & k \end{pmatrix} V^*$$ for unitary $2 \times 2$ matrices $U,V$ and real coefficients $h,k$. If we define $$\begin{aligned}
u_R &= \begin{pmatrix} 1 & 0 \\ 0 & \det U\end{pmatrix}U^* \in SU(2)_R\\
u_L &= \begin{pmatrix} 1 & 0 \\ 0 & \det V\end{pmatrix} V^*\in SU(2)_L\end{aligned}$$ it follows that $$\begin{aligned}
u_R \begin{pmatrix} \phi_{\dot 1}^1 & \phi_{\dot 1}^2 \\[1mm] \phi_{\dot 2}^1 & \phi_{\dot 2}^2
\end{pmatrix} u_L^* & = \begin{pmatrix} h & 0 \\ 0 & k \det UV^* \end{pmatrix}
=: \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix}.\end{aligned}$$ Next, we consider $\Delta_{\dot aI}$ and write $$\left(\Delta_{\dot aI}\right) = \begin{pmatrix} u_1^* \\ u_2^* \end{pmatrix},\qquad \text{with } u_a^* = \begin{pmatrix} \Delta_{\dot a 1}& \Delta_{\dot a 2}& \Delta_{\dot a 3}& \Delta_{\dot a 4} \end{pmatrix}$$ for $a=1,2$. We may suppose that the vectors $u_1,u_2$ are such that their inner product $u_1^* u_2$ is a real number. Indeed, if this is not the case, then multiply $\Delta_{\dot aI}$ by a matrix in $SU(2)_R$ as follows: $$\begin{pmatrix} u_1^* \\ u_2^* \end{pmatrix}\mapsto
\begin{pmatrix} \alpha & 0 \\ 0 & \alpha^* \end{pmatrix}\begin{pmatrix} u_1^* \\ u_2^* \end{pmatrix} = \begin{pmatrix} \alpha u_1^* \\ \alpha^* u_2^* \end{pmatrix}.$$ Now the inner product is $(\alpha^* u_1)^* \alpha u_2 = (\alpha)^2 u_1^* u_2$ and we may choose $\alpha$ so as to cancel the phase of $u_1^* u_2$. Moreover, this transformation respects the above form of $\phi_{\dot a}^b$ after a $SU(2)_L$-transformation of exactly the same form: $$\begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix} \mapsto
\begin{pmatrix} \alpha & 0 \\ 0 & \alpha^* \end{pmatrix} \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix} \begin{pmatrix} \alpha & 0 \\ 0 & \alpha^* \end{pmatrix} ^* = \begin{pmatrix} h & 0 \\ 0 & \chi \end{pmatrix} .$$ Thus let us continue with the vectors $u_1,u_2$ satisfying $u_1^* u_2 \in \R$. We apply Gramm-Schmidt orthonormalization to $u_1$ and $u_2$, to arrive at the following orthonormal set of vectors $\{e_1,e_2\}$ in $\C^4$: $$e_1 = \frac{u_1 }{\| u_1\|}; \qquad e_2 = \frac{u_2 - \frac{u_1^* u_2}{\|u_1\|} u_1}{\|u_2 - \frac{u_1^* u_2}{\|u_1\|} u_1\|}.$$ We complete this set by choosing two additional orthonormal vectors $e_3$ and $e_4$ and write a unitary $4 \times 4$ matrix: $$U = \begin{pmatrix} e_1 & e_2 & e_3 & e_4 \end{pmatrix}$$ The sought-for matrix $u \in SU(4)$ is determined by $$u^t = U \begin{pmatrix} 1_3 & 0 \\ 0 & \det U^* \end{pmatrix}$$ so as to give $$\left(\Delta_{\dot aI}\right) u^t = \begin{pmatrix} u_1^* e_1 & 0 & 0 & 0 \\ u_2^* e_1 & u_2^* e_2 & 0 & 0 \end{pmatrix} =: \begin{pmatrix} 1+\delta_0 & 0 & 0 & 0 \\ \delta_1 & \eta_1 & 0 & 0 \end{pmatrix}\qedhere$$
Note that this is compatible with the dimension of the quotient of the space of field values by the group. Indeed, the fields $\phi_{\dot a}^b$ and $\Delta_{\dot a I}$ span a real 24-dimensional space (at each manifold point). The dimension of the orbit space is then $24- \dim P$ with $P$ a principal orbit of the action of $SU(2)_R \times SU(2)_L \times SU(4)$ on the space of field values. This dimension $ \dim P$ is determined by the dimension of the group and of a principal isotropy group.
First, we see that up to conjugation there is always a $SU(2)$-subgroup of $SU(4)$ leaving $\Delta_{\dot aI}$ invariant: it corresponds to $SU(2)$-transformations in the space orthogonal to the vectors $\Delta_{\dot 1 I}$ and $\Delta_{\dot 2I}$ in $\C^4$. Moreover, one can compute that the isotropy subgroup of the field values $$\begin{pmatrix} \phi_{\dot a}^b \end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}; \qquad \begin{pmatrix} \Delta_{\dot a I} \end{pmatrix} = \begin{pmatrix}1 & 0& 0 & 0 \\ 1 & 1 & 0 & 0 \end{pmatrix}$$ is given by $\mathbb Z_2 \times SU(2)$. Hence, the dimension of the principal orbit is $21 - 3 = 18$ so that the orbit space is 6-dimensional. This corresponds to the 4 real fields $h,\delta_0,\delta_1,\eta_1$ and the complex field $\chi$.
We allow for the colour $SU(3)$-symmetry not to be broken spontaneously, hence we only choose unitarity gauge in the $SU(2)_R \times SU(2)_L \times U(1)$-representations. That is, we retain the row vector $\Delta_{\dot 2 I}$ for $I=1,\ldots, 4$ as a variable and write $$\begin{pmatrix} \Delta_{\dot a I} \end{pmatrix} = \begin{pmatrix}\sqrt{w}+\delta_0/\sqrt w & 0& 0 & 0 \\ \delta_1/\sqrt{w} & \eta_1/\sqrt{w} & \eta_2/\sqrt{w} & \eta_3/\sqrt{w} \end{pmatrix}$$ so that $(\eta_i)$ forms a scalar $SU(3)$-triplet field (so-called [*scalar leptoquarks*]{}). The reason for the rescaling with $\sqrt{w}$ is that it yields the right kinetic terms for $\delta_0,\delta_1$ and $\eta$. Indeed, from the spectral action we then have $$\begin{aligned}
\frac{1}{2}\partial_{\mu}H_{\overset{.}{a}I\overset{.}{b}J}\partial^{\mu}H^{\overset{.}{a}I\overset{.}{b}J}
&=\frac{1}{2} \partial_\mu \left(\Delta_{\overset{.}{a}J}\Delta_{\overset{.}{b}I}\right) \partial^\mu \left(\Delta^{\overset{.}{a}J}\Delta^{\overset{.}{b}J}\right)
\\
&\sim \sum_{a=0}^1 \partial_\mu \delta_a \partial^\mu \delta^a + \partial_\mu \eta \partial^\mu \eta^* + \text{ higher order}\end{aligned}$$ The scalar potential becomes in terms of the fields $h,\chi,\delta_0,\delta_1, \eta_i$: $$\begin{aligned}
&\L_{pot}(h,\chi, \delta_0,\delta_1,\eta) = - \mu^2 ( h^2 + |\chi|^2) -\nu^2 \left( (w+ \delta_0)^2 + \delta_1^2 + |\eta|^2 \right)^2 /w^2
\\
\nn &
\quad + \lambda_{H\Sigma} \left( (w+\delta_0)^2 h^2 + (\delta_1^2 + |\eta|^2) |\chi|^2 \right)\left( (w+ \delta_0)^2 + \delta_1^2 + |\eta|^2 \right) /w^2 \\
\nn &
\quad + \lambda_H \left( (w+\delta_0)^4 + 2 (w+ \delta_0)^2 \delta_1^2+ (\delta_1^2 +|\eta|^2)^2 \right)^2/w^4 + \lambda_\Sigma (h^4+ |\chi|^4) \end{aligned}$$ As we are interested in the truncation to the Standard Model, we look for extrema with $\langle \delta_1\rangle =\langle \eta_i\rangle=0$, whilst setting $\langle h \rangle =v, \langle \delta_0 \rangle = 0, \langle \chi \rangle = x$. Note that the symmetry of these vevs is $$\begin{gathered}
\left\{
\left(\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^* \end{pmatrix},
\begin{pmatrix} \lambda^* & 0 \\ 0 & \lambda \end{pmatrix},
\begin{pmatrix} \lambda^* & 0 \\ 0 & m \end{pmatrix}\right): \lambda \in U(1) , m \in SU(3) \right\}
\\\subset SU(2)_R \times SU(2)_L \times SU(4)\end{gathered}$$ In other words, $SU(2)_R \times SU(2)_L \times SU(4)$ is broken by the above vevs to $U(1) \times SU(3)$.
The first derivative of $V$ vanishes for these vevs precisely if $$\begin{aligned}
2v (w^2 \lambda_{H \Sigma} +2v^2 \lambda_\Sigma - \mu^2 ) = 0,\\
4 x^3 \lambda_\Sigma - 2x \mu^2 = 0,\\
4w (2w^2 \lambda_H + v^2 \lambda_{H \Sigma} - \nu^2) =0.\end{aligned}$$ This gives rise to the fine-tuning of $v,w$ as in [@CC12]: $$w^2 \lambda_{H \Sigma} +2v^2 \lambda_\Sigma - \mu^2 , \qquad 2w^2 \lambda_H + v^2 \lambda_{H \Sigma} - \nu^2$$ choosing $\mu$ and $\nu$ such that the solutions $v,w$ are of the desired orders. Moreover, we find that the vev for $\chi$ either vanishes or is equal to $x=\sqrt{\mu^2/2\lambda_\Sigma}$. Note that this latter vev appears precisely at the entry $k^d h$ (or $k^e h$) of the finite Dirac operator, which we have disregarded by setting $\rho=1$.
If $\langle \chi \rangle = x =0$ then the Hessian is (derivatives with respect to $h,\chi,\delta_0,\delta_1, \eta$): $$\left(
\begin{smallmatrix}
8 v^2 \lambda_\Sigma & 0 & 8 v w \lambda_{H \Sigma} & 0 & 0 \\
0 & -2 w^2\lambda_{H\Sigma} -4 v^2 \lambda \Sigma & 0 & 0 & 0 \\
8 v w \lambda_{H\Sigma} & 0 & 32 w^2 \lambda_H & 0 & 0 \\
0 & 0 & 0 & -2 v^2 \lambda_{H\Sigma} & 0 \\
0 & 0 & 0 & 0 & -8 \lambda_H w^2-2 v^2 \lambda_{H\Sigma} w^2 {\bf 1}_3 \\
\end{smallmatrix}
\right)$$ where the ${\bf 1}_3$ is the identity matrix in colour space, corresponding to the $\eta$-field. This Hessian is not positive definite so we disregard the possibility that $\langle \chi \rangle =0$.
If $x=\sqrt{\mu^2/2\lambda_\Sigma}$ then the Hessian is $$\left(\begin{smallmatrix}
8v^2 \lambda_\Sigma & 0 & 8vw \lambda_{H\Sigma} & 0 & 0 \\
0 & 4w^2 \lambda_{H \Sigma}+8v^2 \lambda_\Sigma & 0 & 0 & 0\\
8vw \lambda_{H\Sigma} & 0 & 32 w^2 \lambda_H & 0 & 0 \\
0 & 0 & 0 & w^2 \frac{\lambda_{H \Sigma}^2}{\lambda_\Sigma} & 0 \\
0 & 0 & 0 & 0 & w^2 \frac{ \lambda_{H\Sigma}^2 - 8 \lambda_H \lambda_\Sigma}{\lambda_\Sigma}{\bf 1}_3
\end{smallmatrix}\right)$$ which is positive-definite if $$\label{eq:positive-mass}
\lambda_{H\Sigma}^2 \geq 8 \lambda_H \lambda_\Sigma.$$ Note that this relation may hold only at high-energies. The masses for $\chi$, $\delta_1$ and $\eta$ are then readily found to be: $$\begin{aligned}
m_\chi^2 &= 4w^2 \lambda_{H \Sigma}+8v^2 \lambda_\Sigma,\\
m_{\delta_1}^2 &= w^2 \frac{\lambda_{H \Sigma}^2}{\lambda_\Sigma} ,\\
m_{\eta}^2 &=w^2 \frac{ \lambda_{H\Sigma}^2 - 8 \lambda_H \lambda_\Sigma}{\lambda_\Sigma} .\end{aligned}$$ Under the assumption that $v^2 \approx 10^2 \operatorname{GeV}, w^2 \approx 10^{11} \operatorname{GeV}$ we have $m_\chi^2 \approx 10^{11} \operatorname{GeV}$ and $m_{\delta_1}^2, m_{\eta} \approx 10^{11}\operatorname{GeV}$.
The (non-diagonal) $h$ and $\delta_0$ sector has mass eigenstates as in [@CC12]: $$\begin{gathered}
m_\pm^2 = 16 w^2\lambda_H +4 v^2 \lambda_\Sigma \\\pm 4\sqrt{16 w^{4} \lambda_H^2 + v^4 \lambda_\Sigma^2 + 4v^2 w^2 \left(\lambda_{H \Sigma}^2 - 2 \lambda_H \lambda_\Sigma \right)}\end{gathered}$$ Under the assumption that $v^2 \ll w^2$ we can expand the square root: $$\begin{aligned}
&4 \sqrt{ 16 \lambda_H^2 w^{4} \left(1 + \frac{\lambda_\Sigma^2}{\lambda_H^2}\frac{v^4}{w^{4}} + \frac{\lambda_{H \Sigma}^2 - 2 \lambda_H \lambda_\Sigma}{4\lambda_H^2} \frac{v^2}{w^2}\right)}\\ \nn
&\qquad\approx 16 \lambda_H w^2 \left(1 +\frac{\lambda_{H \Sigma}^2 - 2 \lambda_H \lambda_\Sigma}{8\lambda_H^2} \frac{v^2}{w^2}\right)
\\\nn
&\qquad= 16\lambda_H w^2 +\frac{2\lambda_{H \Sigma}^2}{\lambda_H} v^2
- 4\lambda_\Sigma v^2.\end{aligned}$$ Consequently, $$\begin{aligned}
m_+ &\approx 32 \lambda_H w^2 + 2 \frac{2\lambda_{H \Sigma}^2}{\lambda_H} v^2, \\
m_- &\approx 8 \lambda_\Sigma v^2 \left(1- \frac{\lambda_{H \Sigma}^2}{4\lambda_H \lambda_\Sigma} \right).\end{aligned}$$ which are of the order of $10^{11}$ and $10^2 \operatorname{GeV}$, respectively. This requires that we have at low energies $$\label{eq:positive-higgs-mass}
4\lambda_H \lambda_\Sigma \geq \lambda_{H \Sigma}^2,$$ which fully agrees with [@CC12] when we identify $\delta_0 \equiv \sigma$ and with the couplings related via $$\lambda_H = \frac14 \lambda_\sigma , \qquad \lambda_{H \Sigma} = \frac12 \lambda_{h\sigma}, \qquad \lambda_\Sigma = \frac14 \lambda_h$$ Note the tension between Equations and , calling for a careful study of the running of the couplings in order to guarantee positive mass eigenstates at their respective energies.
We have summarized the scalar particle content of the above model in Table \[table:part-cont\].
$$\begin{array}{c||ccc}
& U(1)_Y & SU(2)_L & SU(3) \\
\hline\hline
\begin{pmatrix}\phi_1^0 \\ \phi_1^+ \end{pmatrix} = \begin{pmatrix} \phi_{\dot 1}^1\\ \phi_{\dot 1}^2 \end{pmatrix} & 1 & 2 & 1 \\[4mm]
\begin{pmatrix}\phi_2^- \\ \phi_2^0 \end{pmatrix} = \begin{pmatrix} \phi_{\dot 2}^1\\ \phi_{\dot 2}^2 \end{pmatrix}& -1 & 2 & 1 \\[4mm]
\delta_0 & 0 & 1 & 1 \\
\delta_1 & -2 & 1 & 1 \\
\eta & -\frac 23& 1 & 3
\end{array}$$
In terms of the original scalar fields $\phi_{\dot a}^b$ and $\Delta_{\dot aI}$ the vevs are of the following form: $$\begin{aligned}
\begin{pmatrix} \phi_{\dot a}^b \end{pmatrix} &= \begin{pmatrix} v & 0 \\ 0 & \sqrt{\mu^2/2 \Lambda_\Sigma}\end{pmatrix}\\
\begin{pmatrix} \Delta_{\dot a I} \end{pmatrix} &= \begin{pmatrix} w & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix} .\end{aligned}$$ This shows that there are two scales of spontaneous symmetry breaking: at $10^{11}-10^{12} \operatorname{GeV}$ we have $$SU(2)_R \times SU(2)_L \times SU(4) \to U(1)_Y \times SU(2)_L \times SU(3)$$ and then at electroweak scale (both $v$ and $\mu$) we have $$U(1)_Y \times SU(2)_L \times SU(3) \to U(1)_Q \times SU(3)$$
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[^1]: The case $N=1$ was exploited successfully in [@DS11] for a noncommutative description of abelian gauge theories.
[^2]: Also other algebras that appear in the classification of irreducible geometries of KO-dimension have been considered in the literature: besides the case $N=4$ that we consider here the simplest case $N=1$ is relevant for the noncommutative geometric description of quantum electrodynamics [@DS11] and the case $N = 8$ leads to the ‘grand algebra’ of [@DLM14; @DLM14b].
[^3]: The case $k=3$ was ruled out by physical considerations [@DLM14].
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Inchan Ji, Sébastien Peirani, and Sukyoung K. Yi'
date: Accepted 2014 3 May
title: 'On the lifetime of merger features of equal-mass disk mergers'
---
Introduction
============
Since the seminal work of @TT72 illustrating the peculiar features of nearby galaxies, galaxy merger has been considered a fundamental phenomenon in understanding galaxy formation. The Lambda Cold Dark Matter ($\Lambda$CDM) model predicts the hierarchical galaxy formation which suggests that present-day galaxies are formed from successive accretions and mergers of smaller entities [@White78]. In this context, large early-type galaxies are believed to have experienced mergers between disk galaxies in the past [@Toomre77].
Theoretical perspectives of galaxy interactions help us understand galaxy formation and evolution. In particular, star formation histories and the dynamical properties of galaxy mergers have been investigated through numerical simulations [e.g. @Gerhard81; @Barnes91; @Hernquist92; @Naab03; @Cox06a; @Cox06b; @DiMatteo07; @Peirani10]. The numerical simulations suggest that the properties of merger remnants, such as their isophotal shapes, surface density profiles, and the ratio between local rotation to velocity dispersion (v/$\sigma$), are consistent with those of observed early-type galaxies. In addition, dissipative mergers can account for the central concentration of starbursts between merger remnants [@Barnes91; @Mihos94; @Springel00].
However, the gap between observation and theoretical study is still considerable. The properties of galaxies beyond their internal structures, e.g., magnitudes and colors, have rarely been studied primarily due to the difficulty of modeling dust effects in the interstellar medium of galaxies. Various methods have recently been developed to overcome this issue.
One method is the use of Monte Carlo radiative transfers assuming idealized galaxy models or using distributions of gas and stars from numerical simulations [@Silva98; @Jonsson06]. Although this method provides detailed and extensive panchromatic data for galaxies, it requires high computational costs. The other method is to directly assign spectral data to a single stellar population using population synthesis models. This method is effective because of its simplicity and applicability. Several studies have examined the observable properties of merger remnants by assuming the total amount of dust extinction from empirical data [@Kaviraj09; @Gabor11]. Therefore, their approaches are well-suited to studying the integrated magnitudes and colors of each galaxy, but not to studying regional differences.
The data provided by large surveys and improvements in numerical modeling give us an opportunity to study all physical processes that occur during the interactions of galaxies. As an illustration, large surveys such as the CfA2 Redshift Survey, the Two-Degree Field (2dF) Galaxy Redshift Survey, the Deep Extragalactic Evolutionary Probe (DEEP), and the Sloan Digital Sky Survey (SDSS) have collected detailed galaxy properties, while hydrodynamic simulations and population synthesis models have been employed to produce the observable properties of merger remnants. Recent studies have tried to compare the photometric quantities derived from numerical simulations to data from large surveys [@Springel05d; @Gabor12].
Most massive galaxies are expected to undergo a series of merger events [@Stewart08]. It is then necessary to constrain timescales such as first passage, maximum separation, and final coalescence when we predict the physical properties of galaxies. Observations help identifying the timescales using angular separation, line-of-sight radial velocity, and correlation functions [@Patton00; @Barton00; @Lin04; @dePropris05; @Bell06; @Li08]. Recent studies have shown that the timescales before the final coalescence can be formulated analytically based on numerical simulations [@Boylan-Kolchin08; @Jiang08]. However, only a few studies have measured timescales beyond the final coalescence. One such timescale is the merger-feature time, the moment when faint and disturbed features of merger remnants (post-merger features) disappear. @Lotz08 assumed that the merger-feature time (“post-merger” stage in the literature) ends 1 Gyr after the final coalescence. However, the merger-feature time obviously depends on observational conditions such as surface brightness limits.
Figure \[merger\_feature\] shows three early-type galaxies in Abell 2670 observed from the SDSS ($\mu_{\rm limit} \sim$ 25 mag arcsec$^{-2}$) and from the CTIO $r'$ band ($\mu_{\rm limit} =$ 28 mag arcsec$^{-2}$) by @Sheen12. The images from the SDSS show only the central regions of the galaxies and suggest elliptical-like galaxies. However, the deeper images from the CTIO observations clearly show faint structures around the galaxies and thus characterize more accurately the dynamical and relaxation state of the galaxies. Therefore, this suggests that the merger-feature time could be estimated more accurately with the deeper images.
In this work, our aim is to study the merger-feature time after galaxy merger while considering both reasonable dynamical evolutions of galaxy mergers and dust attenuation. We produce a set of merger events using N-body/hydrodynamic simulations from which we extract the dust-attenuated spectra of galaxies using both @BC03 and @Calzetti00. This modeling allows us to construct mock images of galaxies mimicking observational environments to be compared with observational data.
This paper is organized as follows: the numerical modeling is summarized in Section \[section\_numerical\], while the methodology used to extract observable properties is described in Section \[analysis\]. In Section \[results\], we compare the merger-feature times of merger remnants. Lastly, we discuss the results in Section \[discussion\].
![The SDSS images (top) and CTIO $r'$ band images (bottom) of galaxies in Abell 2670. The images are courtesy of Yun-Kyeong Sheen. The galaxies in each column are the same object. The surface bright limits in the SDSS and CTIO observations are $\mu_{\rm limit} \sim$ 25 and 28 mag arcsec$^{-2}$, respectively. One can see that the deep images reveal faint tidal features around the galaxies.[]{data-label="merger_feature"}](f1.jpg){width="45.00000%"}
Numerical modeling {#section_numerical}
===================
Galaxy modeling and simulations
-------------------------------
The numerical methodology used in this paper is described in detail in @Peirani09 to which we refer the reader for more information. For the sake of clarity, we summarize the major steps below.
Our study is focused on 1:1 mass-ratio mergers where the final coalescence time is the shortest in the large parameter space [@Boylan-Kolchin08 for example]. This fact is also useful for measuring various timescales as we will discuss below. Galaxies are created following the prescription described by @Springel05b. We use a compound galaxy model consisting of a spherical dark matter halo, a rotationally-supported disk (of gas and star), and a bulge, with independent parameters describing each of the structural components. The distribution of the dark matter halo follows a Hernquist profile [@Hernquist90], with a concentration parameter of $C = 14$ in both galaxies following @Dolag04. We also assume a thin disk structure of gas and stars with an exponential surface density. The scale length of disk $R_{\rm d}$ is determined following @Graham08, and the bulge distribution is assumed to be spherical while also following a Hernquist profile. We adopt a conventionally determined disk scale height $z_0$, and a bulge scale length $b$ as $b = z_0 = 0.2~R_{\rm d}$. The baryon fraction derived from @Komatsu11 is used. The gas fraction of our model galaxies is consistent with the estimation in @BM98 and @Kannappan04. The maximum speed of the rotation curve is about 150 km/s as suggested by the baryonic Tully-Fisher relation [@McGaugh05]. We limit our study to Sa and Sb model galaxies. To consider unequal-mass mergers, we constructed two satellite galaxies by changing the total mass of the Sb galaxy while fixing the other parameters. They are one-third and one-sixth of the Sb galaxy in total mass and are labelled as ${\rm Sb_{3}}$, ${\rm Sb_{6}}$, and ${\rm Sb_{10}}$, respectively. Table \[initgal\] summarizes the parameters of each galaxy.
Sa Sb ${\rm Sb}_3$ ${\rm Sb}_6$ ${\rm Sb}_{10}$
---------------------------------------------------------- ------- ------ -------------- -------------- -----------------
Total mass, $M_{\rm vir}~({\rm 10^{10} M}_{\odot})$ 17.0 17.0 5.7 2.8 1.7
DM mass, $M_{\rm DM}~({\rm 10^{10} M}_{\odot})$ 14.1 14.1 4.7 2.4 1.4
Stellar disk mass, $M_{\rm d}~({\rm 10^{10} M}_{\odot})$ 1.61 1.81 0.60 0.30 0.18
Stellar bulge mass, $M_{\rm b}~({\rm 10^{9} M}_{\odot})$ 10.74 4.52 1.51 0.75 0.45
Gas mass, $M_{\rm gas}~({\rm 10^{9} M}_{\odot})$ 1.41 5.65 1.88 0.94 0.57
B/T ratio 0.4 0.2 0.2 0.2 0.2
Spin parameter, $\lambda$ 0.05 0.05 0.05 0.05 0.05
Disk scale length, $R_{\rm d}~{\rm (kpc)}$ 2.58 3.29 2.28 1.81 1.52
\[initgal\]
The simulations were performed using the publicly-available code GADGET2 [@Springel05a] with added prescriptions for gas cooling, star formation, and feedback from Type Ia and II supernovae (SNe). Note that gas particles with $T< 2\times 10^4~{\rm K}$, number density $n > 0.1~{\rm cm}^{-3}$, overdensity $\Delta \rho_{gas}> 100$, and ${\bf \nabla . \upsilon} <0$ form stars according to the standard star formation prescription: $d\rho_*/dt = c_* \rho_{\rm gas}/t_{\rm dyn}$, where $\rho_*$ refers to the stellar density, $t_{\rm dyn}$ is the local dynamical timescale of gas, and $c_*$ is the star formation efficiency. More than a million particles were used for each run. Consequently, the particle masses are $m_{\rm DM} = 4 \times
10^{5}~{\rm M}_{\odot}$, $m_{\rm disk} =
m_{\rm gas} = 5 \times 10^{4}~{\rm M}_{\odot}$, and $m_{\rm bulge} =
1.5 \times 10^{5} ~{\rm M}_{\odot}$ for dark matter, disk, gas, and bulge, respectively. The gravitational softening lengths are $\epsilon_{\rm DM}$ = 0.1 kpc, $\epsilon_{\rm disk} = \epsilon_{\rm gas}$ = 0.2 kpc, and $\epsilon_{\rm bulge}$ = 0.1 kpc.
Simulation$^{\mathrm{a}}$ Pair$^{\mathrm{b}}$ $e^{\mathrm{{d}}}$ $R_{\rm peri}^{\mathrm{d}}$ $i_{\rm host}^{\mathrm{e}}$
--------------------------- --------------------- --------------------- ----------------------------- -----------------------------
SbSb0p Sb-Sb 1 5 $0$
SbSb45p Sb-Sb 1 5 $45$
SbSb90p Sb-Sb 1 5 $90$
SbSb135p Sb-Sb 1 5 $135$
SbSb180p Sb-Sb 1 5 $180$
SaSb45p Sa-Sb 1 5 $45$
SaSb90p Sa-Sb 1 5 $90$
SaSb135p Sa-Sb 1 5 $135$
SbSb45r Sb-Sb 1 0 $45$
SbSb45p+ Sb-Sb 1 10 $45$
SbSb45e Sb-Sb 0.95 5 $45$
SbSb45h Sb-Sb 1.05 5 $45$
${\rm SbSb_{3}45h}$ Sb-${\rm Sb_3}$ 1.05 5 $45$
${\rm SbSb_{6}45h}$ Sb-${\rm Sb_6}$ 1.05 5 $45$
SbSb45hC Sb-Sb 0.95$^{\mathrm{f}}$ 472$^{\mathrm{f}}$ -
\[fiducial\]
: Initial orbital configurations
Types of galaxies, initial encounter angle, orbital shape (“p” for parabolic, “r” for radial, “e” for elliptical, “h” for hyperbolic, and “+” for larger pericentric distance), and additional cluster potential (“C” for cluster potential) are marked in the simulation name.
Host-companion pair.
Eccentricity of orbit.
Pericentric distance (kpc).
Inclination of host galaxy ($^{\circ}$).
They are the orbital parameters with respect to a cluster potential.
The stability of our model galaxies is measured by investigating the star formation rate (SFR) of each galaxy in isolation as shown in Figure \[sfr\_consti\]. We found SFRs of, on average, $\sim 0.22$ M$_{\odot}$yr$^{-1}$ and $\sim 1.3~{\rm M}_{\odot}~$yr$^{-1}$ over 1 Gyr for the Sa and Sb models, respectively. Note that the SFRs are consistent with the observational sample of @James04.
The difference between the SFR amplitudes of the two model galaxies is due to the difference between the initial masses of gas in the disks (see Table \[initgal\]). The gradual decay of SFR is as expected since the available gas is progressively converted into stars. External accretion of gas could be considered to reproduce the constant average SFRs found in observations of spiral galaxies in the middle of the Hubble sequence [@Kennicutt83; @Kennicutt94] but is not included in our study to simplify our simulation.
As shown in Figure \[sfr\_consti\], initial bursts of new stars occur as a result of gas compression into density waves early in the simulations. However, the initial bursts are suppressed after supernova feedbacks are exerted on gas [see @DiMatteo07]. Each model galaxy is quite stable as no spurious effects are seen in the evolution of the SFRs due to specific choices of gravitational smoothing or the phase of the initial condensation of gas.
![Star formation rate for isolated Sa and Sb models at intervals of 10 Myr to 1 Gyr. The Sb model with a higher gas content shows star formation rates roughly an order of magnitude higher than the Sa model. Initial bursts of new stars in both models are contributed from the gas compression of gas into density waves and the absence of supernova feedback from young stellar populations.[]{data-label="sfr_consti"}](f2.jpg){width="50.00000%"}
![Overview of merger timescales: first passage time ($t_{\rm FP}$) when a companion or satellite galaxy first passes the closest distance, maximum separation time ($t_{\rm MS}$) after the first passage, final coalescence time ($t_{\rm FC}$) when two galaxies finally merge, and merger-feature time ($t_{\rm MF}$) when post-merger features disappear given a surface brightness limit. []{data-label="merger_step"}](f3.jpg){width="45.00000%"}
Definition of Merger Timescales {#time_scales}
-------------------------------
Figure \[merger\_step\] shows characteristic timescales of a merger event. Each merger event is characterized by several successive and distinct phases. We identify several critical timescales that show distinctive features during the merger: the first passage time ($t_{\rm FP}$) when the companion galaxy first passes the closest distance, the maximum separation time ($t_{\rm MS}$) after the first passage time, and the final coalescence time ($t_{\rm FC}$) when the two galaxies finally merge. In addition, we adopt the merger-feature time ($t_{\rm MF}$) when the merger remnant hides post-merger features viewed perpendicular to and parallel to the orbital plane. We consider two merger-feature times by visual inspection based on two surface brightness limits in the SDSS $r$ band: $\mu_{\rm r,limit}$ = 25 ($t_{\rm MF,25}$) and 28 mag arcsec$^{-2}$ ($t_{\rm MF,28}$). The $\mu_{\rm r,limit}$ = 25 mag arcsec$^{-2}$ is comparable to the observational limit of the SDSS. The characteristic times are listed in Table \[mu28\_time\].
Initial orbital configurations
------------------------------
In all simulations, the initial separation between the host and the companion is 90 kpc, the virial radius of the host galaxy. The pericentric distances adopted here are consistent with those of dark matter haloes in cosmological simulations [see @Khochfar06]. The definition of inclination with respect to the orbital plane is adopted from @TT72.
Table \[fiducial\] summarizes all information related to the specific orbital configuration of each merger. For the convenience of comparison between the simulations, we first constructed five fiducial mergers (top five mergers in Table \[fiducial\]). The progenitor galaxies are Sb models. The eccentricity is e = 1, and the pericentric distance is $R_{\rm peri}$ = 5 kpc. The inclinations of the host galaxies are set to $i_{\rm host} = 0^{\circ}$ (prograde)$,~45^{\circ},~90^{\circ},~135^{\circ}$, and $180^{\circ}$ (retrograde).
Modifying the fiducial mergers, we constructed ten additional mergers. First, the host galaxy was altered from an Sb to an Sa galaxy, and the inclinations of the host galaxies are $i_{\rm host} =
45^{\circ},~90^{\circ}$, and $135^{\circ}$. Second, the pericentric distance of the SbSb45p merger was changed so that $R_{\rm peri}$ = 0 and 10 kpc. Third, the orbital eccentricity of the SbSb45p merger was varied to consider elliptical and hyperbolic orbits as well: we chose e = 0.95 and 1.05. Forth, we changed the total mass of the companion galaxy in the SbSb45h merger so that mass ratios are 3:1, 6:1, and 10:1. We chose the SbSb45h merger for comparison because its merger timescales are relatively shorter than those in other cases and so better suited for demonstrating the effects of merger conditions in question.
Lastly, we placed a merger remnant in a rigid dark matter potential following a Hernquist profile with a Virgo-like mass of ${M_{\rm vir}}
\simeq 4.2 \times {\rm 10^{14}~M}_{\odot}$, a concentration parameter of $C = 9$, and a scale length of $a = 236$ kpc. We chose the merger remnant of the SbSb45h merger at the final coalescence time and let the merger remnant orbit in the cluster potential; the initial distance from the center is 1200 kpc which is the virial radius of the cluster halo potential, and the pericentric distance is twice the scale length of the potential. The choice of the merger in a cluster halo environment is largely arbitrary, and we would only like to check out the environmental effect in rough senses. In all, we simulated and analyzed fifteen merger events considering different host-companion pairs, inclinations, and orbits. All equal-mass mergers are simulated until 8.5 Gyr corresponding to $\sim$ 8 times the dynamical timescale at the virial radius of the host galaxy ($t_{\rm dyn}\sim$ 1.1 Gyr). Since unequal-mass mergers evolve more slowly than equal-mass mergers, they are run for 13 Gyr, instead.
Star Formation
--------------
![Evolution of star formation surface density versus gas column density for the Sa model (filled triangle), Sb model (filled square), and SbSb45p merger (filled circle). Each symbol corresponds to the center of a galaxy at 5 and 20 Myr time intervals for each isolated progenitor (Sa and Sb) and the SbSb45p merger. For the SbSb45p merger, the merger timescales associated with star formation are labeled with arrows: i.e., the first passage time ($t_{\rm FP}$), the maximum separation time ($t_{\rm MS}$), and the final coalescence time ($t_{\rm FC}$). The solid line is the best fit given by @Kennicutt98, and the dashed line represents a factor of 7 range which covers all observations in the same literature.[]{data-label="Kennicutt"}](f4.jpg){width="50.00000%"}
As mentioned above, star formation in this study is intended to follow the empirical relation described in @Kennicutt98. Thus, to investigate the validity of our models, we also test the time evolution of the surface density of the gas and the star formation surface density averaged in the central region of each galaxy.
Figure \[Kennicutt\] shows the evolution of our model galaxies in isolation and one merger process (SbSb45p merger). Each symbol denotes the properties of a galaxy at an epoch given by the color key. In the isolated model galaxies, the surface density of gas and the star formation surface density decrease continuously starting from (log $\Sigma_{\rm gas}$, log $\Sigma_{\rm SFR}$) $\sim$ (1.2 ${\rm M_\odot~pc^{-2}}$, -2 ${\rm M_\odot~yr^{-1}~pc^{-2}}$) and (log $\Sigma_{\rm gas}$, log $\Sigma_{\rm SFR}$) $\sim$ (1.7 ${\rm M_\odot~pc^{-2}}$, -1.3 ${\rm M_\odot~yr^{-1}~pc^{-2}}$) for the Sa and Sb models, respectively. Theses values closely track the empirical relation in @Kennicutt98.
The SbSb45p merger, on the other hand, shows that the surface density of gas and the star formation surface density increase as the progenitor galaxies interact and merge with each other. During the merger-induced starburst, the gas surface density and the star formation surface density reach (log $\Sigma_{\rm gas}$, log $\Sigma_{\rm SFR}$) $\sim$ (2.3 ${\rm M_\odot~pc^{-2}}$, -0.2 ${\rm M_\odot~yr^{-1}~pc^{-2}}$) which are comparable to the “starburst” galaxies with (log $\Sigma_{\rm gas}$, log $\Sigma_{\rm SFR}$) $\gtrsim$ (2 ${\rm M_\odot~pc^{-2}}$, -1 ${\rm M_\odot~yr^{-1}~pc^{-2}}$) in @Kennicutt98. Figure \[Kennicutt\] also indicates that the SbSb45p merger evolves within the observational envelope. In addition, we confirm that there is good accordance between all merger simulations and the observational envelope. Therefore, we conclude that the model galaxies and the mergers discussed in this study closely follow the empirical star formation law, hence passing a sanity check regarding the sensibility of the star formation prescription in our models.
Analysis
========
Modeling 2D Synthetic Images
----------------------------
In this section, we describe our ray-tracing algorithm which allows the extraction of observable quantities from snapshots of merger simulations. This method requires information about the positions of all particles, the ages of stellar components, gas properties as well as chemical composition.
First, we constructed 2D regular Cartesian grids using the positional information of the particles. The spectral energy distribution (SED) is assigned to all stellar particles (both disk and bulge) following @BC03. Since our aim is to study the time evolution of merging galaxies at low redshift, potentially displaying resolved features, we assume a solar metallicity and 2 Gyr-old stellar components following @Gallazzi05. It is worth mentioning that dust has a critical role in diminishing the flux emitted by stars as the wavelength becomes shorter. Although the gas column density determines the strength of dust attenuation, the line-of-sight distribution of gas and stars should be considered for a more realistic approach. We use the empirical fitting formula from @Bohlin78 and the dust extinction curve provided by @Calzetti00 to calculate the amount of dust extinction from hydrogen column density. In their studies, the dust-to-gas ratios convolved with metal abundances are similar to those of Milky Way [@Alton98; @Davies99].
To construct a mock image, we determined the field-of-view (FOV), distance to a galaxy, and plate scale. We assume a luminosity distance, ${\rm d_L} =$ 100 Mpc, which represents a typical distance to nearby galaxies in the widely-used large-scale surveys such as the SDSS. Also, we assume a plate scale of 0.5 arcsecond per pixel which is comparable to the SDSS plate scale ($\sim 0.4$ arcsecond per pixel), and a FOV of 6.8 arcmin which can physically cover the merging area. In this case, 0.5 arcsecond per pixel and 6.8 arcmin correspond to 0.24 kpc and 200 kpc, respectively. In the following sections, we analyze the observational properties of galaxy mergers using the SDSS filter system.
Convergence tests
-----------------
Before presenting our main results, it is necessary to conduct a series of convergence tests to estimate their robustness.
### Number of particles
First, we check to what extent the results are affected by the mass resolution. For this purpose, we ran the SaSb45p merger three times while varying the total number of particles used to model each galaxy/halo component, decreasing the number by a factor of five and increasing by a factor two. We changed the softening lengths to be inversely proportional to the number of particles, $N$. We used $N^{\rm -1/3}$ relation in this study [see @Merritt96]. The gravitational softening lengths in the lower-resolution SaSb45p are $\epsilon_{\rm DM}$ = 0.17 kpc, $\epsilon_{\rm disk} = \epsilon_{\rm gas}$ = 0.34 kpc, and $\epsilon_{\rm bulge}$ = 0.71 kpc, while those in the higher-resolution SaSb45p are $\epsilon_{\rm DM}$ = 0.079 kpc, $\epsilon_{\rm disk} =
\epsilon_{\rm gas}$ = 0.16 kpc, and $\epsilon_{\rm bulge}$ = 0.079 kpc. Figure \[resolution\] shows the SFR and the time evolution of the amount of newly-formed stars for each simulation. Note that, despite slight differences, the SFRs all follow the same trend. Moreover, the difference between the final masses of new stars at $t=3$ Gyr is less than six percent, which suggests that our results are stable.
![Star formation rate and cumulative amount of newly-formed stars for various resolutions of the SaSb45p merger: the SaSb45p merger (black solid line), five times fewer particles than the SaSb45p merger (red dashed line), and two times more particles than the SaSb45p merger (blue dashed dotted line). []{data-label="resolution"}](f5.jpg){width="50.00000%"}
![Amount of dust attenuation in $B$ bands (top) and $J$ bands (bottom) with respect to the ratio between the major (a) and minor axis (b) of the Sb model galaxy. Each symbol represents the attenuated magnitude in each model galaxy with a specific plate grid resolution. The red dashed line in the top panel is the fitting function obtained from @Tully98, and the blue dashed line in the bottom panel is the bilinear fit obtained from [@Masters03].[]{data-label="BJtest"}](f6.jpg){width="45.00000%"}
### Plate scale
It is also important to determine the optimal resolution at which we should perform the ray-tracing for the best accuracy in our predictions. Increased resolution requires more memory space and a longer computation time. In addition, finding a reasonable grid scale is the key to success in ray-tracing since, in this study, the physical size of the gas particle is ignored while we are able to assume stellar components as point sources.
Several studies on the internal extinction of spiral galaxies have been performed in the past decades [@Giovanelli94; @Tully98; @Xilouris99; @Masters03]. The amount of dust attenuation is quantified through modeling an isolated galaxy. Methods for this computation can be put into two categories: analytical approximations [@Byun94; @Baes01; @Tuffs04] and Monte Carlo calculations [@Bianchi96; @Matthews01; @Pierini04]. However, models presented in these previous studies do not include galactic substructures such as spiral arms and clumps of gas which are expected to affect the total amount of dust attenuation [@Corradi96; @Rocha08]. Therefore, taking into account the presence of these substructures in the ray-tracing would be a necessary step toward realism.
We estimated the amount of dust attenuation using isolated late-type galaxies where the dependences of inclinations of galaxies are clear [see e.g. @Byun94]. Figure \[BJtest\] shows the amount of dust extinction in the $B$ and $J$ bands in the Sb model galaxy as a function of inclination and grid resolution. In particular, we compare the extinction slopes for six different plate scales as a function of axial ratio assuming the same luminosity distance.
First, we quantified the amount of extinction in the $B$ band with respect to the grid resolution and the ratio between the major (a) and minor axes (b) of the Sb model galaxy. We found that, apart from the grid with the lowest resolution (5 arcsecond per pixel), the other grid resolutions (0.3 to 3 arcsecond per pixel) led to similar variations in dust attenuation in the $B$ bands, $\Delta M_{\rm B}$ with respect to the axial ratio log$(a/b)$. When compared to the results from @Tully98, who measured $B$ band extinctions of late-type galaxies with various inclinations, we found that our choice of a grid resolution of 0.5 arcsecond per pixel agreed well with observations. Similar trends were found in the $J$ band. @Masters03 argued that dust attenuation is not a linear function of the axial ratio, but is bilinear or has a quadratic form. @Masters03 also suggest a bilinear equation of dust attenuation as a function of inclination compared to the simple linear formulation in @Tully98.
Figure \[BJtest\] shows that high grid resolutions lead to a nice agreement with the observations. Low grid resolutions poorly resolve the structures which result in deviation from the fitting formula suggested by the observations. In the following, we then computed fitting functions by adopting grids of 0.5 arcseconds per pixel. We fit the $B$ and $J$ band extinction with a quadratic curve defined as follows:
$${\Delta M_{\rm X}} = a_0 + a_1 \log (a / b) + a_2 [\log (a / b)]^2
\label{Jband}$$
where $a_0 = 0.01 \pm 0.01$, $a_1 = -0.22 \pm 0.14$, and $a_2 = 2.96
\pm 0.21$ for the $B$ band, $a_0 = 0.00 \pm 0.01$, $a_1 = -0.23 \pm 0.08$, $a_2 = 1.49 \pm 0.12$ for the $J$ band, and $a/b$ is the axial ratio between the semimajor axis and the semiminor axis of a galaxy.
![ Mock image and surface column density of SbSb45h merger at three times the final coalescence time ($t_{\rm FC}$ = 1.57 Gyr). Top panels show mock images viewed perpendicular to (face-on) and parallel to (edge-on) orbital plane of a companion galaxy in left-hand column and in right-hand column, respectively. Color scheme is surface brightness in the SDSS $r$ band assuming luminosity distance, ${\rm d_L}$ = 100 Mpc and observational limit, $\mu_{\rm limit}$ = 28 mag arcsec$^{-2}$. Bottom panels show surface column density of stellar and gaseous mass viewed perpendicular to (face-on) and parallel to (edge-on) orbital plane of companion galaxy in left-hand column and in right-hand column, respectively. Color scheme represents column density of stellar mass. Contour is gaseous surface density, and its values are labeled in logarithmic scale. []{data-label="mu_SD"}](f7.jpg){width="45.00000%"}
{width="100.00000%"}
{width="100.00000%"}
. \[mock\_3:1\]
{width="80.00000%"}
{width="100.00000%"}
Results
=======
Detectability of tidal features
-------------------------------
For collisionless particles, the dynamical timescale in the central region of a merger remnant is relatively short resulted from the violent mixture of the particles [@Mihos96; @Springel00; @Cox06b]. However, the outer region has a larger dynamical timescale which leads to long-lasting post-merger features such as shells, loops, and long tidal arms. For gas particles, they undergo shocks or dissipation of energy and angular momentum during the interaction. The gravitational torques resulted from perturbed structures induce gas to flow inwards [see also @Negroponte83]. The gas inflows end up with the formation of a rotating rings or blobs.
Figure \[mu\_SD\] shows the mock images in the SDSS $r$ band (top panels) and stellar and gaseous column density (bottom panels) of the SbSb45h merger at three times the final coalescence time ($t_{\rm FC}$ = 1.57 Gyr). Although a few giga years elapsed after the final coalescence, post-merger features are clearly visible in the column density. Substructures of merging galaxies are stretched out along with the orbital plane forming characteristic post-merger features such as shells and loops. Although the position and shape of the features are dependent upon a viewing angle, they are clearly identifiable.
However, some of the post-merger features are invisible in mock images (see Figure \[mu\_SD\], top panels). This is mainly because the features have low surface column density. In our simulations, stellar surface density lower than $\sim$ 2 ${\rm M_{\odot}~pc^{-2}}$ exceeds the surface brightness limit of $\mu_{\rm limit}$ = 28 mag arcsec$^{-2}$.
Merger-feature time of equal-mass mergers
-----------------------------------------
Figure \[mock\_1:1\] shows the mock images of the SbSb45h merger at different characteristic times and according to the observational limits of the SDSS $r$ band. The images using the SDSS $\mu_{\rm limit}$ reveal only the central part of each galaxy, resulting in the merger remnant reaching a spheroidal shape in a relatively short period of time. For example, the faint features of the SbSb45h merger become invisible already at twice the final coalescence time ($t_{\rm FC}~\sim$ 1.57 Gyr) under the SDSS conditions. However, the mock images with a deeper $\mu_{\rm limit}$ indicate that the merger remnant exhibits extended faint structures for more than two times the final coalescence time. This confirms the fact that measuring the merger-feature time of merger remnants strongly depends on $\mu_{\rm limit}$.
To compare two merger-feature times, we have estimated the ratio of the merger-feature time to the final coalescence time, ${t_{\rm MF} / t_{\rm FC}}$ (see Table \[mu28\_time\]). Note that in contrast to the merger-feature time based on $\mu_{\rm limit}$ = 25 mag arcsec$^{-2}$, $t_{\rm MF,25}$, not all merger-feature times based on $\mu_{\rm limit}$ = 28 mag arcsec$^{-2}$, $t_{\rm MF,28}$ could be estimated since faint features survive through the end of the simulation. Table \[mu28\_time\] shows that ${t_{\rm MF,25} / t_{\rm FC}}$ is always lower than three. The faint features in all merger simulations disappear by 2.16 $\pm$ 0.80 times the final coalescence time, or 1.38 $\pm$ 0.88 Gyr after the final coalescence, which is comparable to the finding of @Lotz08. However, ${t_{\rm MF,28} / t_{\rm FC}}$ is approximately twice greater than ${t_{\rm MF,25} / t_{\rm FC}}$ and exceeds three for all mergers.
We also study how merger scenarios affect ${t_{\rm MF} / t_{\rm FC}}$. The relaxation timescale of a merger remnant depends not only on orbital configurations [@Hernandez04] but also on the density of the merger remnant [@Conselice06]. Therefore, the relaxation timescales of the inner and outer regions of merger remnants are different, and the timescales are related to ${t_{\rm MF,25} / t_{\rm FC}}$ or ${t_{\rm MF,28} / t_{\rm FC}}$. First, the prograde-prograde merger takes longer to have its substructures relaxed and thus to hide post-merger features than the retrograde-prograde merger does. For example, one can see in Table \[mu28\_time\] that the SbSb0p merger shows a larger value of ${t_{\rm MF,28} / t_{\rm FC}}$ than the SbSb180p merger does, that is, 6.22 $\pm$ 0.62 vs. 5.25 $\pm$ 0.41, respectively. [^1] Second, mergers having a smaller orbital angular momentum (SbSb45r and SbSb45e mergers) tend to have shorter merging times ($t_{\rm FC}$, Column 4 in Table \[mu28\_time\]) and longer merger-feature times ($t_{\rm MF}$, Column 6 in Table \[mu28\_time\]) and hence resulting in larger values of ${t_{\rm MF} / t_{\rm FC}}$ (Column 8 in Table \[mu28\_time\]) than the larger-orbital-angular-momentum mergers (SbSb45p+ and SbSb45h). We can perhaps interpret this as follows. Mergers with low orbital angular momentum (for example, direct head-on collisions) causes a rapid relaxation in the central region of the merger remnant. This makes the central region virtually act like a point source. The outer region of the merger remnant feels more loose and displays merger features for a longer period of time as a result. The essence in this interpretation is to realize that the merger timescale is determined by the structure of the central parts of the merger remnant whereas the merger feature timescale for deep imaging conditions is derived from the outskirts. Lastly, the bulge-to-total mass ratio (B/T) affects the relaxation timescales of at least the inner regions of the merger remnants. For this reason, all values of ${t_{\rm MF,25} / t_{\rm FC}}$ for the SaSb mergers are smaller than for the SbSb mergers.
![ Surface column density of SbSb45hC merger at three times the final coalescence time ($t_{\rm FC}$ = 1.57 Gyr) viewed perpendicular to (face-on) and parallel to (edge-on) orbital plane of a companion galaxy in left-hand column and in right-hand column, respectively. Color scheme represents column density of stellar mass. Contour is gaseous surface density, and its values are labeled in logarithmic scale.[]{data-label="SD"}](f12.jpg){width="45.00000%"}
![Same as Figure \[mock\_1:1\], but for SbSb45hC merger. Note that the evolution of the SbSb45hC merger before the final coalescence time is as the same as the SbSb45h merger.[]{data-label="mu_limits4"}](f13.jpg){width="40.00000%"}
[lccccccc]{}
Simulation & ${t_{\rm FP}}$ (Gyr)$^{\mathrm{a}}$ & ${t_{\rm MS}}$ (Gyr)$^{\mathrm{b}}$ & ${t_{\rm FC}}$ (Gyr)$^{\mathrm{c}}$& ${t_{\rm MF,25}}$ (Gyr)$^{\mathrm{d}}$ & ${t_{\rm MF,28}}$ (Gyr)$^{\mathrm{e}}$& ${t_{\rm MF,25} / t_{\rm FC}}^{\mathrm{f}}$& ${t_{\rm MF,28} / t_{\rm FC}}^{\mathrm{g}}$\
\
\
SbSb0p &0.41 &0.71 &1.07 & 2.43 $\pm$ 0.34 & 6.65 $\pm$ 0.66 & 2.27 $\pm$ 0.32 & 6.22 $\pm$ 0.62\
SbSb45p &0.41 &0.65 &1.07 &2.31 $\pm$ 0.15 & 7.52 $\pm$ 0.74 & 2.16 $\pm$ 0.14 &7.03 $\pm$ 0.69\
SbSb90p &0.41 &0.76 &1.30 &3.37 $\pm$ 0.15 & 5.70 $\pm$ 0.45 & 2.59 $\pm$ 0.11 & 4.38 $\pm$ 0.34\
SbSb135p &0.41 &0.60 &1.13 &2.44 $\pm$ 0.12 & 6.13 $\pm$ 0.84 & 2.16 $\pm$ 0.10 & 5.43 $\pm$ 0.84\
SbSb180p &0.41 &0.65 &1.03 &2.10 $\pm$ 0.24 & 5.41 $\pm$ 0.42 & 2.04 $\pm$ 0.24 & 5.25 $\pm$ 0.41\
SaSb45p &0.41 &0.67 &1.09 &2.40 $\pm$ 0.33 & 5.63 $\pm$ 0.65 & 2.20 $\pm$ 0.30 & 5.17 $\pm$ 0.59\
SaSb90p &0.41 &0.90 &1.41 &2.59 $\pm$ 0.22 & 4.82 $\pm$ 0.25 & 1.83 $\pm$ 0.16 & 3.42 $\pm$ 0.18\
SaSb135p &0.41 &0.63 &1.08 &2.26 $\pm$ 0.20 & 4.99 $\pm$ 0.24 & 2.09 $\pm$ 0.19 & 4.62 $\pm$ 0.22\
SbSb45r &0.38 &0.51 &0.73 &2.25 $\pm$ 0.26 & 7.01 $\pm$ 0.26 & 3.08 $\pm$ 0.35 & 9.60 $\pm$ 0.35\
SbSb45p+ &0.43 &1.00 &1.93 &3.04 $\pm$ 0.18 & 6.69 $\pm$ 0.32 & 1.58 $\pm$ 0.09 & 3.47 $\pm$ 0.17\
SbSb45e &0.49 &0.61 &0.84 &2.68 $\pm$ 0.32 & $> 8.50^{\mathrm{h}}$ & 3.19 $\pm$ 0.38 & $> 10.12^{\mathrm{h}}$\
SbSb45h &0.35 &0.80 &1.57 &2.99 $\pm$ 0.20 & 5.60 $\pm$ 0.47 & 1.90 $\pm$ 0.13 & 3.57 $\pm$ 0.30\
Mean &0.41$\pm$ 0.03 &0.71 $\pm$ 0.14 &1.19 $\pm$ 0.33 &2.57 $\pm$ 0.82 &6.22 $\pm$ 1.73 & 2.16 $\pm$ 0.80 & 5.24 $\pm$ 1.46\
\
\
${\rm SbSb_{3}45h}$ &0.44 &1.11 &2.13 &4.59 $\pm$ 0.17 & 7.26 $\pm$ 0.66 & 2.15 $\pm$ 0.08 & 3.41 $\pm$ 0.31\
${\rm SbSb_{6}45h}$ &0.47 &1.48 &3.75 &5.80 $\pm$ 0.14 & 8.30 $\pm$ 0.39 & 1.55 $\pm$ 0.04 & 2.21 $\pm$ 0.11\
${\rm SbSb_{10}45h}$ &0.48 &2.8 &12.9 &- &- &- &-\
Mean$^{\mathrm{i}}$ &0.46$\pm$ 0.02 &1.30 $\pm$ 0.26 &2.94 $\pm$ 1.15 &5.19 $\pm$ 0.44 &7.78 $\pm$ 0.77 & 1.77 $\pm$ 0.15 & 2.65 $\pm$ 0.26\
\
\
SbSb45hC &0.35 &0.80 &1.57 &2.81$\pm$ 0.17 &4.02 $\pm$ 0.17 & 1.79 $\pm$ 0.11 & 2.56 $\pm$ 0.04\
\[mu28\_time\]
First passage time.
Maximum separation time.
Final coalsecence time.
Merger-feature time based on $\mu_{\rm r,limit}$ = 25 mag arcsec$^{-2}$.
Merger-feature time based on $\mu_{\rm r,limit}$ = 28 mag arcsec$^{-2}$.
Ratio between merger-feature time to final coalescence time based on $\mu_{\rm r,limit}$ = 25 mag arcsec$^{-2}$.
Ratio between merger-feature time to final coalescence time timescales based on $\mu_{\rm r,limit}$ = 28 mag arcsec$^{-2}$.
Disturbed structures are clearly visible until the end of simulation (6.5 Gyr).
In this case, we consider only ${\rm SbSb_{3}45h}$ and ${\rm SbSb_{6}45h}$ mergers.
Effect of mass ratio
--------------------
We run two additional mergers to qualitatively discuss the effect of mass ratio on the merger-feature time. Figure \[mock\_3:1\] and \[mock\_6:1\] show the mock images of the ${\rm SbSb_{3}45h}$ and the ${\rm SbSb_{6}45h}$ mergers, respectively. Their evolution is shown at different characteristic times and according to the observational limits of the SDSS $r$ band. For equal-mass mergers, two long and symmetric tidal arms are well developed as shown in Figure \[mock\_1:1\]. Unequal-mass mergers, however, show asymmetric tidal structures [see also @Barnes92; @Springel05b; @Lotz08]. As the mass ratio between two galaxies becomes larger (see Figure \[mock\_6:1\]), tidal bridges are unseen and tidal arms are tightly wound.
We investigate the effect of mass ratio on the merger feature time. The ratio of the merger-feature time to the final coalescence time has no clear difference from major mergers (defined as mass ratios are less than 3:1). For the ${\rm SbSb_{3}45h}$ merger, both ${t_{\rm MF,25} / t_{\rm FC}}$ and ${t_{\rm MF,28} / t_{\rm FC}}$ are comparable to those of the SbSb45h merger (see Table \[mu28\_time\]). On the other hand, although the merger-feature times of the ${\rm SbSb_{6}45h}$ merger (i.e. ${t_{\rm MF,25}}$ and ${t_{\rm MF,28}}$) are greater than those of the SbSb45h merger, both ${t_{\rm MF,25} / t_{\rm FC}}$ and ${t_{\rm MF,28} / t_{\rm FC}}$ of the ${\rm SbSb_{6}45h}$ merger are less than those of the SbSb45h merger. In essence, unequal-mass mergers show merger features longer mainly because they take longer to merge, compared to equal-mass mergers.
Figure \[mock\_10:1\] shows a reference case of 10:1 merger. It is interesting to note that this “minor” merger case does not exhibit merger features clearly even before $t_{\rm FC}$. We are tempted to interpret this to suggest that such minor mergers, while they may be much more abundant than major mergers, may be very difficult to detect even in deep imaging conditions. Conversely, most of the merge features in today’s typical surveys may very well be results of relatively “major” mergers. Confirmation of this conjecture requires a much more thorough modeling effort.
It is also hinted in Figures \[mock\_1:1\] – \[mock\_10:1\] that long symmetric tidal tails and large loops are found more frequently in major mergers [see e.g. @Wang12]. It seems as well that the degree of symmetry scales to the mass ratio between the merging galaxies. It again requires many more runs filling a possible parameter space to confirm the statement. But if it is confirmed, it would imply that such features can be used to find and characterize major merger remnants.
Effect of tidal force
---------------------
We also qualitatively examined the effect of tidal force on the merger-feature time. Figure \[SD\] shows the stellar and gaseous column density of the SbSb45hC merger at three times the final coalescence time ($t_{\rm FC}$ = 1.57 Gyr). Stars and gas in low density region of the SbSb45hC merger are stripped off having undergone tidal force of a cluster potential. As a result, the stars and gas are concentrated in the central region unlike in isolated environment (see Figure \[mu\_SD\]).
Figure \[mu\_limits4\] shows the mock images of the SbSb45hC merger at twice and three times the final coalescence time and according to the observational limits of the SDSS r band. We found that the merger-feature time of $\mu_{\rm limit}$ = 25 mag arcsec$^{-2}$ for the SbSb45hC merger (${t_{\rm MF,25}}=2.81\pm0.17$) is comparable to the SbSb45h merger (${t_{\rm MF,25}}=2.99\pm0.20$). Shallow mock images of $\mu_{\rm limit}$ = 25 mag arcsec$^{-2}$ mainly reveal the central region of each merger remnant which is less affected by tidal force because of its high density.
On the other hand, the effect of tidal force on post-merger features are better examined in deeper images. The merger-feature time of the SbSb45hC merger based on $\mu_{\rm r,limit}$ = 28 mag arcsec$^{-2}$ (${t_{\rm MF,28}}=2.56\pm0.04$) is $\sim$ 30% smaller than the SbSb45h merger (${t_{\rm MF,28}}=3.57\pm0.30$).
An empirical test on the SDSS database
--------------------------------------
To test the depth dependence of merger feature lifetime, we performed a simplistic test using the SDSS database. We used the images of galaxies that are present both in the standard SDSS DR7 database and in the SDSS Stripe82 database [@Abazajian09]. The surface brightness limit of Stripe82 (27mag in r’ band) is roughly 2 mag deeper than that of DR7 instead of 3 mag which was the difference we made calculations for, and mergers that can be found in SDSS galaxies are not necessarily equal-mass mergers. Thus this comparison is not exactly suited to test our prediction; but it is the easiest test and probably serves as a useful milestone at least. We performed visual inspection on a volume-limited sample of 1,453 galaxies. Among early-type galaxies, 59$\pm$4 and 160$\pm$14 galaxies were classified as post-merger galaxies in DR7 and Stripe82, respectively, where the errors are from multiple inspections. While it is trivial that merger features are more frequent in deeper images, we are encouraged by the fact that roughly a factor of 3 larger number of merger-feature galaxies are found in Stripe82 as suggested by the simulations.
Discussion
==========
In this paper, we investigate the merger-feature time of equal-mass disk mergers using N-body/hydrodynamic simulations. To cover a realistic range of merger scenarios, we consider different orbit types, host galaxy inclinations, and the morphological properties of the two main galaxies. We ran additional merger simulations considering different mass ratios and putting the mergers in a cluster halo environment to help understand the features of equal-mass mergers.
The post-merger signatures of nearly all samples in isolation survived for more than three times the final coalescence time when we see galaxies with a deep surface brightness limit of 28 mag arcsec$^{-2}$. The lifetime of merger features is shorter by 30% when mergers happen in a large cluster environment. So, for random environments, the merger feature lifetime can be said to be $\sim$ 3.5 times the final coalescence time. This has an obvious impact on the galaxy morphology classification, especially on post-merger remnants. @Sheen12 recently found from their deep imaging ($\mu_{\rm limit} \sim$ 28 mag arcsec$^{-2}$) campaign that a large fraction ($\sim$ 40%) of bulge-dominant galaxies in massive clusters show major post-merger features. Using a semi-analytic model of @Lee13, @Yi13 interpreted it as a result of merger relics from the merging events from the past halo environments by adopting the merger-feature time estimates from the preliminary result of our study.
Even though we found and discussed noticeable differences in merger-feature time between a variety of proposed merger scenarios, our sample is still limited. For example, although equal-mass mergers are influential phenomena that clearly exhibit transitions of galaxy property, they are rare cases in the universe. Minor mergers are expected to dominate galaxy evolution over major mergers because they occur more frequently [@Kaviraj09]. However, our reference models suggest hints to that tidal features induced by relatively minor mergers (e.g., $m2/m1 \lesssim 1/6$) might be proved to be difficult to detect in deep images.
Moreover, mergers of gas-rich galaxies are expected to be frequent at high redshifts. Without considering gas-rich mergers, we cannot see the full picture of galaxy evolution. Furthermore, gas-rich mergers should accompany one of the most important effects on galaxy growth, namely AGN feedback. Recent studies have shown that super massive black holes (SMBHs) in the centers of galaxies are connected to the growth of galaxies. The velocity dispersion of the bulge component is proportional to the BH mass of the galaxy [@Tremaine02]. This coevolution of galaxies and SMBHs has been suggested by recent studies. For examples, the correlation between star formation and BH accretion [@Zheng09], quasar activity of merging galaxies [@Sanders88; @Hopkins08; @Li07; @Sijacki09], and the quenching of star formation due to AGN feedback [@Springel05b; @DiMatteo05; @Schawinski06; @Schawinski07a; @Schawinski07b; @Sijacki07; @Dubois10; @Teyssier11; @Dubois13]. For a more realistic and comprehensive study on merging galaxies, we will include the effects of AGNs in our future studies. However, we think that the inclusion of AGN feedback would not affect our analysis on post-merger features much because most of the star formation induced by merger is concentrated in the central regions of the merger remnant while post-merger features are mostly made up of older pre-existing stars in our simulations [see also @Peirani10].
We performed most of numerical simulations assuming an isolated environment. Galaxy interactions in reality occur in various environments including galaxy clusters. In addition to galaxy mergers, there are physical mechanisms affecting the evolution of galaxies in clusters: ram-pressure stripping [@Gunn72], high-speed galaxy encounters [galaxy “harassment”, @Moore96], tidal stripping by a cluster halo potential [@Byrd90; @Valluri93], and thermal evaporation of cold gas inside galaxies [@Cowie77]. Among these mechanisms, tidal stripping plays a particularly important role in shaping merger remnants. Although we confirmed that the tidal structures of merger remnants are stripped off rapidly in a cluster potential [e.g. @Mihos04], we have not checked the effect of orbital parameters of merger remnants falling into galaxy clusters yet. Therefore, the quantification of ${t_{\rm PM} / t_{\rm FC}}$ in various cluster environments should be investigated in future studies.
Ram pressure stripping, and other stripping processes, affect the gas content in galaxies in deep potential wells which will in turn affect any gas-dependent properties of galaxies, e.g. star formation. However, it would not significantly affect our main predictions on merger-feature times. This is because, as our simulations suggest, most of the faint merger features that are easily detected are composed of pre-existing old stars (which behave following dissipationless processes and angular momentum conservation) rather than newly formed stars [see for instance Figure 4 of @Peirani10]. Moreover, we do not expect that Ram pressure stripping has a significant impact on removing gas in our model galaxies (M $\sim$ 1.7 $\times 10^{11} {\rm M}_{\odot}$ for equal-mass mergers) because it is thought to be more efficient for lower mass objects, i.e., M < 6 $\times 10^{10} {\rm M}_{\odot}$ [@Nickerson11]. We admit that more robust calculations must include all important physical processes, such as Ram pressure stripping, in the end. But for the focus of this paper at the moment, current calculations should be sufficiently elaborate.
We thank Volker Spingel for making the Gadget code available to us. SKY acknowledges support from National Research Foundation of Korea (NRF-2009-0078756; NRF- 2010-0029391) and DRC Grant of Korea Research Council of Fundamental Science and Technology (FY 2012). Numerical simulation was performed using the KISTI supercomputer under the program of KSC-2012-C2-11 and KSC-2012-C3-10 and the KASI supercomputer. Much of this manuscript was written during the visit of SKY to University of Nottingham and University of Oxford under the general support by LG Yon-Am Foundation. we would like to thank the referee for his/her comments that have improved the quality of the original manuscript.
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[^1]: Much of our interpretation is based on the $\mu_{\rm limit} =$ 28 mag arcsec$^{-2}$ cases, because $\mu_{\rm limit} =$ 25 mag arcsec$^{-2}$ cases usually reveal only the central regions of mergers which are insufficient to illustrate the whole merger effects.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Standard analyses of the reionization history of the universe from Planck cosmic microwave background (CMB) polarization measurements consider only the overall optical depth to electron scattering ($\tau$), and further assume a step-like reionization history. However, the polarization data contain information beyond the overall optical depth, and the assumption of a step-like function may miss high redshift contributions to the optical depth and lead to biased $\tau$ constraints. Accounting for its full reionization information content, we reconsider the interpretation of Planck 2015 Low Frequency Instrument (LFI) polarization data using simple, yet physically-motivated reionization models. We show that these measurements still, in fact, allow a non-negligible contribution from metal-free (Pop-III) stars forming in mini-halos of mass $M \sim 10^5-10^6 M_\odot$ at $z \gtrsim 15$, provided this mode of star formation is fairly inefficient. Our best fit model includes an early, self-regulated phase of Pop-III star formation in which the reionization history has a gradual, plateau feature. In this model, $\sim$20% of the volume of the universe is ionized by $z \sim 20$, yet it nevertheless provides a good match to the Planck LFI measurements. Although preferred when the full information content of the data is incorporated, this model would spuriously be disfavored in the standard analysis. This preference is driven mostly by excess power from E-mode polarization at multipoles of $10 \lesssim \ell \lesssim 20$, which may reflect remaining systematic errors in the data, a statistical fluctuation, or signatures of the first stars. Measurements from the Planck High Frequency Instrument (HFI) should be able to confirm or refute this hint and future cosmic-variance limited E-mode polarization surveys can provide substantially more information on these signatures.'
author:
- |
Vinicius Miranda,$^{1}$[^1] Adam Lidz,$^{1}$ Chen He Heinrich$^{2}$ and Wayne Hu$^{2,3}$\
$^{1}$Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104\
$^{2}$Kavli Institute for Cosmological Physics, Enrico Fermi Institute, University of Chicago, Chicago IL 60637\
$^{3}$Department of Astronomy & Astrophysics, University of Chicago, Chicago IL 60637
bibliography:
- 'ReiPopIII.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Signatures of metal-free star formation in Planck 2015 Polarization Data'
---
\[firstpage\]
reionization – cosmic background radiation – star formation
Introduction {#sec:intro}
============
Measurements of the CMB polarization on large angular scales probe the probability that CMB photons scatter off of free electrons produced during the Epoch of Reionization (EoR) [@1982RSPTA.307...97H; @1987MNRAS.227P..33E; @1997PhRvD..55.1822Z]. The EoR is the time period during which the first stars, galaxies, and accreting black holes formed, emitted ultraviolet photons, and ionized surrounding hydrogen and helium gas [@Loeb13]. Recent measurements from the Planck LFI and HFI suggest lower values for the optical depth to electron scattering than implied by earlier Wilkinson Microwave Anisotropy Probe (WMAP) observations [@2015arXiv150201589P; @2016arXiv160503507P; @2013ApJS..208...19H]. These new measurements have important implications for our understanding of the EoR, and have been studied extensively in the literature [e.g. @2015ApJ...802L..19R; @2015MNRAS.454L..76M; @Becker:2015lua; @2016MNRAS.460..417S; @2016arXiv160505374G].
In general, previous theoretical studies have emphasized the lower values of the optical depth preferred by Planck. This is in broad agreement with direct censuses of high redshift galaxy populations – which suggest fairly low ionizing emissivities – and other reionization observables [@2015ApJ...802L..19R].
However, it is still plausible that reionization is a very extended process, and that a small fraction of the volume of the intergalactic medium (IGM) remains ionized out to rather high redshift. In this context, it is important to make use of the full information content of the CMB E-mode polarization power spectrum, which cannot be reduced to a single number quantifying the overall optical depth. It is also important to ensure that CMB E-mode inferences on the overall optical depth are not biased by implicit assumptions on the reionization history itself.
Toward this end, we further consider the analysis of Planck LFI data in [@2016arXiv160904788H]. These authors use a principal component (PC) analysis methodology, in which any given reionization history between $6 < z < 30$ is characterized by five numbers. These eigenmode amplitudes completely capture the impact of reionization in this redshift range on the CMB polarization power spectrum observables [@Hu:2003gh; @Mortonson:2007hq; @Mortonson:2008rx]. Interestingly, that study finds a two sigma preference for a contribution to the optical depth from $z \geq 15$ in the LFI data.
This result is intriguing because it may indicate the existence of ionizing sources at very high redshift, and this runs somewhat counter to the current conventional wisdom that Planck requires late reionization. For example, this high redshift $\tau$ contribution may arise from metal-free (Pop-III) stars forming in minhalos ($M \sim 10^5-10^6 M_\odot$) at $z \sim 20$ [@2009Natur.459...49B]. In fact, previous work suggested that Planck data may be used to detect such signatures [@2012ApJ...756L..16A], following-up in part on earlier work from e.g. [@2003ApJ...595....1H] that considered the high $\tau$ indicated by early WMAP data, but this possibility has not yet been investigated with the actual Planck data. The Pop-III mode of star formation at high redshift must be fairly inefficient, otherwise these sources would rapidly reionize the universe and overproduce the observed optical depth [@Visbal15], but a small fraction of the gas in these early halos may nevertheless be converted into massive metal-free stars.
To explore this scenario, we use the fast likelihood technique developed and tested in [@2016arXiv160904788H] to constrain simple reionization models. Although the PC methodology can not be used to reconstruct the reionization history directly, it is ideal for forward-modeling. This paper hence provides an illustrative example of how the PC technique can be used to constrain the parameters of reionization models. Software implementing the fast likelihood calculations of [@2016arXiv160904788H] is available upon request, and should provide a useful tool for reionization modelers.
Reionization Models {#sec:popII_popIII}
===================
In the reionization models considered here, we assume that metal-free stars form exclusively in mini-halos that rely on molecular hydrogen for cooling, while we suppose that normal (Pop-II) star formation occurs in more massive halos where the gas cools by atomic line emission [e.g. @Haiman06]. The rationale for this split is that atomic cooling halos likely have one or more mini-halo progenitors that formed Pop-III stars. Supernovae from these short-lived stars subsequently enrich the surrounding gas with heavy elements and this shuts off the Pop-III star-formation mode before the atomic cooling “descendent” halos collapse. The redshift evolution of the average fraction of ionized hydrogen, ${\ensuremath{\langle x_i \rangle}}$, is determined by the following “photon-counting” equation [@Shapiro87; @Madau99; @Loeb13]: $$\begin{aligned}
\label{eq:dxdt}
\frac{d{\ensuremath{\langle x_i \rangle}}}{dt} = & \frac{d}{dt} \left( \zeta_{\rm II} f_{c,{\rm II}} + \zeta_{\rm III} f_{c,{\rm III}}\right) - \frac{{\ensuremath{\langle x_i \rangle}}}{\bar{t}_{\rm rec}(z)}. \end{aligned}$$ The first two terms describe the rate at which Pop-II and Pop-III stars ionize surrounding hydrogen atoms, while the final term accounts for recombinations. In our model, the ionizing photon production in each population of stars traces the rate at which matter collapses into their respective dark matter host halos. Hence, $f_{c,{\rm II}}$ denotes the collapse fraction in halos above the minimum mass required to host Pop-II stars $M_{\rm min,II}$, while $f_{c,{\rm III}}$ denotes the collapse fraction in the lower mass halos between $M_{\rm min,{\rm III}} \leq M \leq M_{\rm min,II}$ in which Pop-III stars reside. To compute these collapse fractions, we adopt the halo mass function of [@Sheth:1999su].
The recombination term depends on the average time required for ionized hydrogen to recombine, $\bar{t}_{\rm rec}(z)$. This in turn depends on the temperature and the clumpiness of the ionized gas that resides in the intergalactic medium, wth $\bar{t}_{\rm rec} = C/[\alpha_B(T) \bar{n}_e(z)]$. Here $C$ is the clumping factor of the ionized gas, $\alpha_B(T)$ is the recombination coefficient for gas at temperature $T$ and $\bar{n}_e(z)$ is the average electron number density at redshift $z$. We assume case-B recombination and evaluate $\alpha_B$ at a redshift independent temperature of $T=2 \times 10^4$K [@Hui:1997dp]. We choose a redshift-independent clumping factor of $C=2$, in agreement with recent simulations that find small values for the clumping factor [@Pawlik:2008mr; @McQuinn:2011aa].
The minimum mass of the source host halos is set such that Pop-III stars form only in halos where molecular hydrogen cooling is efficient, while normal star formation is assumed to take place in atomic cooling halos. On this basis, we follow [@Haiman06] and assume that metal-free stars form in halos with virial temperatures between $400 \,{\rm K} < T_{\rm vir} < 10^4$K, while Pop-II stars form in halos with virial temperature above $10^4$K. The lower virial temperature is an optimistic value for efficient molecular hydrogen cooling and so we subsequently test the impact of raising the minimum host virial temperature to $10^3$K [@2003ApJ...592..645Y] (see §\[likelihood\]). These considerations set the redshift-dependent values of the minimum host halo masses, $M_{\rm min,III}$, $M_{\rm min,II}$.
The ionizing efficiency parameters $\zeta_{\text{II}}$ and $\zeta_{\text{III}}$ can each be written as the product of several uncertain factors, with $\zeta = A_{\text{He}} \times N_\gamma \times f_{\ast} \times f_{\text{esc}}$ [@Loeb13]. Here, $N_\gamma$ is the number of ionizing photons produced per baryon converted into stars, $f_\ast$ is the star-formation efficiency – i.e., it is the fraction of halo baryons that are converted into stars, while $f_{\text{esc}}$ is the fraction of ionizing photons that escape the host halo and ionize atoms in the IGM. Furthermore, $A_{\text{He}} = 4/(4 - 3Y_p)$ is a rescaling factor to account for (singly-ionized) helium and $Y_p$ is the primordial helium mass fraction. Here we assume that metal-free stars have large masses and high surface temperatures, and so produce copious numbers of ionizing photons: our fiducial Pop-III star model adopts $A_{\text{He}} \times N_{\gamma,\text{III}} = 40,000$ [@Bromm01; @Schaerer02]. Note that the expected yield of ionizing photons depends on the Initial Mass Function for this mode of star formation [@2000ApJ...528L..65T], which is still uncertain. In addition, numerical simulations of Pop-III star formation suggest that a high fraction of ionizing photons are able to escape from their host halos and ionize atoms in the IGM [@2006ApJ...639..621A]. Our model therefore takes $f_{\text{esc}, \text{III}}=0.5$. Finally, we vary the Pop-III star-formation efficiency over a broad range of values from $10^{-4} < f_{\ast,\text{III}} < 0.1$.
![Reionization history in the various models from fits to the Planck 2015 data with parameters from Tab. \[tab:models\]. In order of increasing high redshift ionization contributions these are: step-like Tanh model (black solid line), model with only Pop-II stars (purple dot-short-dashed line), model with additional Pop-III contributions (red long-dashed line) and where those contributions are self-regulated (with $x_{\text{max}}=0.2$, green dot-dashed line). A broad range of ionization histories remain consistent with the Planck 2015 data. []{data-label="fig:xe"}](5.pdf){width="0.95\linewidth"}
In the case of Pop-II stars in atomic cooling halos, the expected efficiency parameters take rather different values. In this case, typical values considered in the literature are $N_{\gamma,\text{II}} \sim 4,000$, $f_{\text{esc},\text{II}}=0.1$, $f_{\ast,\text{II}} = 0.1$, corresponding to $\zeta_{\text{II}} \sim 50$ [@Lidz16]. Here, we conservatively allow the Pop-II star-formation efficiency coefficient to vary over the range of $5 < \zeta_{\text{II}} < 500$. In Fig. \[fig:xe\] we show an example of a Pop-II dominated ionization model and one where the Pop-III contribution dominates at high redshift. We also compare this history to the step-like Tanh reionization history that is assumed in the standard Planck analysis. As we shall see in the next section, the parameters of these models are chosen to best fit the Planck 2015 data and are listed in Tab. \[tab:models\]. The Tanh history has no ionization at high redshift by assumption of form whereas the Pop-III model has a broader tail than the Pop-II model though both have a sharp decline with redshift.
Model $\tau$ $\zeta_{\text{II}}$ $ f_{\ast,\text{III}}$ $ \chi_\text{model}^2-\chi_\text{Pop-II}^2 $
------------------- -------- --------------------- ------------------------ -----------------------------------------------
Tanh 0.079 – – 0.97
Pop-II 0.081 19.6 – 0
Pop-III 0.089 6.91 $0.00045 $ -1.0
Pop-III self-reg. 0.010 12.1 $0.0011$ -$2.2$
: Model Parameters and Fits
\[tab:models\]
The sharp decline of the Pop-III contribution is itself dependent on our model assumptions. The Pop-III star formation mode is likely fragile, and disrupted by a range of feedback effects including that from dissociating ultraviolet radiation, photo-ionization feedback, supernova feedback, and chemical enrichment [e.g. @2009Natur.459...49B and references therein]. In our fiducial Pop-III model, we assume that the impact of feedback is roughly encapsulated in the efficiency parameter, $\zeta_{\text{III}}$. While an overall efficiency parameter may capture internal feedback, “external feedback” from surrounding sources is likely important as well; this should impact the overall progression of reionization, and is unlikely adequately captured by our single efficiency parameter.
A detailed modeling of feedback is well beyond the scope of this paper, but to explore its plausible impact we consider a second Pop-III model in which we multiply the Pop-III term in Equation \[eq:dxdt\] by $(1 -{\ensuremath{\langle x_i \rangle}}/x_{\text{max}})$ if ${\ensuremath{\langle x_i \rangle}} \leq x_{\text{max}}$, while we set this to zero when ${\ensuremath{\langle x_i \rangle}} > x_{\text{max}}$. This form is meant to mimic that Pop-III star formation may be “self-regulating” [@2012ApJ...756L..16A]. Here metal-free star formation is allowed to continue only in neutral regions, with an increasing suppression factor, while it is suppressed completely when the average ionization fraction exceeds a threshold value of $x_{\text{max}}$. The suppression is meant to roughly capture two separate feedback effects. First, Pop-III star formation will be truncated or less efficient in ionized regions where the gas is photo-heated and unable to collapse into mini-halos. Next, and more important, is that early Pop-III stars will produce a dissociating ultraviolet radiation background that will prevent or suppress molecular hydrogen cooling and the formation of additional stars in mini-halos [@1997ApJ...476..458H].
Although this is only a toy model, we find that $x_{\text{max}}=0.2$ produces a plateau-feature in the reionization history similar to that seen in the simulations of [@2012ApJ...756L..16A], although our plateau is slightly more pronounced. In this model, we also boost the ionizing efficiency factor to start reionization as early as possible to $A_{\text{He}} \times N_{\gamma,\text{III}} \times f_{\text{esc,III}} = 10^5$, although this is of course degenerate with $f_{\ast,\text{III}}$. We show an example self-regulated Pop-III model in Fig. \[fig:xe\]. If the reader prefers to compare models at fixed $A_{\text{He}} \times N_{\gamma,\text{III}} \times f_{\text{esc,III}} $ one can rescale the self-regulated star formation efficiency, $f_{\ast,\text{III}}$ upwards by a factor of five. We show below that this model is a marginally better fit to the Planck LFI data.
We can further consider the plausibility of our model value, $x_{\text{max}} \sim 0.2$, by calculating the average specific intensity of the dissociating ultraviolet background at the redshifts and ionized fractions of interest. Ultraviolet photons in the portion of the Lyman-Werner band between $11.2\,{\rm eV} \le h\nu \le 13.6\,{\rm eV}$ can dissociate molecular hydrogen, while even the pre-reionization IGM is largely transparent to such photons, since they lie beneath the hydrogen photoionization edge. However, below the ionization edge photons can still be absorbed out of the dissociating background in Lyman series lines from neutral hydrogen atoms: this imprints a sawtooth feature on the spectrum of the dissociating background [@1997ApJ...476..458H].
A rough estimate of the average specific intensity of dissociating ultraviolet radiation in the Lyman-Werner band, $J_{\text{LW}}$, in our model can be made according to [e.g. @Visbal15]: $$\begin{aligned}
J_{\text{LW}}(z) = \frac{c}{4\pi} \frac{\bar{\rho}_b(z)}{m_p} f_{\ast,\text{III}} \frac{N_\text{LW} h \nu_\text{LW}}{\Delta \nu_{\text{LW}}}
& \left[f_{c,\text{III}}(z) - f_{c,\text{III}}(z_{\text{hor}})\right].
\label{eq:jlw}
\end{aligned}$$ Here $c$ is the speed of light, $\bar{\rho}_b(z)/m_p$ is the baryonic number density at redshift $z$, and $f_{\ast,\text{III}}$ is the Pop-III star formation efficiency, while $N_{\text{LW}}$ is the number of dissociating Lyman-Werner photons per baryon converted into stars, $h \nu_{\text{LW}}$ is their typical energy, and $\Delta \nu_{\text{LW}}$ is the bandwidth of such photons. As in [@Visbal15], we assume that the Lyman-Werner background at redshift $z$ is set by sources out to $z_{\text{hor}} = 1.015 z$. This “Lyman-Werner (LW) horizon” redshift, $z_\text{hor}$, reflects the typical distance a Lyman-Werner photon travels before redshifting into a Lyman series line and being absorbed out of the background. Equation \[eq:jlw\] therefore ignores any absorption between $z$ and $z_\text{hor}$ and neglects any contribution from sources beyond this horizon; it is intended to capture an average suppression from the Lyman series absorption without tracking the full sawtooth effect. In this estimate, we have ignored the model truncation of Pop-III sources in ionized regions and from the build-up of dissociating radiation. We have also neglected any Pop-II contribution, which should be a good approximation for present purposes. We adopt $N_{\text{LW}}=3400$ and $\nu/\Delta \nu_{\text{LW}} = 4.9$ following [@2014MNRAS.445..107V]. According to the estimate of Equation \[eq:jlw\], the intensity of dissociating radiation reaches values of $J_{\text{LW}}/J_{21} \sim 0.1-0.2$ – where $J_{21}$ is the specific intensity in units of $10^{-21}$ ergs cm$^{-2}$ s$^{-1}$ Hz$^{-1}$ str$^{-1}$ – at $z \sim 25$ for $f_{\ast,\text{III}} =10^{-3}$. For this redshift and efficiency factor, ${\ensuremath{\langle x_i \rangle}} \sim 0.1$. Since the contribution of Pop-III stars is suppressed in our self-regulated model by a factor of $1 - {\ensuremath{\langle x_i \rangle}}/x_\text{max}$, this amounts to a factor of two at this redshift and stage of reionization. In other words, our model reduces the contribution of Pop-III stars by a factor of two at $J_\text{LW}/J_{21}\sim 0.1-0.2 $. Note that although $J_\text{LW}$ at a given redshift depends on $f_{\ast,\text{III}}$, the intensity at a given ${\ensuremath{\langle x_i \rangle}}$ is insensitive to the star-formation efficiency since both $J_\text{LW}$ and ${\ensuremath{\langle x_i \rangle}}$ scale in proportion to $f_{\ast,\text{III}}$.
This is a bit on the high side of the threshold specific intensities at which previous studies suggest that Lyman-Werner background photons will largely suppress molecular hydrogen cooling: for example, [@2012ApJ...756L..16A] quote plausible threshold values of $J_{\text{LW,th}}/J_{21}= 0.01-0.1$ (see also the references in that work). This may partly reflect that Equation \[eq:jlw\] only provides a rough estimate: [@2012ApJ...756L..16A] find a broadly similar plateau feature at a comparable ionized fraction, although they adopt a lower dissociation threshold.
In any case, it is likely that any ionization plateau is less well-defined than in our toy model. For one, larger mass halos will generally contain more molecular gas and more intense ultraviolet radiation should therefore be required to suppress cooling in such halos. For example, in the analytic model of @Visbal15, the authors self-consistently model the build-up of the average Lyman-Werner background along with the ionization history, while incorporating a halo-mass dependent ultraviolet background threshold (above which cooling is completely suppressed). Their model ionization histories do not include a plateau feature, in contrast to the study of [@2012ApJ...756L..16A], which adopts a mass-independent threshold. In addition, some halos will turn around early and be largely self-shielded before the ultraviolet background is intense enough to dissociate molecular hydrogen, while the ultraviolet background will itself be somewhat inhomogeneous.[^2] The mass dependence and inhomogeneities may act to soften any plateau feature. Significantly more detailed models than considered here will be required to understand these issues better.
![The effective $\chi^2$ of various Pop-III models to the Planck LFI data relative to $\chi_\text{Pop-II}^2$ for the best fit model with Pop-II stars only. The red dashed line shows $\chi^2$ as a function of the Pop-III star formation efficiency parameter $f_{\ast,\text{III}}$ for the fiducial model after minimization with respect to the Pop-II efficiency parameter. The green dot-dashed line shows the same for our self-regulated Pop-III model.[]{data-label="fig:posterior"}](1_top.pdf){width="0.95\linewidth"}
Planck 2015 Analysis {#likelihood}
====================
In order to compare these models with Planck 2015 data, we use the complete analysis of the reionization information from [@2016arXiv160904788H]. Each reionization history is projected onto a basis of principal components $S_a(z)$ of a cosmic variance limited polarization measurement around some fixed but otherwise arbitrary fiducial ionization history $\langle x^{\text{fix}}_i\rangle(z)$. Only modes $a=1,\ldots, 5$ are needed to specify the entire information content of the E-mode polarization power spectrum for $6 < z < 30$ [@Hu:2003gh]. For any given ionization history, ${\ensuremath{\langle x_i \rangle}}(z)$, the amplitude of each mode is determined according to: $$m_{a}=
\frac{1}{24} \int _{6}^{30} dz\, {S_{a}(z) [\langle x_i \rangle(z)-\langle x^{\text{fix}}_i\rangle(z)]}.
\label{eq:xetommu}$$ We assume that $\langle x_i \rangle(z)=\langle x^{\text{fix}}_i\rangle(z)$ outside of the range of the integral. The fixed ionization history is specified in [@2016arXiv160904788H] such that hydrogen is fully ionized and that helium is singly ionized at $z \le 6$ and helium is fully ionized at $z \lesssim 3.5$. At $z\ge 30$, the ionization history returns to that given by recombination.
Given $m_a$, the 5 principal component amplitudes of a model, we evaluate the effective likelihood ${\cal L}(m_a)$ of that model using the technique of [@2016arXiv160904788H]. Here we interpret this as an effective $\chi^2 = -2\ln {\cal L}$. In order to set the baseline values for comparison, we first minimize $\chi^2$ without any Pop-III contribution to find $\chi^2_{\text{Pop-II}}$. This model is specified as “Pop-II” in Tab. \[tab:models\] and is itself a better fit than the best fit Tanh model. Likewise, for each Pop-III model parameterized by the Pop-III star formation efficiency parameter $f_{\ast,\text{III}}$, we minimize $\chi^2$ over the Pop-II efficiency parameter, $\zeta_\text{II}$. Note that this minimization is equivalent to marginalization over $\zeta_\text{II}$ if the joint posterior probability distribution is a multivariate Gaussian.
![Cumulative optical depth $\tau(z,30)$ in the Planck 2015 analysis. Blue shaded regions are the $68\%$ and $95\%$ constraints from the complete PC analysis. This is compared with the best fit models from Tab. \[tab:models\]: Tanh (black solid line), Pop-II only (purple dot-short-dashed line), additional Pop-III fiducial (red long-dashed line), additional Pop-III self-regulated (green dot-dashed line) models.[]{data-label="fig:cum_tau"}](3b.pdf){width="0.95\linewidth"}
In Fig. \[fig:posterior\], we show the difference $\chi^2_{\rm model} - \chi^2_{\text{Pop-II}}$ as a function of $f_{\ast,\text{III}}$ (with $A_{\text{He}} \times N_\gamma \times f_{\text{esc}}=2 \times 10^4$ in the “Pop-III” fiducial model and $A_{\text{He}} \times N_\gamma \times f_{\text{esc}}=10^5$ in the “Pop-III, self-regulated” model). The model at the minimum of this curve for each case is given in Tab. \[tab:models\]. Let us consider first the fiducial model (red dashed line, Figure \[fig:posterior\]). This shows that including Pop-III stars in our fiducial model with low efficiency, $f_{\ast,{\rm III}} \sim$ a few $\times$ $10^{-4}$ only slightly improves the fit to the Planck data, with $\Delta \chi^2\approx1$. Furthermore, in qualitative agreement with previous work [@Visbal15; @2016MNRAS.460..417S], if Pop-III stars form with too great an efficiency they overproduce the E-mode polarization power and so the Planck data imply an interesting upper limit on the efficiency of metal-free star formation in minhalos.
However, our fiducial model assumes a redshift-independent efficiency factor which is too simplistic, as we discussed previously. In our self-regulated Pop-III model, including metal-free star formation improves the fit further with $\Delta \chi^2 =2.2$ compared to the best-fit model with no Pop-III component (see the green dot-dashed line in the figure). In this case, the preferred efficiency factor is $f_{\ast,{\rm III}} \sim 10^{-3}$ which corresponds to roughly one $\sim 100 M_\odot$ star per halo of mass $M \sim 6 \times 10^5 M_\odot$ (fairly typical of the mini-halos), before this mode of star-formation is shutoff (completely at ${\ensuremath{\langle x_i \rangle}} =0.2$ in this model).[^3]
Although the preference for a Pop-III contribution is weak – and we are not claiming that the Planck data demand these sources – the more important point here is that models with significant high redshift contributions to the ionization history are still viable. The ionization history for these best fit models are shown in Figure \[fig:xe\]. In the self-regulated Pop-III model the extended ionization plateau at ${\ensuremath{\langle x_i \rangle}} \sim 0.2$ near $z \sim 20$ is strikingly different than the best fit Pop-II model, in which the ionization fraction is negligible at such high redshifts. Furthermore even the Pop-II only model has a more extended high redshift tail than the Tanh model upon which the standard Planck analysis of the overall optical depth $\tau$ is based. Tab. \[tab:models\] also gives $\tau$ for the various models which are higher for those with high redshift contributions.
![Impact of incorrectly or incompletely assessing the reionization information content of Planck 2015 LFI data. Effective $\chi^2$ as in Fig. \[fig:posterior\] for the self-regulated Pop-III model but employing the likelihood from: i) the standard Tanh $\tau$ constraint (black dotted line), which is incorrect in the Pop-III context; ii) the overall $\tau$ constraint from the PC analysis (blue dashed line), which is incomplete; iii) the full PC analysis from Fig. \[fig:posterior\] (green dot-dashed line) which reveals the preference for the model. []{data-label="fig:posterior2"}](1_bottom.pdf){width="0.95\linewidth"}
In order to interpret these changes in the overall optical depth, it is useful to compute the cumulative optical depth between redshift $z$ and $30$, $\tau(z,30)$. Rather than directly compute this from $\langle x_i \rangle(z)$ and cosmological parameters, we sum over contributions from each component $\tau^a(z,30)$ $$\tau(z,30) = \tau^{\rm fix}(z,30) + \sum_{a=1}^{5} m_a \tau^a(z,30).$$ This provides a smoothed representation of the cumulative optical depth that can be compared directly to the constraints derived in [@2016arXiv160904788H]. To evaluate $\tau^a(z,30)$ and the contribution from the fixed ionization history $\tau^{\rm fix}(z,30)$, we take the cosmological parameters ${\Omega_b h^2}=0.02224$, ${\Omega_m h^2}=0.1426$, and ${Y}_p=0.24534$.
In Figure \[fig:cum\_tau\], we compare the best fit models to the constraints on the cumulative optical depth from [@2016arXiv160904788H]. This gives some intuition as to what is driving the slight preference in the Planck LFI data for high redshift contributions from, for example, Pop-III stars. As discussed in [@2016arXiv160904788H], the data favor a non-negligible contribution to $\tau(15, 30)=0.033\pm 0.016$. The reionization histories are not extended enough in the Tanh or the Pop-II only models to match the central value. In the case of the Pop-II only model, one needs to wait until lower redshift for the atomic cooling halos to collapse. If one supposes that the efficiency is extremely high in the atomic cooling halos, one can get a slightly earlier start in these models, but in this case reionization generally completes too early and one overproduces $\tau$.
On the other hand, the Pop-III contribution helps extend reionization out to higher redshift since the molecular hydrogen cooling mini-halos collapse at higher redshift. In our redshift-independent Pop-III efficiency model, this contribution only helps slightly, however. In the self-regulated model, one gets a larger early contribution from Pop-III stars – since the formation of these sources is truncated early, the ones that form early can do so at higher – yet still reasonable – efficiency without overproducing $\tau$. In the self-regulated model with $x_{\text{max}}=0.2$, the model $\tau(z,30)$ always lies within the $68\%$ confidence band preferred by the Planck data. In principle, the Pop-II component could be more extended than in our model, which assumes a redshift independent efficiency factor, but the atomic cooling halos are rare at high redshifts and the efficiency factor would need to be uncomfortably large for these sources to have a non-negligible impact at $z \sim 20$, for example.
Figure \[fig:cum\_tau\] also illustrates a final important point. Note that the constraint on the overall optical depth $\tau(0,30)=0.092\pm 0.015$ is shifted upwards from that inferred assuming the Tanh reionization history $\tau=0.079\pm 0.017$ [@2016arXiv160904788H]. The latter does not correspond to the overall optical depth in these Pop-III models and so applying this constraint would spuriously disfavor such models. This point is made explicit in Figure \[fig:posterior2\], which shows the likelihood for the Pop-III component using the Tanh $\tau$ constraint, rather than our PC methodology. Clearly this would disfavor our best-fit model (at $\sim$1-$\sigma$) and rule-out many other viable Pop-III models. Using the overall $\tau(z=0,z_{\rm max})$ from the PC analysis restores compatibility but the small preference for the self-regulated model is only revealed in the full PC analysis (green dot-dashed line). Correspondingly only when one makes use of the full information content of the Planck 2015 data is the hint for high redshift ionization apparent. Incorporating the full PC analysis, the best fit self-regulated model has a notably better $\chi^2$ than expected for the same model using the Tanh $\tau$ constraint: the fit improves by $\Delta \chi^2 \sim 3.2$. Although this improvement does not represent a detection given the number of additional parameters, the best fit Pop-III models are clearly still allowed.
![E-mode polarization power spectra, $C_{\ell}^{\text{EE}}$, of the various best fit models as in Fig. \[fig:cum\_tau\]. The black solid line and gray shaded regions show the Tanh model, along with the $68\%$ and $95\%$ confidence regions expected in the cosmic variance (CV) limit. In particular the two Pop-III models should be clearly distinguishable from the step-like Tanh model for a CV limited measurement.[]{data-label="fig:clee"}](2b.pdf){width="0.95\linewidth"}
The origin of these different inferences can be traced back to the E-mode power spectrum. In Figure \[fig:clee\], we show the power spectra of the best fit models. The most striking difference between the model power spectra is the excess power in the self-regulated Pop-III model at $10 \lesssim \ell \lesssim 20$ when compared to the other models. This is the main feature of the Planck 2015 LFI data that drives the preference for this model. If one tries to generate excess power in a step-like Tanh model at $10 \lesssim \ell \lesssim 20$, one will overproduce the power at lower multipoles and so high redshift contributions to the ionization history are effectively hidden from the usual constraints [@2016arXiv160904788H].
This excess may reflect remaining systematic errors in the Planck 2015 LFI data from, for example, residual foregrounds. Alternatively, it may be an interesting signature of Pop-III star formation or other high redshift sources of ionization. Ultimately, if systematics are under control, the low $\ell$ polarization can be measured to the cosmic variance (CV) limit. The shaded regions in Figure \[fig:clee\] show that if the true model is the Tanh one, CV limited measurements would test the Pop-III models at high significance. This may be of interest for future space-based polarization missions such as the Cosmic Origins Explore (COrE) satellite [@2011arXiv1102.2181T].
Finally, we briefly comment on the sensitivity of our results to some of our model assumptions. First, we also explored self-regulated Pop-III models where we raised the minimum host virial temperature to $T_\text{vir}=10^3$K. After maximizing over the efficiency parameters in this model, the resulting ionization history looks nearly identical to the best fit in Figure \[fig:xe\], which adopts $T_\text{vir}=400$K. However, the best-fit star formation efficiency goes up to $f_{\ast,\text{III}} = 0.0023 $: in this model, the higher star formation efficiency compensates for the boosted minimum host mass. Note also that we have ignored the effect pointed out in [@2010PhRvD..82h3520T]: while this should lead to interesting spatial variations, recent studies suggest a fairly small average suppression for star formation in mini-halos [@2012MNRAS.424.1335F], and this may be compensated again by boosting our star formation efficiency parameter. Lastly, we investigated a range of values of $x_\text{max}$. While slightly lower values of $x_\text{max}$ still match the data, the star formation efficiency needs to be increased in this case since the Pop-III phase is briefer in these models.
Discussion {#sec:discussion}
==========
We have given an example illustrating how one may reach qualitatively different conclusions regarding the reionization history of the universe and the nature of the ionizing sources, when one accounts for the full information content of the Planck 2015 LFI data, rather than assuming a step-like ionization history. For example, our best fit ionization history has ${\ensuremath{\langle x_i \rangle}}=0.2$ at $z \sim 20$: contrary to conventional wisdom, that data still allow an extended tail of ionization out to high redshift and a non-negligible contribution from metal-free stars in minihalos. It will be interesting to see if the hint for high redshift contributions to $\tau$ is sharpened by future CMB polarization data. We eagerly await upcoming results from the new Planck HFI large scale polarization data – which is still proprietary – to see if these observations strengthen the hint seen in the LFI data. Alternatively, these may reflect remaining systematics in the LFI data that do not also apply to the HFI data. Based on Figure \[fig:clee\], we expect that a future cosmic-variance limited experiment should be able to confirm or refute the high redshift contribution to $\tau$, although we postpone a more detailed investigation of these prospects to future work.
It will be extremely challenging to test the hint for non-negligible ionization fractions at $z \gtrsim 15$ by other means. The best alternative is likely redshifted 21 cm surveys but these face challenges at high redshift owing to the bright galactic emission at the frequencies of interest. However, these surveys may find that the average neutral fraction rises more slowly towards high redshift than expected in models with Pop-II stars alone (see Figure \[fig:xe\]). There may be differences in the sizes of the ionized regions in these scenarios as well, which may provide a potential signature [@2016arXiv160904400K]. However, the possibility of a significant contribution from Pop-III stars at high redshift may also somewhat complicate the interpretation of the 21 cm power spectrum measurements [e.g. @2013MNRAS.432.2909F]. Another possibility may be to extract signatures of early phases of patchy reionization using the Kinetic Sunyaev-Zel’dovich effect, but these will have to push to smaller angular scales which is challenging given foreground contamination. An extended self-regulated phase likely evades current constraints [@2012ApJ...756...65Z; @2015ApJ...799..177G] on the duration of reionization [@2013ApJ...769...93P].
Finally, it is worth commenting on the implications of our findings for the goal of determining $\tau$ from the ionization history inferred from redshifted 21 cm observations [@2016PhRvD..93d3013L]. This is an appealing idea, because the optical depth is an important nuisance parameter that can limit, for instance, inferences regarding the sum of the neutrino masses from upcoming CMB lensing measurements. However, this prospect becomes difficult if there are significant high redshift contributions to the optical depth, as hinted at in the Planck LFI data, which will be hard to extract from upcoming redshifted 21 cm surveys.
We hope that this work will encourage reionization modelers to adopt this PC analysis methodology of [@Hu:2003gh] and effective likelihood of [@2016arXiv160904788H]. Here we have considered rather simple models for the ionization history in an effort to explore the broad-brush implications of the Planck data, but it would be interesting to explore more detailed reionization models and to combine the full information content of the CMB data with other reionization observables. It will also be interesting to explore whether dark matter annihilations [@2015arXiv151200526K] or early accreting black holes [@2004MNRAS.352..547R] provide interesting alternative sources of high redshift ionization to the Pop-III stars considered here.
[*Acknowledgments*]{}: We thank Tom Abel, James Aguirre, and Kyungjin Ahn for useful discussions. WH thanks the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293, where part of this work was completed. CH and WH were supported by NASA ATP NNX15AK22G, U.S. Dept. of Energy contract DE-FG02-13ER41958, and the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-0114422 and NSF PHY-0551142. Computing resources were provided by the University of Chicago Research Computing Center. VM was supported in part by the Charles E. Kaufman Foundation, a supporting organization of the Pittsburgh Foundation.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: Note, however, that the study of [@2012ApJ...756L..16A] does model inhomogeneities in the Lyman-Werner background.
[^3]: Note again that we assume a higher ionizing efficiency in the self-regulated model. If one prefers the lower efficiency assumed in our fiducial model, one can rescale the star-formation efficiency in the self-regulated model upwards by a factor of five.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We report on preliminary results from our XMM-Newton observation of the Low Mass X-ray Binary 4U 1850-087, located in the galactic globular cluster NGC 6712. It is an ultracompact binary system, with an orbital period of 20.6 min. In previous low-resolution X-ray observations 4U 1850-087 displayed a soft excess residual around 0.7 keV, possibly indicative of the presence of a high Ne/O ratio, as already found in other three ultracompact LMXBs containing a neutron star (4U 0614+091, 4U 0918-549, and 4U 1543-624). We discuss here our preliminary analysis of high resolution X-ray spectra of 4U 1850-087, and timing results.'
author:
- 'L. Sidoli'
- 'A.N. Parmar'
- 'T. Oosterbroek'
title: 'XMM-Newton results on the ultracompact Low Mass X–ray Binary 4U 1850-087 in the Globular Cluster NGC 6712'
---
\[2001/04/25 1.1 (PWD)\]
Introduction
============
X1850-087 is an ultracompact Low Mass X-ray Binary (orbital period, P$_{orb}$, of 20.6 minutes, Homer et al. 1996) located in the globular cluster NGC6712 (at a distance of 6.8 kpc).
Ultracompact X-ray binaries (systems with orbital period less than 80 minutes) are believed to contain hydrogen–poor or degenerate companions (e.g. Verbunt & van den Heuvel, 1995).
Recent X-ray spectral evidence indicates that these degenerate companions contain anomalously high abundance of neutral Neon (e.g. Schulz et al., 2001; Juett et al., 2001). Indeed, a broad line-like feature at 0.7 keV, found in ASCA observations of ultracompact binaries (including X1850-087), is suggestive of the presence of an excess in the neutral neon absorption local to the systems (Juett et al., 2001). High resolution spectroscopy of the ultracompact LMXRBs 2S0918-549 and 4U 1543-624 confirmed an enhanced Ne/O number ratio with respect to the expected ISM ratio Ne/O=0.18 (Juett & Chakrabarty, 2003).
The aim of our XMM-Newton (Jansen et al. 2001) observation is to deeply investigate the presence of Neon excess suggested by previous ASCA observation, through high spectral resolution observation with RGS (see den Herder eet al. 2001 for the description of the instrument, covering the energy range 0.3–2 keV).
Observations and Data Analysis
==============================
XMM-Newton data consist of two observations, performed in 2003, September/October, 12 days apart, for a net EPIC PN exposure of 8.1ks and 5.9ks, respectively. PN operated in Small Window Mode in order to minimize pile-up problems. The source rates in the 2 PN observations were $\sim$46 $^{-1}$ and 49 $^{-1}$ in the 0.3-12 keV band. We will concentrate here only on the RGS spectra. The analysis of MOS and PN spectra is still in progress, because they are particularly complex and cannot be fit with simple models, because of the presence of structures in the residuals at low energies, below 2 keV. Indeed, the EPIC PN XMM-Newton spectrum cannot be accounted for nor by the ASCA best-fit (Juett et al. 2001) nor by the BeppoSAX model (Sidoli et al. 2001).
RGS spectral Results
====================
An accurate visual inspection of the RGS spectra does not show any evidence for the presence of bright and narrow lines, neither in emission nor in absorption (see Fig. 1).
In order to search for neutral Neon excess local to the source, we fit the RGS spectrum with a black-body, absorbed with a variable-absorption model ([vphabs]{} in [xspec]{}) and fixing to zero the Neon and Oxygen abundances. Then, we included the edges from neutral Neon and Oxygen and after fitting, we obtained the Ne and O column densities. We got the the following Ne/O number ratio:
Ne/O = 0.17 $\pm{0.09}$
-0.5cm to be compared with the ISM value of 0.18. Thus we can conclude that there is no evidence for a neutral Neon excess local to X1850-087.
Preliminary Timing Results: a possible candidate X–ray period at 1258 sec
=========================================================================
A search for periodicity has been performed with epoch folding techniques around the optical period (1233$\pm{4}$ s, Homer et al. 1996). Our preliminary results shows that the $\chi^2$ distribution reveales a peak at 1258$\pm{20}$ s, but only in the second PN observation. The fact that the X-ray period has been detected only in one of the XMM-Newton observations is puzzling and deserves further investigation. The double-peaked shape of the folded lightcurve (see Fig. 2) is possibly indicative of the presence of some symmetric structure in the accretion disk.
den Herder, J. W., Brinkman, A. C., Kahn, S. M., et al. 2001, A&A, 365, L7
Homer, L., Charles, P. A., Naylor, T., et al., 1996, MNRAS 282, L37
Jansen, F., Lumb, D., Altieri, B., et al. 2001, A&A, 365
Juett, A.M., Psaltis, D., Chakrabarty, D., 2001, ApJ 560, L59
Juett, A.M., Chakrabarty, D., 2003, ApJ, 599, 498
Schulz, N.S., Chakrabarty, D., Marshall, H., et al., 2001, ApJ, 563, 941
Sidoli L., Parmar, A.N., Oosterbroek, T., et al., 2001, A&A 368, 451
Verbunt F., van den Heuvel E. P. J., 1995, in Lewin W. H. G., van Paradijs J. van den Heuvel E. P. J., eds, X-ray Binaries. Cambridge, Cambridge Univ. Press, p. 457
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
The [*disjointness graph*]{} $G=G({\cal S})$ of a set of segments ${\cal S}$ in ${{{\mathbb R}}}^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of $G$ satisfies $\chi(G)\le(\omega(G))^4+(\omega(G))^3$, where $\omega(G)$ denotes the clique number of $G$. It follows, that $\cal S$ has $\Omega(n^{1/5})$ pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.
We show that computing $\omega(G)$ and $\chi(G)$ for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of $G$ in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free ($\omega(G)=2$), but whose chromatic numbers are arbitrarily large.
author:
- '[*János Pach*]{}[^1]'
- '[*Gábor Tardos*]{}[^2]'
- '[*Géza Tóth*]{}[^3]'
date:
-
-
title: Disjointness graphs of segments
---
Introduction
============
Given a set of (geometric) objects, their [*intersection graph*]{} is a graph whose vertices correspond to the objects, two vertices being connected by an edge if and only if their intersection is nonempty. Intersection graphs of intervals on a line [@H57], more generally, chordal graphs [@B61; @Di61] and comparability graphs [@D50], turned out to be [*perfect graphs*]{}, that is, for them and for each of their induced subgraph $H$, we have $\chi(H)=\omega(H)$, where $\chi(H)$ and $\omega(H)$ denote the chromatic number and the clique number of $H$, respectively. It was shown [@HS58] that the complements of these graphs are also perfect, and based on these results, Berge [@B61] conjectured and Lovász [@Lo72] proved that the complement of every perfect graph is perfect.
Most geometrically defined intersection graphs are not perfect. However, in many cases they still have nice coloring properties. For example, Asplund and Grünbaum [@AG60] proved that every intersection graph $G$ of axis-parallel rectangles in the plane satisfies $\chi(G)=O((\omega(G))^2)$. It is not known if the stronger bound $\chi(G)=O(\omega(G))$ also holds for these graphs. For intersection graphs of chords of a circle, Gyárfás [@G85] established the bound $\chi(G)=O((\omega(G))^24^{\omega(G)})$, which was improved to $O(2^{\omega(G)})$ in [@KoK97]. Here we have examples of $\chi(G)$ slightly superlinear in $\omega(G)$ [@Ko88]. In some cases, there is no functional dependence between $\chi$ and $\omega$. The first such example was found by Burling [@Bu65]: there are sets of axis-parallel boxes in ${{{\mathbb R}}}^3$, whose intersection graphs are [*triangle-free*]{} ($\omega=2$), but their chromatic numbers are arbitrarily large. Following Gyárfás and Lehel [@GL83], we call a family $\cal G$ of graphs [*$\chi$-bounded*]{} if there exists a function $f$ such that all elements $G\in\cal G$ satisfy the inequality $\chi(G)\le
f(\omega(G))$. The function $f$ is called a [*bounding function*]{} for $\cal
G$. Heuristically, if a family of graphs is $\chi$-bounded, then its members can be regarded “nearly perfect". Consult [@GL85; @G87; @Ko04] for surveys.
At first glance, one might believe that, in analogy to perfect graphs, a family of intersection graphs is $\chi$-bounded if and only if the family of their complements is. Burling’s above mentioned constructions show that this is not the case: the family of complements of intersection graphs of axis-parallel boxes in ${{{\mathbb R}}}^d$ is $\chi$-bounded with bounding function $f(x)=O(x\log^{d-1}x)$, see [@Ka91]. More recently, Pawlik, Kozik, Krawczyk, Lasoń, Micek, Trotter, and Walczak [@PKK14] have proved that Burling’s triangle-free graphs can be realized as intersection graphs of segments in the plane. Consequently, the family of these graphs is [*not*]{} $\chi$-bounded either. On the other hand, the family of their complements is, see Theorem 0.
To simplify the exposition, we call the complement of the intersection graph of a set of objects their [*disjointness graph*]{}. That is, in the disjointness graph two vertices are connected by an edge if and only if the corresponding objects are disjoint. Using this terminology, the following is a direct consequence of a result of Larman, Matoušek, Pach, and Törőcsik.
[**Theorem 0.**]{} [@LMPT94] [*The family of disjointness graphs of segments in the plane is $\chi$-bounded. More precisely, every such graph $G$ satisfies the inequality $\chi(G)\le(\omega(G))^4$.*]{}
For the proof of Theorem 0, one has to introduce [four]{} partial orders on the family of segments, and apply Dilworth’s theorem [@D50] four times. Although this method does not seem to generalize to higher dimensions, the statement does. We establish the following.
[**Theorem 1.**]{}
*The disjointness graph $G$ of any system of segments in ${{{\mathbb R}}}^d, d\ge 2$ satisfies the inequality $\chi(G)\le(\omega(G))^4+(\omega(G))^3$.*
Moreover, there is a polynomial time algorithm that, given the segments corresponding to the vertices of $G$, finds a complete subgraph $K\subseteq G$ and a proper coloring of $G$ with at most $|V(K)|^4+|V(K)|^3$ colors.
If we consider full lines in place of segments, we obtain stronger bounds.
[**Theorem 2.**]{}
*(i) Let $G$ be the disjointness graph of a set of lines in ${{{\mathbb R}}}^d, \; d\ge 3.$ Then we have $\;\; \chi(G)\le(\omega(G))^3.$*
\(ii) Let $G$ be the disjointness graph of a set of lines in the projective space ${{{\mathbb P}}}^d, \; d\ge 3.$ Then we have $\;\; \chi(G)\le(\omega(G))^2.$
In both cases, there are polynomial time algorithms that, given the lines corresponding to the vertices of $G$, find complete subgraphs $K\subseteq G$ and proper colorings of $G$ with at most $|V(K)|^3$ and $|V(K)|^2$ colors, respectively.
Note that the difference between the two scenarios comes from the fact that parallel lines in the Euclidean space are disjoint, but the corresponding lines in the projective space intersect.
Most computational problems for geometric intersection and disjointness graphs are hard. It was shown by Kratochvíl and Nešetřil [@KrN90] and by Cabello, Cardinal, and Langerman [@CaCL13] that finding the clique number $\omega(G)$ resp. the independence number $\alpha(G)$ of disjointness graphs of segments in the plane are NP-hard. It is also known that computing the chromatic number $\chi(G)$ of disjointness and intersection graphs of segments in the plane is NP-hard [@EET86]. Our next theorem shows that some of the analogous problems are also NP-hard for disjointness graphs of lines in space, while others are tractable in this case. In particular, according to Theorem 3(i), in a disjointness graph $G$ of lines, it is NP-hard to determine $\omega(G)$ and $\chi(G)$. In view of this, it is interesting that one can design polynomial time algorithms to find proper colorings and complete subgraphs in $G$, where the number of colors is bounded in terms of the size of the complete subgraphs, in the way specified in the closing statements of Theorems 1 and 2.
[**Theorem 3.**]{}
*(i) Computing the clique number $\omega(G)$ and the chromatic number $\chi(G)$ of disjointness graphs of lines in ${{\mathbb R}}^3$ or in ${{\mathbb P}}^3$ are NP-hard problems.*
\(ii) Computing the independence number $\alpha(G)$ of disjointness graphs of lines in ${{\mathbb R}}^3$ or in ${{\mathbb P}}^3$, and deciding for a fixed $k$ whether $\chi(G)\le k$, can be done in polynomial time.
The bounding functions in Theorems 0, 1, and 2 are not likely to be optimal. As for Theorem 2 (i), we will prove that there are disjointness graphs $G$ of lines in $\mathbb{R}^3$ for which $\frac{\chi(G)}{\omega(G)}$ are arbitrarily large. Our best constructions for disjointness graphs $G'$ of lines in the projective space satisfy $\chi(G')\ge2\omega(G')-1$; see Theorem 2.3.
The proof of Theorem 1 is based on Theorem 0. Any strengthening of Theorem 0 leads to improvements of our results. For example, if $\chi(G)=O((\omega(G))^\gamma)$ holds with any $3\le \gamma\le 4$ for the disjointness graph of every set of segments in the plane, then the proof of Theorem 1 implies the same bound for disjointness graphs of segments in higher dimensions. In fact, it is sufficient to verify this statement in $3$ dimensions. For $d\ge 4$, we can find a projection in a generic direction to the $3$-dimensional space that does not create additional intersections and then we can apply the $3$-dimensional bound. We focus on the case $d=3$.
It follows immediately from Theorem 0 that the disjointness (and, hence, the intersection) graph of any system of $n$ segments in the plane has a clique or an independent set of size at least $n^{1/5}$. Indeed, denoting by $\alpha(G)$ the maximum number of independent vertices in $G$, we have $$\alpha(G)\ge \frac{n}{\chi(G)}\ge\frac{n}{(\omega(G))^4},$$ so that $\alpha(G)(\omega(G))^4\ge n$. Analogously, Theorem 1 implies that $\max(\alpha(G),\omega(G))\ge (1-o(1))n^{1/5}$ holds for disjointness (and intersection) graphs of segments in any dimension $d\ge 2$. For disjointness graphs of $n$ lines in $\mathbb{R}^d$ (respectively, in $\mathbb{P}^d$), we obtain that $\max(\alpha(G),\omega(G))$ is $\Omega(n^{1/4})$ (resp., $\Omega(n^{1/3})$). Using more advanced algebraic techniques, Cardinal, Payne, and Solomon [@CPS16] proved the stronger bounds $\Omega(n^{1/3})$ (resp., $\Omega(n^{1/2})$).
If the order of magnitude of the bounding functions in Theorems 0 and 1 are improved, then the improvement carries over to the lower bound on $\max(\alpha(G),\omega(G))$. Despite many efforts [@LMPT94; @KaPT97; @Ky12] to construct intersection graphs of planar segments with small clique and independence numbers, the best known construction, due to Kynčl [@Ky12], gives only $$\max(\alpha(G),\omega(G))\le n^{\log 8/\log 169}\approx n^{0.405},$$ where $n$ is the number of vertices. This bound is roughly the square of the best known lower bound.
Our next theorem shows that any improvement of the lower bound on $\max(\alpha(G),\omega(G))$ in the plane, even if it was not achieved by an improvement of the bounding function in Theorem 0, would also carry over to higher dimensions.
[**Theorem 4.**]{} [*If the disjointness graph of any set of $n$ segments in the plane has a clique or an independent set of size $\Omega(n^\beta)$ for some fixed $\beta\le1/4$, then the same is true for disjointness graphs of segments in ${{{\mathbb R}}}^d$ for any $d>2$.*]{}
A continuous arc in the plane is called a [*string*]{}. One may wonder whether Theorem 0 can be extended to disjointness graphs of strings in place of segments. The answer is no, in a very strong sense.
[**Theorem 5.**]{} [*There exist triangle-free disjointness graphs of $n$ strings in the plane with arbitrarily large chromatic numbers. Moreover, we can assume that these strings are simple polygonal paths consisting of at most $4$ segments.*]{}
Very recently, Mütze, Walczak, and Wiechert [@MWW17] improved this result. They proved that the statement holds even if the strings are simple polygonal paths of at most $3$ segments, moreover, any two intersect at most once.
The following problems remain open.
[**Problem 6.**]{}
*(i) Is the family of disjointness graphs of polygonal paths, each consisting of at most two segments, $\chi$-bounded?*
\(ii) Is the previous statement true under the additional assumption that any two of the polygonal paths intersect in at most one point?
[**Problem 7.**]{} [*Is the family of intersection graphs of lines in ${{{\mathbb R}}}^3$ $\chi$-bounded?*]{}
By Theorem 2, the family of [*complements*]{} of intersection graphs of lines in ${{{\mathbb R}}}^3$ is $\chi$-bounded.
This paper is organized as follows. In the next section, we prove Theorem 2, which is needed for the proof of Theorem 1. Theorem 1 is established in Section 3. The proof of Theorem 4 is presented in Section 4. In Section 5, we construct several examples of disjointness graphs whose chromatic numbers are much larger than their clique numbers. In particular, we prove Theorem 5 and some similar statements. The last section contains the proof of Theorem 3 and remarks on the computational complexity of related problems.
Disjointness graphs of lines–Proof of Theorem 2
===============================================
[**Claim 2.1.**]{} [*Let $G$ be the disjointness graph of a set of $n$ lines in ${{\mathbb P}}^d$. If $G$ has an isolated vertex, then $G$ is perfect.*]{}
[**Proof.**]{} Let $\ell_0\in V(G)$ be a line representing an isolated vertex of $G$. Consider the bipartite multigraph $H$ with vertex set $V(H)=A\cup B$, where $A$ consists of all points of $\ell_0$ that belong to at least one other line $\ell\in V(G)$, and $B$ is the set of all ($2$-dimensional) planes passing through $\ell_0$ that contain at least one other line $\ell\in V(G)$ different from $\ell_0$. We associate with any line $\ell\in V(G)$ different from $\ell_0$ an edge $e_\ell$ of $H$, connecting the point $p=\ell\cap\ell_0\in A$ to the plane $\pi\in B$ that contains $\ell$. Note that there may be several parallel edges in $H$. See Figure 1.
Observe that two lines $\ell,\ell'\in V(G)\setminus\{\ell_0\}$ intersect if and only if $e_\ell$ and $e_{\ell'}$ share an endpoint. This means that $G$ minus the isolated vertex $\ell_0$ is isomorphic to the complement of the line graph of $H$. The line graphs of bipartite multigraphs and their complements are known to be perfect. (For the complements of line graphs, this is the König-Hall theorem; see, e. g., [@L93].) The graph $G$ can be obtained by adding the isolated vertex $\ell_0$ to a perfect graph, and is, therefore, also perfect. $\Box$
[**Proof of Theorem 2.**]{} We start with the proof of part (ii). Let $G$ be a disjointness graph of lines in ${{\mathbb P}}^d$. Let $C\subseteq G$ be a maximal clique in $G$. Clearly, $|C|\le\omega(G)$. By the maximality of $C$, for every $\ell\in V(G)\setminus C$, there exists $c\in C$ that is not adjacent to $\ell$ in $G$. Hence, there is a partition of $V(G)$ into disjoint sets $V_c, c\in C,$ such that $c\in V_c$ and $c$ is an isolated vertex in the induced subgraph $G[V_c]$ of $G$. Applying Claim 2.1 separately to each subgraph $G[V_c]$, we obtain $$\chi(G)\le\sum_{c\in C}\chi(G[V_c])=\sum_{c\in C}\omega(G[V_c])
\le|C|\omega(G)\le(\omega(G))^2.$$
Now we turn to the proof of part (i) of Theorem 2. Let $G$ be a disjointness graph of lines in ${{\mathbb R}}^d$. Consider the lines in $V(G)$ as lines in the projective space ${{\mathbb P}}^d$, and consider the disjointness graph $G'$ of these projective lines. Clearly, $G'$ is a subgraph of $G$ with the lines $\ell$, $\ell'\in V(G)$ adjacent in $G$ but not adjacent in $G'$ if and only if $\ell$ and $\ell'$ are parallel. Thus, an independent set in $G'$ induces a disjoint union of complete subgraphs in $G$, where the vertices of each complete subgraph correspond to pairwise parallel lines. If $k$ is the maximal number of pairwise parallel lines in $V(G)$, then $k\le\omega(G)$ and each independent set in $G'$ can be partitioned into at most $k$ independent sets in $G$. Applying part (ii), we obtain $$\chi(G)\le k\chi(G')\le \omega(G)(\omega(G'))^2\le(\omega(G))^3.$$
Finally, we prove the last claim concerning polynomial time algorithms. In the proof of part (ii), we first took a maximal clique $C$ in $G$. Such a clique can be efficiently found by a greedy algorithm. The partition of $V(G)$ into subsets $V_c, c\in C,$ such that $c\in V_c$ is an isolated vertex in the subgraph $G[V_c]$, can also be done efficiently. It remains to find a clique of maximum size and a proper coloring of each perfect graph $G[V_c]$ with the smallest number of colors. It is well known that for perfect graphs, both of these tasks can be completed in polynomial time. See e.g. Corollary 9.4.8 on page 298 of [@GLS88]. Alternatively, notice that in the proof of Claim 2.1 we showed that $G[V_c]$ is, in fact, the complement of the line graph of a bipartite multigraph (plus an isolated vertex). Therefore, finding a maximum size complete subgraph corresponds to finding a maximum size matching in a bipartite graph, while finding an optimal proper coloring of $G[V_c]$ corresponds to finding a minimal size vertex cover in a bipartite graph. This can be accomplished by much simpler and faster algorithms than the general purpose algorithms developed for perfect graphs.
To finish the proof of the algorithmic claim for part (ii), we can simply output as $K$ the set $C$ or one of the largest maximum cliques in $G[V_c]$ over all $c\in C$, whichever is larger. We color each $V_c$ optimally, with pairwise disjoint sets of colors.
For the algorithmic claim about part (i), first color the corresponding arrangement of projective lines, and then refine the coloring by partitioning each color class into at most $k$ smaller classes, where $k$ is the maximum number of parallel lines in the arrangement. It is easy to find the value of $k$, just partition the lines into groups of parallel lines. Output as $K$ the set we found for the projective lines, or a set of $k$ parallel lines, whichever is larger. $\Box$
[**Theorem 2.3.**]{}
*(i) There exist disjointness graphs $G$ of families of lines in $\mathbb{R}^3$ for which the ratio $\chi(G)/\omega(G)$ is arbitrarily large.*
\(ii) For any $k$ one can find a system of lines in $\mathbb{P}^3$ whose disjointness graph $G$ satisfies $\omega(G)=k$ and $\chi(G)=2k-1$.
[**Proof.**]{} First, we prove (i). For some $m$ and $d$ to be determined later, consider the set $W_m^d$ of integer points in the $d$-dimensional hypercube $[1,m]^d$. That is, $W_m^d=\{ 1, 2, \ldots , m\}^d$. A [*combinatorial line*]{} is a sequence of $m$ distinct points of $x^1,\ldots x^m\in W_m^d$ such that for every $1\le i\le d$, their $i$th coordinates $(x^j)_i$ are either the same for all $1\le j\le m$ or we have $(x^j)_i=j$ for all $1\le j\le m$. Note that the points of any combinatorial line lie on a geometric straight line. Let ${\cal L}$ denote the set of these geometric lines.
Let $G$ denote the disjointness graph of $\cal L$. Since each line in $\cal L$ passes through $m$ points of $W_m^d$, and $|W_m^d|=m^d$, we have $\omega(G)\le m^{d-1}$. (It is easy to see that equality holds here, but we do not need this fact for the proof.)
Consider any proper coloring of $G$. The color classes are families of pairwise crossing lines in $\cal L$. Observe that any such family has a common point in $W_m^d$, except some families consisting of $3$ lines. Take an optimal proper coloring of $G$ with $\chi(G)$ colors, and split each $3$-element color class into two smaller classes. In the resulting coloring, there are at most $2\chi(G)$ color classes, each of which has a point of $W_m^d$ in common. This means that the set of at most $2\chi(G)$ points of $W_m^d$ (the “centers” of the color classes) “hits” every combinatorial line. By the density version of the Hales-Jewett theorem, due to Furstenberg and Katznelson [@Bo98; @FK91], if $d$ is large enough relative to $m$, then any set containing fewer than half of the points of $W_m^d$ will miss an entire combinatorial line. Choosing any $m$ and a sufficiently large $d$ depending on $m$, we conclude that $2\chi(G)\ge
m^d/2$ and $\chi(G)/\omega(G)\ge m/4$.
Note that the family $\cal L$ consists of lines in ${{\mathbb R}}^d$. To find a similar family in 3-space, simply take the image of $\cal L$ under a projection to ${{\mathbb R}}^3$. One can pick a generic projection that does not change the disjointness graph $G$. This completes the proof of part (i). Note that the same construction does not work for projective lines, as the combinatorial lines in $W_m^d$ fall into $2^d-1$ parallel classes, so the chromatic number of the corresponding projective disjointness graph is smaller than $2^d$.
To establish part (ii), fix a positive integer $k$, and consider a set $S$ of $2k+1$ points in general position (no four in a plane) in $\mathbb{R}^3\subseteq
\mathbb{P}^3$. Let $\cal L$ denote the set of $\binom{2k+1}{2}$ lines determined by them. Note that by the general position assumption, two lines in $\cal L$ intersect if and only if they have a point of $S$ in common. This means that the disjointness graph $G$ of $\cal L$ is isomorphic to the [*Kneser graph*]{} $G^{*}(2k+1,2)$ formed by all $2$-element subsets of a $(2k+1)$-element set. Obviously, $\omega(G^{*}(n,m))=\lfloor n/m\rfloor$, so $\omega(G)=k$. By a celebrated result of Lovász [@Lo78], $\chi G^*(n,m)=n-2m+2$ for all $n\ge2m-1$. Thus, we have $\chi(G)=2k-1$, as claimed. $\Box$
Disjointness graphs of segments–Proof of Theorem 1
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If all segments lie in the same plane, then by Theorem 0 we have $\chi(G)\le(\omega(G))^4$. Our next theorem generalizes this result to the case where the segments lie in a bounded number of distinct planes.
[**Theorem 3.1.**]{}
*Let $G$ be the disjointness graph of a set of segments in $\mathbb{R}^d, d>2,$ that lie in the union of $k$ two-dimensional planes. We have $$\chi(G)\le(k-1)\omega(G)+(\omega(G))^4.$$*
Given the segments representing the vertices of $G$ and $k$ planes containing them, there is a polynomial time algorithm to find a complete subgraph $K\subseteq G$ and a proper coloring of $G$ with at most $(k-1)|V(K)|+|V(K)|^4$ colors.
[**Proof.**]{} Let $\pi_1, \pi_2, \ldots, \pi_k$ be the planes containing the segments. Partition the vertex set of $G$ into the classes $V_1, V_2, \ldots, V_k$ by putting a segment $s$ into the class $V_i$, where $i$ is the [*largest*]{} index for which $\pi_i$ contains $s$.
For $i=1, 2,\ldots, k,$ we define subsets $W_i, Z_i\subseteq V_i$ with $Z_i\subseteq W_i\subseteq V_i$ by a recursive procedure, as follows. Let $W_1=V_1$ and let $Z_1\subseteq W_1$ be a [*maximal size*]{} clique in $G[W_1]$.
Assume that the sets $W_1,\dots, W_i$ and $Z_1,\ldots, Z_i$ have already been defined for some $i<k$. Let $W_{i+1}$ denote the set of all vertices in $V_{i+1}$ that are adjacent to every vertex in $Z_1\cup Z_2\cup\ldots\cup
Z_i$, and let $Z_{i+1}$ be a maximal size clique in $G[W_{i+1}]$. By definition, $\bigcup_{i=1}^kZ_i$ induces a complete subgraph in $G$, and we have $$\sum_{i=1}^k|Z_i|\le\omega(G).$$
Let $s$ be a segment belonging to $Z_i$, for some $1\le i< k$. A point $p$ of $s$ is called a [*piercing point*]{} if $p\in \pi_j$ for some $j>i$. Notice that in this case, $s$ “pierces” the plane $\pi_j$ in a single point, otherwise we would have $s\subset\pi_j$, contradicting our assumption that $s\in V_i$. Letting $P$ denote the set of piercing points of all segments in $\bigcup_{i=1}^kZ_i$, we have $$|P|\le\sum_{i=1}^k(k-i)|Z_i|\le(k-1)\sum_{i=1}^k|Z_i|\le(k-1)\omega(G).$$
Let $V_0=V(G)\setminus\bigcup_{i=1}^kW_i$. We claim that every segment in $V_0$ contains at least one piercing point. Indeed, if $s\in V_i\setminus W_i$ for some $i\le k$, then $s$ is not adjacent in $G$ to at least one segment $t\in Z_1\cup\ldots\cup Z_{i-1}$. Thus, $s$ and $t$ are not disjoint, and their intersection point is a piercing point, at which $t$ pierces the plane $\pi_i$.
Assign a color to each piercing point $p\in P$. Coloring every segment in $V_0$ by the color of one of its piercing points, we get a proper coloring of $G[V_0]$ with $|P|$ colors, so that $\chi(G[V_0])\le|P|.$
For every $i\le k$, all segments of $W_i$ lie in the plane $\pi_i$. Therefore, we can apply Theorem 0 to their disjointness graph $G[W_i]$, to conclude that $\chi(G[W_i])\le(\omega(G[W_i]))^4$. By definition, $Z_i$ induces a maximum complete subgraph in $G[W_i]$, hence $|Z_i|=\omega(G[W_i])$ and $\chi(G[W_i])\le|Z_i|^4$.
Putting together the above estimates, and taking into account that $\bigcup_{i=1}^kZ_i$ induces a complete subgraph in $G$, we obtain $$\chi(G)\le\chi(G[V_0])+\sum_{i=1}^k\chi(G[W_i])\le|P|+\sum_{i=1}^k|Z_i|^4$$ $$\le(k-1)\omega(G)+(\sum_{i=1}^k|Z_i|)^4\le(k-1)\omega(G)+(\omega(G))^4,$$ as required.
We can turn this estimate into a polynomial time algorithm as required, using the fact that the proof of Theorem 0 is constructive. In particular, we use that, given a family of segments in the plane, one can efficiently find a subfamily $K$ of pairwise disjoint segments and a proper coloring of the disjointness graph with at most $|K|^4$ colors. This readily follows from the proof of Theorem 0, based on the four easily computable (semi-algebraic) partial orders on the family of segments, introduced in [@LMPT94].
Our algorithm finds the sets $V_i$, as in the proof. However, finding $W_i$ and a maximum size clique $Z_i\subseteq W_i$ is a challenge. Instead, we use the constructive version of Theorem 0 to find $Z_i\subseteq W_i$ and a proper coloring of $G[W_i]$. The definition of $W_i$ remains unchanged. Next, the algorithm identifies the piercing points.
The algorithm outputs the clique $K=\bigcup Z_i$ and the coloring of $G$. The latter one is obtained by combining the previously constructed colorings of the subgraphs $G[W_i]$ (using disjoint sets of colors for different subgraphs), and coloring each remaining vertex by a previously unused color, associated with one of the piercing points the corresponding segment passes through. $\Box$
[**Proof of Theorem 1.**]{} Consider the set of all lines in the [ *projective*]{} space $\mathbb{P}^d$ that contain at least one segment belonging to $V(G)$. Let $\bar{G}'$ denote the disjointness graph of these lines. Obviously, we have $\omega(\bar{G}')\le\omega(G)$. Thus, Theorem 2(ii) implies that $$\chi(\bar{G}')\le (\omega(\bar{G}'))^2\le(\omega(G))^2.$$
Let $C$ be the set of lines corresponding to the vertices of a maximum complete subgraph in $\bar{G}'$. Fix an optimal proper coloring of $\bar{G}'$. Suppose that we used $k$ “planar” colors (each such color is given to a set of lines that lie in the same plane) and $\chi(\bar{G'})-k$ “pointed” colors (each given to the vertices corresponding to a set of lines passing through a common point).
Consider now $G$, the disjointness graph of the segments. Let $G_0$ denote the subgraph of $G$ induced by the set of segments whose supporting lines received one of the $k$ planar colors in the above coloring of $\bar{G}'$. These segments lie in at most $k$ planes. Therefore, applying Theorem 3.1 to $G_0$, we obtain $$\chi(G_0)\le (k-1)\omega(G_0)+(\omega(G_0))^4
\le (k-1)\omega(G)+(\omega(G))^4.$$
For $i, 1\le i\le \chi(\bar{G}')-k,$ let $G_i$ denote the subgraph of $G$ induced by the set of segments whose supporting lines are colored by the $i$th pointed color. It is easy to see that $G_i$ is the complement of a chordal graph. That is, the complement of $G_i$ contains no induced cycle of length larger than $3$. According to a theorem of Hajnal and Surányi [@HS58], any graph with this property is perfect, so that $$\chi(G_i)=\omega(G_i)\le\omega(G).$$
Putting these bounds together, we obtain that $$\chi(G)\le \chi(G_0)+\sum_{i=1}^{\chi(\bar{G}')-k}\chi(G_i)
\le(k-1)\omega(G)+(\omega(G))^4+\sum_{i=1}^{\chi(\bar{G}')-k}\omega(G)$$ $$\le((\omega(\bar{G}'))^2-1)\omega(G)+(\omega(G))^4<(\omega(G))^3
+(\omega(G))^4.$$
To prove the algorithmic claim in Theorem 1, we first apply the algorithm of Theorem 2 to the disjointness graph $\bar G'$. We distinguish between planar and pointed color classes and find the subgraphs $G_i$. We output a coloring of $G$, where for each $G_i, i>0$ we use the smallest possible number of colors ($G_i$ is perfect, so its optimal coloring can be found in polynomial time), and we color $G_0$ by the algorithm described in Theorem 3.1. The subgraphs $G_i$ are colored using pairwise disjoint sets of colors. We output the largest clique $K$ that we can find. This may belong to a subgraph $G_i$ with $i>0$, or may be found in $G_0$ or in $\bar G'$ by the algorithms given by Theorem 3.1 or Theorem 2, respectively. (In the last case, we need to turn a clique in $\bar G'$ into a clique of the same size in $G$, by picking an arbitrary segment from each of the pairwise disjoint lines.) $\Box$
Ramsey-type bounds in $\mathbb{R}^2$ vs. $\mathbb{R}^3$–Proof of Theorem 4
==========================================================================
As we have pointed out in the Introduction, it is sufficient to establish Theorem 4 in $\mathbb{R}^3$. We rephrase Theorem 4 for this case in the following form.
[**Theorem 4.1.**]{}
*Let $f(m)$ be a function with the property that for any disjointness graph $G$ of a system of segments in $\mathbb{R}^2$ with $\max(\alpha(G),\omega(G))\le m$ we have $|V(G)|\le f(m).$*
Then for any disjointness graph $G$ of a system of segments in $\mathbb{R}^3$ with $\max(\alpha(G),\omega(G))$ $\le m$ we have $|V(G)|\le f(m)+m^4.$
Applying Theorem 4.1 with $f(k)=ck^{1/\beta}$, Theorem 4 immediately follows. We prove Theorem 4.1 by adapting the proof of Theorem 3.1.
[**Proof of Theorem 4.1.**]{} Let $G$ be the disjointness graph of a set of segments in $\mathbb{R}^3$ with $\omega(G)\le m$ and $\alpha(G)\le m$.
First, assume that all segments lie in the union of $k$ planes, for some $k\ge 1$. Define the sets of vertices $V_i$, $W_i$, and $Z_i$ for every $1\le i\le k$, as in the proof of Theorem 3.1, and let $V_0=V(G)\setminus\bigcup_{i=1}^kW_i$. Since all elements of $W_i$ lie in the same plane, the subgraph induced by them is a planar segment disjointness graph for every $i\ge1$. We can clearly represent these graphs by segments in a common plane $\pi$ such that two segments intersect if and only they come from the same set $W_i$ and there they intersect. In this way, we obtain a system of segments in the plane whose disjointness graph $G^*$ is the [*join*]{} of the graphs $G[W_i]$, i.e., $G^*$ is obtained by taking the disjoint union of $G[W_i]$ (for all $i\ge1$) and adding all edges between $W_i$ and $W_j$ for every pair $i\ne
j$. Clearly, we have $$\omega(G^*)=\sum_{i=1}^k\omega(G[W_i])=\sum_{i=1}^k|Z_i|\le\omega(G)\le m,$$ and $$\alpha(G^*)=\max_{i=1}^k\alpha(G[W_i])\le\alpha(G)\le m.$$
By our assumption, $G^*$ has at most $f(m)$ vertices, so that $\sum_{i=1}^k|W_i|\le f(m).$ As we have seen in the proof of Theorem 3.1, the total number of piercing points is at most $(k-1)\sum_{i=1}^k|Z_i|\le(k-1)\omega(G)<km$, and each segment in $V_0$ contains at least one of them. Each piercing point is contained in at most $m$ segments, because these segments induce an independent set in $G$. Thus, we have $|V_0|<km^2$ and $$|V(G)|=|V_0|+\bigcup_{i=1}^k|W_i|<km^2+|V(G^*)|\le km^2+f(m).$$
Now we turn to the general case, where there is no bound on the number of planes containing the segments. As in the proof of Theorem 1, we consider the disjointness graph $\bar{G}'$ of the supporting lines of the segments in the projective space $\mathbb{P}^3$. Clearly, we have $\omega(\bar{G}')\le\omega(G)\le m$, so by Theorem 1 we have $\chi(\bar{G}')\le m^2$. Following the proof of Theorem 1, take an optimal coloring of $\bar{G}'$, and let $G_0$ denote the subgraph of $G$ induced by the segments whose supporting lines received one of the planar colors. Letting $k$ denote the number of planar colors, for every $i, 1\le
i\le \omega(\bar{G}')-k,$ let $G_i$ denote the subgraph of $G$ induced by the set of segments whose supporting lines received the $i$th pointed color. As in the proof of Theorem 1, every $G_i, i\ge 1$ is perfect and, hence, its number of vertices satisfies $$|V(G_i)|\le\chi(G_i)\alpha(G_i)\le\omega(G_i)\alpha(G_i)\le\omega(G)\alpha(G)\le
m^2.$$ The segments belonging to $V(G_0)$ lie in at most $k$ planes. In view of the previous paragraph, $|V(G_0)|\le km^2+f(m)$ vertices. Combining the above bounds, we obtain $$|V(G)|=|V(G_0)|+\sum_{i=1}^{\chi(\bar{G}')-k}|V(G_i)|
\le km^2+f(m)+(\chi(\bar{G}')-k)m^2$$ $$\le km^2+f(m)+(m^2-k)m^2\le f(m)+m^4,$$ which completes the proof. $\Box$
Constructions–Proof of Theorem 5
================================
The aim of this section is to describe various arrangements of geometric objects in 2, 3, and 4 dimensions with triangle-free disjointness graphs, whose chromatic numbers grow logarithmically with the number of objects. (This is much faster than the rate of growth in Theorem 2.3.) Our constructions can be regarded as geometric realizations of a sequence of graphs discovered by Erdős and Hajnal.
[**Definition 5.1.**]{} [@EH64]. [*Given $m>1$, let $H\noindent_m$, the [*$m$-th shift graph*]{}, be a graph whose vertex set consists of all ordered pairs $(i, j)$ with $1\le i < j\le m$, and two pairs $(i, j)$ and $(k, l)$ are connected by an edge if and only if $j=k$ or $l=i$.*]{}
Obviously, $H_m$ is triangle-free for every $m>1$. It is not hard to show (see, e.g., [@L93], Problem 9.26) that $\chi(H_m) = \lceil \log_2m\rceil$. Therefore, Theorem 5 follows directly from part (vii) of the next theorem.
[**Theorem 5.2.**]{}
*For every $m$, the shift graph $H_m$ can be obtained as a disjointness graph, where each vertex is represented by*
\(i) a line minus a point in ${{\mathbb R}}^2$;
\(ii) a two-dimensional plane in ${{\mathbb R}}^4$;
\(iii) the intersection of two general position half-spaces in ${{\mathbb R}}^3$;
\(iv) the union of two segments in ${{\mathbb R}}^2$;
\(v) a triangle in ${{\mathbb R}}^4$;
\(vi) a simplex in ${{\mathbb R}}^3$;
\(vii) a polygonal curve in ${{\mathbb R}}^2$, consisting of four line segments.
[**Proof.**]{} (i) Let $\ell_1, \ldots , \ell_m$ be lines in general position in the plane. For any $1\le i < j\le m$, let us represent the pair $(i, j)$ by the “pointed line” $p_{ij}=\ell_i\setminus\ell_j$.
Fix $1\le i < j\le m$, $1\le k < l\le m$, and set $X=p_{ij}\cap
p_{kl}=(\ell_i\cap \ell_k)\setminus(\ell_j\cup\ell_l)$. If $i=k$, then $X$ is an infinite set.
Otherwise, $\ell_i\cap\ell_k$ consists of a single point. In this case, $X$ is empty if and only if this point belongs to $\ell_j\cup\ell_l$. By the general position assumption, this happens if and only if $j=k$ or $l=i$. Thus, the disjointness graph of the sets $p_{ij},\; 1\le i<j\le m,$ is isomorphic to the shift graph $H_m$.
\(ii) Let $h_1, \ldots h_m$ be hyperplanes in general position in ${{\mathbb R}}^4$. For every $i$, fix another hyperplane $h'_i$, parallel (but not identical) to $h_i$. For any $1\le i < j\le m$, represent the pair $(i, j)$ by the two dimensional plane $p_{ij}=h_i\cap h'_j$.
Given $1\le i < j\le m$, $1\le k < l\le m$, the set $X=p_{ij}\cap p_{k,l}=h_i\cap h'_j\cap h_k\cap h'_l$ is the intersection of four hyperplanes. If the four hyperplanes are in general position, then $X$ consists of a single point.
If the hyperplanes are not in general position, then some of the four indices must coincide. If $i=k$ or $j=l$, then two of the hyperplanes coincide and $X$ is a line. In the remaining cases, when $j=k$ or $l=i$, among the four hyperplanes two are parallel, so their intersection $X$ is empty.
\(iii) For $i=1,\dots,m$, define the half-space $h_i$ as $$h_i=\{(x,y,z)\in{{\mathbb R}}^3\mid ix+i^2y+i^3z<1\}.$$ Note that the bounding planes of these half-spaces are in general position. For any $1\le i < j\le m$, represent the pair $(i, j)$ by $p_{ij}=h_j\setminus h_i$.
Now let $1\le i < j\le m$, $1\le k < l\le m$. If $j=k$ or $l=i$, the sets $p_{ij}$ and $p_{kl}$ are obviously disjoint. If $i=k$ or $j=l$, then $p_{ij}\cap p_{kl}$ is the intersection of at most 3 half-spaces in general position, so it is unbounded and not empty.
It remains to analyze the case when all four indices are distinct. This requires some calculation. We assume without loss of generality that $j<l$. Consider the point $P=(x,y,z)\in{{\mathbb R}}^3$ with $x=\frac1i+\frac1j+\frac1k$, $y=-\frac1{ij}-\frac1{jk}-\frac1{ki}$ and $z=\frac1{ijk}$. This is the intersection point of the bounding planes of $h_i$, $h_j$ and $h_k$. Therefore, the polynomial $zu^3+yu^2+xu-1$ vanishes at $u=i,j,k$, and it must be positive at $u=l$, as $l>i,j,k$ and the leading coefficient is positive. This means that $P$ lies in the open half-space $h_l$. As the bounding planes of $h_i$, $h_j$ and $h_k$ are in general position, one can find a point $P'$ arbitrarily close to $P$ (the intersection point of these half-planes) with $P'\in
h_j\setminus(h_i\cup h_k)$. If we choose $P'$ close enough to $P$, it will also belong to $h_l$. Thus, $P'\in p_{ij}\cap p_{kl}$, and so $p_{ij}$ and $p_{kl}$ are not disjoint.
(iv), (v), and (vi) directly follow from (i), (ii) and (iii), respectively, by replacing the unbounded geometric objects representing the vertices with their sufficiently large bounded subsets.
\(vii) Let $C$ be an almost vertical, very short curve (arc) in the plane, convex from the right (that is, the set of points to the right of $C$ is convex) lying in a small neighborhood of $(0,1)$. Let $p_1, p_2, \ldots , p_{m}$ be a sequence of $m$ points on $C$ such that $p_j$ is above $p_i$ if and only if $j>i$. For every $1\le i\le m$, let $T_i$ be an equilateral triangle whose base is horizontal, whose upper vertex is $p_{i}$, and whose center is on the $x$-axis. Let $q_i$ and $r_i$ be the lower right and lower left vertices of $T_i$, respectively. It is easy to see that $T_j$ contains $T_i$ in its interior if $j>i$. Let $s_i$ be a point on $r_ip_{i}$, very close to $p_i$.
Let us represent the vertex $(i,j)$ of the shift graph $H_m$ by the polygonal curve $p_{ij}=t_{ij}p_jq_jr_js_j$, where the point $t_{ij}$ is on the $x$-axis slightly to the left of the line $p_ip_j$. Note that if $C$ is short enough and close enough to vertical, then $t_{ij}$ can be chosen so that it belongs to the interior of all triangles $T_k$ for $1\le k\le m$. In particular, the entire polygonal path $p_{ij}$ belongs to $T_j$.
It depends on our earlier choices of the vertices $p_{i'}$, how close we have to choose $s_i$ to $p_i$. Analogously, it depends on our earlier choices of $p_{i'}$ and $s_{i'}$, how close we have to choose $t_{ij}$ to the line. Instead of describing an explicit construction, we simply claim that with proper choices of these points, we obtain a disjointness representation of the shift graph.
To see this, let $1\le i < j\le m$, $1\le k < l\le m$. If $j=l$, then three of the four line segments in $p_{ij}$ and $p_{kl}$ are the same, so they intersect. Otherwise, assume without loss of generality that $j<l$. As noted above, $p_{ij}$ belongs to the triangle $T_j$, which, in turn, lies in the interior of $T_l$. Three segments of $p_{kl}$ lie on the edges of $T_l$, so if $p_{ij}$ and $p_{kl}$ meet, the fourth segment, $t_{kl}p_l$, must meet $p_{ij}$. This segment enters the triangle $T_j$, so it meets one of its edges. Namely, for $j>k$ it follows from the convexity of the curve $C$ that the segment $t_{kl}p_l$ intersects the edge $p_jq_j$ and, hence, also $p_{ij}$. Analogously, if $j<k$, then $t_{kl}p_l$ intersects the interior of the edge $r_jp_j$. This is true even if $t_{kl}$ were chosen [*on*]{} the line $p_kp_l$, so choosing $s_j$ close enough to $p_j$, one can make sure that $t_{kl}p_l$ intersects $r_js_j$ and, hence, also $p_{ij}$. On the other hand, if $j=k$, we choose $t_{kl}$ so that $t_{kl}p_l$ is just slightly to the left of $p_j=p_k$, so it enters $T_j$ through the interior of the segment $s_jp_j$ that is [*not*]{} contained in $p_{ij}$. To see that in this case $p_{ij}$ and $p_{kl}$ are disjoint, it is enough to check that $t_{kl}p_l$ and $t_{ij}p_j$ are disjoint. This is true, because $p_j$ is on the right of $t_{kl}p_l$ and (from the convexity of $C$) the slope of the segments is such that $p_j$ is the closest point of the segment $t_{ij}p_j$ to $t_{kl}p_l$. $\Box$
Complexity issues–Proof of Theorem 3
====================================
The aim of this section is to outline the proof of Theorem 3 and to establish some related complexity results. For simplicity, we only consider systems of lines in the [*projective*]{} space ${{\mathbb P}}^3$. It is easy to see that by removing a generic hyperplane (not containing any of the intersection points), we can turn a system of projective lines into a system of lines into ${{\mathbb R}}^3$ without changing the corresponding disjointness graph.
It is more convenient to speak about intersection graphs rather than their complement in formulating the next theorem.
[**Theorem 6.1**]{}
*(i) If $G$ is a graph with maximum degree at most $3$, then $G$ is an intersection graph of lines in ${{\mathbb P}}^3$.*
\(ii) For an arbitrary graph $G$ the [*line graph*]{} of $G$ is an intersection graph of lines in ${{\mathbb P}}^3$.
[**Proof.**]{} (i) Suppose first that $G$ is triangle-free. Let $V(G)=\{v_1,\dots,v_k\}$. Let vertex $v_1$ be represented by an arbitrary line $\ell_1$. Suppose, recursively, that the line $\ell_j$ representing vertex $j$ has already been defined for every $j<i$. We will maintain the “general position” property that no doubly ruled surface contains more than $3$ pairwise disjoint lines. We must choose $\ell_i$ representing $v_i$ such that
\(a) it intersects the lines representing the neighbors $v_j$ of $v_i$ with $j<i$,
\(b) it does not intersect the lines representing the non-neighbors $v_j$ with $j<i$, and
\(c) we maintain our general position conditions.
These are simple algebraic conditions. The vertex $v_i$ has at most $3$ neighbors among $v_j$ for $j<i$, and they must be represented by pairwise disjoint lines. Thus, the Zariski-closed conditions from (a) determine an irreducible variety of lines, so unless they force the violation of a specific other (Zariski-open) condition from (b) or (c), all of those conditions can be satisfied with a generic line through the lines representing the neighbors. In case $v_i$ has three neighbors $v_j$ with $j<i$, the corresponding condition forces $\ell_i$ to be in one of the two families of lines on a doubly ruled surface $\Sigma$. This further forces $\ell_i$ to intersect [*all*]{} lines of the other family on $\Sigma$, but due to the general position condition, none of the vertices of $G$ is represented by lines there, except the three neighbors of $v_i$. We would violate the general position condition with the new line $\ell_i$ if the family we choose it from already had three members representing vertices. However, this would mean that the degrees of the neighbors of $v_i$ would be at least $4$, a contradiction. In case $v_i$ has fewer than $3$ neighbors, the requirement of $\ell_i$ intersecting the corresponding lines does not force $\ell_i$ to intersect any further lines or to lie on any doubly ruled surface.
We prove the general case by induction on $|V(G)|$. Suppose that $a,b,c\in
V(G)$ form a triangle in $G$ and that the subgraph of $G$ induced by $V(G)\setminus\{a,b,c\}$ can be represented as the intersection graph of distinct lines in ${{\mathbb P}}^3$. Note that each of $a$, $b$ and $c$ has at most a single neighbor in the rest of the graph. We extend the representation of the subgraph by adding three lines $\ell_a$, $\ell_b$ and $\ell_c$, representing the vertices of the triangle. We choose these lines in a generic way so that they pass through a common point $p$, and $\ell_a$ intersects the line representing the neighbor of $a$ (in case it exists), and similarly for $\ell_b$ and $\ell_c$. It is clear that we have enough degrees of freedom (at least six) to avoid creating any further intersection. For instance, it suffices to choose $p$ outside all lines in the construction and all planes determined by intersecting pairs of lines.
\(ii) Assign distinct points of ${{\mathbb P}}^3$ to the vertices of $G$ so that no four points lie in a plane. Represent each edge $xx'\in E(G)$ by the line connecting the points assigned to $x$ and $x'$. As no four points are coplanar, two lines representing a pair of edges will cross if and only if the edges share an endpoint. Therefore, the intersection graph of these lines is isomorphic to the edge graph of $G$. $\Box$
The following theorem implies Theorem 3, as the disjointness graph $H=\bar{G}$ is the complement of the intersection graph $G$, and we have $\omega(G)=\alpha(H)$, $\alpha(G)=\omega(H)$, $\chi(G)=\theta(H)$, and $\theta(G)=\chi(H)$. Here $\theta(H)$ denotes the [*clique covering number*]{} of $H$, that is, the smallest number of complete subgraphs of $H$ whose vertex sets cover $V(H)$.
[**Theorem 6.2.**]{}
*Let $H$ be an intersection graph of $n$ lines in the Euclidean space ${{\mathbb R}}^3$ or in the projective space ${{\mathbb P}}^3$.*
\(i) Computing $\alpha(H)$, the independence number of $H$, is NP-hard.
\(ii) Computing $\theta(H)$, the clique covering number of $H$, is NP-hard.
\(iii) Deciding whether $\chi(H)\le3$, that is, whether $H$ is $3$-colorable, is NP-complete.
\(iv) Computing $\omega(H)$, the clique number of $H$, is in P.
\(v) Deciding whether $\theta(H)\le k$ for a fixed $k$ is in P.
\(vi) All the above statements remain true if $H$ is not given as an abstract graph, but with its intersection representation with lines.
[**Proof.**]{} We only deal with the case where the lines are in ${{\mathbb P}}^3$. The reduction of the Euclidean case to this case is easy.
\(i) The problem of determining the independence number of 3-regular graphs is $NP$-hard; see [@AK00]. By Theorem 6.1(i), all 3-regular graphs are intersection graphs of lines in ${{\mathbb P}}^3$.
\(ii) The [*vertex cover number*]{} of a graph $H$ is the smallest number of vertices with the property that every edge of $H$ is incident to at least one of them. Note that the vertex cover number of $H$ is $|V(H)|-\alpha(H)$. In [@P74], it was shown that the problem of determining the [*vertex cover number*]{} is $NP$-hard even for triangle-free graphs. We can reduce this problem to the problem of determining the clique covering number of an intersection graph of lines. For this, note that each complete subgraph of the line graph $H'$ of $H$ corresponds to a star of $H$ and thus $\theta(H')$ is the vertex cover number of $H$. The reduction is complete, as $H'$ is the intersection graph of lines in ${{\mathbb P}}^3$, by Theorem 6.1(ii).
\(iii) Deciding whether the [*chromatic index*]{} (chromatic number of the line graph) of a $3$-regular graph is $3$ is NP-complete, see [@Ho81]. Using that the line graph of any graph is an intersection graph of lines in ${{\mathbb P}}^3$ (Theorem 6.1(ii)), the statement follows.
\(iv) A maximal complete subgraph corresponds to a set of lines passing through the same point $p$ or lying in the same plane $\Pi$. Any such point $p$ or plane $\Pi$ is determined by two lines, and in both cases we can verify for each remaining line whether it belongs to the corresponding complete subgraph (whether it passes through $p$ or belongs to $\Pi$, respectively). This gives an $O(n^3)$-time algorithm, but we suspect that the running time can be much improved.
\(v) As we have seen in part (iv), there are polynomially many maximal complete subgraphs in $H$. We can check all $k$-tuples of them, and decide whether they cover all vertices in $H$.
\(vi) For this, we need to consider the constructions of lines in the representations described in the proof of Theorem 6.1, and show that they can be built in polynomial time. This is obvious in part (ii) of the theorem. For part (i), the situation is somewhat more complex. To find many possible representations of the next vertex intersecting the lines it should, is an algebraically simple task. In polynomial time, we can find one of them that is generic in the sense needed for the construction. However, if the coordinates of each line would be twice as long as those of the preceding line (a condition that is hard to rule out [*a priori*]{}), then the whole construction takes more than polynomial time.
A simple way to avoid this problem is the following. First, color the vertices of the triangle-free graph $G$ of maximal degree at most $3$ by at most $4$ colors, by a simple greedy algorithm. Find the lines representing the vertices in the following order: first for the first color class, next for second color class, etc. The coordinates of each line will be just slightly more complex than the coordinates of the lines representing vertices in [*earlier color classes*]{}. Therefore, the construction can be performed in polynomial time. A similar argument works also for graphs $G$ with triangles: First we find a maximal subset of pairwise vertex-disjoint triangles in $G$. Let $G_0$ be the graph obtained from $G$ by removing these triangles. Then we construct an auxiliary graph $G'$ with these triangles as vertices by connecting two of them with an edge if there is an edge in $G$ between the triangles. The graph $G'$ has maximum degree at most $3$, so it can be greedily $4$-colored. If we construct $G$ by adding back the triangles to $G_0$, in the order determined by their colors, then the procedure will end in polynomial time. $\Box$
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[^1]: Ecole Polytechnique Fédérale de Lausanne and Rényi Institute, Hungarian Academy of Sciences, P.O.Box 127 Budapest, 1364, Hungary; `[email protected]`. ; `[email protected]`. Supported by Swiss National Science Foundation Grants 200021-165977 and 200020-162884.
[^2]: Rényi Institute, Hungarian Academy of Sciences, P.O.Box 127 Budapest, 1364, Hungary; `[email protected]`. Supported by the Cryptography “Lendület” project of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office, NKFIH, projects K-116769 and SNN-117879.
[^3]: Rényi Institute, Hungarian Academy of Sciences, P.O.Box 127 Budapest, 1364, Hungary; `[email protected]`. Supported by National Research, Development and Innovation Office, NKFIH, K-111827.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider graph states of arbitrary number of particles undergoing generic decoherence. We present methods to obtain lower and upper bounds for the system’s entanglement in terms of that of considerably smaller subsystems. For an important class of noisy channels, namely the Pauli maps, these bounds coincide and thus provide the exact analytical expression for the entanglement evolution. All the results apply also to (mixed) graph-diagonal states, and hold true for any convex entanglement monotone. Since any state can be locally depolarized to some graph-diagonal state, our method provides a lower bound for the entanglement decay of any arbitrary state. Finally, this formalism also allows for the direct identification of the robustness under size scaling of graph states in the presence of decoherence, merely by inspection of their connectivities.'
author:
- Daniel Cavalcanti
- Rafael Chaves
- Leandro Aolita
- Luiz Davidovich
- Antonio Acín
title: 'Open-system dynamics of graph-state entanglement'
---
*Introduction.–* Graph states [@graph_review] constitute an important class of entangled states with broad-reaching applications in quantum information, including measurement-based quantum computation [@Brie_review; @RausBrie], quantum error correction [@SchWer], and secure quantum communication [@DurCasBrie-ChenLo]. Moreover, instances of this family, such as the Greenberger-Horne-Zeilinger states, play a crucial role in fundamental tests of quantum non-locality [@GHZ]. Consequently, a great effort has been made both to theoretically understand their properties [@graph_review; @HeinEisBrie] and to create and coherently manipulate them experimentally [@ClusterExp]
Needless to say, it is crucial to understand the dynamics of their entanglement in realistic scenarios, where the system unavoidably decoheres due to experimental errors or to the interaction with its environment. Previous studies on the robustness of graph-state entanglement in the presence of decoherence observed a disentanglement time (or lower bounds thereof) insensitive to the system size . However, the disentanglement time on its own is not in general able to provide any faithful assessment about the entanglement’s robustness, since it can grow with the number $N$ of particles and yet the entanglement can get closer to zero the faster, the larger $N$ [@us]. The full dynamical evolution of entanglement must then be studied to draw conclusions on its fragility. Taking the latter into account, the entanglement of the linear-cluster states, an example of graph states, was shown to be robust with the size of the system against the particular case of collective dephasing decoherence [@Guhne].
The present work provides a general framework for the study of the entanglement evolution of graph states under decoherence. Our techniques apply to (i) any graph, and graph-diagonal, states; (ii) arbitrary kinds of noise, individual or collective; and (iii) any convex (bi- or multi-partite) entanglement quantifier that does not increase under local operations and classical communication (LOCC). In the developed formalism we consider local measurement protocols to efficiently obtain lower and upper bounds for the entanglement of the whole system contained in any given partition in terms of that of a considerably smaller subsystem consisting only of those qubits lying on the boundary of the partition. No optimization on the full system’s parameter space is required throughout. For an important class of noisy channels – namely arbitrary Pauli maps, to be defined below – the lower and upper bounds coincide, providing thus the exact entanglement evolution. With the same methods we also establish a second family of lower bounds that, despite less tight, depend only on the connectivity of the graph and not on its size. This allows us to assess the robustness based on the full dynamics of the entanglement and not just its disentanglement time. Our approach can also be used to establish lower bounds to the entanglement behavior of any initial quantum state.
*Graph states.–* Consider a mathematical graph $G_{(\mathcal{V},\mathcal{E})}\equiv\{\mathcal{V},\mathcal{E}\}$, composed of a set $\mathcal{V}$, of $N$ vertices $i \in
\mathcal{V}$, and a set $\mathcal{E}$, of edges $\{i,j\}\in
\mathcal{E}$ connecting each vertex $i$ to some other $j$. The associated physical state is operationally defined as follows: to each vertex $i$ associate a qubit, initialize all $N$ qubits in the product state ${| {g_{(\mathcal{V})}}_0 \rangle}\equiv\bigotimes_{i
\in\mathcal{V}} {| +_i \rangle}$, being ${| +_i \rangle}=({| 0_i \rangle}+{| 1_i \rangle})/\sqrt{2}$, and to all pairs $\{i,j\}$ of qubits joined by an edge apply a maximally-entangling control-$Z$ ($CZ$) gate, $CZ_{ij}={| 0_{i}0_{j} \rangle}{\langle 0_{i}0_{j} |}+{| 0_{i}1_{j} \rangle}{\langle 0_{i}1_{j} |}+{| 1_{i}0_{j} \rangle}{\langle 1_{i}0_{j} |}-{| 1_{i}1_{j} \rangle}{\langle 1_{i}1_{j} |}$. The resulting $N$-qubit graph state is $${| {G_{(\mathcal{V},\mathcal{E})}}_0 \rangle}=\bigotimes_{\{i,j\} \in \mathcal{E}} CZ_{ij}{| {g_{(\mathcal{V})}}_0 \rangle}.$$
![\[Graph\] (Color online) Example of a mathematical graph associated to a physical graph state. We have displayed a possible partition of this graph, splitting the system in three parts $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. The vertices and edges in grey corresponds to the *boundary qubits* and the *boundary-crossing edges* respectively.](graph2){width="0.7\linewidth"}
An example of such graph is shown in Fig. \[Graph\], where the system is divided into three regions, $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$. We call all edges that go from one region to the other the *boundary-crossing edges* and label the subset of all such edges by $\mathcal{X}$. All qubits connected by the boundary-crossing edges are in turn called the *boundary qubits* and the subset composed of all of these is called $\mathcal{Y}$.
*Open-system dynamics.–* Our ultimate goal is to quantify the entanglement in any partition of arbitrary graph states undergoing a generic physical process during a time interval $t$. The action of such process on an initial density operator $\rho$ can be described by a completely-positive trace-preserving map $\Lambda$ as $\rho_t=\Lambda(\rho)$, where $\rho_t$ is the evolved density matrix after time $t$. All such maps can be expressed in a Kraus representation, $\Lambda(\rho)=\sum_\mu p_\mu K_\mu \rho
K_\mu^\dag$, where $\sqrt{p_\mu}K_\mu$ are called the Kraus operators (each of which appearing with probability $p_\mu$), which satisfy the normalization conditions Tr$[K_\mu^\dag K_\mu]=1$ and $\sum_\mu p_\mu =1$ [@nielsen]. The Kraus representation guarantees that the map is (completely) positive and preserves trace normalization. When the map can be factorized as the composition of individual maps acting independently on each qubit, the noise is said to be individual (or independent); if not, it is said to be collective.
A very important class of processes is described by the Pauli maps, separable (non-entangling) maps whose Kraus operators are given by tensor products of Pauli operators $X$, $Y$, $Z$, and the identity. Examples of these are the collective or individual depolarizing, dephasing or bit-flip channels [@nielsen]. As we show next, it is possible to determine the exact entanglement evolution of graph and graph-diagonal states (whose formal definition is provided below) subject to individual Pauli maps.
*Exact entanglement of graph states under Pauli maps.–*
Let us start by recalling that a graph state is the simultaneous eigenvector – of eigenvalue 1 – of the $N$ generators of the stabilizer group, that is, of the $N$ operators consisting each of which of one $X$ acting on each single qubit and $Z$’s on all its neighboring ones [@graph_review]. Therefore, the application of an $X$ or $Y$ operator on a qubit $k$ of a graph state is equivalent to the application of $Z$ operators on all neighboring qubits of $k$, or on all of its neighboring qubits and on $k$ itself, respectively. The action of any Pauli map $\Lambda$ on a graph state is thus equivalent to that of another separable map, $\tilde{\Lambda}$, whose Kraus operators $\tilde{K}_\mu$ are obtained from $K_\mu$ replacing in the latter each $X$ and $Y$ operators by tensor products of $Z$ and identity operators according to the rule just described [@comment]. Thus we need to consider how a general combination of $Z$ operators acts on a graph state. We use the multi-index $\tilde{\mu}=(\mu_1,...,\mu_N)$, with $\mu_i=\{0,1\}$, to denote such a combination through $Z^{\mu_1} \otimes Z^{\mu_2}\otimes...\otimes Z^{\mu_N}$. The action of such operator on a graph state ${| {G_{(\mathcal{V},\mathcal{E})}}_0 \rangle}$ generates another graph state ${| {G_{(\mathcal{V},\mathcal{E})}}_{\tilde{\mu}} \rangle}$, orthogonal to the former one [@graph_review; @HeinDurBrie]. These considerations imply that $\rho_t$ can be expressed as $$\begin{aligned}
\label{graph diagonal} \nonumber
\rho_t&=&\Lambda({| {G_{(\mathcal{V},\mathcal{E})}}_0 \rangle})=\tilde{\Lambda}({| {G_{(\mathcal{V},\mathcal{E})}}_0 \rangle})\\
&=&\sum_{\tilde{\mu}} \tilde{p}_{\tilde{\mu}}
{| {G_{(\mathcal{V},\mathcal{E})}}_{\tilde{\mu}} \rangle}{\langle {G_{(\mathcal{V},\mathcal{E})}}_{\tilde{\mu}} |}.\end{aligned}$$ All possible $2^N$ graph states ${| {G_{(\mathcal{V},\mathcal{E})}}_{\tilde{\mu}} \rangle}$ associated to the graph $G_{(\mathcal{V},\mathcal{E})}$ form a complete orthonormal basis of the $N$-qubit Hilbert space. State is a [*graph-diagonal state*]{}. Calculating the exact entanglement in any partition of the such state is in general a problem that involves an optimization over the entire parameter space of $\rho_t$. In what follows we will show that it is possible to greatly reduce the complexity of this optimization problem. Consider any partition of the state $\rho_t$. We now factor out explicitly all the $CZ$ gates but those corresponding to the boundary-crossing edges and write the state as $$\begin{aligned}
\label{rho_t} \rho_t&=&\bigotimes_{\{i,j\} \in \mathcal{E}/\mathcal{X}}
CZ_{ij}\sum_{\gamma,\delta} \tilde{p}_{\gamma,\delta}
{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\nonumber\\
&\otimes&{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}{\langle {g_{(\mathcal{V}/\mathcal{Y})}}_\delta |}\bigotimes_{\{k,l\} \in \mathcal{E}/\mathcal{X}}CZ_{kl},\end{aligned}$$ Here we have grouped together all indices inside $\tilde{\mu}$ into two new multiple indices, $\gamma$ and $\delta$. Multiple index $\gamma$ accounts for all possible graph states ${| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}$ generated by applying tensor products of $Z$ and identity operators to the graph state ${| {G_{(\mathcal{Y},\mathcal{X})}}_0 \rangle}\equiv\bigotimes_{\{i,j\}
\in\mathcal{X}} CZ_{ij}\otimes{| {g_{(\mathcal{Y})}}_0 \rangle}$, associated to the [*boundary graph*]{} $G_{(\mathcal{Y},\mathcal{X})}=\{\mathcal{Y},\mathcal{X}\}$, with ${| {g_{(\mathcal{Y})}}_0 \rangle}\equiv\bigotimes_{i \in\mathcal{Y}}
{| +_i \rangle}$. Multiple index $\delta$ on the other hand accounts for all states ${| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}$ generated from $Z$ or identity operators on the state ${| {g_{(\mathcal{V}/\mathcal{Y})}}_0 \rangle}\equiv\bigotimes_{i
\in\mathcal{V}/\mathcal{Y}} {| +_i \rangle}$ of the non-boundary qubits $\mathcal{V}/\mathcal{Y}$. Probability $\tilde{p}_{\gamma,\delta}$ is defined as the sum of all $p_{\mu}$ such that $\tilde{K}_\mu{| {G_{(\mathcal{Y},\mathcal{X})}}_0 \rangle}\otimes
{| {g_{(\mathcal{V}/\mathcal{Y})}}_0 \rangle}={| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}\otimes{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}$. Because the $CZ$ gates explicitly factored out in state are local unitary operations with respect to the partition of interest, the entanglement of $\rho_t$, $E(\rho_t)$, reads $$\begin{aligned}
\label{ent}
E\big(\sum_{\gamma,\delta}
\tilde{p}_{\gamma,\delta}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}
{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\otimes{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}{\langle {g_{(\mathcal{V}/\mathcal{Y})}}_\delta |}\big).
\end{aligned}$$ where $E$ is any convex entanglement quantifier not increasing under LOCC. In what follows, we first establish a lower and upper bound to this expression and then show that these bounds coincide, obtaining the exact expression of the graph-state entanglement evolution.
First, consider an LOCC protocol consisting of measuring all the non-boundary qubits $\mathcal{V}/\mathcal{Y}$ of the state within brackets in Eq. in the product basis composed by all orthonormal states $\{{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}\}$ and tracing out the measured subsystem after communicating the outcomes. The remaining subsystem $\mathcal{Y}$ is [*flagged*]{} by each measurement outcome $\delta$ – meaning that outcome $\delta$ provides full information about to which state $\mathcal{Y}$ has been projected after each measurement run. The final entanglement after the entire protocol is then given by the average entanglement over all measurement runs. Since $E$ is non-increasing under LOCC, $E(\rho_t)$ must satisfy $$\label{perfectlower} E(\rho_t)\geq\sum_\delta\tilde{p}_\delta E\Big(\sum_{\gamma}
\tilde{p}_{(\gamma|\delta)}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}
{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\Big),$$ where $\tilde{p}_{\delta}\equiv\sum_{\gamma}\tilde{p}_{\gamma,\delta}$ is the total probability of occurrence of an event $\delta$ and $\tilde{p}_{(\gamma|\delta)}$ is the conditional probability of an event $\gamma$ given that event $\delta$ has happened.
On the other hand, convexity of $E$ implies that $E(\rho_t)$, as given by , must necessarily be smaller or equal to $\sum_\delta \tilde{p}_\delta E\Big(\sum_{\gamma}
\tilde{p}_{(\gamma|\delta)}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}
{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\otimes{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}{\langle {g_{(\mathcal{V}/\mathcal{Y})}}_\delta |}\Big)$, which, since locally added ancillary systems do not change the entanglement, is in turn equal to the right-hand side of . This means that the right-hand side of Eq. provides at the same time an upper and a lower bound to $E(\rho_t)$ and therefore yields its [*exact*]{} value, [[*[i.e. ]{}*]{}]{}:
$$\label{exact}
E(\rho_t)=\sum_\delta\tilde{p}_\delta E\Big(\sum_{\gamma}
\tilde{p}_{(\gamma|\delta)}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}
{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\Big).$$
A comment on the implications of this exact result on the computational-cost is now in place. The calculation of the entanglement of systems composed by $N=N_\mathcal{Y}+N_{\mathcal{V}/\mathcal{Y}}$ qubits (being $N_\mathcal{Y}$ and $N_{\mathcal{V}/\mathcal{Y}}$ the number of boundary and non-boundary qubits respectively) is a problem that, in general, involves an optimization over $O\big(2^{2N}\big)$ real parameters. Through Eq. such calculation is reduced to that of the average entanglement over a sample of $2^{N_{\mathcal{V}/\mathcal{Y}}}$ states (one for each measurement outcome $\delta$) of $N_\mathcal{Y}$ qubits, which involves at most $2^{N_{\mathcal{V}/\mathcal{Y}}}$ optimizations over $O\big(2^{2N_\mathcal{Y}}\big)$ real parameters. Thus the present method provides an exponential decrease in the computational power needed to calculate $E(\rho_t)$, since only the boundary qubits appear in the computation of Eq. .
In order to illustrate the power of the method we have calculated, using , the exact entanglement of formation $E_F$ [@Wootters] of 1-D graph states under the action of independent depolarizing channels, which mix, with probability $p$, any one-qubit state with the maximally mixed state $\openone/2$ [@nielsen]. In Fig. \[Fig2\] we display the curves corresponding to the bipartition first qubit versus the rest, although other partitions can be considered. The 1-D graph state (also called the linear cluster state), given by ${| LC \rangle}=\bigotimes_{i=1}^{N-1} CZ_{i,i+1} \bigotimes_k^N{| +_k \rangle}$, evolves from $p=0$ towards a final maximally mixed state at $p=1$. Not only this calculation would have been impossible had we attempted a brute-force optimization approach, but also, since in this particular case the boundary qubits are just two, the use of allows to perform the calculation with no optimization at all, for an explicit formula for the entanglement of formation exists for arbitrary two-qubit systems [@Wootters].
![ \[Fig2\] (Color online.) Entanglement of formation ($E_F$) in the partition of the first particle versus the rest for 1-D graph states of 2 (black), 4 (grey) and 7 (red) particles undergoing individual depolarization as a function of the depolarization probability $p$. The dashed curve represents a size-independent lower bound. ](UptoSevennotDashed){width=".80\linewidth"}
*Beyond graph states and Pauli maps.–* The expression is actually a method for calculating the entanglement of any graph-diagonal state as the one in . Since Pauli maps acting on initial graph-diagonal states also produce graph-diagonal states, all the arguments used so far are also valid for this class of initial states. Furthermore, any quantum state can be depolarized to a graph-diagonal state by means of LOCC [@ADB]. Using again the fact that the entanglement of a state does not increase if an LOCC protocol is applied, one can see that the present method also provides (in general non-tight) lower bounds to the decay of the entanglement of any initial state subject to any decoherence process.
*Robustness of graph-state entanglement.–* The developed techniques can be further simplified to obtain new lower bounds to graph-state entanglement during all the evolution that, despite not being tight, can be calculated in a much more efficient way than and often turn out to be independent of the total number of qubits. This dramatically simplifies the study of the entanglement robustness of graph states as a function of the system’s size, a central question for the applicability of these states as quantum information resources. As an illustration, we compare next graph states of different sizes under the action of general $N$-qubit Pauli maps $\Lambda$ that scale with $N$ in a way such that, for each $\sqrt{p_{\mu}}K_\mu$, the Kraus operators of the map acting on $M$ more qubits are obtained as tensor products of $\sqrt{p_{\mu}}K_\mu$ with Pauli or identity operators on the other $M$ qubits, weighted with some new probabilities that sum up to one (for each $\mu$). That is, so that the total probability of event $\mu$ on the $N$ first qubits, $p_{\mu}$, remains the same. All the individual or collective Pauli maps mentioned above fall into this category. The state between brakets in Eq. can then be written also as $\sum_{\gamma}
\tilde{p}_{\gamma}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\otimes\sum_\delta
\tilde{p}_{(\delta|\gamma)}{| {g_{(\mathcal{V}/\mathcal{Y})}}_\delta \rangle}{\langle {g_{(\mathcal{V}/\mathcal{Y})}}_\delta |}\big)$, where $\tilde{p}_{\gamma}\equiv\sum_{\delta}\tilde{p}_{\gamma,\delta}$ and $\tilde{p}_{(\delta|\gamma)}$ is the conditional probability of $\delta$ given $\gamma$. By tracing out the state of the non-boundary qubits ([*i.e.*]{}, by disregarding the flag that lead to above) and using again the fact that $E$ does not increase under LOCC, we arrive at $$\label{size-independent LB}
E(\rho_t)\geq E\big(\sum_{\gamma}
\tilde{p}_{\gamma}{| {G_{(\mathcal{Y},\mathcal{X})}}_\gamma \rangle}{\langle {G_{(\mathcal{Y},\mathcal{X})}}_\gamma |}\big).$$ Now, notice that – for the maps here-considered – probability $\tilde{p}_{\gamma}$ depends only on the boundary graph $G_{(\mathcal{Y},\mathcal{X})}$ and the number of non-boundary qubits directly connected to it (the boundary graph is affected by the noise on up to its first neighbors), not on the total system size $N$. Bound is unaffected by the addition of $M$ extra particles if these new particles are not connected to the boundary subsystem. In the latter sense, and for the considered noise scenario, noisy graph-state entanglement is thus robust with respect to the variation of the system size provided $G_{(\mathcal{Y},\mathcal{X})}$ and its connectivity to the rest do not vary.
Size-independent bound (for the case of $E=E_F$) is compared with the exact entanglement, again for a linear cluster and the individual depolarizing channel, in Fig. \[Fig2\]
*Discussion.–* To summarize, in this work we have presented a general framework to study the entanglement decay of graph states under decoherence. It is important to emphasize that any function that satisfies the requirements of convexity and monotonicity under LOCC falls into the range of applicability of the machinery here-developed. This includes genuine multipartite entanglement quantifiers, as well as those functions aiming at quantifying the usefulness of quantum states for given quantum informational tasks.
To conclude with, let us make the following observations. First, the techniques developed to obtain perfect bounds can also be applied to tackle some cases other than Pauli maps. For example, for graph states in the presence of individual thermal baths at arbitrary temperature, an LOCC procedure similar to the one used to obtain the bound , but using general measurements instead of orthogonal ones, can be used to obtain highly non-trivial entanglement lower bounds.
Second, bound , when restricted to bipartite entanglement, provides the same type of lower bound as the one used in section V-B of Ref. [@HeinDurBrie] to find lower bounds to the entanglement lifetime for the case of $E$ being the negativity. The present bound has the advantage of dealing with other possible partitions and general entanglement quantifiers. All these topics will be touched upon elsewhere.
We thank J. Eisert, M. Plenio, F. Brandão, and A. Winter, for inspiring conversations, and the CNPq, the Brazilian Millenium Institute for Quantum Information, the PROBRAL CAPES/DAAD, the European QAP, COMPAS and PERCENT projects, the Spanish MEC FIS2007-60182 and Consolider-Ingenio QOIT projects, and the Generalitat de Catalunya, for financial support.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Peter Beelen[^1]'
title: 'A note on the generalized Hamming weights of Reed–Muller codes'
---
Preliminaries {#sec:in}
=============
Let ${\mathbb{F}_q}$ be the finite field with $q$ elements and denote by $\AA^m:=\AA^m({\mathbb{F}_q})$ the $m$-dimensional affine space defined over ${\mathbb{F}_q}$. This space consists of $q^m$ points $(a_1,\dots,a_m)$ with $a_1,\dots,a_m \in {\mathbb{F}_q}$. Let $T(m):={\mathbb{F}_q}[x_1, \dots , x_m]$ denote the ring of polynomials in $m$ variables and coefficients in ${\mathbb{F}_q}.$ Further let $T_{\le d}(m)$ be the set of polynomials in $T(m)$ of total degree at most $d$. A monomial $X_1^{\alpha_1}\cdots X_m^{\alpha_m}$ is called reduced if $(\alpha_1,\dots,\alpha_m) \in \{0,1,\dots,q-1\}^m$. Similarly a polynomial $f \in T(m)$ is called reduced if it is an ${\mathbb{F}_q}$-linear combination of reduced monomials. We denote the set of reduced polynomials by $T^{\mathrm{red}}(m)$ and define $T^{\mathrm{red}}_{\le d}(m):=^T_{\le d}(m) \cap T^{{\mathrm{red}}}(m)$.
One reason for considering reduced polynomials comes from coding theory. Indeed Reed–Muller codes are obtained by evaluating certain polynomials in the points of $\AA^m$, but the evaluation map $${\mathrm{Ev}}: T(m) \to {\mathbb{F}_q}^{q^m}, \ \makebox{defined by} \ {\mathrm{Ev}}(f)=(f(P))_{P \in \AA}$$ is not injective. However, its restriction to $T^{\mathrm{red}}(m)$ is. In fact the kernel of ${\mathrm{Ev}}$ consists precisely of the ideal $I \subset T(m)$ generated by the polynomials $x_i^q-x_i$ ($1 \le i \le m$). Working with reduced polynomials is simply a convenient way to take this into account, since for two reduced polynomials $f_1,f_2 \in T(m)$ the equality $f_1+I=f_2+I$ holds if and only if $f_1=f_2$.
The Reed–Muller code ${\mathrm{RM}_q}(d,m)$ is the set of vectors from ${\mathbb{F}_q}^{q^m}$ obtained by evaluating polynomials of total degree up to $d$ in the $q^m$ points of $\AA^m$, that is to say: $${\mathrm{RM}_q}(d,m):=\{(f(P))_{P \in \AA^m} \, : \, f \in T_{\le d}(m)\}.$$ By the above, we also have ${\mathrm{RM}_q}(d,m):=\{(f(P))_{P \in \AA^m} \, : \, f \in T_{\le d}^{\mathrm{red}}(m)\}$ and moreover, we have $$\label{eq:dimRM0}
\dim {\mathrm{RM}_q}(d,m)=\dim T^{\mathrm{red}}_{\le d}(m).$$ Reed–Muller codes ${\mathrm{RM}_q}(d,m)$ have been studied extensively for their elegant algebraic properties. Their generalized Hamming weights $d_r({\mathrm{RM}_q}(d,m))$ have been determined in [@HP] by Heijnen and Pellikaan. For a general linear code $C \subseteq {\mathbb{F}_q}^n$ these are defined as follows: $$d_r(C):=\min_{D \subseteq C: \dim D=r}|\mathrm{supp}(D)|,$$ where the minimum is taken over all $r$-dimensional ${\mathbb{F}_q}$-linear subspaces $D$ of $C$ and where $\mathrm{supp}(D)$ denotes the support size of $D$, that is to say $$\mathrm{supp}(D):=\#\{i \, : \, \exists \, (c_1,\dots,c_n) \in D, \ c_i \not = 0\}.$$ In case of Reed–Muller codes, there is a direct relation between generalized Hamming weights and the number of common solutions to systems of polynomial equations. Indeed, if $D \subset {\mathrm{RM}_q}(d,m)$ is spanned by $(f_i(P))_{P \in \AA}$ for $f_1,\dots,f_r \in T_{\le d}^{\mathrm{red}}(m)$, then $\mathrm{supp}(D)=q^m-\#{\mathsf{Z}}(f_1,\dots,f_r)$ where ${\mathsf{Z}}(f_1, \dots , f_r):=\{P \in \AA^m \,:\, f_1(P)=\cdots =f_r(P)=0\}$ denotes the set of common zeros of $f_1, \dots , f_r$ in the $m$-dimensional affine space $\AA^m$ over ${\mathbb{F}_q}$. Therefore, if we define $$\label{erdmA}
\bar e^{\AA}_r(d,m): = \max\left\{ \left| {\mathsf{Z}}(f_1, \dots , f_r) \right| \,:\, f_1, \dots , f_r \in T^{\mathrm{red}}_{\le d}(m) \; \text{ linearly independent}\right\} ,$$ then $d_r({\mathrm{RM}_q}(d,m))=q^m-\bar e^{\AA}_r(d,m).$ Note that $T^{\mathrm{red}}(m)$ is a vector space over ${\mathbb{F}_q}$ of dimension $q^m$ and that a reduced polynomial has total degree at most $m(q-1)$. Therefore $T^{\mathrm{red}}(m)=T^{\mathrm{red}}_{\le m(q-1)}(m)$. This implies in particular that ${\mathrm{RM}_q}(d,m)={\mathbb{F}_q}^{q^m}$ for $d \ge m(q-1)$. Therefore, we will always assume that $d \le m(q-1).$
The result of Heijnen–Pellikaan in [@HP] on the value of $d_r({\mathrm{RM}_q}(d,m))$ can now be restated as follows, see for example [@BD]. $$\label{Hrdm}
\bar e^{\AA}_r(d,m)= \sum_{i=1}^m \mu_i q^{m-i},$$ where $(\mu_1, \dots, \mu_{m})$ is the $r$-th $m$-tuple in descending lexicographic order among all $m$-tuples $(\beta_1, \dots , \beta_{m})\in \{0,1,\dots,q-1\}^m$ satisfying $\beta_1+ \cdots + \beta_{m} \le d$.
Following the notation in [@HP], we denote with $\rho_q(d,m)$ the dimension of ${\mathrm{RM}_q}(d,m)$. Equation implies that $\rho_q(d,m)=\dim(T_{\le d}^{\mathrm{red}}(m)).$ In particular, we have $$\rho_q(d,m)=\dim(T_{\le d}(m))=\binom{m+d}{d}, \ \makebox{if $d \le q-1$},$$ since $T_{\le d}(m)=T_{\le d}^{\mathrm{red}}(m)$ if $d<q.$ Here as well as later on we use the convention that $\binom{a}{b}=0$ if $a<b$. In particular we have $\rho_q(d,m)=0$ if $d<0$. As shown in [@AK §5.4], for the general case $d \le m(q-1)$, we have $$\label{eq:dimRM}
\rho_q(d,m)=\dim(T^{\mathrm{red}}_{\le d}(m))=\sum_{i=0}^d \sum_{j=0}^{m} (-1)^j \binom{m}{j} \binom{m-1+i-qj}{m-1}.$$
In this note, we will present an easy-to-obtain expression for $\bar e^{\AA}_r(d,m)$ involving a certain representation of the number $\rho_q(d,m)-r$ that we introduce in the next section.
The $d$-th Macaulay representation with respect to $q$
======================================================
Let $d$ be a positive integer. The $d$-th Macaulay (or $d$-binomial) representation, of a nonnegative integer $N$ is a way to write $N$ as sum as certain binomial coefficients. To be precise $$N=\sum_{i=1}^d \binom{s_i}{i},$$ where the $s_i$ integers satisfying $s_d>s_{d-1}> \cdots > s_1 \ge 0$. The usual convention that $\binom{a}{b}=0$ if $a<b$, is used. For example, the $d$-th Macaulay representation of $0$ is given by $0=\sum_{i=1}^d \binom{i-1}{i}.$ Given $d$ and $N$ the integers $s_i$ exist and are unique. The Macaulay representation is among other things used for the study of Hilbert functions of graded modules, see for example [@green]. It is well known (see for example [@green]) that if $N$ and $M$ are two nonnegative integers with Macaulay representations given by $(k_d,\dots,k_1)$ and $(\ell_d,\dots,\ell_1)$ then $N \le M$ if and only if $(k_d,\dots,k_1) \preccurlyeq (\ell_d,\dots,\ell_1)$, where $\preccurlyeq$ denotes the lexicographic order.
For our purposes it is more convenient to define $m_i:=s_i-i$. We then obtain $$\label{eq_Macaulay}
N=\sum_{i=1}^d \binom{m_i+i}{i},$$ where $m_i$ are integers satisfying $m_d \ge m_{d-1} \ge \cdots \ge m_1 \ge -1.$ The reason for this is that for $d \le q-1$ we have $\rho_q(d,m)=\binom{m+d}{d}$. Therefore, we can interpret Equation as a statement concerning dimensions of the Reed–Muller codes ${\mathrm{RM}_q}(i,m_i)$. For a suitable choice of $N$, it turns out that the $m_i$ completely determine the value of $\bar e^{\AA}_r(d,m)$ if $d \le q-1$. For $d \ge q$, even though the dimension $\rho_q(d,m)$ is not longer given by $\binom{m+d}{d}$, there exists a variant of the usual $d$-th Macaulay representation that turns out to be equally meaningful for Reed–Muller codes. Before stating this representation, we give a lemma.
\[lem:help\] Let $m \ge 1$ be an integer. We have $$\rho_q(d,m) = \sum_{i=0}^{\min\{d,q-1\}}\rho_q(d-i,m-1).$$
Any polynomial $f \in T(m)$ can be seen as a polynomial in the variable $X_m$ with coefficients in $T(m-1)$. This implies that $T(m) = \sum_{i \ge 0} X_m^i T(m)$, where the sum is a direct sum. Similarly we can write $$T^{\mathrm{red}}_{\le d}(m)=\sum_{i=0}^{\min\{d,q-1\}}X_m^iT^{\mathrm{red}}_{\le d-i}(m-1).$$ The result now follows.
A consequence of this lemma is the following.
\[cor:help\] Let $d=a(q-1)+b$ for integers $a$ and $b$ satisfying $a \ge 0$ and $1 \le b \le q-1.$ Further suppose that $m \ge a$. Then $$\rho_q(d,m)-1=\sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m-j-1) + \sum_{i=1}^b\rho_q(i,m-a-1).$$
This follows using Lemma \[lem:help\] repeatedly. First applying the lemma to each sum within the double summation on the right-hand side, we see that $$\begin{gathered}
\sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m-j-1) = \\
\sum_{j=0}^{a-1} \left( \rho_q(d-j(q-1),m-j)-\rho_q(d-(j+1)(q-1),m-j-1) \right) = \\
\rho_q(d,m)-\rho_q(d-a(q-1),m-a) = \rho_q(d,m)-\rho_q(b,m-a).\end{gathered}$$ Using the same lemma to rewrite the single summation on the right-hand side in Equation we see that if $m>a$ $$\sum_{i=1}^b\rho_q(i,m-a-1)=\rho_q(b,m-a)-\rho_q(0,m-a-1)=\rho_q(b,m-a)-1,$$ while if $m=a$, the single summation equals $0$ and the double summation simplifies to $\rho_q(d,m)-1$. In either case, we obtain the desired result
We can now show the following.
\[thm:genrepMac\] Let $N \ge 0$ and $d \ge 1$ be integers and $q$ a prime power. Then there exist uniquely determined integers $m_1,\dots,m_d$ satisfying
1. $N=\sum_{i=1}^d \rho_q(i,m_i),$
2. $-1 \le m_1 \le \cdots \le m_d,$
3. for all $i$ satisfying $1 \le i \le d-q+1$, either $m_{i+q-1} > m_i$ or $m_{i+q-1}=m_i=-1$.
We start by showing uniqueness. Suppose that $$\label{eq:uniqueness}
N=\sum_{i=1}^d \rho_q(i,m_i)=\sum_{i=1}^d \rho_q(i,n_i)$$ and the integers $n_1,\dots,n_d$ and $m_1,\dots m_d$ satisfy the conditions from the theorem. First of all, if $m_d=-1$ or $n_d=-1$ then $N=0$. Either assumption implies that $(m_d,\dots,m_1)=(-1,\dots,-1)=(n_d,\dots,n_1)$. Indeed $n_i\ge 0$ or $m_i \ge 0$ for some $i$ directly implies that $N>0$. Therefore we from now on assume that $m_d\ge 0$ and $n_d \ge 0$. To arrive at a contradiction, we may assume without loss of generality that $n_d \le m_d-1$.
Define $e$ to be the smallest integer such that $n_e \ge 0$. Equation can then be rewritten as $$\label{eq:uniqueness1}
N=\sum_{i=1}^d \rho_q(i,m_i)=\sum_{i=e}^d \rho_q(i,n_i)$$ Condition 3 from the theorem implies that $n_{i-q+1}<n_i$ for all $i$ satisfying $e \le i \le d $. Now write $d-e+1=a(q-1)+b$ for integers $a$ and $b$ satisfying $a \ge 0$ and $1 \le b \le q-1$. With this notation, we obtain that for any $0 \le j \le a-1$ and $0 \le \ell \le q-2$ we have that $$n_{d-j(q-1)-\ell} \le n_d-j \le m_d-j-1.$$
In particular choosing $j=a-1$ and $\ell=0$, this implies that $m_d \ge a+n_{q-1+b} \ge a+1+n_b \ge a$. Using these observations, we obtain from Equation that $$\label{eq:uniqueness2}
\rho_q(d,m_d) \le N = \sum_{i=e}^d \rho_q(i,n_i) \le \sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m_d-j-1) + \sum_{i=1}^b\rho_q(e+i-1,m_d-a-1).$$ Applying the same technique as in the proof of Corollary \[cor:help\], we derive that $$\sum_{j=0}^{a-1} \sum_{\ell=0}^{q-2} \rho_q(d-j(q-1)-\ell,m_d-j-1)=\rho_q(d,m_d)-\rho_q(b+e-1,m_d-a)$$ and Equation can be simplified to $$\label{eq:uniqueness3}
\rho_q(d,m_d) \le \rho_q(d,m_d)-\rho_q(b+e-1,m_d-a) + \sum_{i=1}^b\rho_q(e+i-1,m_d-a-1).$$ For $m_d=a$ the right-hand side equals $\rho_q(d,m_d)-1$, leading to a contradiction. If $m_d>q$, Equation implies $$\begin{array}{rcl}
\rho_q(b+e-1,m_d-a) &\le & \sum_{i=1}^b\rho_q(e+i-1,m_d-a-1)\\
\\
&= & \sum_{j=0}^{b-1}\rho_q(e+b-1-j,m_d-a-1)\\
\\
&< & \sum_{j=0}^{\min\{e+b-1,q-1\}}\rho_q(e+b-1-j,m_d-a-1)
\\
& = & \rho_q(b+e-1,m_d-a),
\end{array}$$ where in the last equality we used Lemma \[lem:help\]. Again we arrive at a contradiction. This completes the proof of uniqueness of the $d$-th Macaulay representation with respect to $q$.
Now we show existence. Let $d$, $N$ and $q$ be given. We will proceed with induction on $d$. For $d=1$, note that $\rho_q(1,m)=m+1$ for any $m \ge -1$. Therefore, for a given $N \ge 0$, we can write $N=\rho_q(1,N-1)$.
Now assume the theorem for $d-1$. There exists $m_d \ge -1$ such that $$\label{eq:existence}
\rho_q(d,m_d) \le N < \rho_q(d,m_d+1).$$ Applying the induction hypothesis on $N-\rho_q(d,m_d)$, we can find $m_{d-1},\dots,m_1$ satisfying the conditions of the theorem for $d-1$. In particular we have that
1. $N-\rho_q(d,m_d)=\sum_{i=1}^{d-1} \rho_q(i,m_i),$
2. $-1 \le m_1 \le \cdots \le m_{d-1},$
3. $m_{i+(q-1)} > m_i$ for all $1 \le i \le d-q.$
Clearly this implies that $N=\sum_{i=1}^{d} \rho_q(i,m_i),$ but it is not clear a priori that $m_1,\dots,m_d$ satisfy conditions 2 and 3 as well. Conditions 2 and 3 would follow once we show that $m_d \ge m_{d-1}$ and either $m_d > m_{d-q+1}$ or $m_d=m_{d-q+1}=-1$. First of all, if $m_d=-1$, then $N=0$ and $(m_d,\dots,m_1)=(-1,\dots,-1)$. Hence there is nothing to prove in that case. Assume $m_d \ge 0$. From Equation and Lemma \[lem:help\] we see that $$\label{eq:existence1}
N-\rho_q(d,m_d)<\rho_q(d,m_d+1)-\rho_q(d,m_d)=\sum_{i=1}^{\min\{d,q-1\}}\rho_q(d-i,m_d).$$ First suppose that $d \le q-1$. First of all, Condition 3 is empty in that setting. Further, Equation implies $$N-\rho_q(d,m_d) < \sum_{i=1}^{d}\rho_q(d-i,m_d) = \sum_{i=1}^{d-1}\rho_q(d-i,m_d) +1$$ and hence $$N-\rho_q(d,m_d) \le \sum_{i=1}^{d-1}\rho_q(d-i,m_d)= \sum_{j=0}^{d-2}\rho_q(d-1-j,m_d) < \rho_q(d-1,m_d+1).$$ This shows that $m_{d-1} \le m_d$ as desired.
Now suppose that $d \ge q$. In this situation Equation implies $$N-\rho_q(d,m_d) < \sum_{i=1}^{q-1}\rho_q(d-i,m_d) = \sum_{j=0}^{q-2}\rho_q(d-1-j,m_d)< \rho_q(d-1,m_d+1).$$ Hence $m_{d-1} \le m_d$ as before. Finally assume that $m_d \le m_{d-q+1}$. Then by the previous and Condition 2, we have $m_{d}= m_{d-1}=\cdots = m_{d-q+1}$. Hence $N \ge \sum_{i=0}^{q-1}\rho_q(d-i,m_d)=\rho_q(d,m_d+1)$ which is in contradiction with Equation . This concludes the induction step and hence the proof of existence.
We call the representation of $N$ in the above theorem the $d$-th Macaulay representation of $N$ with respect to $q$. One retrieves the usual $d$-th Macaulay representation letting $q$ tend to infinity. We refer to $(m_d,\dots,m_1)$ as the coefficient tuple of this representation. A direct corollary of the above is the following.
\[cor:greedy\] The coefficient tuple $(m_d,\dots,m_1)$ of the $d$-th Macaulay representation with respect to $q$ of a nonnegative integer $N$ can be computed using the following greedy algorithm: The coefficient $m_{d-i}$ can be computed recursively (starting with $i=0$) as the unique integer $m_{d-i} \ge -1$ such that $$\rho_q(d-i,m_{d-i}) \le N-\sum_{j=d-i+1}^{d}\rho_q(j,m_j) <\rho_q(d-i,m_{d-i}+1).$$
From the existence-part of the proof of Theorem \[thm:genrepMac\] it follows directly that the given greedy algorithm finds the desired coefficients.
A further corollary is the following. As before $\preceq$ denotes the lexicographic order.
\[cor:lexorder\] Suppose the $N$ and $M$ are two nonnegative integers whose respective coefficient tuples are $(n_d,\dots,n_1)$ and $(m_d,\dots,m_1)$. Then $$N \le M \ \makebox{if and only if} \ (n_d,\dots,n_1) \preceq (m_d,\dots,m_1).$$
Assume $(n_d,\dots,n_1) \preceq (m_d,\dots,m_1).$ It is enough to show the corollary in case $n_d < m_d$. We know from the previous corollary that $n_d$ and $m_d$ may be determined using the given greedy algorithm. In particular this implies that $n_d < m_d$ implies $$N < \rho_q(d,n_d+1) \le \rho_q(d,m_d) \le M.$$
Assume that $N \le M$. We use induction on $d$. The induction basis is trivial: If $d=1$, then $m_1=M-1$ and $n_1=N-1$. For the induction step, note that $N \le M < \rho_q(d,m_d+1)$ implies by the greedy algorithm that $n_d \le m_d$. If $n_d < m_d$, we are done. If $n_d=m_d$, we replace $N$ with $N-\rho_q(d,m_d)$ and $M$ with $M-\rho_q(d,m_d)$ and use the induction hypothesis to conclude that $(n_d,\dots,n_1) \preceq (m_d,\dots,m_1)$.
A simple expression for $\bar e^{\AA}_r(d,m)$
=============================================
We are now ready to state and prove the relation between the Macaulay representation with respect to $q$ and $\bar e^{\AA}_r(d,m)$.
\[thm:main\] For $1 \le r \le \rho_q(d,m)$, let the $d$-th Macaulay representation of $\rho_q(d,m)-r$ with respect to $q$ be given by $$\rho_q(d,m)-r=\sum_{i=1}^d \rho_q(i,m_i).$$ Denoting the floor function as $\lfloor \cdot \rfloor$, we have $$\bar e^{\AA}_r(d,m)=\sum_{i=1}^d \lfloor q^{m_i} \rfloor.$$
We know from Equation that we need to show that $$\sum_{i=1}^d \lfloor q^{m_i} \rfloor=\sum_{i=1}^m \mu_i q^{m-i},$$ with $(\mu_1, \dots, \mu_{m})$ is the $r$-th element in descending lexicographic order among all $m$-tuples $(\beta_1,\dots,\beta_m)$ in $\{0,1,\dots,q-1\}^m$ satisfying $\beta_1+ \cdots + \beta_{m} \le d$. First of all note that since $r \ge 1$, we have $\rho_q(d,m)-r < \rho_q(d,m)$. In particular this implies that $m_d \le m-1$. Therefore the coefficients of the $d$-tuple $(m_d,\dots,m_1)$ are in $\{-1,0,\dots,m-1\}$. Now for $1 \le i \le m+1$ define $\mu_i:=|\{j \, : \, m_j=m-i\}|.$ Since the $d$-tuple $(m_d,\dots,m_1)$ is nonincreasing by Condition 2 from Theorem \[thm:genrepMac\], we can reconstruct it uniquely from the $(m+1)$-tuple $(\mu_{1},\mu_2,\dots,\mu_{m+1}).$ Moreover, Condition 3 from Theorem\[thm:genrepMac\], implies that $(\mu_1,\dots,\mu_m) \in \{0,1,\dots,q-1\}^m$, but note that $\mu_{m+1}$ could be strictly larger than $q-1$. Further by construction we have $\mu_1+\cdots+\mu_m+\mu_{m+1}=d$, implying that $\mu_1+\cdots+\mu_m \le d$. Note that $\mu_{m+1}$ is determined uniquely by $(\mu_1,\dots,\mu_m)$, since $\mu_0=d-\mu_1-\cdots-\mu_m$. Therefore the correspondence between the $d$-tuples $(m_d,\dots,m_1)$ of coefficients of the $d$-th Macaulay representations with respect to $q$ of integers $0 \le N < \rho_q(d,m)$ and the $m$-tuples $(\mu_1, \dots , \mu_{m})\in \{0,1,\dots,q-1\}^m$ satisfying $\mu_1+ \cdots + \mu_{m} \le d$, is a bijection. Moreover by construction we have $$\sum_{i=1}^d\lfloor q^{m_i} \rfloor =\sum_{j=1}^{m+1} \mu_j \lfloor q^{m-j} \rfloor = \sum_{j=1}^{m} \mu_j q^{m-j}.$$
What remains to be shown is that the constructed $m$-tuple coming from the integer $\rho_q(d,m)-r$ is in fact the $r$-th in descending lexicographic order. First of all, by Corollary \[cor:help\] we see that for $r=1$ and $d=aq+b$ that the $m$-tuple associated to $\rho_q(d,m)-1$ equals $(q-1,\dots,q-1,b,0,\dots,0)$, which under the lexicographic order is the maximal $m$-tuple among all $m$-tuples $(\beta_1, \dots , \beta_{m})\in \{0,1,\dots,q-1\}^m$ satisfying $\beta_1+ \cdots + \beta_{m} \le d$. Next we show that the conversion between $d$-tuples $(m_d,\dots,m_1)$ to $m$-tuples $(\mu_1,\dots,\mu_m)$ preserves the lexicographic order. Suppose therefore that $1\le r \le s \le \rho_q(d,m)$. We write $N:=\rho_q(d,m)-s$ and $M:=\rho_q(d,m)-r.$ and denote their Macaulay coefficient tuples with $(n_d,\dots,n_1)$ and $(m_d,\dots,m_1)$. Since $N \le M$, Corollary \[cor:lexorder\] implies that $(n_d,\dots,n_1) \preceq (m_d,\dots,m_1)$. Also, since these $d$-tuples are nonincreasing, this implies that their associated $m$-tuples $(\nu_1,\dots,\nu_m)$ and $(\mu_1,\dots,\mu_m)$ satisfy $(\nu_1,\dots,\nu_m) \preceq (\mu_1,\dots,\mu_m)$. Indeed assuming without loss of generality that $\nu_1<\mu_1$ we see that $m_i=n_i=m-1$ for $d-\nu_1 \le i \le d$ but $n_i < m_i=m-1$ for $i=\nu_1+1$. Now the desired result follows immediately.
Combining this theorem with the greedy algorithm in Corollary \[cor:greedy\], it is very simple to compute values of $\bar e^{\AA}_r(d,m)$ or equivalently of $d_r({\mathrm{RM}_q}(d,m))$. We illustrate this in the two following examples. The parameters in these example also occur in examples from [@HP].
Let $q=4$, $r=8$, $d=m=3$. Since $d\le q-1$, we may work with the usual Macaulay representation when applying Theorem \[thm:main\]. We have $\rho_q(d,m)=\binom{6}{3}=20$ and hence $$\rho_q(d,m)-r=12=\binom{5}{3}+\binom{2}{2}+\binom{1}{1}=\rho_4(3,2)+\rho_4(2,0)+\rho_4(1,0)$$ is the $3$-rd Macaulay representation of $12$. Theorem \[thm:main\] implies that $\bar e^{\AA}_8(3,3)=4^2+4^0+4^0=18$ and hence $d_8(\mathrm{RM}_4(3,3))=64-18=46$ in accordance with Example 6.10 in [@HP].
Let $q=2$, $r=10$, $d=3$ and $m=5$. We have $\rho_2(3,5)=26$ by Equation and hence applying the greedy algorithm from Corollary \[cor:greedy\], we compute that $$\rho_q(d,m)-r=16=15+1+0=\rho_2(3,4)+\rho_2(2,0)+\rho_2(1,-1)$$ is the $3$rd Macaulay representation of $16$ with respect to $2$. Theorem \[thm:main\] implies that $\bar e^{\AA}_{10}(3,3)=2^4+2^0=17$ and hence $d_8(\mathrm{RM}_2(3,5))=32-17=15$ in accordance with Example 6.12 in [@HP].
Theorem \[thm:main\] is somewhat similar in spirit as Theorem 6.8 from [@HP] in the sense that in both theorems a certain representation in terms of dimensions of Reed–Muller codes is used to give an expression for $d_r({\mathrm{RM}_q}(d,m))$. Where we studied decompositions of $\rho_q(d,m)-r$, in [@HP] the focus was on $r$ itself. This suggest there may exist a duality between the two approaches, but the similarities seem to stop there. The representation in [@HP] is not the Macaulay representation with respect to $q$ that we have used here. For us it is for example very important that each degree $i$ between $1$ and $d$ occurs once in Theorem \[thm:genrepMac\] (implying that the greedy algorithm terminates after at most $d$ iterations), while this is not the case in Theorem 6.8 [@HP]. It could be interesting future work to determine if a deeper lying relationship between the two approaches exists.
[22]{} E.F. Assmus Jr. and J.D. Key, Designs and their Codes, Cambridge University Press, 1992.
P. Beelen and M. Datta, Generalized Hamming weights of affine Cartesian codes, Finite Fields and Applications 51, 130–145, 2018.
M. Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, In: Algebraic Curves and Projective Geometry, Lecture Notes in Mathematics 1389, 76–86, 2006.
P. Heijnen and R. Pellikaan, Generalized [H]{}amming weights of [$q$]{}-ary [R]{}eed-[M]{}uller codes, IEEE Trans. Inform. Theory 44(1), 181–196, 1998.
[^1]: Department of Applied Mathematics and Computer Science, Technical University of Denmark, DK-2800, Kongens Lyngby, Denmark, [email protected]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'High redshift blazars are among the most powerful objects in the Universe. Although they represent a significant fraction of the extragalactic hard X-ray sky, they are not commonly detected in $\gamma$-rays. High redshift (z$>$2) objects represent $<$10% of the AGN population observed by [*Fermi*]{} so far, and $\gamma$-ray flaring activity from these sources is even more uncommon. The characterization of the radio-to-$\gamma$-ray properties of high redshift blazars represent a powerful tool for the study of both the energetics of such extreme objects and the Extragalactic Background Light. We present results of a multi-band campaign on TXS0536+145, which is the highest redshift flaring $\gamma$-ray blazar detected so far. At the peak of the flare the source reached an apparent isotropic $\gamma$-ray luminosity of 6.6$\times$10$^{49}$ erg/s, which is comparable with the luminosity observed from the most powerful blazars. The physical properties derived from the multi-wavelength observations are then compared with those shown by the high redshift population. In addition preliminary results from the high redshift flaring blazar PKS2149-306 will be discussed.'
author:
- 'M. Orienti, F. D’Ammando, M. Giroletti, D. Dallacasa, T. Venturi'
- 'J. Finke'
- 'M. Ajello'
title: 'Flaring $\gamma$-ray emission from high redshift blazars'
---
Introduction
============
The population of high redshift ($z>2$) blazars represents a small fraction ($<$10%) of the extragalactic $\gamma$-ray sky. They are mainly associated with flat spectrum radio quasars (FSRQ), although a few BL Lacs with $z>2$ are present in the third catalog of active galactic nuclei detected by the Large Area Telescope (LAT) on board the [*Fermi*]{} satellite after the first four years of scientific observations [@ackermann15]. The number counts drop when higher energies are considered. In the first LAT catalog of $\gamma$-ray sources above 10 GeV (1FHL) only seven objects with $z>2$ are detected [@ackermann13].\
Although the detection of high redshift blazars during a $\gamma$-ray flare is even more uncommon, the characterization of the radio-to-$\gamma$-ray properties of high redshift blazars represent a powerful tool for the study of both the energetics of such extreme objects and the Extragalactic Background Light (EBL). During $\gamma$-ray flaring episodes the spectra of FSRQ sometimes show a moderate hardening [@pacciani14], allowing us to explore energies that are usually strongly attenuated due to the intrinsic source spectrum.\
So far, 10 blazars at $z > 2$ have been detected during $\gamma$-ray flaring activity. Among these objects there are TXS0536+145 at $z=2.69$, and PKS2149-306 at $z=2.34$. TXS0536+145 was not part of the [*Fermi*]{}-LAT first (1FGL) and second source (2FGL) catalogs [@abdo10b; @nolan12], indicating its low activity state during the first two years of [*Fermi*]{}-LAT observations. On 2012 March 22 it underwent a $\gamma$-ray flare, becoming the $\gamma$-ray flaring object at the highest redshift observed so far [@orienti14].\
PKS2149-306 was detected by [*Fermi*]{}-LAT in a flaring state on 2013 January 4 [@dammando13], with a daily $\gamma$-ray flux about 25 times higher than the average source flux reported in the 2FGL catalog [@nolan12].\
The high activity states observed in both sources triggered multiwavelength monitoring observations aimed at characterizing the variability in the various bands of the electromagnetic spectrum and at determining the spectral energy distribution of these extreme objects.\
TXS0536+145
===========
[*Fermi*]{}-LAT data
--------------------
We analyzed [*Fermi*]{}-LAT data collected during the first five years of scientific observations, from 2008 August 4 (MJD 54682) to 2013 August 4 (MJD 56508). We considered an energy range between 0.1 and 100 GeV, and we followed the standard LAT analysis procedures (for more details see [@orienti14]).\
TXS0536+145 was not detected during the first two years of observations. The 2$\sigma$ upper limit estimated over this period is 10$^{-8}$ ph cm$^{-2}$ s$^{-1}$. During the third and fourth years of observations, the source was detected with a flux of (4.2 $\pm$ 0.6)$\times$10$^{-8}$ ph cm$^{-2}$ s$^{-1}$ and a photon index $\Gamma=2.37 \pm 0.09$. On 2012 March 22 the source was observed during a $\gamma$-ray flare, when it reached a flux of (1.0 $\pm$ 0.3)$\times$10$^{-6}$ ph cm$^{-2}$ s$^{-1}$ and a photon index $\Gamma=2.05 \pm 0.08$, indicating a hardening of the spectrum. This flux corresponds to an apparent isotropic luminosity of 6.6$\times$10$^{49}$ erg/s. Before this flare, the source was first detected in $\gamma$-rays on 2012 January showing an enhancement of its high-energy activity, but without reaching a similar peak flux (Fig. \[fermi\_0536\]).\
The analysis of the [*Fermi*]{}-LAT data with E $>$ 10 GeV collected between 2011 August and 2013 August could not detect the source at such high energies. We evaluated the 2$\sigma$ upper limit as 9.3$\times$10$^{-11}$ ph cm$^{-2}$ s$^{-1}$ (assuming $\Gamma=2.37$).
Radio properties
----------------
Monitoring campaigns of TXS0536+145 with the Very Long Baseline Array (VLBA) at 8.4, 15, and 24 GHz, and with the European VLBI Network (EVN) at 22 GHz were triggered by the $\gamma$-ray flare with the aim of studying changes in the parsec-scale structure and the flux density variability related to the central region of the source. The observations were performed between 2012 April and 2013 October. The source has a core-jet structure (Fig. \[morpho\]). The radio emission is dominated by the compact bright core component, which accounts for about 90 per cent of the total flux density at 8.4 GHz, and about 95 per cent at 15 and 24 GHz. The jet emerges from the main component with a position angle of about 180$^{\circ}$, then at $\sim$ 1.5 mas (i.e. $\sim$ 12 pc) it slightly changes orientation to about 160$^{\circ}$ and extends to $\sim$ 6 mas (i.e. $\sim$ 48 pc).\
The flux density variability is ascribed to the core region, while the jet is not variable. The radio light curves show a flux density increase about 2-3 months after the $\gamma$-ray flare, with longer delay occurring at lower frequencies. The spectral index of the core computed between 8.4 and 15 GHz shows a softening of the spectrum from $\alpha \sim -1.0$ just after the flare, to $\alpha \sim 0.1$[^1] a few months later (upper panel of Fig. \[radio\]). The light curve at 15 GHz shows a possible double hump similar to that observed in $\gamma$-ray light curve (bottom panel of Fig. \[radio\]).\
No new superluminal component was observed after the flare. This may be related to the high redshift of the target. In fact, only superluminal components with a speed higher than 35$c$ would have been picked up during the 16-month monitoring campaign.\
[*Swift*]{} data and SED
------------------------
Triggered by the flaring activity, [*Swift*]{} observed TXS0536+145 a few days after the 2012 March $\gamma$-ray flare, and the source was found in a high state in X-rays. The X-ray flux decreases of a factor of two a couple of weeks after the peak. An additional observation was carried out a few months later, when the source was in a similar low activity state. In the past, the source was not detected by the ROSAT all-sky survey. Therefore, this is the first detection of TXS0535+145 in X-rays.\
Due to severe Galactic absorption and the short exposures, the source was not detected by UVOT in any filter.\
The hard X-ray flux of this source turned out to be below the sensitivity of the BAT instrument for such short exposures, and therefore the source was not detected. The source was not present in the [*Swift*]{} BAT 70-month hard X-ray catalogue [@baumgartner13].\
The spectral energy distribution (SED) of TXS0536+145 in flaring activity is well fitted by a synchrotron/external Compton model where the seed photons upscattered to high energies may be those from the dusty torus (see e.g., [@orienti14]). Due to the rather poor optical coverage, the model parameter are not well constrained.\
PKS2149-306
===========
The FSRQ PKS2149-306 was observed by [*Fermi*]{}-LAT during a flaring episode on 2013 January 4 and preliminary results were reported in [@dammando13]. This source was part of the 2LAC [@ackermann11], indicating on average a higher level of $\gamma$-ray activity with respect to TXS0536+145.\
We analyzed [*Fermi*]{}-LAT data collected during the first six years of scientific observations, from 2008 August 4 (MJD 54682) to 2014 August 4 (MJD 56873). As in the case of TXS0536+145, we considered an energy range between 0.1 and 100 GeV, and we followed the standard LAT analysis procedures.\
The source was clearly detected by [*Fermi*]{}-LAT for most of the period with one-month integration time (Fig. \[fermi\_2149\]). A first significant increase of activity was observed in 2011 February. During the strong flaring activity observed in 2013 January, the source reached a daily peak flux of (3.0$\pm$0.4)$\times$10$^{-6}$ ph cm$^{-2}$ s$^{-1}$, and showed a hardening of the spectrum. This value corresponds to an apparent peak luminosity of 1.5$\times$10$^{50}$ erg/s. A dedicated multiwavelength analysis of PKS2149-306 is ongoing aiming at studying the high-energy SED, also in perspective of the next generation of high-energy telescopes.\
Discussion and conclusions
==========================
High redshift flaring blazars are among the most luminous objects in the Universe. The high redshift FSRQ TXS0536+145 and PKS2149-306 underwent a huge $\gamma$-ray flare, reaching an apparent isotropic luminosity (0.1-100 GeV) of 6.6$\times$10$^{49}$ erg/s and 1.5$\times$10$^{50}$ erg/s, respectively. Such values are comparable to the luminosity observed in the high-redshift gravitationally lensed blazar PKS1830-211 detected during a flare ($L_{\gamma} \sim 3\times
10^{49}$ erg/s; [@abdo15]), as well as in the brightest flaring blazars, like 3C454.3 ($L_{\gamma} \sim 2\times 10^{50}$ erg/s; [@abdo11]), PKS1510-089 ($L_{\gamma} \sim 4\times 10^{48}$ erg/s; [@orienti13]), and PKS1622-297 ($L_{\gamma} \sim 4\times 10^{48}$ erg/s; [@mattox97]).\
We compared the $\gamma$-ray properties of TXS0536+145 and PKS2149-306 with those shown by the population of high redshift ($z>2$) $\gamma$-ray sources from the 2LAC [@ackermann11]. The photon index and the luminosity in the low activity state of the targets are in agreement with those of the other high-z objects. During the flaring state both sources showed a hardening of the spectrum. A similar behaviour was observed in the high-z flaring blazar 4C+71.07 [@akyuz13].\
Despite the harder spectrum, no significant emission above 10 GeV is observed for TXS0536+145. Although this value is consistent with current EBL models (e.g. [@finke10]), the low statistics do not allow us to attribute the spectral curvature to this effect [@orienti14]. The improved sensitivity of the LAT at a few GeV with Pass 8 data will be important for characterizing in more detail the $\gamma$-ray spectrum of the high-redshift blazar population.
The *Fermi* LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l’Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d’Études Spatiales in France. The VLBA is operated by the US National Radio Astronomy Observatory which is a facility of the National Science Foundation operated under a cooperative agreement by Associated Universities, Inc. The European VLBI Network is a joint facility of European, Chinese, South African, and other radio astronomy institutes funded by their national research councils.
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[^1]: The radio spectral index is defined as $S_{\nu} \propto \nu^{-
\alpha}$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the irreducible components of the moduli space of instanton sheaves on ${{\mathbb{P}^{3}}}$, that is rank 2 torsion free sheaves $E$ with $c_1(E)=c_3(E)=0$ satisfying $h^1(E(-2))=h^2(E(-2))=0$. In particular, we classify all instanton sheaves with $c_2(E)\le4$, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space ${{\mathcal T}}(d)$ of stable sheaves on ${{\mathbb{P}^{3}}}$ with Hilbert polynomial $P(t)=dt$, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity $d$; we describe all the irreducible components of ${{\mathcal T}}(d)$ for $d\le4$.'
address:
- |
IMECC - UNICAMP\
Departamento de Matemática\
Rua Sérgio Buarque de Holanda, 651\
13083-970 Campinas-SP, Brazil
- 'Institute of Mathematics of the Romanian Academy, Calea Grivitei 21, Bucharest 010702, Romania'
- |
Department of Mathematics\
National Research University Higher School of Economics\
6 Usacheva Street\
119048 Moscow, Russia
author:
- Marcos Jardim
- Mario Maican
- 'Alexander S. Tikhomirov'
title: Moduli spaces of rank 2 instanton sheaves on the projective space
---
Introduction
============
Instanton bundles on ${\mathbb{C}}{{\mathbb{P}^{3}}}$ were introduced by Atiyah, Drinfeld, Hitchin and Manin in the late 1970’s as the holomorphic counterparts, via twistor theory, to anti-self-dual connections with finite energy (instantons) on the 4-dimensional round sphere $S^4$. To be more precise, an *instanton bundle of charge $n$* is a $\mu$-stable rank 2 bundle $E$ on ${{\mathbb{P}^{3}}}$ with $c_1(E)=0$ and $c_2(E)=n$ satisfying the cohomological condition $h^1(E(-2))=0$; equivalently, an instanton bundle of charge $n$ is a locally free sheaf which arises as cohomology of a linear monad of the form $$\label{instanton monad}
0 \to n\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(-1) \longrightarrow (2+2n)\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \longrightarrow n\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(1) \to 0 .$$
The moduli space ${{\mathcal I}}(n)$ of such objects has been thoroughly studied in the past 35 years by various authors and it is now known to be an irreducible [@T1; @T2], nonsingular [@JV] affine [@CO] variety of dimension $8n-3$.
The closure of ${{\mathcal I}}(n)$ within the moduli space ${{\mathcal M}}(n)$ of semistable rank 2 sheaves with Chern classes $c_1=0$, $c_2=n$ and $c_3=0$ contains non locally free sheaves which also arise as cohomology of monads of the form (\[instanton monad\]). Such *instanton sheaves* can alternatively be defined as rank 2 torsion free sheaves satisfying the cohomological conditions $$h^0(E(-1)) = h^1(E(-2)) = h^2(E(-2)) = h^3(E(-3)) = 0.$$ We prove that such sheaves are always stable, see Theorem \[instanton stability\] below, so they admit a moduli space ${{\mathcal L}}(n)$ regarded as an open subset of ${{\mathcal M}}(n)$ which, of course, contains ${{\mathcal I}}(n)$.
The spaces ${{\mathcal L}}(1)$ and ${{\mathcal L}}(2)$ were essentially known to be irreducible, see details in the first few paragraphs of Section \[L(n)\] below. However, ${{\mathcal L}}(3)$ was observed to have at least two irreducible components [@JMT1 Remark 8.6], while several new components of ${{\mathcal L}}(n)$ were constructed in [@JMT2].
The main goal of this paper is to characterize the irreducible components of ${{\mathcal L}}(3)$ and ${{\mathcal L}}(4)$. We prove:
\[mthm1\]
- ${{\mathcal L}}(3)$ is a connected quasi-projective variety consisting of exactly two irreducible components each of dimension 21;
- ${{\mathcal L}}(4)$ is a connected quasi-projective variety consisting of exactly four irreducible components, three of dimension 29 and one of dimension 32.
For every instanton sheaf $E$, the quotient $E^{\vee\vee}/E$ is a semistable sheaf with Hilbert polynomial $d\cdot(t+2)$ (see Section \[SIS\] below), therefore an essential ingredient for the proof of Main Theorem \[mthm1\] is the study of the moduli space ${{\mathcal T}}(d)$ of semistable sheaves on ${{\mathbb{P}^{3}}}$ with Hilbert polynomial $P(t)=d\cdot t$. Since these spaces are also interesting in their own right, we prove:
\[mthm2\]
- ${{\mathcal T}}(1)$ is an irreducible projective variety of dimension 5;
- ${{\mathcal T}}(2)$ is a connected projective variety consisting of exactly two irreducible components of dimension 8;
- ${{\mathcal T}}(3)$ is a connected projective variety consisting of exactly four irreducible components, two of dimension 12 and two of dimension 13.
- ${{\mathcal T}}(4)$ is a connected projective variety consisting of exactly eight irreducible components, four of dimension 16, two of dimension 17, one of dimension 18 and one of dimension 20.
We also give a precise description of a generic point in each of the irreducible components mentioned in the statement of the theorem, see Section \[1d chi=0\].
[**Acknowledgements.**]{} MJ is partially supported by the CNPq grant number 303332/2014-0, and the FAPESP grants number 2014/14743-8 and 2016/03759-6; this work was completed during a visit to the University of Edinburgh, and he is grateful for its hospitality. MJ also thanks Daniele Faenzi and Simone Marchesi for their help in the proof of Theorem \[instanton stability\] below. AST was supported by a subsidy to the HSE from the Government of the Russian Federation for the implementation of the Global Competitiveness Program. AST also acknowledges the support from the Max Planck Institute for Mathematics in Bonn, where this work was finished during the winter of 2017.
Stability of instanton sheaves {#SIS}
==============================
Recall from [@J-i] that a torsion free sheaf $E$ on ${{\mathbb{P}^{3}}}$ is called an *instanton sheaf* if $c_1(E)=0$ and the following cohomological conditions hold $$h^0(E(-1))=h^1(E(-2))=h^{2}(E(-2))=h^3(E(-3))=0.$$ The integer $n:=-\chi(E(-1))$ is called the charge of $E$; it is easy to check that $n=h^1(E(-1))=c_2(E)$, and that $c_3(E)=0$. The trivial sheaf $r\cdot{{\mathcal O}_{\mathbb{P}^{3}}}$ of rank $r$ is considered as an instanton sheaf of charge zero. In this paper, we will only be interested in rank 2 instanton sheaves.
Recall that the singular locus ${\rm Sing}(G)$ of a coherent sheaf $G$ on a nonsingular projective variety $X$ is given by $${\rm Sing}(G) := \{ x\in X ~|~ G_x ~~\text{is not free over}~~ \mathcal{O}_{X,x} \} ,$$ where $G_x$ denotes the stalk of $G$ at a point $x$ and $\mathcal{O}_{X,x}$ is its local ring. The following result, proved in [@JG Main Theorem], provides a key piece of information regarding the singular loci of rank 2 instanton sheaves.
\[jg-thm\] If $E$ is a non locally free instanton sheaf of rank $2$ on ${{\mathbb{P}^{3}}}$, then
- its singular locus has pure dimension $1$;
- $E^{\vee\vee}$ is a (possibly trivial) locally free instanton sheaf.
\[remark 4\] In fact, the quotient sheaf $Q_E:=E^{\vee\vee}/E$ is a *rank 0 instanton sheaf*, in the sense of [@hauzer Section 6.1]; see also [@JG Section 3.2]. More precisely, a *rank $0$ instanton sheaf* is a coherent sheaf $Q$ on ${{\mathbb{P}^{3}}}$ such that $h^0(Q(-2))=h^1(Q(-2))=0$; the integer $d:=h^0(Q(-1))$ is called the *degree* of $Q$.
The Hilbert polynomial of a rank 2 instanton sheaf $E$ (in fact, of any coherent sheaf on ${{\mathbb{P}^{3}}}$ of rank 2 with $c_1=0$, $c_2=n$ and $c_3=0$) is given by $$\label{pe(t)}
P_E(t) = \frac{1}{3}(t+3)\cdot(t+2)\cdot(t+1) - n\cdot(t+2) = 2\cdot\chi({{\mathcal O}_{\mathbb{P}^{3}}}(t)) - n\cdot(t+2) .$$ Let $n':=c_2(E^{\vee\vee})\ge 0$; it follows from the standard sequence $$\label{std dual sqc}
0 \to E \to E^{\vee\vee} \to Q_E \to 0$$ that $$P_{Q_E}(t)=d\cdot(t+2) ~~ {\rm where} ~~ d:=n-n' ~.$$ Note that the $d=n-n'$ is precisely the multiplicity of $Q_E$ as a rank 0 instanton sheaf.
Rank 0 instanton sheaves can be characterized in the following way.
\[generic\_case\] Every rank 0 instanton sheaf $Q$ admits a resolution of the form $$\label{generic_resolution}
0 {\longrightarrow}d\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(-1) {\longrightarrow}2d\cdot{{\mathcal O}_{\mathbb{P}^{3}}} {\longrightarrow}d\cdot{{\mathcal O}_{\mathbb{P}^{3}}}(1) {\longrightarrow}Q {\longrightarrow}0.$$
Consider the Beilinson spectral sequence from [@choi_chung_maican Section 6], applied to the sheaf $Q':=Q(-2)$. We have $\H^0(Q') = 0$, hence also $\H^0(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) = 0$ and $\H^0(Q'(-1)) = 0$. We adopt the notations of [@choi_chung_maican Section 6]. Since $\ker({\varphi}_5)/{{\mathcal Im}}({\varphi}_4) = 0$, we deduce that $\H^0(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) = 0$. Thus, the bottom row of the $E^1$-term of the spectral sequence vanishes. Since ${\varphi}_7$ is an isomorphism, we deduce that ${\varphi}_1$ is injective. Since ${\varphi}_8$ is injective, we deduce that $\ker({\varphi}_2) = {{\mathcal Im}}({\varphi}_1)$. The top row of the $E^1$-term of the spectral sequence yields the resolution $$0 {\longrightarrow}\H^1(Q'(-1)) {\otimes}{{\mathcal O}_{\mathbb{P}^{3}}}(-3) \overset{{\varphi}_1}{{\longrightarrow}}
\H^1(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) {\otimes}{{\mathcal O}_{\mathbb{P}^{3}}}(-2) \overset{{\varphi}_2}{{\longrightarrow}}$$ $$\overset{{\varphi}_2}{{\longrightarrow}} \H^1(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) {\otimes}{{\mathcal O}_{\mathbb{P}^{3}}}(-1) {\longrightarrow}Q' {\longrightarrow}0.$$ We have $$\chi(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) = -d, \qquad \chi(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) = -2d, \qquad \chi(Q'(-1)) = -d,$$ hence $${\operatorname{h}}^1(Q' {\otimes}\Omega^1_{{\mathbb{P}^{3}}}(1)) = d, \qquad {\operatorname{h}}^1(Q' {\otimes}\Omega^2_{{\mathbb{P}^{3}}}(2)) = 2d, \qquad {\operatorname{h}}^1(Q'(-1)) = d.$$ The above exact sequence yields (\[generic\_resolution\]).
Let now examine the stability properties of instanton sheaves.
\[instanton stability\] Every nontrivial rank 2 instanton sheaf $E$ is stable. In addition, a nontrivial instanton sheaf $E$ is $\mu$-stable if and only if its double dual $E^{\vee\vee}$ is nontrivial.
Since rank 2 instanton sheaves have no global sections [@J-i Prop. 11], every nontrivial locally free rank 2 instanton sheaf is $\mu$-stable; therefore, if $E^{\vee\vee}$ is nontrivial, then $E$ is also $\mu$-stable. Conversely, if $E$ is $\mu$-stable, then so is $E^{\vee\vee}$, hence it must be nontrivial.
Therefore, in order to prove the first claim of the Theorem, it is enough to consider *quasi-trivial instanton sheaves*, i.e. rank 2 instanton sheaves $E$ with $E^{\vee\vee}\simeq 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}$; note that the multiplicity of $Q_E$ is exactly $n=c_2(E)$.
Since $E$ has no global sections, it can only be destabilized by the ideal sheaf ${I_{C/{{\mathbb{P}^{3}}}}}$ of a subscheme $C\subset{{\mathbb{P}^{3}}}$. Moreover, we can assume that the quotient sheaf $E/{I_{C/{{\mathbb{P}^{3}}}}}$ is torsion free, thus it is also the ideal sheaf ${I_{D/{{\mathbb{P}^{3}}}}}$ of another subscheme $D\subset{{\mathbb{P}^{3}}}$. We obtain two exact sequences $$\xymatrix{
& 0 \ar[d] & & & \\
& {I_{C/{{\mathbb{P}^{3}}}}}\ar[d] & & & \\
0 \ar[r] & E \ar[d]\ar[r] & 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \ar[r] & Q_E \ar[r] & 0 \\
& {I_{D/{{\mathbb{P}^{3}}}}}\ar[d] & & & \\
& 0 & & &
}$$
Taking the double dual of the top vertical morphisms we obtain, using also the Snake Lemma, the following commutative diagram $$\label{big diag}
\xymatrix{
& 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\
0 \ar[r] & {I_{C/{{\mathbb{P}^{3}}}}}\ar[d]\ar[r] & {{\mathcal O}_{\mathbb{P}^{3}}} \ar[d]\ar[r] & {{\mathcal O}_{C}}\ar[d]\ar[r] & 0 \\
0 \ar[r] & E \ar[d]\ar[r] & 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \ar[d]\ar[r] & Q_E \ar[d]\ar[r] & 0 \\
0 \ar[r] & {I_{D/{{\mathbb{P}^{3}}}}}\ar[d]\ar[r] & {{\mathcal O}_{\mathbb{P}^{3}}} \ar[d]\ar[r] & {{\mathcal O}_{D}}\ar[d]\ar[r] & 0 \\
& 0 & 0 & 0 &
}$$ Since $h^0(Q_E(-2))=0$, then also $h^0({{\mathcal O}_{C}}(-2))=0$, hence $C$ must have pure dimension 1. Moreover, note also that $h^1(Q_E(-2))=0$ implies $h^1({{\mathcal O}_{D}}(-2))=0$.
We show that $\dim D=0$. Indeed, assume that $D$ has dimension 1. Let $U$ be the maximal 0-dimensional subsheaf of ${{\mathcal O}_{D}}$, and set ${\mathcal O}_{D'}:={{\mathcal O}_{D}}/U$; clearly, $D'$ has pure dimension 1. Next, let $D'':=D'_{\rm red}$ be the underlying reduced scheme. We end up with two exact sequences $$0 \to U \to {{\mathcal O}_{D}}\to {\mathcal O}_{D'} \to 0 ~~{\rm and}~~$$ $$0 \to T \to {\mathcal O}_{D'} \to {\mathcal O}_{D''} \to 0,$$ so that the vanishing of $h^1({{\mathcal O}_{D}}(-2))$ forces $h^1({\mathcal O}_{D''}(-2))=0$.
Still, $D''$ may be reducible, so let $D'' := D''_1 \cup\dots\cup D''_p$ be its decomposition into irreducible components. For each index $j=1,\dots,p$ we obtain a sequence: $$0 \to S_j \to {\mathcal O}_{D''} \to {\mathcal O}_{D''_j} \to 0,$$ thus also $h^1({\mathcal O}_{D''_j}(-2))=0$. Let $d_j$ and $p_j$ denote the degree and arithmetic genus of $D''_j$, respectively. It follows that $$0\le h^0({\mathcal O}_{D''_j}(-2))=\chi({\mathcal O}_{D''_j}(-2))=-2d_j+1-p_j$$ hence $p_j\le-2d_j+1\le-1$, which is imposible for a reduced and irreducible curve.
Now let $\delta=h^0({{\mathcal O}_{D}})$ be the length of $D$; since $\deg(C)=n$, we have $$P_{{I_{C/{{\mathbb{P}^{3}}}}}}(t) = \chi({{\mathcal O}_{\mathbb{P}^{3}}}(t)) - \chi({{\mathcal O}_{C}}(t)) = \chi({{\mathcal O}_{\mathbb{P}^{3}}}(t)) - nt + (\delta-2n) .$$ Comparing with equation (\[pe(t)\]), we have $$\label{pe-pic}
\frac{P_E(t)}{2} - P_{{I_{C/{{\mathbb{P}^{3}}}}}}(t) = \frac{n}{2}t + n - \delta ,$$ which is positive for $n$ sufficiently large, and $E$ contains no destabilizing subsheaves.
As a consequence of the proof above, we also obtain the following interesting fact.
Every rank 2 quasi-trivial instanton on ${{\mathbb{P}^{3}}}$ is an extention of the ideal of a 0-dimensional scheme $D$ by the ideal of a pure 1-dimensional scheme containing $D$.
On the other hand, it is easy to check that every rank 0 instanton sheaf is semistable.
\[rk 0 semistable\] Every rank 0 instanton sheaf is semistable.
Let $Z$ be a rank 0 instanton sheaf, and let $T$ be a subsheaf of $Z$ with Hilbert polynomial $P_Z(t)=a\cdot t+\chi(Z)$. Since $h^0(Z(-2))=0$, then $h^0(T(-2))=0$ and $-2a+\chi(Z)=-h^1(T(-2))\leq0$. It follows that $$\frac{\chi(Z)}{a} \leq 2 = \frac{\chi(Q)}{d} . \qedhere$$
Clearly, not every rank 0 instanton sheaf is stable: if $Q_1$ and $Q_2$ are rank 0 instanton sheaves, then so is any extention of $Q_1$ by $Q_2$, and this cannot possibly be stable.
Conversely, there are semistable sheaves with Hilbert polynomial $dt+2d$ which are not rank 0 instanton sheaves: just consider $Q:={{\mathcal O}}_\Sigma(2)$ for an elliptic curve $\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$, so that $h^0(Q(-2))\ne0$.
Moduli space of instanton sheaves {#L(n)}
=================================
Let ${{\mathcal L}}(n)$ denote the open subscheme of the Maruyama moduli space ${{\mathcal M}}(n)$ of semistable rank 2 torsion free sheaves with Chern classes $c_1=0$, $c_2=n$ and $c_3=0$ consisting of instanton sheaves of charge $n$. Let also ${{\mathcal I}}(n)$ denote the open subscheme of ${{\mathcal M}}(n)$ consisting of locally free instanton sheaves. Finally, let $\overline{{{\mathcal L}}(n)}$ and $\overline{{{\mathcal I}}(n)}$ denote the closures within ${{\mathcal M}}(n)$ of ${{\mathcal L}}(n)$ and ${{\mathcal I}}(n)$, respectively. We also consider the set ${{\mathcal I}}^{0}(n):=\overline{{{\mathcal I}}(n)}\cap {{\mathcal L}}(n)$, which consists of those instanton sheaves which either are locally free, or can be deformed into locally free ones.
It was shown in [@T1; @T2] that ${{\mathcal I}}(n)$ is irreducible for every $n>0$; its closure $\overline{{{\mathcal I}}(n)}$ is called the *instanton component* of ${{\mathcal M}}(n)$. However, the same is not true for ${{\mathcal L}}(n)$ as soon as $n\ge 3$. Indeed, it is well known that $$\overline{{{\mathcal I}}(1)}={{\mathcal L}}(1)={{\mathcal M}}(1)\simeq {{\mathbb{P}^{5}}} ,$$ see for instance [@JMT2 Section 6].
The case $n=2$ has also been understood.
\[L(2)\] $\overline{{{\mathcal L}}(2)} = \overline{{{\mathcal I}}(2)}$.
In particular, ${{\mathcal L}}(2)$ possesses a single irreducible component of dimension 13.
Le Potier showed in [@LeP] that ${{\mathcal M}}(2)$ has exactly 3 irreducible components; according to the description of these components provided in [@JMT2 Section 6], only the instanton component $\overline{{{\mathcal I}}(2)}$ contains instanton sheaves.
Let us now describe the irreducible components of ${{\mathcal L}}(n)$ for $n\ge 3$ introduced in [@JMT2 Section 3].
Let $\Sigma$ be an irreducible, nonsingular, complete intersection curve in ${{\mathbb{P}^{3}}}$, given as the intersection of a surface of degree $d_1$ with a surface of degree $d_2$, with $1\le d_1\le d_2$; denote by $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ the inclusion morphism. Let also $L\in\operatorname{{Pic}}^{g-1}(\Sigma)$ such that $h^0(\Sigma,L)=h^1(\Sigma,L)=0$. Given a (possibly trivial) locally free instanton sheaf $F$ of charge $c\ge0$ and an epimorphism $\varphi:E{\twoheadrightarrow}(\iota_*L)(2)$, the kernel $F:=\ker\varphi$ is an instanton sheaf of charge $c+d_1d_2$. Thus we may consider the set $$\label{c-comp's}
{{\mathcal C}}(d_1,d_2,c) := \left\{ [E]\in{{\mathcal M}}(c+d_1d_2) ~|~ E^{\vee\vee}\in{{\mathcal I}}(c) ~,~
E^{\vee\vee}/E \simeq (\iota_*L)(2) \right\}$$ as a subvariety of ${{\mathcal M}}(c+d_1d_2)$. The following result is proved in [@JMT2], cf. Theorems 15, 17 and 23.
\[dim C thm\] For each $c\ge0$ and $1\le d_1\le d_2$ such that $(d_1,d_2)\ne(1,1),(1,2)$, $\overline{{{\mathcal C}}(d_1,d_2,c)}$ is an irreducible component of ${{\mathcal M}}(c+d_1d_2)$ of dimension $$\label{dim C}
\dim \overline{{{\mathcal C}}(d_1,d_2,c)} = 8c-3 + \frac{1}{2}d_1d_2(d_1+d_2+4) + h,$$ where $$h = \left\{ \begin{array}{l}
2{{d_1+3}\choose{3}} - 4 , ~~{\rm if}~~ d_1=d_2 \\ ~~ \\
{{d_1+3}\choose{3}} + {{d_2+3}\choose{3}} - {{d_2-d_1+3}\choose{3}} - 2, ~~{\rm if}~~ d_1<d_2
\end{array} \right.$$ In addition, $\overline{{{\mathcal C}}(d_1,d_2,c)}\cap{{\mathcal I}}^{0}(c+d_1d_2)\ne
\emptyset$.
We do not know whether the families ${{\mathcal C}}(d_1,d_2,c)$ exhaust all components of ${{\mathcal L}}(n)$, though we prove that this holds for $n=3,4$ in Sections \[L(3) section\] and \[L(4) section\] below, respectively.
However, we remark that the previous result allows for a partial count of the number of components of ${{\mathcal L}}(n)$. Indeed, let $\tau(n)$ denote the number of irreducible components of the union $$\overline{{{\mathcal I}}(n)} \bigcup \left( \bigcup_{d_1d_2+c=n} \overline{{{\mathcal C}}(d_1,d_2,c)} \right).$$ To estimate $\tau(n)$, we must count the different ways in which an integer $n\ge3$ can be written as $n=d_1d_2+c$ with $c\ge0$, and $1\le d_1\le d_2$ excluding the pairs $(d_1,d_2)=(1,1),(1,2)$. Consider the function $$\delta(p) = \left\{ \begin{array}{l}
\frac{1}{2} ( d(p) + 1 ), \mbox{ if } p \mbox{ is a perfect square} \\ ~~ \\
\frac{1}{2} d(p), \mbox{ otherwise}
\end{array} \right.$$ where $d(p)$ is the *divisor function*, i.e. the number of divisors of a positive integer $p$, including $p$ itself. Note that $\delta(p)$ is the number of different ways in which we can write $p$ as a product $d_1d_2$ with $1\le d_1\le d_2$. Adding the instanton component, we have that the number of irreducible components of ${{\mathcal L}}_0(n)$ is given by: $$\label{ell0}
\tau(n) = 1 + \sum_{p=3}^{n} \delta(p) =
\frac{1}{2}\left( \sum_{p=3}^{n} d(p) + \left\lfloor \sqrt{n} \right\rfloor + 1 \right),$$ since $\left\lfloor\sqrt{n}\right\rfloor - 1$ accounts for the number of perfect squares between 3 and $n$.
Let $l(n)$ be the number of irreducible components of the moduli space of instanton sheaves of charge $n$. Then, for $n$ sufficiently large $l(n)> \frac{1}{2} n\cdot\log(n)$.
Determining the asymptotic behaviour of the sum of divisors function is a relevant problem in Number Theory called the *Dirichlet divisor problem*; indeed, it is known that $$\sum_{p=1}^{n} d(p) = n\cdot\log(n) + (2\gamma - 1)n + O(n^\theta),$$ where $\gamma$ denotes the Euler–Mascheroni constant, and $1/4\le \theta\le 131/416$, cf. [@Huxley]. Comparing with equation (\[ell0\]), we easily obtain the desired estimate.
Also relevant for us is a class of instanton sheaves studied in [@JMT1]; more precisely, for $n>0$ and each $m=1,\dots,n$, consider the subset ${{\mathcal D}}(m,n)$ of ${{\mathcal M}}(n)$ consisting of the isomorphism classes $[E]$ of the sheaves $E$ obtained in this way: $${{\mathcal D}}(m,n) :=
\{ [E]\in{{\mathcal M}}(n) ~|~ [E^{\vee\vee}]\in{{\mathcal I}}(n-m), ~~
\Gamma=\mathrm{Supp}(E^{\vee\vee}/E)\in {{\mathcal R}}^*_{0}(m)_{E^{\vee\vee}}~,$$ $$~~{\rm and}~~
E^{\vee\vee}/E\simeq{{\mathcal O}}_\Gamma(2m-1) \},$$ where the space ${{\mathcal R}}^*_{0}(m)_{E^{{{\scriptscriptstyle \operatorname{D}}}{{\scriptscriptstyle \operatorname{D}}}}}$ is decribed as follows: first, let ${{\mathcal R}}^*_{0}(m)$ denote the space of nonsingular rational curves $\Gamma\hookrightarrow{{\mathbb{P}^{3}}}$ of degree $m$ whose normal bundle $N_{\Gamma/{{\mathbb{P}^{3}}}}$ is given by $2\cdot{{\mathcal O}}_\Gamma((2m-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$; then, for any instanton sheaf $F$ we set $${{\mathcal R}}^*_{0}(m)_F := \{\:\Gamma\in {{\mathcal R}}^*_{0}(m)~|~ F|_\Gamma\simeq2\cdot{{\mathcal O}}_\Gamma\ \}.$$ One can show that for every rank 2 instanton sheaf $F$, the space ${{\mathcal R}}^*_{0}(m)_F$ is a nonempty open subset of ${{\mathcal R}}^*_{0}(m)$, cf. [@JMT1 Lemma 6.2].
Let $\overline{{{\mathcal D}}(m,n)}$ denote the closure of ${{\mathcal D}}(m,n)$ within ${{\mathcal M}}(n)$. Note that since $E^{\vee\vee}$ is a locally free instanton sheaf of charge $n-m$, and ${{\mathcal O}}_\Gamma(2m-1)$ is a rank 0 instanton sheaf of degree $m$, then $E$ is an instanton sheaf of charge $n$, so that ${{\mathcal D}}(m,n)\subset{{\mathcal L}}(n)$. In fact, it is shown in [@JMT2 Theorem 7.8] that $\overline{{{\mathcal D}}(m,n)}\subset{{\mathcal I}}^0(n)$. In addition, we prove:
\[disjoint rat curves\] Let $\Gamma_1,\dots,\Gamma_r$ be disjoint, smooth irreducible rational curves in ${{\mathbb{P}^{3}}}$ of degrees $m_1,\dots,m_r$, respectively; set $Q:=\bigoplus_{j=1}^{r}{{\mathcal O}}_{\Gamma_j}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$. If $F$ is a locally free instanton sheaf of charge $c$ such that $F|_{\Gamma_j}\simeq2\cdot{{\mathcal O}}_{\Gamma_j}$ for each $j=1,\dots,r$, and $\varphi:F{\twoheadrightarrow}Q(2)$ is an epimorphism, then $[\ker\varphi]\in{{\mathcal I}}^0(c+m_1+\cdots+m_r)$.
The proof of the previous proposition requires the following technical lemma, proved in [@JMT1 Lemma 7.1].
\[F,G\] Let $C$ be a smooth irreducible curve with a marked point $0$, and set $\mathbf{B}:=C \times{{\mathbb{P}^{3}}}$. Let $\mathbf{F}$ and $\mathbf{G}$ be $\mathcal{O}_{\mathbf{B}}$-sheaves, flat over $C $ and such that $\mathbf{F}$ is locally free along $\mathrm{Supp}(\mathbf{G})$. Denote $$\mathbf{G}_t:=\mathbf{G}|_{\{t\}\times{{\mathbb{P}^{3}}}} ~~{\rm and}~~
\mathbf{F}_t=\mathbf{F}|_{\{t\}\times{{\mathbb{P}^{3}}}}\ \ {\rm for}\ t\in C .$$ Assume that, for each $t\in C$, $$\label{vanish Hi}
H^i({{\mathcal H}{\it om}}(\mathbf{F}_t,\mathbf{G}_t))=0,\ \ \ i\ge1.$$ If $s:\mathbf{F}_0\to\mathbf{G}_0$ is an epimorphism, then, after possibly shrinking $C$, $s$ extends to an epimorhism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$.
We argue by induction on $r$; the case $r=1$ is just the aforementioned result, namely [@JMT2 Theorem 7.8].
Let $Q':=\bigoplus_{j=1}^{r-1}{{\mathcal O}}_{\Gamma_j}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$, so that $Q=Q'\oplus{{\mathcal O}}_{\Gamma_r}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$. Let $E:=\ker\varphi$, and let $E'$ denote the kernel of the composition $F\stackrel{\varphi}{{\twoheadrightarrow}} Q(2) {\twoheadrightarrow}Q'(2) $. We obtain the following exact sequence: $$0 \to E \to E' \stackrel{\varphi'}{\rightarrow} {{\mathcal O}}_{\Gamma_r}((2m_r-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}}) \to 0.$$ By the induction hypothesis, $[E']\in{{\mathcal I}}^0(c+m_1+\cdots+m_{r-1})$, thus one can find an affine open subset $0\in U\subset\mathbb{A}^1$ and a coherent sheaf $\mathbf{E}$ on ${{\mathbb{P}^{3}}}\times U$, flat over $U$, such that $\mathbf{E}_0=E'$ and $\mathbf{E}_t$ is a locally free instanton sheaf of charge $c+m_1+\cdots+m_{r-1}$ satisfying $\mathbf{E}_t|_{\Gamma_r}\simeq 2\cdot{{\mathcal O}}_{\Gamma_r}$ for every $t\in U\setminus\{0\}$. Setting $\mathbf{G}=:\pi^*Q'$ where $\pi:{{\mathbb{P}^{3}}}\times U\to U$ is the projection onto the first factor, note that $$H^i({{\mathcal H}{\it om}}(\mathbf{E}_t,\mathbf{G}_t)) = H^i(2\cdot{{\mathcal O}}_{\Gamma_r}((2m_r-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})) = 0 ~~{\rm for}~~ i\ge1 ~~{\rm and}~~ t\in U.$$ This claim is clear for $t\ne0$; when $t=0$, simply observe that the sequence $0\to E'\to F \to Q'(2)\to 0$ implies that $E'_{\Gamma_r}\simeq F_{\Gamma_r}$, since the support of $Q'$ is disjoint from $\Gamma_r$.
By Lemma \[F,G\], there exists an epimorphism $\mathbf{s}:\mathbf{E}\twoheadrightarrow\mathbf{G}$ extending $\varphi':E'\to {{\mathcal O}}_{\Gamma_r}((2m_r-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$, so that $[\ker\mathbf{s}_t]\in{{\mathcal D}}(m_r,c+m_1+\cdots+m_r)$, by construction. It then follows that $[E]\in\overline{{{\mathcal D}}(m_r,c+m_1+\cdots+m_r)}$, hence, by [@JMT2 Theorem 7.8], $[E]\in{{\mathcal I}}^0(c+m_1+\cdots+m_r)$, as desired.
Next, we consider the following situation: let $\Sigma$ be an irreducible, nonsingular, complete intersection curve in ${{\mathbb{P}^{3}}}$, given as the intersection surfaces of degrees $d_1$ and $d_2$, with $1\le d_1\le d_2$ and $(d_1,d_2)\ne(1,1),(1,2)$, and let $\Gamma$ be a smooth irreducible rational curve in ${{\mathbb{P}^{3}}}$ of degree $m$ disjoint from $\Sigma$. Set $Q:=L\oplus{{\mathcal O}}_{\Gamma}(-{{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$ for some $L\in\operatorname{{Pic}}^{g-1}(\Sigma)$ such that $h^0(\Sigma,L)=h^1(\Sigma,L)=0$, where $g$ is the genus of $\Sigma$.
\[disjoint curves\] If $F$ is a locally free instanton sheaf of charge $c$ such that $F|_{\Gamma}\simeq2\cdot{{\mathcal O}}_{\Gamma}$, and $H^1(F^\vee|_{\Gamma}\otimes L(2))=0$. If $\varphi:F{\twoheadrightarrow}Q(2)$ is an epimorphism, then $[\ker\varphi]\in\overline{{{\mathcal C}}(d_1,d_2,c+m)}$.
The idea is the same as in the proof of Proposition \[disjoint rat curves\]. Let $E'$ be the kernel of the composition $F\stackrel{\varphi}{{\twoheadrightarrow}}Q(2){\twoheadrightarrow}{{\mathcal O}}_{\Gamma}((2m-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}})$, so that $E:=\ker\varphi$ and $E'$ are related via the following exact sequence: $$0 \to E \to E' \stackrel{\varphi'}{\rightarrow} L(2) \to 0.$$ By [@JMT2 Theorem 7.8], one can find an affine open subset $0\in U\subset\mathbb{A}^1$ and a coherent sheaf $\mathbf{E}$ on ${{\mathbb{P}^{3}}}\times U$, flat over $U$, such that $\mathbf{E}_0=E'$ and $\mathbf{E}_t$ is a locally free instanton sheaf of charge $c+m$ for every $t\in U\setminus\{0\}$.
Setting $\mathbf{G}:=\pi^*L(2)$, we must, in order to apply Lemma \[F,G\], check that $$H^i({{\mathcal H}{\it om}}(E_t,G_t))=0 ~~{\rm for}~~ i\ge1 ~~{\rm and}~~ t\in U.$$ Indeed, since $\dim G_t=1$, it is enough to show that $H^1({{\mathcal H}{\it om}}(E_t,G_t))=0$. Note that $${{\mathcal H}{\it om}}(E_0,G_0) = {{\mathcal H}{\it om}}(E',L(2)) \simeq {{\mathcal H}{\it om}}(F,L(2)) \simeq F^\vee|_{\Sigma}\otimes L(2) ,$$ where the middle isomorphism follows from applying the functor ${{\mathcal H}{\it om}}(\cdot,L(2))$ to the sequence $$0\to E' \to F \to {{\mathcal O}}_{\Gamma}((2m-1){{\mathrm{\hspace{0.2ex}p\hspace{-0.23ex}t}}}) \to0,$$ also exploring the fact that $\Sigma$ and $\Gamma$ are disjoint. It follows that $H^1({{\mathcal H}{\it om}}(E_0,G_0))=H^1(F^\vee|_{\Sigma}\otimes L(2))=0$ by hypothesis. By semicontinuity of $h^1({{\mathcal H}{\it om}}(E_t,G_t))$, we can shrink $U$ to another affine open subset $U'\subset\mathbb{A}^1$, if necessary, to guarantee that $H^1({{\mathcal H}{\it om}}(E_t,G_t))=0$ for every $t\in U'$.
By Lemma \[F,G\], there exists an epimorphism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$ extending $\varphi':E'\to L(2)$, so that $[\ker\mathbf{s}_t]\in{{\mathcal C}}(d_1,d_2,c+m)$, by construction. Since $E\simeq \ker\mathbf{s}_0$, it follows that $[E]\in\overline{{{\mathcal C}}(d_1,d_2,c+m)}$.
Moduli of sheaves of dimension one and Euler characteristic zero {#1d chi=0}
================================================================
Given two integers $d$ and $\chi$, $d\ge1$, let ${{\mathcal T}}(d,\chi)$ be the moduli space of semistable coherent sheaves on ${{\mathbb{P}^{3}}}$ with Hilbert polynomial $P(t)=d\cdot t+\chi$. In this section, we focus on the space ${{\mathcal T}}(d):={{\mathcal T}}(d,0)$.
Apart from its intrinsic interest, the space ${{\mathcal T}}(d)$ is also relevant for the study of instanton sheaves, and the description of ${{\mathcal T}}(d)$ for $d\le4$ provided in this section will be a key ingredient for the proof of the Main Theorem \[mthm1\].
In addition, let ${{\mathcal Z}}(d)$ denote the set of rank 0 instanton sheaves of degree $d$ *modulo S-equivalence* (which makes sense, since, by Lemma \[rk 0 semistable\], every rank 0 instanton sheaf is semistable). After a twist by ${{\mathcal O}_{\mathbb{P}^{3}}}(-2)$, ${{\mathcal Z}}(d)$ can be regarded as an open subscheme of the moduli space ${{\mathcal T}}(d)$ consisting of those sheaves $Q$ satisfying $h^0(Q)=0$.
The space ${{\mathcal T}}(d)$ has several distinguished subsets, which we now describe.
First, let ${{\mathcal P}}_d \subset {{\mathcal T}}(d)$ be the subset of planar sheaves; it is a fiber bundle over $({{\mathbb P}}^3)^*$ with fiber being the moduli space of semistable coherent sheaves on ${{\mathbb{P}^{2}}}$ with Hilbert polynomial $P=d\cdot t$. In view of [@lepotier Theorem 1.1], ${{\mathcal P}}_d$ is a projective irreducible variety of dimension $d^2+4$. In particular, ${{\mathcal P}}_d$ is closed.
Next, consider the subsets ${{\mathcal R}}^o_d,~ {{\mathcal E}}^o_d \subset {{\mathcal T}}(d)$ of sheaves supported on smooth rational curves of degree $d$, respectively, on smooth elliptic curves of degree $d$. Let ${{\mathcal R}}_d$ and ${{\mathcal E}}_d$ denote their closures.
Given a partition $(d_1,\ldots,d_s)$ of $d$ such that $d_1 \ge \cdots \ge d_s$, we denote by ${{\mathcal T}}_{d_1, \ldots, d_s} \subset {{\mathcal T}}(d)$ the locally closed subset of points of the form $$\label{polystable}
[Q_1 \oplus \cdots \oplus Q_s],$$ where $Q_i$ gives a stable point in ${{\mathcal T}}(d_i)$; in particular, ${{\mathcal T}}_d$ is the open subset of stable points in ${{\mathcal T}}(d)$. Let ${{\mathcal T}}^o_{d_1, \ldots, d_s} \subset {{\mathcal T}}_{d_1, \ldots, d_s}$ be the open dense subset given by the condition that ${\operatorname{supp}}(Q_i)$ be mutually disjoint. Clearly, each irreducible component of ${{\mathcal T}}^o_{d_1, \ldots, d_s}$ is an open dense subset of an irreducible component of ${{\mathcal T}}_d$. Hence irreducible components of $\overline{{{\mathcal T}}}_{d_1, \ldots, d_s}$ are also irreducible components of ${{\mathcal T}}(d)$. On the other hand, each point of ${{\mathcal T}}(d)$ is an $S$-equivalence class of a polystable (e. g. stable) sheaf of the form (\[polystable\]). Hence, the following result follows.
\[irred comp\] (i) All irreducible components of ${{\mathcal T}}(d)$ are exhausted by the irreducible components of the union $$\label{union}
\underset{(d_1,...,d_s)}{\bigcup }\overline{{{\mathcal T}}}_{d_1, \ldots, d_s},$$ this union being taken over all the partitions $(d_1,...,d_s)$ of $d$.\
(ii) For a given partition $(d_1,...,d_s)$ of $d$, each irreducible component of $\overline{{{\mathcal T}}}_{d_1, \ldots, d_s}$ is birational to a symmetric product $$({{\mathcal X}}_1 \times \cdots \times {{\mathcal X}}_s)/\Sigma$$ of irreducible components ${{\mathcal X}}_i$ of ${{\mathcal T}}_{d_i}$, where $\Sigma$ is the subgroup of the full symmetric group $\Sigma_s$ of degree $s$ generated by the transpositions $(i, j)$ for which $d_i = d_j$ and ${{\mathcal X}}_i = {{\mathcal X}}_j$.
We have only to prove statement (ii). Indeed, let $\Sigma' \subset \Sigma_s$ be the subgroup generated by the transpositions $(i, i+1)$ for which $d_i = d_{i+1}$. We have a bijective morphism $$({{\mathcal T}}_{d_1} \times \cdots \times {{\mathcal T}}_{d_s})/\Sigma' {\longrightarrow}{{\mathcal T}}_{d_1, \ldots, d_s}, \qquad ([Q_1], \ldots, [Q_s]) \longmapsto [Q_1 \oplus \cdots \oplus Q_s],$$ which is an isomorphism over ${{\mathcal T}}^o_{d_1, \ldots, d_s}$, because over this set we can construct local inverse maps. Whence, the statement (ii) follows.
\[irred of Td\] Lemma \[irred comp\] implies that the problem of finding the irreducible components of ${{\mathcal T}}(d)$ is reduced to the problem of finding the irreducible components of ${{\mathcal T}}_2, \ldots, {{\mathcal T}}_d$.
It also follows from Lemma \[irred comp\] that the number of irreducible components of ${{\mathcal T}}(d)$ is at least as large as the number of partitions of $d$, usualy denoted $p(d)$. A well-known formula by Hardy and Ramanujan gives the following asymptotic expression $$p(d) \sim \frac{1}{4\sqrt{3}\cdot d}\exp\left(\pi\sqrt{\frac{2d}{3}}\right).$$ Therefore, the number of irreducible components of ${{\mathcal T}}(d)$ grows at least exponentially on $\sqrt{d}$. However, as we shall see below in the cases $d=3$ and $d=4$, $p(d)$ is just a rough underestimate of the number of irreducible components of ${{\mathcal T}}(d)$.
Given a coherent sheaf $Q$ on ${{\mathbb{P}^{3}}}$, we define $Q^{{\scriptscriptstyle \operatorname{D}}}:= {\mathcal Ext}^c (Q, \omega_{{{\mathbb{P}^{3}}}})$, where $c = \operatorname{codim} (Q)$. We will use below the following general result regarding stable sheaves in ${{\mathcal T}}(d)$.
\[trivial\_lemma\] Assume that ${{\mathcal F}}$ gives a stable point in ${{\mathcal T}}(d)$ and that $P \in {\operatorname{supp}}({{\mathcal F}})$ is a closed point. Then there are exact sequences $$\label{trivial_extension}
0 {\longrightarrow}{{\mathcal E}}{\longrightarrow}{{\mathcal F}}{\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0$$ and $$\label{trivial_sequence}
0 {\longrightarrow}{{\mathcal F}}{\longrightarrow}{{\mathcal G}}{\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0$$ for some sheaves ${{\mathcal E}}\in {{\mathcal T}}(d,-1)$ and ${{\mathcal G}}\in {{\mathcal T}}(d,+1)$.
Choose a surjective morphism ${{\mathcal F}}\to {{\mathbb C}}_P$ and denote its kernel by ${{\mathcal E}}$. Since ${{\mathcal F}}$ is stable, ${{\mathcal E}}$ is semi-stable, so we have sequence (\[trivial\_extension\]). According to [@rendiconti Theorem 13], the dual sheaf ${{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}$ gives a stable point in ${{\mathcal T}}(d)$. Thus, we have an exact sequence $$0 {\longrightarrow}{{\mathcal E}}_1 {\longrightarrow}{{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}{\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0$$ with ${{\mathcal E}}_1 \in {{\mathcal T}}(d,-1)$. According to [@rendiconti Remark 4], ${{\mathcal F}}$ is reflexive. According to [@rendiconti Theorem 13], the sheaf ${{\mathcal G}}= {{\mathcal E}}_1^{{\scriptscriptstyle \operatorname{D}}}$ gives a point in ${{\mathcal T}}(d,1)$. Since ${{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}$ is pure, we can apply [@huybrechts_lehn Proposition 1.1.10] to deduce that ${\mathcal Ext}^3({{\mathcal F}}^{{\scriptscriptstyle \operatorname{D}}}, \omega_{{{\mathbb P}}^3}) = 0$. The long exact sequence of extension sheaves associated to the above exact sequence yields (\[trivial\_sequence\]).
The goal of this section is to describe the irreducible components of ${{\mathcal T}}(d)$ for $d\le4$. According to [@dedicata], for ${{\mathcal F}}\in {{\mathcal T}}(d)$ we have the following cohomological conditions $$\begin{aligned}
& {\operatorname{h}}^0 ({{\mathcal F}}) = 0 \qquad \text{if $d = 1$ or $2$}, \\
& {\operatorname{h}}^0 ({{\mathcal F}}) \le 1 \qquad \text{if $d = 3$ or $4$}.
$$
Moduli of sheaves of degree 1 and 2
-----------------------------------
The case $d=1$ is easy: clearly, ${{\mathcal T}}(1)\simeq {{\mathcal R}}_1$, being isomorphic to the Grassmanian of lines in ${{\mathbb{P}^{3}}}$.
The moduli space ${{\mathcal T}}(1)$ is an irreducible projective variety of dimension 4.
In addition, it is easy to see that ${{\mathcal Z}}(1)={{\mathcal T}}(1)$.
\[components\_2\] The moduli space ${{\mathcal T}}(2)$ is connected, and has two irreducible components, each of dimension 8: ${{\mathcal P}}_2$ (which coincides with ${{\mathcal R}}_2$) and $\overline{{{\mathcal T}}}_{1, 1}$.
If ${{\mathcal F}}\in {{\mathcal T}}_2$, then we have the exact sequence (\[trivial\_sequence\]) in which ${{\mathcal G}}\in {{\mathcal T}}(2,1)$. Thus, ${{\mathcal G}}$ is the structure sheaf of a conic curve, hence ${{\mathcal G}}$ is planar, and hence ${{\mathcal F}}$ is planar. We conclude that $ {{\mathcal T}}(2) = {{\mathcal P}}_2 \cup \overline{{{\mathcal T}}}_{1,1}$. The intersection ${{\mathcal P}}_2 \cap \overline{{{\mathcal T}}}_{1,1}$ consists of those points of the form $[{{\mathcal O}}_{\ell_1}(-1)\oplus{{\mathcal O}}_{\ell_2}(-1)]$ where $\ell_1$ and $\ell_2$ are two intersecting (and possibly coincident) lines.
Note also that ${{\mathcal Z}}(2)={{\mathcal T}}(2)$; the fact that ${{\mathcal Z}}(2)$ consists of two irreducible components of dimension 8 should be compared with [@hauzer Corollary 6.12], where Hauzer and Langer prove that the moduli space of *framed* rank 0 instanton sheaves also consists of two irreducible components of dimension 8.
Moduli of sheaves of degree 3
-----------------------------
\[components\_3\] The moduli space ${{\mathcal T}}(3)$ has four irreducible components ${{\mathcal P}}_3$, ${{\mathcal R}}_3$, $\overline{{{\mathcal T}}}_{2,1}$ and $\overline{{{\mathcal T}}}_{1,1,1}$, of dimension 13, 13, 12, respectively, 12. If ${{\mathcal F}}\in {{\mathcal T}}_3$ and $\H^0({{\mathcal F}}) \neq 0$, then ${{\mathcal F}}$ is the structure sheaf of a planar cubic curve.
By Proposition \[components\_2\] we have $\overline{{{\mathcal T}}}_2={{\mathcal P}}_2$, so that in view of Lemma \[irred comp\] we already obtain the irreducible components $\overline{{{\mathcal T}}}_{2,1}$ and $\overline{{{\mathcal T}}}_{1,1,1}$ of ${{\mathcal T}}(3)$. Therefore, by Remark \[irred of Td\], we only have to find the irreducible components of ${{\mathcal T}}_3$.
Thus, given ${{\mathcal F}}\in {{\mathcal T}}_3$, take a point $P \in {\operatorname{supp}}({{\mathcal F}})$. We then have the exact sequence (\[trivial\_sequence\]) for ${{\mathcal G}}\in {{\mathcal T}}(3,1)$. According to [@freiermuth_trautmann Theorem 1.1], ${{\mathcal T}}(3,1)$ has two irreducible components: the subset ${{\mathcal P}}$ of planar sheaves and the subset ${{\mathcal R}}$ that is the closure of the set of structure sheaves of twisted cubics. Moreover, all sheaves in ${{\mathcal R}}\setminus {{\mathcal P}}$ are structure sheaves of curves $R \subset {{\mathbb P}}^3$ of degree $3$ and arithmetic genus zero. If ${{\mathcal G}}$ is planar, then ${{\mathcal F}}$ is planar. If ${{\mathcal G}}= {{\mathcal O}}_R$, then $R = {\operatorname{supp}}({{\mathcal F}})$, where the scheme-theoretic support ${\operatorname{supp}}({{\mathcal F}})$ of the sheaf ${{\mathcal F}}$ is defined by the 0-th Fitting ideal ${{\mathcal F}}itt^0({{\mathcal F}})$: ${{\mathcal I}}_{R/{{\mathbb P}}^3}={{\mathcal F}}itt^0({{\mathcal F}})$. The morphism $$\rho \colon {{\mathcal T}}_3 \setminus {{\mathcal P}}_3 {\longrightarrow}{{\mathcal R}}\setminus {{\mathcal P}}, \qquad \rho ([{{\mathcal F}}]) = [{{\mathcal O}}_{{\operatorname{supp}}({{\mathcal F}})}],$$ is injective. Indeed, if $\rho ([{{\mathcal F}}_1]) = \rho ([{{\mathcal F}}_2])$, then ${\operatorname{supp}}({{\mathcal F}}_1) = {\operatorname{supp}}({{\mathcal F}}_2) = R$. Choose a point $P \in R$. We have exact sequences $$0 {\longrightarrow}{{\mathcal F}}_1 {\longrightarrow}{{\mathcal G}}_1 {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0, \qquad
0 {\longrightarrow}{{\mathcal F}}_2 {\longrightarrow}{{\mathcal G}}_2 {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0,$$ with ${{\mathcal G}}_1, {{\mathcal G}}_2 \in{{\mathcal T}}(3,1)$. Clearly, ${{\mathcal G}}_1$ and ${{\mathcal G}}_2$ are both isomorphic to ${{\mathcal O}}_R$, hence ${{\mathcal F}}_1$ and ${{\mathcal F}}_2$ are both isomorphic to the ideal sheaf ${{\mathcal I}}_{P, R}$ of $P$ in $R$. The image of $\rho$ is a constructible set of the irreducible variety ${{\mathcal R}}\setminus {{\mathcal P}}$ and contains an open subset of ${{\mathcal R}}\setminus {{\mathcal P}}$, namely the subset given by the condition that $R$ be irreducible. Indeed, if $R$ is irreducible, then it is easy to check that ${{\mathcal I}}_{P, R}$ is stable; we have $\rho ([{{\mathcal I}}_{P, R}]) = [{{\mathcal O}}_R]$. We deduce that ${{\mathcal T}}_3 \setminus {{\mathcal P}}_3$ is irreducible. It follows that ${{\mathcal R}}_3^o$ is dense in ${{\mathcal T}}_3 \setminus {{\mathcal P}}_3$. Thus, ${{\mathcal T}}_3$ has two irreducible components, hence ${{\mathcal T}}(4)$ has the four irreducible components announced in the proposition.
Assume now that $\H^0({{\mathcal F}}) \neq 0$. Then ${{\mathcal F}}$ cannot be isomorphic to ${{\mathcal I}}_{P, R}$, hence ${{\mathcal F}}$ is planar. Take a non-zero morphism ${{\mathcal O}}\to {{\mathcal F}}$. This morphism factors through an injective morphism ${{\mathcal O}}_C \to {{\mathcal F}}$, where $C$ is a planar curve. The semi-stability of ${{\mathcal F}}$ implies that $C$ is a cubic. Comparing Hilbert polynomials, we see that ${{\mathcal O}}_C \to {{\mathcal F}}$ is an isomorphism.
Moduli of sheaves of degree 4
-----------------------------
\[components\_4\] The moduli space ${{\mathcal T}}(4)$ has eight irreducible components: ${{\mathcal P}}_4$, ${{\mathcal E}}_4$, ${{\mathcal R}}_4$, $\overline{{{\mathcal T}}}_{2,2}$, $\overline{{{\mathcal T}}}_{2,1,1}$, $\overline{{{\mathcal T}}}_{1,1,1,1}$ and two irreducible components of ${{\mathcal T}}_{3,1}$ that are birational to ${{\mathcal P}}_3 \times {{\mathcal T}}_1$, respectively, to ${{\mathcal R}}_3 \times {{\mathcal T}}_1$. Their dimensions are, respectively, $20$, $18$, $16$, $16$, $16$, $16$, $17$, $17$.
By Propositions \[components\_2\] and \[components\_3\] and Lemma \[irred comp\] we already have 5 irreducible components of ${{\mathcal T}}(4)$ which are $\overline{{{\mathcal T}}}_{2,2}$, $\overline{{{\mathcal T}}}_{2,1,1}$, $\overline{{{\mathcal T}}}_{1,1,1,1}$ and two irreducible components of ${{\mathcal T}}_{3,1}$ that are birational to ${{\mathcal P}}_3 \times {{\mathcal T}}_1$, respectively, to ${{\mathcal R}}_3 \times {{\mathcal T}}_1$. Therefore by Remark \[irred of Td\] we have only to find irreducible components of ${{\mathcal T}}_4$. Thus, given ${{\mathcal F}}\in {{\mathcal T}}_4$, take a point $P \in {\operatorname{supp}}({{\mathcal F}})$. We then have the exact sequence (\[trivial\_sequence\]) for ${{\mathcal G}}\in {{\mathcal T}}(4,1)$. According to [@choi_chung_maican Theorem 4.12], ${{\mathcal T}}(4,1)$ has three irreducible components: the subset ${{\mathcal P}}$ of planar sheaves, the subset ${{\mathcal R}}$ that is the closure of the set of structure sheaves of rational quartic curves, and the set ${{\mathcal E}}$ that is the closure of the set of sheaves of the form ${{\mathcal O}}_E(P')$, where $E$ is a smooth elliptic quartic curve and $P' \in E$. If ${{\mathcal G}}\in {{\mathcal P}}$, then ${{\mathcal F}}\in {{\mathcal P}}_4$. The sheaves in ${{\mathcal R}}\setminus ({{\mathcal P}}\cup {{\mathcal E}})$ are structure sheaves of quartic curves of arithmetic genus zero. The sheaves in ${{\mathcal E}}\setminus {{\mathcal P}}$ are supported on quartic curves of arithmetic genus $1$. Let ${{\mathcal T}}_{4, {\operatorname{rat}}} \subset {{\mathcal T}}_4$ be the subset of sheaves whose support is a quartic curve of arithmetic genus zero. As in Proposition \[components\_3\], we can construct an injective dominant morphism $$\rho \colon {{\mathcal T}}_{4, {\operatorname{rat}}} {\longrightarrow}{{\mathcal R}}\setminus ({{\mathcal P}}\cup {{\mathcal E}}), \qquad \rho [{{\mathcal F}}] = [{{\mathcal O}}_{{\operatorname{supp}}({{\mathcal F}})}].$$ It follows that ${{\mathcal T}}_{4, {\operatorname{rat}}}$ is irreducible, hence ${{\mathcal T}}_{4, {\operatorname{rat}}} \subset {{\mathcal R}}_4$. To finish the proof of the proposition we need to show that ${{\mathcal T}}_4 \setminus ({{\mathcal P}}_4 \cup {{\mathcal T}}_{4, {\operatorname{rat}}})$ is contained in ${{\mathcal E}}_4$.
According to [@space_quartics], discussion after Proposition 8, the sheaves ${{\mathcal G}}$ in ${{\mathcal E}}\setminus {{\mathcal P}}$ are of two kinds:
1. ${{\mathcal O}}_E(P')$ for a curve $E$ of arithmetic genus $1$ given by an ideal of the form $(q_1, q_2)$, where $q_1$, $q_2$ are quadratic forms, and $P' \in E$. Notice that ${\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathbb C}}_{P'}, {{\mathcal O}}_E) {{\settowidth{\rrrr}{$\scriptstyle{xx}$}
\xrightarrow{\makebox[\rrrr]{$\scriptstyle{}$}}
\hspace{-0.5\rrrr }\hspace{-1.1 em}
\raisebox{- 0.5 ex}{$\sim$}\hspace{0.7\rrrr }
}}{{\mathbb C}}$, so the notation ${{\mathcal O}}_E(P')$ is justified. Also note that ${{\mathcal O}}_E$ is stable;
2. non-planar extensions of the form $$0 {\longrightarrow}{{\mathcal O}}_L(-1) {\longrightarrow}{{\mathcal G}}{\longrightarrow}{{\mathcal C}}{\longrightarrow}0,$$ where $L$ is a line and ${{\mathcal C}}$ gives a point in ${{\mathcal T}}_H(3,1)$ for a plane $H$ possibly containing $L$. (Here and below we use the notation ${{\mathcal T}}_S(d,\chi)$ for the moduli space of 1-dimensional sheaves on a given surface $S$ in ${{\mathbb P}}^3$ with Hilbert polynomial $P(t)=dt+\chi$. We also set ${{\mathcal T}}_S(d):={{\mathcal T}}_S(d,0)$.)
\
*Claim 1*. Case (ii) is unfeasible.
\
Assume, firstly, that $P \in H$. Tensoring (\[trivial\_sequence\]) with ${{\mathcal O}}_H$, we get the exact sequence $${{\mathcal F}}_{| H} {\longrightarrow}{{\mathcal G}}_{| H} \overset{\alpha}{{\longrightarrow}} {{\mathcal C}}_P {\longrightarrow}0.$$ Thus, ${{\mathcal Ker}}(\alpha)$ is a quotient sheaf of ${{\mathcal F}}$ of slope zero. This contradicts the stability of ${{\mathcal F}}$. Assume, secondly, that $P \notin H$. According to [@space_quartics Sequence (10)], we have an exact sequence $$0 {\longrightarrow}{{\mathcal E}}{\longrightarrow}{{\mathcal G}}{\longrightarrow}{{\mathcal O}}_L {\longrightarrow}0$$ for some sheaf ${{\mathcal E}}\in {{\mathcal T}}_H (3)$. The composite map ${{\mathcal E}}\to {{\mathcal G}}\to {{\mathcal C}}_P$ is zero, hence ${{\mathcal E}}$ is a subsheaf of ${{\mathcal F}}$. This contradicts the stability of ${{\mathcal F}}$ and proves Claim 1.
\
It remains to deal with the sheaves from (i). We have one of the following possibilities:
1. $E$ is contained in a smooth quadric surface $S$;
2. $E$ is contained in an irreducible cone $\Sigma$ but not in a smooth quadric surface;
3. $\operatorname{span}\{ q_1, q_2 \}$ contains only reducible quadratic forms and $q_1$, $q_2$ have no common factor.
\
*Claim 2*. In case (a), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
Notice that ${{\mathcal F}}\in {{\mathcal T}}_S(4)$. According to [@ballico_huh Proposition 7], ${{\mathcal T}}_S(4)$ has five disjoint irreducible components ${{\mathcal T}}_S (p, q, 4)$, where $(p, q)$ is the type of the support of the 1-dimensional sheaf w.r.t. $\mathrm{Pic}S$. Clearly, ${{\mathcal F}}\in {{\mathcal T}}_S (2, 2, 4)$. Thus, ${{\mathcal F}}$ is a limit of sheaves in ${{\mathcal T}}_S (2, 2, 4)$ supported on smooth curves of type $(2, 2)$, hence ${{\mathcal F}}\in {{\mathcal E}}_4$.
\
It remains to deal with cases (b) and (c). Next we reduce further to the case when $P = P'$. Notice that, if $P \neq P'$, then ${{\mathcal F}}\simeq {{\mathcal O}}_E(P') {\otimes}({{\mathcal O}}_E(P))^{{\scriptscriptstyle \operatorname{D}}}$, hence the notation ${{\mathcal F}}= {{\mathcal O}}_E(P' - P)$ is justified.
\
*Claim 3*. Assume that ${{\mathcal F}}= {{\mathcal O}}_E(P' - P)$ for an elliptic quartic curve $E$ and distinct closed points $P', P \in E$. Then ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
Let $Z_1, \ldots, Z_m$, denote the irreducible components of $E$. Fix $i, j \in \{ 1, \ldots, m \}$. Consider the locally closed subset ${{\mathcal X}}\subset {{\mathcal E}}\times {{\mathcal E}}$ of pairs $([{{\mathcal O}}_{E'}(P_1)], [{{\mathcal O}}_{E'}(P_2)])$, where $E'$ is a quartic curve of arithmetic genus $1$ whose ideal is generated by two quadratic polynomials, and $P_1$, $P_2$ are distinct points on $E'$ such that $P_1 \notin \cup_{k \neq i} Z_k$, $P_2 \notin \cup_{k \neq j} Z_k$. Consider the morphisms $$\xi \colon {{\mathcal X}}{\longrightarrow}{{\mathcal T}}(4), \qquad ([{{\mathcal O}}_{E'}(P_1)], [{{\mathcal O}}_{E'}(P_2)]) \longmapsto [{{\mathcal O}}_{E'}(P_1 - P_2)],$$ $$\sigma \colon {{\mathcal X}}{\longrightarrow}{\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t), \qquad ([{{\mathcal O}}_{E'}(P_1)], [{{\mathcal O}}_{E'}(P_2)]) \longmapsto E',$$ where ${\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$ is the Hilbert scheme of subschemes of ${{\mathbb P}}^3$ with Hilbert polynomial $P(t)=4t$. According to [@chen_nollet Examples 2.8 and 4.8], ${\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$ consists of two irreducible components, denoted ${\mathbf H}_1$ and ${\mathbf H}_2$. The generic member of ${\mathbf H}_1$ is a smooth elliptic quartic curve. The generic member of ${\mathbf H}_2$ is the disjoint union of a planar quartic curve and two isolated points. Note that ${\mathbf H}_2$ lies in the closed subset $\{ E' \mid \ {\operatorname{h}}^0({{\mathcal O}}_{E'}) \ge 3 \}$. Since $E$ lies in the complement of this subset, we deduce that $E \in {\mathbf H}_1$. It follows that there exists an irreducible quasi-projective curve $\Gamma \subset {\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$ containing $E$, such that $\Gamma \setminus \{ E \}$ consists of smooth elliptic quartic curves (see the proof of [@space_quartics Proposition 12]). The fibers of the map $\sigma^{-1} (\Gamma) \to \Gamma$ are irreducible of dimension $2$. By [@shafarevich Theorem 8, p. 77], we deduce that $\sigma^{-1}(\Gamma)$ is irreducible. Thus, $\xi (\sigma^{-1}(\Gamma))$ is irreducible. This set contains $[{{\mathcal O}}_{E}(P'-P)]$ for $P' \in Z_i \setminus \cup_{k \neq i} Z_k$, $P \in Z_j \setminus \cup_{k \neq j} Z_k$. The generic member of $\xi (\sigma^{-1}(\Gamma))$ is a sheaf supported on a smooth elliptic quartic curve. We conclude that $[{{\mathcal O}}_E(P' - P)] \in {{\mathcal E}}_4$. Since $i$ and $j$ are arbitrary, the result is true for all $P'$, $P$ closed points on $E$.
\
*Claim 4*. In case (c), $E$ is a quadruple line supported on a line $L$. More precisely, there are three distinct planes $H$, $H'$, $H''$ containing $L$, such that $E = (H \cup H') \cap (2H'')$.
\
The claim will follow if we can show that there are linearly independent forms $u, v \in V^*$ such that $q_1, q_2 \in {{\mathbb C}}[u, v]$. Indeed, in this case $(q_1, q_2)$ has the normal form $(uv, (u+v)^2)$. We argue by contradiction. Assume that $q_1 = XY$ and $q_2 = Zl$. Consider first the case when $l = aX + bY + cZ$. We will find $\lambda \in {{\mathbb C}}$ such that $f = XY + \lambda Z l$ is irreducible, which is equivalent to saying that $$\frac{\partial f}{\partial X} = Y + a\lambda Z, \qquad \frac{\partial f}{\partial Y} = X + b\lambda Z, \qquad \frac{\partial f}{\partial Z} = \lambda (aX + bY + 2cZ)$$ have no common zero, or, equivalently, $$\left|
{\begin{array}}{ccc}
0 & 1 & a\lambda \\
1 & 0 & b\lambda \\
a\lambda & b\lambda & 2c\lambda
{\end{array}}\right| \neq 0.$$ We have reduced to the inequality $2 a b \lambda^2 - 2c\lambda \neq 0$. If $c \neq 0$ we can find a solution. If $c = 0$, then $a b \neq 0$, otherwise $q_1$ and $q_2$ would have a common factor, and we can choose any $\lambda \in {{\mathbb C}}^*$. Assume now that $l = aX + bY + cZ + dW$ with $d \neq 0$. Note that $f = XY + \lambda Z l$ is irreducible if its image in $${{\mathbb C}}[X, Y, Z, W]/\langle (c-1)Z + dW \rangle \simeq {{\mathbb C}}[X, Y, Z]$$ is irreducible. The above isomorphism sends $f$ to $XY + \lambda Z (aX + bY + Z)$ which brings us to the case examined above.
\
*Claim 5*. In case (c), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
We have ${{{\mathcal O}}_E}_{|H} \simeq {{\mathcal O}}_C$, ${{{\mathcal O}}_E}_{|H'} \simeq {{\mathcal O}}_{C'}$ for conic curves $C$ and $C'$ supported on $L$. The kernel of the map ${{\mathcal O}}_E \to {{\mathcal O}}_C$ has Hilbert polynomial $2t-1$ and is stable, because ${{\mathcal O}}_E$ is stable, hence it is isomorphic to ${{\mathcal O}}_{C'}(-1)$. We have a commutative diagram $$\xymatrix
{
0 \ar[r] & {{\mathcal O}}_E \ar[r] \ar[d] & {{\mathcal O}}_E(P') \ar[r] \ar[d] & {{\mathbb C}}_{P'} \ar[r] \ar@{=}[d] & 0 \\
0 \ar[r] & {{\mathcal O}}_C \ar[r] & {{\mathcal O}}_E(P')_{|H} \ar[r] & {{\mathbb C}}_{P'} \ar[r] & 0
}$$ in which the second row is obtained by restricting the first row to $H$. Applying the snake lemma, we obtain the first row of the following exact commutative diagram: $$\xymatrix
{
0 \ar[r] & {{\mathcal O}}_{C'}(-1) \ar[r] & {{\mathcal O}}_E(P') \ar[r] \ar[d] & {{\mathcal O}}_E(P')_{|H} \ar[r] \ar[d]^-{\alpha} & 0 \\
& & {{\mathbb C}}_P \ar@{=}[r] & {{\mathbb C}}_P
}$$ Applying the snake lemma to this diagram, we get the exact sequence $$0 {\longrightarrow}{{\mathcal O}}_{C'}(-1) {\longrightarrow}{{\mathcal F}}{\longrightarrow}{{\mathcal Ker}}(\alpha) {\longrightarrow}0.$$ Note that ${{\mathcal Ker}}(\alpha)$ has Hilbert polynomial $2t+1$ and is semi-stable, being a quotient of the stable sheaf ${{\mathcal F}}$. It follows that ${{\mathcal Ker}}(\alpha) \simeq {{\mathcal O}}_C$. Thus, ${{\mathcal F}}$ gives a point in the set ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \big)^{{\scriptstyle \operatorname{s}}}$ of stable non-split extensions of ${{\mathcal O}}_C$ by ${{\mathcal O}}_{C'}(-1)$.
Consider the family of planes $H''_t$, $t \in {{\mathbb P}}^1 \setminus \{ 0, \infty \}$, containing $L$ and different from $H$ and $H'$. Denote $E_t = (H \cup H') \cap (2 H''_t)$. We have a two-dimensional family of semi-stable sheaves $$\{ {{\mathcal O}}_{E_t}(P' - P'') \mid t \in {{\mathbb P}}^1 \setminus \{ 0, \infty \}, \ P'' \in L \setminus \{ P' \} \} \subset {{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1))\big).$$ This family is dense in the right-hand-side because ${\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}} ({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \simeq {{\mathbb C}}^3$. To prove this we use the standard exact sequence obtained from Thomas’ spectral sequence $$\begin{gathered}
0 \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{{\mathcal O}}_C}_{| H'}, {{\mathcal O}}_{C'}(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}} ({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \to \\ {\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'}), {{\mathcal O}}_{C'}(-1))
\to {\operatorname{Ext}}^2_{{{\mathcal O}}_{H'}} ({{{\mathcal O}}_C}_{| H'}, {{\mathcal O}}_{C'}(-1)),\end{gathered}$$ see also [@choi_chung_maican Lemma 4.2]. Note that ${{{\mathcal O}}_C}_{| H'} \simeq {{\mathcal O}}_L$. Using Serre duality we obtain the isomorphisms $$\begin{aligned}
{\operatorname{Ext}}^2_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_L, {{\mathcal O}}_{C'}(-1)) & \simeq {\operatorname{Hom}}_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{C'}(-1), {{\mathcal O}}_L(-3))^* = 0, \\
{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_L, {{\mathcal O}}_{C'}(-1)) & \simeq {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{C'}(-1), {{\mathcal O}}_L(-3))^* \simeq {{\mathbb C}}^2.\end{aligned}$$ The last isomorphism follows from the long exact sequence of extension sheaves $$\begin{gathered}
0 = {\operatorname{Hom}}({{\mathcal O}}_{H'}(-1), {{\mathcal O}}_L(-3)) {\longrightarrow}{\operatorname{Hom}}({{\mathcal O}}_{H'}(-3), {{\mathcal O}}_L(-3)) \simeq \H^0 ({{\mathcal O}}_L) \simeq {{\mathbb C}}\\
{\longrightarrow}{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{C'}(-1), {{\mathcal O}}_L(-3)) {\longrightarrow}\\
{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{H'}(-1), {{\mathcal O}}_L(-3)) \simeq \H^1({{\mathcal O}}_L(-2)) \simeq {{\mathbb C}}{\longrightarrow}{\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}} ({{\mathcal O}}_{H'}(-3), {{\mathcal O}}_L(-3)) = 0\end{gathered}$$ derived from the short exact sequence $$0 {\longrightarrow}{{\mathcal O}}_{H'}(-3) {\longrightarrow}{{\mathcal O}}_{H'}(-1) {\longrightarrow}{{\mathcal O}}_{C'}(-1) {\longrightarrow}0.$$ Choose linear forms $u$ and $u'$ defining $H$ and $H'$. Restricting the standard resolution $$0 {\longrightarrow}{{\mathcal O}}(-3) \xrightarrow{\tiny \left[ \!\! {\begin{array}}{l} -u \\ (u')^2 {\end{array}}\!\!\! \right]} {{\mathcal O}}(-2) \oplus {{\mathcal O}}(-1)
\xrightarrow{\tiny \left[ \!\! {\begin{array}}{cc} (u')^2 & \!\! u {\end{array}}\!\! \right]} {{\mathcal O}}{\longrightarrow}{{\mathcal O}}_C {\longrightarrow}0$$ to $H'$, we see that ${{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'})$ is isomorphic to the cohomology of the complex $${{\mathcal O}}_{H'}(-3) \xrightarrow{\tiny \left[ \!\! {\begin{array}}{l} - u_{| H'} \\ \phantom{-} 0 {\end{array}}\!\! \right]} {{\mathcal O}}_{H'}(-2) \oplus {{\mathcal O}}_{H'}(-1)
\xrightarrow{\tiny \left[ \!\! {\begin{array}}{cc} 0 & \!\! u_{| H'} {\end{array}}\!\! \right]} {{\mathcal O}}_{H'}$$ that is, to ${{\mathcal O}}_L(-2)$. Using the fact that ${{\mathcal O}}_{C'}(-1)$ and ${{\mathcal O}}_L(-2)$ are reflexive, we have the isomorphisms $${\operatorname{Hom}}({{\mathcal O}}_L(-2), {{\mathcal O}}_{C'}(-1)) \simeq {\operatorname{Hom}}({{\mathcal O}}_{C'}(-1)^{{\scriptscriptstyle \operatorname{D}}}, {{\mathcal O}}_L(-2)^{{\scriptscriptstyle \operatorname{D}}}) \simeq {\operatorname{Hom}}({{\mathcal O}}_{C'}, {{\mathcal O}}_L) \simeq {{\mathbb C}}.$$ The above discussion shows that $[{{\mathcal F}}]$ is a limit of points in ${\operatorname{M}}_{{{\mathbb P}}^3}(4m)$ of the form $[{{\mathcal O}}_{E_t}(P' - P'')]$, with $P' \neq P''$. Claim 5 now follows from Claim 3.
\
It remains to consider sheaves ${{\mathcal F}}$ given by sequence (\[trivial\_sequence\]) in which ${{\mathcal G}}= {{\mathcal O}}_E(P)$ and $E$ is as at (b). We reduce further to the case when $E$ has no regular points.
\
*Claim 6*. Assume that $E$ has a regular point. Then ${{\mathcal F}}\simeq {{\mathcal O}}_E$, hence ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
The proof of the claim is obvious because $P$ in sequence (\[trivial\_sequence\]) can be chosen arbitrarily on $E$. We choose $P \in \operatorname{reg}(E)$. The kernel of the map ${{\mathcal O}}_E(P) \to {{\mathbb C}}_P$ is ${{\mathcal O}}_E$. Note that $E$ belongs to the irreducible component ${\mathbf H}_1$ of ${\operatorname{Hilb}}_{{{\mathbb P}}^3}(4t)$, hence it is the limit of smooth elliptic quartic curves.
\
*Claim 7*. Let $E \subset {{\mathbb P}}^3$ be a quartic curve of arithmetic genus $1$ which is contained in an irreducible cone $\Sigma$, but not in a smooth quadric surface. Assume that $E$ has no regular points. Then we have one of the following two possibilities:
1. $E = \Sigma \cap (H \cup H')$, where $H$, $H'$ are distinct planes each intersecting $\Sigma$ along a double line;
2. $E = \Sigma \cap (2H)$, where $H$ is a plane intersecting $\Sigma$ along a double line.
\
To fix notations assume that $\Sigma$ has vertex $O$ and base a conic curve $\Gamma$ contained in a plane $\Pi$. Assume first that $E = \Sigma \cap \Sigma'$ for $\Sigma'$ another irreducible cone. If $\Sigma$ and $\Sigma'$ have distinct vertices, then $E$ has regular points. Thus, $\Sigma'$ has vertex $O$ and base an irreducible conic curve $\Gamma'$ contained in $\Pi$. Since $E$ has no regular points, $\Gamma \cap \Gamma'$ is the union of two double points $Q_1$ and $Q_2$. Now $E$ is the cone with vertex $O$ and base $Q_1 \cup Q_2$, so $E$ is as at (b1).
Assume next that $E = \Sigma \cap (H \cup H')$ for distinct planes $H$ and $H'$. If $H$ or $H'$ does not contain $O$, then $E$ has regular points. If $H$ or $H'$ is not tangent to $\Gamma$, then $E$ has regular points. We deduce that $E$ is as in (b1).
Assume, finally, that $E = \Sigma \cap (2H)$ for a double plane $2H$. If $O \notin H$, then it can be shown that $E$ is contained in a smooth quadric surface. Indeed, assume that $\Sigma$ has equation $X^2 + Y^2 + Z^2 = 0$ and $H$ has equation $W=0$. Then $E$ is contained in the smooth quadric surface with equation $X^2 + Y^2 + Z^2 + W^2 = 0$. Thus, $O \in H$. If $\Gamma \cap H$ is the union of two distinct points, then $\Gamma \cap (2H)$ is the union of two double points $Q_1$ and $Q_2$ and $E$ is as in (b1). If $\Gamma \cap H$ is a double point, then $E$ is as in (b2).
\
*Claim 8*. In case (b1), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
We have ${{\mathcal O}}_{E | H} \simeq {{\mathcal O}}_C$, ${{\mathcal O}}_{E | H'} \simeq {{\mathcal O}}_{C'}$ for conic curves $C$, $C'$ supported on lines $L$, respectively, $L'$. Assume that $P \in L$ and choose a point $P' \in L$ not necessarily distinct from $P$. Let ${{\mathcal F}}' \in {{\mathcal T}}_4$ be given by the exact sequence $$0 {\longrightarrow}{{\mathcal F}}' {\longrightarrow}{{\mathcal O}}_E(P') {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0.$$ As in the first paragraph in the proof of Claim 5, we see that ${{\mathcal F}}'$ gives a point in the set ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1))\big)^{{\scriptstyle \operatorname{s}}}$. We have $\dim {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \le 2$. Indeed, start with the exact sequence $$\begin{aligned}
0 \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathcal O}}_{C | H'}, {{\mathcal O}}_{C'}(-1)) & \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \to \\ & \to {\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'}), {{\mathcal O}}_{C'}(-1)).\end{aligned}$$ The group on the second line vanishes because ${{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{H'})$ is supported on $O$ while ${{\mathcal O}}_{C'}(-1)$ has no zero-dimensional torsion. It follows that $${\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1)) \simeq {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathcal O}}_{C | H'}, {{\mathcal O}}_{C'}(-1)).$$ The sheaf ${{\mathcal O}}_{C | H'}$ is the structure sheaf of a double point supported on $O$, hence we have the exact sequence $${{\mathbb C}}\simeq {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathbb C}}_O, {{\mathcal O}}_{C'}(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathcal O}}_{C | H'}, {{\mathcal O}}_{C'}(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{H'}}({{\mathbb C}}_O, {{\mathcal O}}_{C'}(-1)) \simeq {{\mathbb C}}$$ from which we get our estimate on the dimension of the middle vector space.
The one-dimensional family ${{\mathcal O}}_E(P' - P)$, $P' \in L \setminus \{ P \}$, is therefore dense in ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_{C'}(-1))\big)^{{\scriptstyle \operatorname{s}}}$, hence, in view of Claim 3, ${{\mathcal F}}$ is a limit of sheaves in ${{\mathcal E}}_4$. We conclude that ${{\mathcal F}}\in {{\mathcal E}}_4$.
\
*Claim 9*. In case (b2), ${{\mathcal F}}$ belongs to ${{\mathcal E}}_4$.
\
Let $L$ be the reduced support of $\Sigma \cap H$. We have ${{\mathcal O}}_{E | H} \simeq {{\mathcal O}}_C$ for a conic curve supported on $L$. Choose a point $P' \in L$ not necessarily distinct from $P$ and let ${{\mathcal F}}' \in {{\mathcal T}}_4$ be given by the exact sequence $$0 {\longrightarrow}{{\mathcal F}}' {\longrightarrow}{{\mathcal O}}_E(P') {\longrightarrow}{{\mathbb C}}_P {\longrightarrow}0.$$ As in the first paragraph of the proof of Claim 5, we see that ${{\mathcal F}}'$ gives a point in the set ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_C(-1))\big)^{{\scriptstyle \operatorname{s}}}$. We have $\dim {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) = 5$. This follows from the exact sequence $$\begin{gathered}
0 \to {\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \to {\operatorname{Ext}}^1_{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \to \\
{\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_H), {{\mathcal O}}_C(-1)) \to {\operatorname{Ext}}^2_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)).\end{gathered}$$
From Serre duality we get $${\operatorname{Ext}}^2_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \simeq {\operatorname{Hom}}_{{{\mathcal O}}_H}({{\mathcal O}}_C(-1), {{\mathcal O}}_C(-3))^* \simeq \H^0({{\mathcal O}}_C(-2))^* = 0.$$ We have ${{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_H) \simeq {{\mathcal O}}_C(-1)$ hence ${\operatorname{Hom}}({{\mathcal Tor}}_1^{{{\mathcal O}}_{{{\mathbb P}}^3}}({{\mathcal O}}_C, {{\mathcal O}}_H), {{\mathcal O}}_C(-1)) \simeq {{\mathbb C}}$. Applying the functor ${\operatorname{Hom}}(\cdot,{{\mathcal O}}_C(-1))$ to the short exact sequence $$0 {\longrightarrow}{{\mathcal O}}_H(-2) {\longrightarrow}{{\mathcal O}}_H {\longrightarrow}{{\mathcal O}}_C {\longrightarrow}0$$ we obtain the following exact sequence, $$\begin{gathered}
0 \to {\operatorname{Hom}}({{\mathcal O}}_H(-2), {{\mathcal O}}_C(-1)) \simeq \H^0({{\mathcal O}}_C(1)) \simeq {{\mathbb C}}^3 \\
\to {\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_C, {{\mathcal O}}_C(-1)) \to \\
{\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_H, {{\mathcal O}}_C(-1)) \simeq \H^1({{\mathcal O}}_C(-1)) \simeq {{\mathbb C}}\to 0,\end{gathered}$$ since ${\operatorname{Hom}}({{\mathcal O}}_H, {{\mathcal O}}_C(-1)) \simeq \H^0({{\mathcal O}}_C(-1)) = 0$, and ${\operatorname{Ext}}^1_{{{\mathcal O}}_H}({{\mathcal O}}_H(-2), {{\mathcal O}}_C(-1)) \simeq \H^1({{\mathcal O}}_C(1)) = 0$.
Denote $Q = L \cap \Pi$. We have a three-dimensional family $\Gamma_t$ of irreducible conic curves in $\Pi$ that contain $Q$ and are tangent to $H$. Let $\Sigma_t$ be the cone with vertex $O$ and base $\Gamma_t$. Put $E_t = \Sigma_t \cap (2H)$. The four-dimensional family ${{\mathcal O}}_{E_t}(P' - P)$, $P' \in L \setminus \{ P \}$ is dense in ${{\mathbb P}}\big( {\operatorname{Ext}}^1({{\mathcal O}}_C, {{\mathcal O}}_C(-1))\big)^{{\scriptstyle \operatorname{s}}}$, hence, in view of Claim 3, ${{\mathcal F}}$ is the limit of sheaves in ${{\mathcal E}}_4$. We conclude that ${{\mathcal F}}\in {{\mathcal E}}_4$.
The proof of Main Theorem \[mthm2\] is finally complete.
Components and connectedness of ${{\mathcal L}}(3)$ {#L(3) section}
===================================================
We are now ready to prove that the moduli space of rank 2 instanton sheaves of charge 3 on ${{\mathbb{P}^{3}}}$ is connected and has precisely two irreducible components. Indeed, the two components of $\overline{{{\mathcal L}}(3)}$ have already been identified above:
- $\overline{{{\mathcal I}}(3)}$, whose generic point corresponds to locally free instanton sheaves.
- $\overline{{{\mathcal C}}(1,3,0)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(1,3,0)}
0 \to E \to 2\cdot {{\mathcal O}_{\mathbb{P}^{3}}} \to \iota_*L(2) \to 0$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular plane cubic $\Sigma$, and $L\in{\rm Pic}^0(\Sigma)$ is such that $h^0(\Sigma,L)=0$.
Both components have dimension 21; this is classical result for the component ${{\mathcal I}}^{0}(3)$, while the dimension of ${{\mathcal C}}(1,3,0)$ is given by Theorem \[dim C thm\]. In addition, this same result also guarantees that the union ${{\mathcal I}}^{0}(3)\cup {{\mathcal C}}(1,3,0)$ is connected.
Therefore, our task is to prove that $\overline{{{\mathcal L}}(3)}$ has no other irreducible components, i.e. that every instanton sheaf of charge 3 can be deformed either into a locally free instanton sheaf, or into an instanton sheaf given by a sequence of the form (\[C(1,3,0)\]).
So let $E$ be a non locally free instanton sheaf of charge 3, and let $Q_E:=E^{\vee\vee}/E$ be the corresponding rank 0 instanton sheaf; let $d_E$ denote the degree of $Q_E$. There are three possibilities to consider: $d_E=1$, $d_E=2$ and $d_E=3$.
The first possibility is easy to deal with: if $d_E=1$, then $Q_E={{\mathcal O}}_\ell(1)$, where $\ell\hookrightarrow{{\mathbb{P}^{3}}}$ is a line in ${{\mathbb{P}^{3}}}$. It follows that $E$ fits into an exact sequence of the form $$0 \to E \to F \to {{\mathcal O}}_\ell(1) \to 0 ~,$$ where $F$ is a locally free instanton sheaf of charge 2. However, [@JMT1 Proposition 7.2] ensures that $E$ can be deformed in a (’t Hooft) locally free instanton sheaf of charge 3. In other words, if $d_E=1$, then $E$ lies within ${{\mathcal I}}^0(3)$.
Now, if $d_E=2$, then, since $Q_E$ is semistable and by Proposition \[components\_2\] above, one can find an affine open subset $0\in U\subset{{\mathbb A}}^1$ and a coherent sheaf $\mathbf{G}$ on ${{\mathbb{P}^{3}}}\times U$ such that $G_0=Q_E$ and, for $u\ne 0$,
- either $G_u={{\mathcal O}}_\Gamma(3\mathrm{pt})$, where $\Gamma$ is a nonsingular conic in ${{\mathbb{P}^{3}}}$;
- or $G_u={{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)$ where $\ell_1$ and $\ell_2$ are skew lines in ${{\mathbb{P}^{3}}}$.
Since $d_E=2$, $E^{\vee\vee}$ is a locally free instanton sheaf of charge 1 (a.k.a. a null-correlation bundle); we set $N:=E^{\vee\vee}$ denote such sheaf. Take $\mathbf{F}:=\pi^*N$, where $\pi:{{\mathbb{P}^{3}}}\times U \to {{\mathbb{P}^{3}}}$ is the projection onto the first factor. Let $s:N\twoheadrightarrow Q_E$ be the epimorphism given by the standard sequence (\[std dual sqc\]). For every $u\in U$, the sheaf ${{\mathcal H}{\it om}}(F_u,G_u)\simeq N\otimes G_u$ is supported in dimension 1, thus clearly $H^i({{\mathcal H}{\it om}}(F_u,G_u))=0$ for $i=2,3$. For $u\ne 0$ we can, after possibly shrinking $U$, assume that either $N|_\Gamma\simeq 2\cdot{{\mathcal O}}_\Gamma$ or $N|_{\ell_1}\simeq 2\cdot{{\mathcal O}}_{\ell_1}$ and $N|_{\ell_2}\simeq 2\cdot{{\mathcal O}}_{\ell_2}$; in both situations, it is easy to check that $H^1({{\mathcal H}{\it om}}(F_u,G_u))=0$. Finally, for $u=0$, we twist the resolution of $Q_E$ $$0 \to 2\cdot {{\mathcal O}_{\mathbb{P}^{3}}}(-1) \stackrel{\alpha}{\longrightarrow} 4\cdot {{\mathcal O}_{\mathbb{P}^{3}}} \stackrel{\beta}{\longrightarrow} 2\cdot {{\mathcal O}_{\mathbb{P}^{3}}}(1) \to Q_E \to 0$$ by $N$ and check that $H^1({{\mathcal H}{\it om}}(F_0,G_0))=H^1(N\otimes Q_E)\simeq H^2(N\otimes B)=0$, where $B:=\operatorname{im}\beta$.
Therefore, it follows from Lemma \[F,G\] that there exists an epimorphism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$ extending $s:N\twoheadrightarrow Q_E$. Let $\mathbf{E}:=\ker\mathbf{s}$; clearly, $E_0:=\mathbf{E}|_{\{0\}\times{{\mathbb{P}^{3}}}}=E$. For $u\ne 0$, $E_u$ fits into the exact sequence $$0 \to E_u \to N \to G_u \to 0.$$
In the case [**(i)**]{} described above, $G_u$ lies within ${{\mathcal D}}(2,3)$ for $u\ne0$, hence $E=E_0$ lies within $\overline{{{\mathcal D}}(2,3)}$, which is contained in $\overline{{{\mathcal I}}(3)}$ by [@JMT1 Theorem 7.8]. In other words, $E_0$ can be deformed into a locally free instanton sheaf of charge 3, thus it lies within ${{\mathcal I}}^0(3)$.
In the case [**(ii)**]{}, Propostion \[disjoint rat curves\] also implies that $[E_0]\in{{\mathcal I}}^0(3)$.
An argument similar to the one used in the proof of [@JMT1 Proposition 7.2] works to show that $E$ can be deformed into a locally free (’t Hooft) instanton sheaf.
Summing up, we conclude that if $d_E=2$, then $E$ lies within ${{\mathcal I}}^0(3)$.
Finally, consider $d_E=3$, so that $E^{\vee\vee}=2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}$. Since $Q_E$ is semistable, it follows from Proposition \[components\_3\] that one can find an affine open subset $0\in U\subset{{\mathbb A}}^1$ and a coherent sheaf $\mathbf{G}$ on ${{\mathbb{P}^{3}}}\times U$ such that $G_0=Q_E$ and, for $u\ne 0$,
- either $G_u={{\mathcal O}}_\Delta(5\mathrm{pt})$, where $\Delta$ is a nonsingular twisted cubic in ${{\mathbb{P}^{3}}}$;
- or $G_u={{\mathcal O}}_\Gamma(3\mathrm{pt})\oplus
{{\mathcal O}}_{\ell}(1)$, where $\Gamma$ is a nonsingular conic and $\ell$ is a line disjoint from $\ell$;
- or $G_u={{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)\oplus{{\mathcal O}}_{\ell_3}(1)$ where $\ell_j$ are 3 skew lines in ${{\mathbb{P}^{3}}}$;
- or $G_u=L(2)$, where $L\in\operatorname{{Pic}}^0(\Sigma)$, for some nonsingular plane cubic $\Sigma$ in ${{\mathbb{P}^{3}}}$.
Now set $\mathbf{F}:2\cdot\pi^*{{\mathcal O}_{\mathbb{P}^{3}}}$. Note that $H^i({{\mathcal H}{\it om}}(F_u,G_u))=H^i(2\cdot G_u)$, and this vanishes for $i=1,2,3$ in all of the four cases outlined above for $u\ne0$; for $u=0$, $H^i(G_0)=H^i(Q_E)$ and this vanishes by dimension of $Q_E$ when $i=2,3$, and by the vanishing of $h^1(Q_E(-2))$ when $i=1$.
We complete the argument as before; again, it follows from Lemma \[F,G\] that there exists an epimorphism $\mathbf{s}:\mathbf{F}\twoheadrightarrow\mathbf{G}$ extending the epimorphism $s:2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}\twoheadrightarrow Q_E$ obtained from the standard sequence (\[std dual sqc\]) for $E$. Let $\mathbf{E}:=\ker\mathbf{s}$; clearly, $E_0:=\mathbf{E}|_{\{0\}\times{{\mathbb{P}^{3}}}}=E$. For $u\ne 0$, $E_u$ fits into the exact sequence $$0 \to E_u \to 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}} \to G_u \to 0.$$
In the cases [**(i)**]{} through [**(iii)**]{}, we know from [@JMT1 Theorem 7.8] and Proposition \[disjoint rat curves\] above that $[E_0]\in\overline{{{\mathcal D}}(3,3)}$, thus also $[E_0]\in{{\mathcal I}}^0(3)$.
In the case [**(iv)**]{}, $E_u$ lies within ${{\mathcal C}}(1,3,0)$, by definition. This completes the proof of the first part of Main Theorem \[mthm1\].
Components and connectedness of ${{\mathcal L}}(4)$ {#L(4) section}
===================================================
In this section we prove the second part of Main Theorem \[mthm1\], i.e. we enumerate the irreducible components of ${{\mathcal L}}(4)$ and show that ${{\mathcal L}}(4)$ is connected. Note that we already know from Theorem \[dim C thm\] four irreducible components of $\overline{{{\mathcal L}}(4)}$:
- $\overline{{{\mathcal I}}(4)}$, whose generic point corresponds to locally free instanton sheaves;
- $\overline{{{\mathcal C}}(1,3,1)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(1,3,1)}
0 \to E \to N\to \iota_*L(2) \to 0,$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular plane cubic $\Sigma$, and $L\in{\rm Pic}^0(\Sigma)$ is such that $h^0(\Sigma,L)=0$;
- $\overline{{{\mathcal C}}(2,2,0)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(2,2,0)}
0 \to E \to 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}\to \iota_*L(2) \to 0,$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular elliptic space quartic $\Sigma$, and $L\in{\rm Pic}^0(\Sigma)$ is such that $h^0(\Sigma,L)=0$;
- $\overline{{{\mathcal C}}(1,4,0)}$, whose generic point corresponds to instanton sheaves $E$ fitting into exact sequences of the form $$\label{C(1,4,0)}
0 \to E \to 2\cdot{{\mathcal O}_{\mathbb{P}^{3}}}\to \iota_*L(2) \to 0,$$ where $\iota:\Sigma\hookrightarrow{{\mathbb{P}^{3}}}$ is the inclusion of a nonsingular plane quartic $\Sigma$, and $L\in{\rm Pic}^2(\Sigma)$ is such that $h^0(\Sigma,L)=0$.
The first three components have dimension 29, and the last one has dimension 32; this is a classical result for the component ${{\mathcal I}}^0(4)$, while the dimensions of ${{\mathcal C}}(1,3,1)$, ${{\mathcal C}}(2,2,0)$ and ${{\mathcal C}}(1,4,0)$ are given by Theorem \[dim C thm\] above. Furthermore, [@JMT2 Theorem 23] implies that each of the last three components intersects ${{\mathcal I}}^0(4)$. Thus the union of these four components is connected.
To finish the proof of the second part of Main Theorem \[mthm1\], it is again enough to show that there are no other components in ${{\mathcal L}}(4)$, except for those described above. The argument here is the same as before, exploring Theorem \[jg-thm\], Remark \[remark 4\] and Proposition \[components\_4\].
Take any $[E]\in{{\mathcal L}}(4)$ and consider the triple (\[std dual sqc\]). Then, in view of Theorem \[jg-thm\] and Remark \[remark 4\], $Q_E$ is a rank 0 instanton sheaf of multiplicity $1\le d_E\le 4$, and $E^{\vee\vee}$ is an instanton bundle of charge $4-d_E$. Consider the possible cases for $d_E$.
[$\mathbf{d_E=1.}$]{} As in the similar case in Section \[L(3) section\], $Q_E={{\mathcal O}}_{\ell}(1)$ where $l$ is a line in ${{\mathbb{P}^{3}}}$. Respectively, $[E^{\vee\vee}]\in{{\mathcal I}}(3)$. Deforming $\ell$ in ${{\mathbb{P}^{3}}}$ we may assume that $E^{\vee\vee}|_{\ell}\simeq2\cdot{{\mathcal O}}_{\ell}$, so that $[E]\in{{\mathcal D}}(1,4)$. Therefore, $[E]\in{{\mathcal I}}^0(4)$.
[$\mathbf{d_E=2.}$]{} As in the similar case in Section \[L(3) section\], $Q_E$ can be deformed in a flat family either into a sheaf ${{\mathcal O}}_\Gamma(3\mathrm{pt})$, where $\Gamma$ is a nonsingular conic in ${{\mathbb{P}^{3}}}$, or into a sheaf ${{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)$ where $\ell_1$ and $\ell_2$ are skew lines in ${{\mathbb{P}^{3}}}$. Respectively, $[E^{\vee\vee}]\in{{\mathcal I}}(2)$. Now the same argument as in Section \[L(3) section\] shows that $[E]\in{{\mathcal I}}^0(4)$.
[$\mathbf{d_E=3.}$]{} Then $E^{\vee\vee}$ is a null-correlation bundle and, as in the case $d_E=3$ of Section \[L(3) section\], the sheaf $Q_E$ deforms in a flat family to one of the sheaves:
- $L(2)$, where $L\in\operatorname{{Pic}}^0(\Sigma)$, for some nonsingular plane cubic $\Sigma$ in ${{\mathbb{P}^{3}}}$.
- ${{\mathcal O}}_\Delta(5\mathrm{pt})$, where $\Delta$ is a nonsingular twisted cubic in ${{\mathbb{P}^{3}}}$;
- ${{\mathcal O}}_\Gamma(3\mathrm{pt})\oplus{{\mathcal O}}_{\ell}(1)$, where $\Gamma$ is a nonsingular conic and $\ell$ is a line disjoint from $\ell$;
- ${{\mathcal O}}_{\ell_1}(1)\oplus{{\mathcal O}}_{\ell_2}(1)\oplus
{{\mathcal O}}_{\ell_3}(1)$ where $\ell_j$ are 3 skew lines in ${{\mathbb{P}^{3}}}$.
By definition, $[E]\in\overline{{{\mathcal C}}(1,3,1)}$ in case [**(i)**]{}. The same argument as in Section \[L(3) section\], based on [@JMT1 Theorem 7.8] and Proposition \[disjoint rat curves\], shows that $[E]\in\overline{{{\mathcal I}}(4)}$ in cases [**(ii)**]{} through [**(iv)**]{}.
$\mathbf{d_E=4.}$ Then $E^{\vee\vee}\simeq2\cdot{{\mathcal O}}_{{{\mathbb{P}^{3}}}}$ and, according to Proposition \[components\_4\], the sheaf $Q_E$ deforms in a flat family to one of the sheaves:
- $L(2)$, where $L\in\operatorname{{Pic}}^2(\Sigma)$, for some nonsingular plane quartic $\Sigma$ in ${{\mathbb{P}^{3}}}$, and $L$ satisfies an open condition $h^1(L)=0$;
- $L(2)$, where $0\ne L\in\operatorname{{Pic}}^0(\Delta)$, for some nonsingular space elliptic quartic $\Delta$ in ${{\mathbb{P}^{3}}}$;
- ${{\mathcal O}}_{\Delta}(7\mathrm{pt})$ for some nonsingular rational space quartic $\Delta$ in ${{\mathbb{P}^{3}}}$;
- $L(2)\oplus{{\mathcal O}}_{\ell}(1)$, where $L\in\operatorname{{Pic}}^0(\Sigma)$, for some nonsingular plane cubic $\Sigma$ in ${{\mathbb{P}^{3}}}$ and a line $\ell$ disjoint from $\Sigma$;
- ${{\mathcal O}}_\Delta(5\mathrm{pt})\oplus{{\mathcal O}}_{\ell}(1)$, where $\Delta$ is a nonsingular twisted cubic and $\ell$ is a line disjoint from $\Delta$;
- ${{\mathcal O}}_\Gamma(3\mathrm{pt})\oplus{{\mathcal O}}_{\ell_1}(1)
\oplus{{\mathcal O}}_{\ell_2}(1)$, where $\Gamma$ is a nonsingular conic and $\ell_1,\ \ell_2$ are two skew lines disjoint from $\Gamma$;
- ${{\mathcal O}}_{\ell}(1)\oplus{{\mathcal O}}_{\ell_2}(1)
\oplus{{\mathcal O}}_{\ell_3}(1)\oplus{{\mathcal O}}_{\ell_4}(1)$, where $\ell_1,\ \ell_2,\ \ell_3,\ \ell_4$ are four disjoint lines in ${{\mathbb{P}^{3}}}$.
In case [**(i)**]{}, since, in the notation of Lemma $\ref{F,G}$, $F_0=2\cdot{{\mathcal O}}_{{{\mathbb{P}^{3}}}}$ and $G_0=L$, hence $H^i({{\mathcal H}{\it om}}(F_0,G_0))=0,\ \ \ i\ge1,$ and therefore the condition (\[vanish Hi\]) is satisfied by the semicontinuity, so that the deformation argument as above shows that $E\in\overline{{{\mathcal C}}(1,4,0)}$.
In case [**(ii)**]{}, by the same reason, $[E]\in\overline{{{\mathcal L}}(2,2,0)}$.
In case [**(iii)**]{}, a similar argument shows that $[E]\in\overline{{{\mathcal D}}(4,4)}$, and therefore $[E]\in\overline{{{\mathcal I}}(4)}$.
In case [**(iv)**]{}, Proposition \[disjoint curves\] above guarantees that $[E]\in\overline{{{\mathcal L}}(1,3,1)}$.
In the cases remaining, [**(v)**]{} through [**(vii)**]{}, as in cases [**(ii)**]{} and [**(iii)**]{} for $d_E=3$ above, we again obtain $[E]\in\overline{{{\mathcal I}}(4)}$.
Main Theorem \[mthm1\] is finally proved.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $\phi(z)=(\phi_1(z), \ldots,\phi_n(z))$ be a holomorphic self-map of $U^n$ and $\psi(z)$ a holomorphic function on $U^n,$ where $U^n$ is the unit polydisk of $\mbox{\Bbb C}^n.$ Let $p\geq
0,$ $q\geq 0$, this paper gives some necessary and sufficient conditions for the weighted composition operator $W_{\psi,\phi}$ induced by $\psi$ and $\phi$ to be bounded and compact between $p$-Bloch space ${\cal B}^p(U^n)$ and $q$-Bloch space ${\cal
B}^q(U^n).$
author:
- |
[Zehua ZhouRenyu Chen]{}\
Department of Mathematics, Tianjin University\
Tianjin 300072, P. R. China\
E-mail: [email protected]
title: ' Weighted Composition Operators between different Bloch-type Spaces in Polydisk '
---
140 mm
[^1]
Bloch-type space, Weighted composition operator, Composition operator.
2000Mathematics Subject Classification 47B38, 47B33, 32A37, 32A30
Introduction
============
Let $\Omega$ be a bounded homogeneous domain in $\mbox{\Bbb
C}^{n}.$ The class of all holomorphic functions with domain $\Omega$ will be denoted by $H(\Omega).$ Let $\phi $ be a holomorphic self-map of $\Omega,$ the composition operator $C_{\phi}$ induced by $\phi$ is defined by $$(C_{\phi}f)(z)=f(\phi(z)),$$ for $z$ in $\Omega$ and $f\in
H(\Omega)$. If, in addition, $\psi$ is a holomorphic function defined on $\Omega,$ the multiplication operator induced by $\psi$ is defined by $$M_{\psi}f(z)=\psi(z)f(z),$$ and weighted composition operators $W_{\psi,\phi}$ induced by $\psi$ and $\phi$ is defined by $$\left(W_{\psi,\phi}f\right)(z)=\psi(z)f(\phi(z))$$ for $z$ in $\Omega$ and $f\in H(\Omega).$ If let $\psi\equiv 1,$ then $W_{\psi,\phi}=C_{\phi},$ if let $\phi=Id$, then $W_{\psi,\phi}=M_{\psi}.$ So we can regard weighted composition operator as a generalization of a multiplication operator and a composition operator.
Let $K(z,z)$ be the Bergman kernel function of $\Omega$, the Bergman metric $H_{z}(u,u)$ in $\Omega$ is defined by $$H_{z}(u,u)=\displaystyle\frac{1}{2}
\sum\limits^{n}_{j,k=1} \displaystyle\frac{\partial^{2}\log
K(z,z)}{\partial z_{j}
\partial {\overline {z}}_{k}}u_{j}{\overline u}_{k},$$ where $z\in\Omega$ and $u=(u_{1},\ldots,u_{n})\in \mbox{\Bbb
C}^{n}.$
Following Timoney \[1\], we say that $f\in H(\Omega)$ is in the Bloch space ${\cal B}(\Omega),$ if $$\|f\|_{{\cal B}(\Omega)}=\sup\limits_{z\in \Omega}Q_{f}(z)<\infty,$$ where $$Q_{f}(z)=\sup\left\{\displaystyle\frac {|\bigtriangledown
f(z)u|}{H^{\frac{1}{2}}_{z}(u,u)}: u\in \mbox{\Bbb
C}^{n}-\{0\}\right\},$$ and $\bigtriangledown f(z)
=\left(\frac{\partial f(z)}{\partial z_{1}}, \ldots,
\frac{\partial f(z)}{\partial z_{n}} \right), \bigtriangledown
f(z)u =\sum\limits^{n}_{l=1}\frac{\partial f(z)}{\partial
z_{l}}u_{l}.$
The little Bloch space ${\cal B}_0(\Omega)$ is the closure in the Banach space ${\cal B}(\Omega)$ of the polynomial functions.
Let $\partial\Omega$ denote the boundary of $\Omega$. Following Timoney \[2\], for $\Omega=B_n$ the unit ball of $\mbox{\Bbb C}^n$, ${\cal B}_0(B_n)=\left\{f\in{\cal B}(B_n): Q_f(z)\to 0,
\mbox{as}\hspace*{2mm}z\to\partial B_n \right\};$ for $\Omega=\cal
D$ the bounded symmetric domain other than $B_n$, $\left\{
f\in{\cal B}({\cal D}): Q_f(z)\to 0, \mbox{as}\hspace*{2mm}
z\to\partial{\cal D}\right\}$ is the set of constant functions on $\cal D.$ So if $\cal D$ is a bounded symmetric domain other than the ball, we denote the ${\cal B}_{0*}({\cal D})= \left\{f\in{\cal
B}({\cal D}): Q_f(z)\to 0, \mbox{as}\hspace*{2mm}
z\to\partial^*{\cal D}\right\}$ and also call it little Bloch space, here $\partial^*{\cal D}$ means the distinguished boundary of $\cal D$. The unit ball is the only bounded symmetric domain $\cal D$ with the property that $\partial^*{\cal D}=\partial{\cal
D}.$
Let $U^n$ be the unit polydisk of $\mbox{\Bbb C}^n.$ Timony \[1\] shows that $f\in(U^n)$ if and only if $$\|f\|_=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\displaystyle\frac{\partial f} {\partial
z_k}(z)\right|\left(1- |z_k|^2\right)<+\infty.$$ This equality was the starting point for introducing the $p$-Bloch spaces.
Let $p\geq 0,$ a function $f\in H(U^n)$ is said to belong to the $p$-Bloch space ${\cal B} ^p(U^n)$ if $$\|f\|_p=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\frac{\partial f} {\partial z_k}(z)\right|\left(1-
|z_k|^2\right)^p<+\infty.$$ It is easy to show that ${\cal
B}^p(U^n)$ is a Banach space with the norm $\|\cdot\|_p.$
It is easy to see that if $$\lim_{z\to\partial U^n}\sum\limits^n_{k=1}\left|\frac{\partial f}{\partial
z_k}(z)\right|(1- |z_k|^2)^p=0,$$ then $f$ must be a constant. So, there is no sense to introduce the corresponding little $p$-Bolch space in this way. We will say that the little $p$-Bolch space ${\cal B}_0^p(U^n)$ is the closure of the polynomials in the $p$-Bolch space.
In the recent years, there have been many papers focused on studying the composition operators in function spaces (say, for 1-dimensional case see \[3-8\], for n-dimensional case see \[9-12\]). More recently, S.Ohno, K. Stroethoff and R.H.Zhao in \[8\] discuss the weighted composition operators between-type spaces for 1-dimensional case.
In this paper, we discuss the boundedness and compactness of the weighted composition operators between $p$-Bloch space and $q$-Bloch space in polydisk, some new methods and techniques have been used because of the difference between topology boundary $\partial U^n$ and distinguished boundary $\partial^* U^n$ of $U^n$, where $n>1$, especially in the proof of Theorems 2 and 3 (need to be discussed according to the properties of the boundary). The results in this paper will extend corresponding results on the Bloch spaces (see \[4-11\]).
Our main results are the following:
Let $\phi=(\phi_1, \ldots, \phi_n)$ be a holomorphic self-map of $U^n$ and $\psi(z)$ a holomorphic function of $U^n,$ $p\geq 0,$ $q\geq 0.$\
(i) If $p=1$, then $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow
{\cal B}^q(U^n) $ is bounded if and only if $$\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \psi}
{\partial z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\ln
\displaystyle\frac{4}{1-|\phi_l(z)|^2}=O(1) \hspace*{6mm}(z\in U^n
)\label{1}$$ and $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \phi_{l}}
{\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{1-|\phi_l(z)|^2}=O(1)
\hspace*{6mm}( z\in U^n ).\label{2}$$ (ii) If $0\leq
p<1$, then $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow {\cal
B}^q(U^n) $ is bounded if and only if $$\psi\in {\cal B}^q(U^n)\label{3}$$ and $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p}}=O(1)
\hspace*{6mm}( z\in U^n ).\label{4}$$ (iii) If $p>1$, then $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow {\cal
B}^q(U^n) $ is bounded if and only if $$\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \psi}
{\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p-1}}
=O(1) \hspace*{6mm}( z\in U^n)\label{5}$$ and $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p}}=O(1)
\hspace*{6mm}( z\in U^n ).\label{6}$$
Let $\phi=(\phi_1, \ldots, \phi_n)$ be a holomorphic self-map of $U^n$ and $\psi(z)$ a holomorphic function of $U^n$,$p\geq 0,
q\geq 0.$\
(i) If $p=1$, then $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow
{\cal B}^q(U^n) $ is compact if and only if $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n)$ is bounded and $$\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \psi}
{\partial z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\ln
\displaystyle\frac{4}{1-|\phi_l(z)|^2}=o(1)
\hspace*{6mm}(\phi(z)\to\partial U^n),\label{7}$$ and$$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{1-|\phi_l(z)|^2}
=o(1)\hspace*{6mm}(\phi(z)\to\partial U^n).\label{8}$$ (ii) If $p>1$, then $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow
{\cal B}^q(U^n) $ is compact if and only if $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n)$ is bounded and $$\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \psi}
{\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p-1}}
=o(1)\hspace*{6mm}(\phi(z)\to\partial U^n),\label{9}$$ and $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p}}
=o(1)\hspace*{6mm}(\phi(z)\to\partial
U^n).\label{10}$$
[**Remark 1**]{} It is easy to show that if (1) or (5) or (7) or (9) holds, then $\psi\in{\cal B}^q(U^n)$ and $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial \psi(z)}
{\partial z_k}(z)\right|\left(1-|z_k|^2\right)^{q}
=o(1)\hspace*{6mm}( \phi(z)\to\partial
U^n).\label{11}$$ In fact, (1) implies that $$\sum\limits^n_{k}\left|\displaystyle\frac{\partial \psi}
{\partial
z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\sum\limits^n_{l=1}\ln
\displaystyle\frac{4}{1-|\phi_l(z)|^2}\leq C$$ for all $z\in\mbox{\Bbb C}^n.$ The same reason for others.
Let $\phi=(\phi_1, \ldots, \phi_n)$ be a holomorphic self-map of $U^n$ and $\psi(z)$ a holomorphic function of $U^n$, $0\leq p<1,$ $q\geq 0.$ Then $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n) $ is compact if and only if $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow {\cal B}^q(U^n)
$ is bounded and $$\lim\limits_{|\phi_l(z)|\to 1}\sum\limits^n_{k=1}\left|\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}=0\label{12}$$ for each $l\in\{1,...,n\}$ with $|\phi_l(z)|\to 1.$
Let $\phi=(\phi_1,...,\phi_n)$ be a holomorphic self-map of $U^n$, $p,q\geq 0.$ Then $C_{\phi}:{\cal B} ^p(U^n)\to
{\cal B} ^q(U^n)$ is bounded if and only if there exists a constant $C$ such that $$\sum\limits^n_{k,l=1}\left|\frac{\partial \phi_{l}} {\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q} {(1-|\phi_l(z)|^2)^p}\leq C ,$$ for all $z\in U^n.$
[**Proof**]{}Let $\psi(z)\equiv 1, z\in U^n$, then $\displaystyle\frac{\partial \psi} {\partial z_k}(z)=0$ for all $k\in\{1,2,\cdots,n\}.$ It is clear that condition (\[1\]) in Theorem 1, condition (\[7\]) in Theorem 2 and condition (\[9\]) in Theorem 3 hold, note that $W_{\psi,\phi}=C_{\phi}$, the Corollary follows by combining the above Theorems.
Similar to Corollary 1, the following Corollary follows.
Let $\phi=(\phi_1,...,\phi_n)$ be a holomorphic self-map of $U^n.$ If $p\geq 1$ and $q\geq 0,$ then $C_{\phi}:{\cal B} ^p(U^n)\to {\cal B} ^q(U^n)$ is compact if and only if $$\sum\limits^n_{k,l=1}\left|\frac{\partial \phi_{l}} {\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q} {(1-|\phi_l(z)|^2)^p}\leq C ,$$ for all $z\in U^n,$ and $$\sum\limits^n_{k,l=1}\left|\frac{\partial \phi_{l}} {\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p} =o(1)\quad
( \phi(z)\to \partial U^n ).$$ If $0\leq p<1)$ and $q\geq 0,$ Then $C_{\phi}: {\cal B} ^p(U^n)\to {\cal B} ^q(U^n)$ is compact if and only if $$\sum\limits^n_{k,l=1}\left|\frac{\partial \phi_{l}} {\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q} {(1-|\phi_l(z)|^2)^p}\leq C ,$$ for all $z\in U^n,$ and $$\lim\limits_{|\phi_l(z)|\to
1}\sum\limits^n_{k=1}\left|\frac{\partial \phi_{l}} {\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}=0$$for each $l\in\{1,...,n\}$ with $\phi_l(z)|\to 1.$
[**Remark 2**]{} If let $\phi=id: U^n\longrightarrow U^n,$ then we can obtain the corresponding results about multiplication operator$M_{\psi}:{\cal B} ^p(U^n)\to {\cal B}^q(U^n).$
Throughout the remainder of this paper $C$ will denote a positive constant, the exact value of which will vary from one appearance to the next.
Some Lemmas
===========
Let $f\in {\cal B}^p(U^n),$
\(i) If $p=1,$ then $|f(z)|\leq
\left(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{2n\ln
2}\right)\left(\sum\limits^n_{l=1}\ln\displaystyle\frac{4}{1-|z_l|^2}\right)\|f\|_{p};$
\(ii) If $0\leq p<1,$ then $|f(z)|\leq
\left(1+\displaystyle\frac{1}{1-p}\right)\|f\|_{p};$
\(iii) If $p>1$, then $|f(z)|\leq
\left(\displaystyle\frac{1}{n}+\displaystyle\frac{2^{p-1}}{p-1}\right)\left(\sum\limits^n_{l=1}
\displaystyle\frac{1}{(1-|z_l|^2)^{p-1}}\right)
\|f\|_{p}.$
[**Proof**]{} This Lemma can be proved by some integral estimates (if necessary, the proof can be omitted).
By the definition of $\|.\|_{p}$, $$|f(0)|\leq \|f\|_{p},\hspace*{4mm}\left|\displaystyle\frac{\partial f(z)
}{\partial z_l}\right|\leq
\displaystyle\frac{\|f\|_{p}}{(1-|z_l|^2)^p}
\hspace*{4mm}(l\in\{1,2,\cdots,n\})$$ and $$\begin{aligned}
&&f(z)-f(0)=
\int^1_0 \displaystyle\frac{d f(tz)}{d
t}dt=\sum\limits^n_{l=1}\int^1_0 z_l\displaystyle\frac{\partial
f}{\partial\zeta_l}(tz)dt,\end{aligned}$$ So$$\begin{aligned}
&&|f(z)|\leq
|f(0)|+\sum\limits^n_{l=1}|z_l|\int^1_0\displaystyle\frac{\|f\|_p}{\left(1-t^2|z_l|^2\right)^p}dt
\nonumber\\
&&\leq
\|f\|_p+\|f\|_p\sum\limits^n_{l=1}\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt.\label{13}\end{aligned}$$ If $p=1,$ $$\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt=\displaystyle\frac{1}{2}
\ln\displaystyle\frac{1+|z_l|}{1-|z_l|}\leq
\displaystyle\frac{1}{2}
\ln\displaystyle\frac{4}{1-|z_l|^2}.\label{14}$$ It is clear that $\ln\displaystyle\frac{4}{1-|z_l|^2}>\ln 4=2\ln 2,$ so$$1\leq \displaystyle\frac{1}{2\ln 2}
\ln\displaystyle\frac{4}{1-|z_l|^2}\leq
\displaystyle\frac{1}{2n\ln 2}
\sum\limits^n_{l=1}\ln\displaystyle\frac{4}{1-|z_l|^2}.\label{15}$$ Combining (\[13\]),(\[14\]) and (\[15\]), we get $$|f(z)|\leq \left(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{2n\ln
2}\right)\left(\sum\limits^n_{l=1}\ln\displaystyle\frac{4}{1-|z_l|^2}\right)\|f\|_p.$$ If $p\neq 1,$ $$\begin{aligned}
&&\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt
=\int^{|z_l|}_0\displaystyle\frac{1}{(1-t)^p}\cdot
\displaystyle\frac{1}{(1+t)^p}dt\nonumber\\
&&\leq
\int^{|z_l|}_0\displaystyle\frac{1}{(1-t)^p}dt=\displaystyle\frac{1-(1-|z_l|)^{-p+1}}{1-p}.\label{16}\end{aligned}$$ If $0\leq p<1,$ (\[16\]) gives that $\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt\leq\displaystyle\frac{1}{1-p},$ it follows from (\[13\]) that $|f(z)|\leq\left(1+\displaystyle\frac{n}{1-p}\right)\|f\|_p.$
If $p>1,$ (\[16\]) gives that $$\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt\leq \displaystyle\frac{1-(1-|z_l|^{p-1})}{(p-1)(1-|z_l|)^{p-1}}
\leq\displaystyle\frac{2^{p-1}}{(p-1)(1-|z_l^2|)^{p-1}},$$ it follows from (\[13\]) that $$\begin{aligned}
|f(z)|&\leq&\|f\|_p+\displaystyle\frac{2^{p-1}}{p-1}\left(\sum\limits^n_{l=1}
\displaystyle\frac{1}{(1-|z_l|^2)^{p-1}}\right)\|f\|_p\\
&\leq&
\left(\displaystyle\frac{1}{n}+\displaystyle\frac{2^{p-1}}{p-1}\right)\left(\sum\limits^n_{l=1}
\displaystyle\frac{1}{(1-|z_l|^2)^{p-1}}\right)
\|f\|_{p}.\end{aligned}$$Now the Lemma is proved.
Let $\phi$ be a holomorphic self-map of $U^n$ and $\psi(z)$ a holomorphic function on $U^n,$ then $W_{\psi,\phi}:{\cal B} ^p(U^n)\to {\cal B}
^q(U^n)$ is compact if and only if for any bounded sequence $\{f_j\}$ in ${\cal B} ^p(U^n)$ which converges to zero uniformly on compact subsets of $U^n$, we have $\|W_{\psi,\phi}f_j\|_q\to
0,$ as $j\to\infty.$
[**Proof**]{} Assume that $W_{\psi,\phi}$ is compact and suppose $\{f_j\}$ is a sequence in ${\cal B} ^p(U^n)$ with $\sup\limits_{j\in {\bf N}}\|f_j\|_p<\infty$ and $f_j\to 0$ uniformly on compact subsets of $U^n.$ By the compactness of $W_{\psi,\phi}$ we have that $W_{\psi,\phi}(f_j)=\psi
f_j\circ\phi$ has a subsequence $\psi f_{j_m}\circ\phi$ which converges in ${\cal B}^q,$ say, to $g.$ By Lemma 1 we have that for any compact $K\subset U^n$ there is a positive constant $C_K$ independent of $f$ such that $$|\psi(z)f_{j_m}(\phi(z))-g(z)|\leq C_K\|\psi f_{j_m}\circ \phi-g\|_q$$ for all $z\in
K$. This implies that $\psi(z)f_{j_m}(\phi(z))-g(z)\to 0$ uniformly on compacts of $U^n.$ Since $K$ is a compact subset of $U^n,$ $|\psi(z)|\leq C$ for all $z\in K$ and $\phi(K)$ is also a compact subset of $U^n,$ by the hypothesis, $\left|\psi(z)f_{j_m}(\phi(z))\right|\leq
C\left|f_{j_m}(\phi(z))\right|$ converges to zero uniformly on $K$. It follows from the arbitrary of $K$ that the limit function $g$ is equal to $0.$ Since it is true for arbitrary subsequence of $\{f_j\}$ we see that $W_{\psi,\phi}f_j\to 0$ in ${\cal B}^q.$
Conversely, $\{g_j\}$ be any sequence in the ball ${\cal
K}_M=B_{{\cal B} ^p}(0,M)$ of the space ${\cal B} ^p(U^n).$ Since $\|g_j\|_p\leq M<\infty,$ by Lemma 1, $\{g_j\}$ is uniformly bounded on compact subsets of $U^n$ and hence normal by Montel’s theorem. Hence we may extract a subsequence $\{g_{j_m}\}$ which converges uniformly on compact subsets of $U^n$ to some $g\in
H(U^n).$ It follows that $\frac{{\partial g}_{j_m}}{\partial
z_l}\to \frac{\partial g}{\partial z_l}$ uniformly on compacts, for each $l\in\{1,...,n\},$ which implies $g\in {\cal B} ^p(U^n)$ and $\|g\|_p\leq M.$ Hence the sequence $\{g_{j_m}-g\}$ is such that $\|g_{j_m}-g\|_p\leq 2M<\infty,$ and converges to $0$ on compact subsets of $U^n,$ by the hypothesis and Lemma 2, we have that $\psi g_{j_m}\circ\phi\to\psi g\circ\phi$ in ${\cal B}^q.$ Thus the set $W_{\psi,\phi}({\cal K}_M)$ is relatively compact, finishing the proof.
Let $0\leq p<1.$ If $f\in{\cal B} ^p(U^n)$, then $$|f(z)-f(w)|\leq \displaystyle\frac{2}{1-p}\|f\|_p\sum\limits^n_{k=1}|z_k-w_k|^{1-p},
\quad z,w\in U^n.$$
[**Proof**]{} For any $z=(z_1,z_2,\cdots,z_n),$ $w=(w_1,w_2,\cdots,w_n)\in U^n,$ then $$tz+(1-t)w\in U^n$$ for $t\in [0,1].$ Denote $F(t)=f\left(tz+(1-t)w\right),$ then $$\begin{aligned}
&&f(z)-f(w)=F(1)-F(0)=\int^1_0 F'(t)dt=\int^1_0\displaystyle\frac{d}{dt}\left(f(tz+(1-t)w\right)dt
\nonumber\\
&&=\sum\limits^n_{k=1}(z_k-w_k)\int^1_0\displaystyle\frac{\partial
f} {\partial
\zeta_k}\left(tz+(1-t)w\right)dt.\label{17}\end{aligned}$$ $f\in
{\cal B} ^p(U^n),$ so for $k\in\{1,2,\cdots,n\}$ $$(1-|\zeta_k|^2)^p\left|\displaystyle\frac{\partial
f} {\partial \zeta_k}(\zeta)\right|\leq \|f\|_p$$ $$\left|\displaystyle\frac{\partial
f} {\partial \zeta_k}\right|\leq \|f\|_p(1-|\zeta_k|^2)^{-p}\leq
\|f\|_p(1-|\zeta_k|)^{-p}.$$ Let $\zeta_k=tz_k+(1-t)w_k=w_k+t(z_k-w_k),$ note that if $|a|<1,|b|<1$ and $|a+b|<1$ then $|a|+|b|+|a+b|<2,$ so $1-|\zeta_k|=1-|w_k+t(z_k-w_k)|\geq
\left|1-|w_k|-t|z_k-w_k|\right|,$ $$\left|\displaystyle\frac{\partial f} {\partial
\zeta_k}(tz+(1-t)w)\right|\leq \|f\|_p(1-|\zeta_k|^2)^{-p}\leq
\|f\|_p\left|1-|w_k|-t|z_k-w_k|\right|^{-p}.\label{18}$$ It follows from (\[17\]) and (\[18\]) that $$\begin{aligned}
&&|f(z)-f(w)|\leq
\|f\|_p\sum\limits^n_{k=1}|z_k-w_k|\int^1_0\left|1-|w_k|-t|z_k-w_k|\right|^{-p}dt\nonumber\\
&&=\|f\|_p\sum\limits^n_{k=1}\int^{|z_k-w_k|}_0\left|1-|w_k|-t\right|^{-p}dt.\label{a}\end{aligned}$$ If $1-|w_k|\leq |z_k-w_k|,$ then $$\begin{aligned}
&&\int^{|z_k-w_k|}_0\left|1-|w_k|-t\right|^{-p}dt\nonumber\\
&&=\int^{1-|w_k|}_0(1-|w_k|-t)^{-p}dt+\int^{|z_k-w_k|}_{1-|w_k|}\left(t-(1-|w_k|)\right)^{-p}dt\nonumber\\
&&=\displaystyle\frac{1}{1-p}\left((1-|w_k|)^{1-p}+(|z_k-w_k|-(1-|w_k|))^{1-p}\right)\nonumber\\
&&\leq
\displaystyle\frac{2}{1-p}(|z_k-w_k|)^{1-p}.\label{b}\end{aligned}$$ If $1-|w_k|>|z_k-w_k|,$ let $a=1-|w_k|, b=|z_k-w_k|, a>b$ and note that the fundamental inequality $a^{1-p}<(a-b)^{1-p}+b^{1-p},$ where $0\leq p<1,$ so $$\begin{aligned}
&&\int^{|z_k-w_k|}_0\left|1-|w_k|-t\right|^{-p}dt=\int^{|z_k-w_k|}_0\left(1-|w_k|-t\right)^{-p}dt\nonumber\\
&&=\displaystyle\frac{1}{1-p}\left((1-|w_k|)^{1-p}-(1-|w_k|-|z_k-w_k|)^{1-p}\right)\nonumber\\
&&<\displaystyle\frac{1}{1-p}|z_k-w_k|^{1-p}.\label{c}\end{aligned}$$ Combining $(\ref{b}),(\ref{c})$ and $(\ref{a})$, this Lemma is proved.
Let $0\leq p<1.$ Every norm-bounded sequence in ${\cal B} ^p(U^n)$ has a subsequence that converges uniformly on $\overline{U^n}$.
[**Proof**]{} Let $\{f_j\}$ be a sequence in ${\cal
B}^p(U^n),$ and $\|f_j\|_p\leq C $ for $j=1,2,\cdots.$ It follows from Lemma 3 that $|f_j(z)-f_j(w)|\leq
C\sum\limits^n_{k=1}|z_k-w_k|^{1-p}$ for $z,w\in U^n.$ Thus the family $\{f_j: n=1,2,\cdots\}$ is equicontinuous. By Lemma 1, $|f_j(z)|\leq C\|f_j\|_p\leq C, $ the family $\{f_j:
n=1,2,\cdots\}$ is bounded uniformly on $\overline{U^n}$. The statement of the lemma now follows from the Arzela-Ascoli Theorem.
Let $p\in [0,1)$ and $\{f_j\}$ be a norm bounded sequence in ${\cal B} ^p(U^n)$ which converges to zero uniformly on compacts of $U^n.$ Then it converges to zero uniformly on $\overline{U^n}.$
[**Proof**]{} Let $\varepsilon\in (0,1),$ $r\in
(1-\varepsilon,1),$ and $w\in rU^n,$ then from Lemma 3 we have $$\begin{aligned}
&&|f_j(z)|\leq
|f_j(w)|+C\|f_j\|_p\sum\limits^n_{k=1}|z_k-w_k|^{1-p}\\
&&\leq |f_j(w)| +C\sum\limits^n_{k=1}|z_k-w_k|^{1-p},
\hspace*{4mm} z\in U^n.\end{aligned}$$ Since $\sup\limits_{w\in
rU^n}|f_j(w)|\to 0$ and since for each $z\in
\overline{U^n}\setminus{rU^n},$ $1-\varepsilon<r\leq |z_k|\leq 1,$ there exists $\eta$ such that $$1-\varepsilon<\eta<r\leq |z_k|\leq 1.$$ Choosing $w_z=(\eta z_1,\eta z_2,\cdots,\eta z_n),$ then $w_z\in {rU^n}$ and $$|z_k-(w_z)_k|^{1-p}=|z_k-\eta z_k|^{1-p}=(1-\eta)^{1-p}|z_k|<\varepsilon^{1-p}\leq\varepsilon.$$ So $\lim\limits_{k\to\infty}\sup_{z\in \overline{U^n}}|f_j(z)|\leq
C\varepsilon,$ from which the result follows.
Combining Lemma 2 and Lemma 5, we can obtain the following Lemma at once.
Let $\phi$ be a holomorphic self-map of $U^n$ and $\psi(z)$ a holomorphic function on $U^n.$ Let $0\leq p<1$ and $q\geq 0,$ then $W_{\psi,\phi}:{\cal
B}^p(U^n)\to {\cal B} ^q(U^n)$ is compact if and only if for any bounded sequence $\{f_j\}$ in ${\cal B} ^p(U^n)$ which converges to zero uniformly on $\overline{U^n}$ we have $\|W_{\psi,\phi}f_j\|_q\to 0,$ as $j\to\infty.$
The Proof of Theorem 1
======================
First shows the sufficiency. If $f\in {\cal B}^p(U^n),$ a direct calculation gives $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\left(W_{\psi,\phi}f(z)\right)} {\partial z_k}\right|
(1-|z_k|^2)^{q}\nonumber\\
&&=\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi(z)}{\partial
z_k}f(\phi(z))+\psi(z)\sum\limits^n_{l=1}\displaystyle\frac{\partial
f}{\partial w_l}(\phi(z)) \displaystyle\frac{\partial
\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^{q}\label{19}\\
&&\leq \sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi(z)}{\partial
z_k}\right||f(\phi(z))|(1-|z_k|^2)^{q}\nonumber\\
&& +|\psi(z)|\sum\limits^n_{k,
l=1}\left|\displaystyle\frac{\partial f}{\partial w_l}(\phi(z))
\displaystyle\frac{\partial \phi_l}{\partial
z_k}(z)\right|(1-|z_k|^2)^{q}.\label{20}\end{aligned}$$ By Lemma 1, if $p=1$, (\[19\]) gives that $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\left(W_{\psi,\phi}f(z)\right)} {\partial z_k}\right|
(1-|z_k|^2)^{q}\nonumber\\
&&\leq
C\left(\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\psi} {\partial z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\ln
\displaystyle\frac{4}{1-|\phi_l(z)|^2}\right.\nonumber\\
&&\left.+|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{1-|\phi_l(z)|^2}
\right)\|f\|_{p}.\label{21}\end{aligned}$$ Combining (\[1\]), (\[2\]) and (\[21\]), we know $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n) $ is bounded.
If $0\leq p<1$, by Lemma 1, (\[20\]) gives that $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\left(W_{\psi,\phi}f(z)\right)} {\partial z_k}\right|
(1-|z_k|^2)^{q}\nonumber\\
&&\leq
C\left(\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\psi} {\partial
z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\right.\nonumber\\
&&\left.+|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p}}
\right)\|f\|_{p}.\label{22}\end{aligned}$$ Combining (\[3\]), (\[4\]), Lemma 1 and (\[22\]), we know $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n) $ is bounded.
If $p>1,$by Lemma 1, (\[20\]) gives that $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\left(W_{\psi,\phi}f(z)\right)} {\partial z_k}\right|
(1-|z_k|^2)^{q}\nonumber\\
&&\leq
C\left(\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\psi} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p-1}}\right.\nonumber\\
&&\left.+|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p}}
\right)\|f\|_{p}.\label{23}\end{aligned}$$ Combining (\[5\]), (\[6\]) and (\[23\]), we know $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n) $ is bounded.
To show the necessity, assume that $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n) $ is bounded, with $$\|W_{\psi,\phi}f\|_{q}\leq C\|f\|_{p}\label{24}$$ for all $f\in {\cal B}^p(U^n).$ It is clear that $\psi\in{\cal
B}^q(U^n),$ that is, (\[3\]) holds.
If $p=1$, For fixed $l (1\leq l\leq n),$ we will make use of a family of test functions $\{f_{w}: w\in\mbox{\Bbb C}, |w|<1\}$ in ${\cal B}^p(U^n)$ defined as follows: let $$f_w(z)=\displaystyle\frac{z_l}{1-\overline {w}z_l},$$ then $\displaystyle\frac{\partial
f_w(z) }{\partial z_l}=\displaystyle\frac{1}{(1-\overline
{w}z_l)^2},\hspace{4mm}\displaystyle\frac{\partial f_w(z)
}{\partial z_k}=0 \hspace*{4mm}(k\neq l)$, $\|f_w\|_p\leq \displaystyle\frac{4}{1-|w|^2}.$
It follows from (\[19\]) and (\[24\]) that $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi(z)}{\partial z_k}\displaystyle\frac{\phi_l(z)}{1-\overline
{w}\phi_l(z)}+\psi(z)\displaystyle\frac{1}{(1-\overline
{w}\phi_l(z))^2}\displaystyle\frac{\partial \phi_l}{\partial
z_k}(z)\right|(1-|z_k|^2)^{q}\leq C
\displaystyle\frac{1}{1-|w|^2},$$let $w=\phi_l(z),$ then $$|\psi(z)|\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{1-|\phi_l(z)|^2}\leq
C+\|\psi\|_q\leq C.\label{25}$$ So for any $z\in U^n$, $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{1-|\phi_l(z)|^2}\leq
C,$$ that is, (\[2\]) is valid.
If we set $$g_w(z)=\ln\displaystyle\frac{4}{1-\overline
{w}z_l},$$ then $\displaystyle\frac{\partial g_w(z) }{\partial
z_l}=\displaystyle\frac{\overline w}{1-\overline
{w}z_l},\hspace{4mm}\displaystyle\frac{\partial g_w(z) }{\partial
z_k}=0 \hspace*{4mm}(k\neq l)$, $\|g_w\|_p\leq 2+\ln
4 .$
It follows from (\[19\]) and (\[24\]) that $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi(z)}{\partial z_k}\ln\displaystyle\frac{4}{1-\overline
{w}\phi_l(z)}+\psi(z)\displaystyle\frac{\overline
w}{1-\overline{w}\phi_l(z)}\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^{q}\leq C,$$ let $w=\phi_l(z)$, (\[1\]) is valid.
If $0\leq p<1$, $\psi=W_{\psi,\phi}1\in{\cal B}^q(U^n),$ (3) is valid. For fixed $l (1\leq l\leq n),$ we will make use of a family of test functions $\{f_{w}: w\in\mbox{\Bbb C}-\{0\}, |w|<1\}$ in ${\cal B} ^p(U^n$ defined as follows: set $$f_w(z)=\int^{z_l}_0\left(1-\displaystyle\frac{\overline w^2}{|w|^2}z_l^2
\right)^{-p}dz_l$$ with $f_w(0)=0.$ Since for $w\neq 0,$ $$\displaystyle\frac{\partial f_w}{\partial z_l}
=\left(1-\displaystyle\frac{\overline
w^2}{|w|^2}z_l^2\right)^{-p}, \hspace*{4mm}
\displaystyle\frac{\partial f_w}{\partial z_i}=0,
\hspace*{4mm}(i\neq l),$$ it is easy to show $\|f_w\|_{\alpha}=1.$ Lemma 1 gives $|f_w(z)|\leq 1+\displaystyle\frac{n}{1-p},$ so by (\[3\]), $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi}{\partial z_k}(z)f_w(\phi(z)) (1-|z_k|^2)^q\right| \leq
C\sum\limits^n_{k=1} \left|\displaystyle\frac{\partial
\psi}{\partial z_k}(z)\right|\left(1-|z_k|^2 \right)^q\leq
C.\label{26}$$ For $z\in U^n,$ it follows from (\[19\]) and (\[24\]) that $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi}{\partial z_k}(z)f_w(\phi(z))
+\psi(z)\sum\limits^n_{l=1}\displaystyle\frac{\partial
f_w(\phi(z))}{\partial \phi_l} \displaystyle\frac{\partial
\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^q\leq C,$$ combining (\[26\]), we get $$\sum\limits^n_{k=1}\left|\psi(z)
\displaystyle\frac{\partial f_w(\phi(z))}{\partial \phi_l}
\displaystyle\frac{\partial \phi_l}{\partial
z_k}(z)\right|(1-|z_k|^2)^q\leq C.$$ For an arbitrary $z\in U^n$ with $\phi_l(z)\neq 0$ and set $w=\phi_l(z)$ in the above inequality to obtain $$|\psi(z)|\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial z_k}(z)\right|
\displaystyle\frac{\left(1-|z_k|^2\right)^q}{\left(1-|\phi_l(z)|^2\right)^p}
\leq C.$$
If $\phi_l(z)=0,$ let $f(z)=z_l,$ it follows from (\[19\]) and (\[24\]) that $$|\psi(z)|\left|\displaystyle\frac{\partial
\phi_{l}} {\partial z_k}(z)\right|
\displaystyle\frac{\left(1-|z_k|^2\right)^q}{\left(1-|\phi_l(z)|^2\right)^p}
=\left|\displaystyle\frac{\partial\psi}{\partial z_k}(z)\phi_l(z)
+\psi(z)\displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|(1-|z_k|^2)^q \leq C.$$
So for all $z\in U^n,$ $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial z_k}(z)\right|
\displaystyle\frac{\left(1-|z_k|^2\right)^q}{\left(1-|\phi_l(z)|^2\right)^p}\leq
C,$$ that is, (\[4\]) is valid.
If $p>1$, similarly, the test functions would be $$f_w(z)=\displaystyle\frac{1}{\overline w}\left(\displaystyle\frac{1-|w|^2}{\left(1-\overline
{w}z_l\right)^p}-\displaystyle\frac{1}{\left(1-\overline
{w}z_l\right)^{p-1}}\right),\hspace*{4mm}(|w|<1, w\neq 0, 1\leq
l\leq n)$$ for proving (\[6\]), and $$g_w(z)=\left(\displaystyle\frac{p}{\left(1-\overline
{w}z_l\right)^{p-1}}-\displaystyle\frac{(p-1)(1-|w|^2)}{\left(1-\overline
{w}z_l\right)^p}\right),\hspace*{4mm}(|w|<1, 1\leq l\leq n)$$ for proving (\[5\]), we omit the details. Now the proof of Theorem 1 is completed.
The Proof of Theorem 2
======================
To show the sufficiency.
If $p=1$. First assume conditions (\[7\]) and (\[8\]) hold, we need to prove $W_{\psi,\phi}: {\cal B}^p(U^n)\longrightarrow {\cal
B}^q(U^n) $ is compact. According to Lemma 2, assume that $\left\|f_{j}\right\|_{p}\leq C, j=1,2,\ldots,$ and $\{f_{j}\}$ converges to zero uniformly on compact subsets of $U^n,$ we need only prove that $\left\|W_{\psi,\phi}f_{j}\right\|_{q}\to 0,$ as $j\to\infty.$
For every $\varepsilon>0,$ (\[7\]) and (\[8\]) imply that there exists a $r$, $0<r<1$, such that $$\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \psi}
{\partial z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\ln
\displaystyle\frac{4}{1-|\phi_l(z)|^2} <\varepsilon
\label{27}$$ and$$\sum\limits^n_{k,l=1}\left|\psi(z)\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z)\right|
\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{1-|\phi_l(z)|^2}
<\varepsilon,\label{28}$$ whenever $dist(\phi(z),
\partial U^n)<r.$ Since $f_j(\phi(0))$ converges to zero, by (\[27\]) and (\[28\]), it follows from (\[21\]) that for large enough $j$ $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial\left(W_{\psi,\phi}f_j\right)}
{\partial z_k}(z)\right| (1-|z_k|^2)^{q}\leq
C\varepsilon,\label{29}$$ whenever $dist\left(\phi(z),
\partial U^n\right)< r.$ If we set $\varepsilon=1,$ by (\[27\]) and (\[28\]),it is easy to show that (\[1\]) and (\[2\]) holds, from Remark 1, we know $\psi\in{\cal B}^q(U^n).$
On the other hand, if we write $E=\left\{w\in U^n: dist(w,\partial
U^n)\geq r\right\},$ which is a closed subset of $U^n$, then $f_j(w)$ and $\displaystyle\frac{\partial f_j}{\partial w_l}(w)\to
0$ uniformly on $E$. So by (\[19\]) and the boundedness condition (2) in Theorem 1, we obtain for $dist(\phi(z),\partial
U^n)\geq r,$ $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial\left(W_{\psi,\phi}f_j\right)}
{\partial z_k}(z)\right| (1-|z_k|^2)^{q}\leq
C\left(|f_j(\phi(z))|+\sum\limits^n_{l=1}\left|\displaystyle\frac{\partial
f_j}{\partial w_l}(\phi(z))\right|\right)\nonumber\\
&& =
C\left(|f_j(w)|+\sum\limits^n_{l=1}\left|\displaystyle\frac{\partial
f_j}{\partial w_l}(w)\right|\right)\leq C\varepsilon.\label{30}\end{aligned}$$
Since $\psi(0)f_j(\phi(0))\to 0$ for large enough $j,$ combining (\[29\]) and (\[30\]) we know $\left\|W_{\psi,\phi}f_{j}\right\|_{q}\to 0,$ as $j\to\infty.$
If $p>1,$ note that (\[23\]), in a similar manner to the case $p=1$, we can show the sufficiency, omit the details.
To show the necessity.
First we will prove the following Lemma:
Let $p\geq
1$ and $q\geq 0$. If $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n)$ is compact, then $$|\psi(z)|\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^{q}}{\left(1-|\phi_l(z)|^2\right)^{p}}
=o(1)\hspace*{6mm} (\phi(z)\to\partial
U^n).\label{31}$$
[**Proof**]{} For any $\{z^{j}\}$ in $U^n$ with $\phi(z^j)\to\partial U^n$ as $j\to\infty.$ Let $A_j=\phi(z^{j})=(a_{1j},\ldots, a_{nj}),$ where $a_{kj}=\phi_{k}(z^j), 1\leq k\leq n .$ Since $\phi(z^{j})\to\partial U^n,$ as $j\to\infty$, there exists some $s, (1\leq s\leq n),$ with $|a_{sj}|\to 1$ as $j\to\infty.$ Without loss of generality, we may assume that $s=1$.
Case 1: If for some $l (1\leq l\leq n),$ $|a_{lj}|\to 1$ as $j\to\infty,$ then set $$f_{j}(z)=\displaystyle\frac{\left(1-|a_{lj}|^2\right)^2}{\left(1-z_l\overline{a_{lj}}\right)^{p+1}}-
\displaystyle\frac{1-|a_{lj}|^2}{\left(1-z_l\overline{a_{lj}}\right)^p},
\hspace*{4mm}j=1,2,\cdots.\label{32}$$ $$\displaystyle\frac{\partial f_{j}}{\partial
z_{k}}=0,\hspace*{4mm} (k\neq l),$$ $$\displaystyle\frac{\partial
f_{j}}{\partial z_{l}}
=(p+1)\overline{a_{lj}}\displaystyle\frac{\left(1-|a_{lj}|^2\right)^2}{\left(1-z_l\overline{a_{lj}}\right)^{p+2}}-
p\overline{a_{lj}}\displaystyle\frac{1-|a_{lj}|^2}{\left(1-z_l\overline{a_{lj}}\right)^{p+1}},$$ so $\|f_{j}\|_{p}\leq C$ and $\{f_j(z)\}$ tends to zero uniformly on compact subsets of $U^n.$ So by Lemma 2, we know $\left\|W_{\psi,\phi}f_{j}\right\|_{q}\to 0$ as $j\to\infty.$ It follows from (\[19\]) that $$\begin{aligned}
&&\sum\limits^n_{k=1}\left(1-|z^j_k|^2\right)^{q}
\left|\displaystyle\frac{\partial \psi}{\partial
z_k}(z^j)f_j(\phi(z^j))+\psi(z^j)\displaystyle\frac{\partial
f_j}{\partial w_l}(\phi(z^j))\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\nonumber\\
&&\leq \left\|W_{\psi,\phi}f_{j}\right\|_{q}\to 0
.\label{33}\end{aligned}$$ Note that $f_j(\phi(z^j))=0$ and $\displaystyle\frac{\partial f_j}{\partial
z_l}(\phi(z^j))=\overline{a_{lj}}\displaystyle\frac{1}{\left(1-|a_{lj}|^2\right)^p}$, (\[33\]) gives that $$\begin{aligned}
&&|\psi(z^j)|\sum\limits^n_{k=1}
\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{\left(1-|a_{lj}|^2\right)^p}
\left|\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\nonumber\\
&&\leq
C\displaystyle\frac{1}{|a_{lj}|}\left\|W_{\psi,\phi}f_{j}\right\|_{q}
\to 0 \hspace*{4mm}\mbox{as $j\to\infty.$}\label{34}\end{aligned}$$
Case 2: If for some $l (2\leq l\leq n),$$|a_{lj}|\not\to 1$ as $j\to\infty,$ then we assume $|a_{lj}|\leq\rho<1.$
If $p\geq 1$, The compactness of $W_{\psi,\phi}$ implies $W_{\psi,\phi}$ is bounded, by Theorem 1 and Remark 1, (\[11\]) is valid. Set $$f_{j}(z)=z_l\left(\displaystyle\frac{1-|a_{1j}|^2}
{1-z_1\overline{a_{1j}}}\right)^p,
\hspace*{4mm}j=1,2,\cdots.\label{35}$$ It is easy to show that $\|f_j\|_{p}\leq C$, and $\{f_j\}$ tends to zero uniformly on compact subsets of $U^n.$ It follows from (\[19\]) that $$\begin{aligned}
&&|\psi(z^j)|\sum\limits^n_{k=1}
\left(1-|z^j_k|^2\right)^{q}\left|\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\nonumber\\
&&\leq \left\|W_{\psi,\phi}f_{j}\right\|_{q}
+C\sum\limits^n_{k=1}\left(1-|z^j_k|^2\right)^{q}\left|\displaystyle\frac{\partial
\psi }{\partial z_k}(z^j)\right|\nonumber\\
&&+C|\psi(z^j)|\sum\limits^n_{k=1}
\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{1-|a_{1j}|^2}
\left|\displaystyle\frac{\partial \phi_1}{\partial
z_k}(z^j)\right|
\nonumber\\
&&\leq \left\|W_{\psi,\phi}f_{j}\right\|_{q}
+C\sum\limits^n_{k=1}\left(1-|z^j_k|^2\right)^{q}\left|\displaystyle\frac{\partial
\psi }{\partial z_k}(z^j)\right|\nonumber\\
&&+C|\psi(z^j)|\sum\limits^n_{k=1}
\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{(1-|a_{1j}|^2)^p}
\left|\displaystyle\frac{\partial \phi_1}{\partial
z_k}(z^j)\right|\to 0.\label{36}\end{aligned}$$ So $$\begin{aligned}
&&|\psi(z^j)|\sum\limits^n_{k=1}
\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}
{(1-|a_{lj}|^2)^p}\left|\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\nonumber\\
&&\leq
\displaystyle\frac{1}{(1-\rho^2)^p}|\psi(z^j)|\sum\limits^n_{k=1}
\left(1-|z^j_k|^2\right)^{q}\left|\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\to 0.\label{37}\end{aligned}$$
Now we return to prove the necessary of Theorem 2.
If $p=1$, by Lemma 7, (\[8\]) is necessary, now we prove that (\[7\]) is also necessary.
In case 1, set$$f_j(z)=\left(\ln\displaystyle\frac{4}{1-|a_{lj}|^2}\right)^{-1}
\left(\ln\displaystyle\frac{4}{1-z_l\overline{a_{lj}}}\right)^2,\label{38}$$ It is easy to show that $\|f_j\|_{p}\leq C$, and $\{f_j\}$ tends to zero uniformly on compact subsets of $U^n.$ It follows from (\[19\]) that $$\sum\limits^n_{k=1}\left(1-|z^j_k|^2\right)^{q}
\left|\displaystyle\frac{\partial \psi}{\partial
z_k}(z^j)f_j(\phi(z^j))+\psi(z^j)\displaystyle\frac{\partial
f_j}{\partial w_l}(\phi(z^j))\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\leq
\left\|W_{\psi,\phi}f_{j}\right\|_{q}\to 0 .$$ By a direct calculation and by (\[31\]), we obtain $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial \psi}
{\partial z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\ln
\displaystyle\frac{4}{1-|\phi_l(z)|^2}\nonumber\\
&&\leq
\left\|W_{\psi,\phi}f_{j}\right\|_{q}+2|\psi(z^j)|\sum\limits^n_{k=1}
\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{\left(1-|a_{lj}|^2\right)^{p}}
\left|\displaystyle\frac{\partial \phi_l}{\partial
z_k}(z^j)\right|\to 0.\label{39}\end{aligned}$$
In case 2, by remark 1, $$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial \psi}
{\partial z_k}(z^j)\right|\left(1-|z^j_k|^2\right)^{q}\ln
\displaystyle\frac{4}{1-|\phi_l(z^j)|^2}\leq
\ln\displaystyle\frac{4}{1-\rho^2}\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi} {\partial z_k}(z^j)\right|\left(1-|z^j_k|^2\right)^{q}\to
0.$$ Combining case 1 and case 2, we know (\[7\]) is necessary.
If $p>1.$ by Lemma 7, we know (\[10\]) is necessary, now we prove (\[9\]) is also necessary.
In case 1, set $$f_{j}(z)=(p+1)\displaystyle\frac{1-|a_{lj}|^2}{\left(1-z_l\overline{a_{lj}}\right)^p}-
p\displaystyle\frac{\left(1-|a_{lj}|^2\right)^2}{\left(1-z_l\overline{a_{lj}}\right)^{p+1}},
\hspace*{4mm}j=1,2,\cdots.$$ It is easy to show that $\|f_j\|_{p}\leq C$, and $\{f_j\}$ tends to zero uniformly on compact subsets of $U^n.$ Note that $f_j(\phi(z^j))=\displaystyle\frac{1}{\left(1-|a_{lj}|^2\right)^{p-1}}$ and $\displaystyle\frac{\partial f_j}{\partial z_l}(\phi(z^j))=0,$ it follows from (\[19\]) that $$\begin{aligned}
&&\sum\limits^n_{k=1}\left(1-|z^j_k|^2\right)^{q}
\left|\displaystyle\frac{\partial \psi}{\partial
z_k}(z^j)f_j(\phi(z^j))+\psi(z^j)\displaystyle\frac{\partial
f_j}{\partial w_l}(\phi(z^j))\displaystyle\frac{\partial
\phi_l}{\partial
z_k}(z^j)\right|\nonumber\\
&&=\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi}{\partial
z_k}(z^j)\right|\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{(1-|a_{lj}|^2)^{p-1}}\leq
\left\|W_{\psi,\phi}f_{j}\right\|_{q}\to
0.\label{40}\end{aligned}$$
In case 2, by remark 1, $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial \psi}
{\partial
z_k}(z^j)\right|\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{\left(1-|\phi_l(z^j)|^2\right)^{p-1}}\\
&&\leq
\displaystyle\frac{1}{\left(1-\rho^2\right)^{p-1}}\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi} {\partial z_k}(z^j)\right|\left(1-|z^j_k|^2\right)^{q} \to
0.\end{aligned}$$ Combining cas1 and case 2, we know (\[9\]) is necessary.
Now the proof of Theorem 2 is completed.
Proof of Theorem 3
==================
Proof of Theorem 3. First assume $W_{\psi,\phi}: {\cal
B}^p(U^n)\longrightarrow {\cal B}^q(U^n)$ is bounded and (\[12\]) holds, we prove that $W_{\psi,\phi}:{\cal B} ^p(U^n)\to
{\cal B} ^q(U^n)$ is compact. By Lemma 6, assume that $\left\|f_{j}\right\|_p\leq C,\, j=1,2,...,$ and $\{f_{j}\}$ converges to zero uniformly on $\overline{U^n},$ we need to prove that $\left\|W_{\psi,\phi}f_{j}\right\|_q\to 0,$ as $j\to\infty.$ Note that the boundedness of $W_{\psi,\phi}$ and Theorem 1, (\[1\]) and (\[2\]) hold. So $$|\psi(z)|\leq C\|\psi\|_q\leq C,\hspace*{4mm} |\psi(z)|\sum\limits^n_{k=1}\left|\frac{\partial
\phi_l}{\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\leq C
\label{41}$$ for $l\in\{1,2,\cdots,n\}.$
By the assumption as $j\to\infty$ $$\sup\limits_{z\in\overline{U^n}}|f_j(z)|\to 0\label{42}$$ $$\begin{aligned}
&&\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\left(W_{\psi,\phi}f_j(z)\right)} {\partial z_k}\right|
(1-|z_k|^2)^{q}\nonumber\\
&&=\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi(z)}{\partial
z_k}f_j(\phi(z))+\psi(z)\sum\limits^n_{l=1}\displaystyle\frac{\partial
f_j}{\partial w_l}(\phi(z)) \displaystyle\frac{\partial
\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^{q}\nonumber\\
&&\leq
|f_j(\phi(z))|\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\psi} {\partial
z_k}(z)\right|\left(1-|z_k|^2\right)^{q}\nonumber\\
&&+\sum\limits^n_{k=1}\sum\limits^n_{l=1}\left|\frac{\partial
f_j}{\partial w_l}(\phi(z))\right||\psi(z)|\left|\frac{\partial
\phi_l} {\partial
z_k}(z)\right|\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\nonumber\\
&&\leq
C\left|f_j(\phi(z))\right|+\sum\limits^n_{k=1}\sum\limits^n_{k=1}\left|\frac{\partial
f_j}{\partial w_l}(\phi(z))\right||\psi(z)|\left|\frac{\partial
\phi_l} {\partial
z_k}(z)\right|\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\nonumber\\
&&\leq
C\varepsilon+\sum\limits^n_{k=1}\sum\limits^n_{l=1}\left|\frac{\partial
f_j}{\partial w_l}(\phi(z))\right||\psi(z)|\left|\frac{\partial
\phi_l} {\partial
z_k}(z)\right|\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}.\label{43}\end{aligned}$$
For every $\varepsilon>0$, $l\in\{1,...,n\},$ (\[12\]) implies that there exists an $r$, $0<r<1$, such that $$|\psi(z)|\sum\limits^n_{k=1}\left|\frac{\partial
\phi_l}{\partial
z_k}(z)\right|\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}<\varepsilon,\label{44}$$ whenever $|\phi_l(z)|>r.$
Note that $$U^n= \left\{z\in U^n:
|\phi_l(z)|>r\right\}\bigcup\left\{z\in U^n: |\phi_l(z)|\leq
r\right\}.$$
For each $l\in\{1,2,\cdots,n\}.$ If $|\phi_l(z)|>r,$ from (\[44\]) and $\|f_j\|_p\leq C$, we have $$\left|\frac{\partial f_j}{\partial
w_l}(\phi(z))\right||\psi(z)|\left|\frac{\partial \phi_l}
{\partial
z_k}(z)\right|\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}<C\varepsilon.\label{45}$$
If $|\phi_l(z)|\leq r,$ by the Cauchy’s estimate applied to the function $$g(w_l)=f(w_1,...,w_{l-1},w_l,w_{l+1}...,w_n),$$ of one variable, if $|w_l|\leq r,$ we have $$\left|\frac{\partial f_j}{\partial w_l}(w)\right|\leq C\sup_{|w_l|\leq \frac{1+r}2}|f_j(w)|\leq C\, \sup_{z\in \overline{U^n}}|f_j(z)|,$$ for some $C>0$ independent of $f.$ From this, (\[41\]) and (\[42\]) we have $$\begin{aligned}
&&\left|\frac{\partial
f_j}{\partial w_l}(\phi(z))\right||\psi(z)|\left|\frac{\partial
\phi_l} {\partial
z_k}(z)\right|\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\nonumber\\
&&\leq C\sup_{z\in \overline{U^n}}|f_j(z)|\to 0
\label{46}\end{aligned}$$ as $j\to\infty.$ So
From (\[43\]),(\[45\]) and (\[46\]), and since $\lim\limits_{j\to\infty}f_j(\phi(0))=0,$ we obtain $\left\|C_{\phi}f_{j}\right\|_q\to 0,$ as $j\to\infty,$ from which the result follows.
Now suppose that $W_{\psi,\phi}: {\cal B} ^p(U^n)\to {\cal
B}^q(\mbox{\Bbb C}^n)$ is compact. Then $W_{\psi,\phi}$ is bounded, and by Theorem 1 we know that (\[4\]) holds. Now we prove that condition (\[12\]) holds. Let $\{z^{j}\}$ be a sequence in $U^n$ and $w^j=\phi(z^{j})=(w^j_1,..., w^j_n).$ For each $l\in\{1,2,\cdots,n\}$ with $\lim\limits_{j\to\infty}|w^j_l|\to 1.$
Let $$f_{j}(z)=\frac{1-|w^j_l|^2}{(1-z_l\overline{w^j_l})^p},\quad j=1,2,...\;,$$ then$$\displaystyle\frac{\partial f_j}{\partial
z_l}(z)=p\overline{w^j_l}\displaystyle\frac{1-|w^j_l|^2}{(1-z_l\overline{w^j_l|^2})^{p+1}}.$$ As in the proof of Theorem 2 we can prove that $\|f_{j}\|_p\leq
1+2^{p+1}p,$ for all $j\in {\bf N},$ i.e. $\{f_j\}$ is bounded on ${\cal B} ^p(U^{n}).$ It is easy to see that $f_j$ tends to zero uniformly on compact subsets of $\mbox{\Bbb C}^n.$ So by Lemma 2, we know $\left\|W_{\phi}f_{j}\right\|_q\to 0$ as $j\to\infty$ and notice that $$f_j(\phi(z^j))=(1-|w^j_l|^2)^{1-p},\hspace*{4mm}\displaystyle\frac{\partial
f_j}{\partial
w_l}(\phi(z^j))=p\overline{w^j_l}\displaystyle\frac{1}{(1-|w^j_l|^2)^p},$$ consequently, by (\[19\]) $$\begin{aligned}
&&\sum\limits^n_{k=1}\left(1-|z^j_k|^2\right)^{q}
\left|\displaystyle\frac{\partial \psi}{\partial
z_k}(z^j)f_j(\phi(z^j))+\psi(z^j)\displaystyle\frac{\partial
f_j}{\partial w_l}(\phi(z^j))\displaystyle\frac{\partial
\phi_l}{\partial z_k}(z^j)\right|\nonumber\\
&&\leq
\left\|W_{\psi,\phi}f_{j}\right\|_{q}.\label{47}\end{aligned}$$ So $$\sum\limits^n_{k=1}p|w^j_l||\psi(z^j)\left|\displaystyle\frac{\partial
\phi_l}{\partial
z_k}(z^j)\right|\displaystyle\frac{\left(1-|z^j_k|^2\right)^{q}}{(1-|w^j_l|^2)^p}\leq
\|\psi\|_q\left(1-|w^j_l|^2\right)^{1-p} +
\left\|W_{\psi,\phi}f_{j}\right\|_{q}.$$ Hence $$|\psi(z^j)|\sum\limits^n_{k=1}\frac{(1-|z^j_k|^2)^q}{(1-|w^j_l|^2)^p}
\left|\frac{\partial \phi_l}{\partial z_k}(z^j)\right|\leq
\frac{1}{p|w^j_l|}\left(C(1-|w^j_l|^2)^{1-p} +
\left\|W_{\psi,\phi}f_{j}\right\|_{q}\right)\to 0$$ as $j\to\infty.$ Now the proof of Theorem 3 is finished.
[99]{} R.Timoney, Bloch Function in Several Complex Variables, I, Bull.London Math.Soc. 12(1980),241-267. R.Timoney, Bloch Function in Several Complex Variables, II, J. Reine Angew Math., 319(1980):1-22. C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL,1995. K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347(1995): 2679-2687. K. Madigan, Compact composition operators on analytic Lipschitz space, Proc. Amer. Math. Soc. 119(1993): 465-473. M. D. Contreras and A. G. Hernandez-Diaz, Weighted composition operators in weighted Banach spaces of ananytic functions, J. Austral. Math. Soc. (Series A), 69(2000):41-46. Alfonso Montes-Rodriguez, Weighted composition operators on weighted banach spaces of analytic functions, J. London Math. Soc. (2)61(2000):872-884. S.Ohno, K. Stroethoff and R.H.Zhao, Weighted composition operators between-type spaces, Rocky Mountain Journal of Mathematics,Vol33, No.1(2003):191-215. J. H. Shi and L. Luo, Composition operators on the Bloch space of several complex variables, Acta Math. Sinica, English Series, 16(2000):85-98. Z. H. Zhou and J. H. Shi, Compact composition operators on the Bloch space in polydisk, Science in China (Series A), 44(2001): 286-291. Z. H. Zhou, Composition operators on the Lipschitz spaces in polydisk. Science in China (Series A), 46(2003): 33-38. Z. H. Zhou and J. H. Shi, Compactness of composition operators on the Bloch space in classical bounded symmetric domains, The Michigan Mathematical Journal, 50(2002): 381-405. J. P. D’Angelo, Several complex variables and geometry of real hypersurfaces, CRC Press, 1993.
[^1]: Zehua Zhou, Corresponding author. Supported in part by the National Natural Science Foundation of China (Grand No.10371091), and LiuHui Center for Applied Mathematics, Nankai University and Tianjin University
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study strongly correlated phases of a pseudo-spin-1/2 Bose gas in an artificial gauge field using the exact diagonalization method. The atoms are confined in two dimensions and interact via a two-body contact potential. In Abelian gauge fields, pseudo-spin singlets are favored by pseudo-spin independent interactions. We find a series of incompressible phases at fillings $\nu=2k/3$. By comparison with the non-Abelian spin singlet (NASS) states, constructed as zero-energy eigenstates of a $(k+1)$-body contact interaction, we classify the non-trivial topology of the states. An additional spin-orbit coupling is shown to switch between NASS-like states and spin-polarized phases from the Read-Rezayi series.'
author:
- 'T. Graß$^1$, B. Juliá-Díaz$^{1}$, N. Barberán$^2$, and M. Lewenstein$^{1,3}$'
bibliography:
- 'lett.bib'
title: 'Non-Abelian spin singlet states of two-component Bose gases in artificial gauge fields'
---
Topological phases of matter attract a great deal of attention since the late 1980s, when the fractional quantum Hall (FQH) effect of electrons in two dimensions (2D) subjected to a strong perpendicular magnetic field was recognized as a paradigm of such phases [@wen-niu]. It was then pointed out that amongst these states there might be some which support non-Abelian excitations, where the exchange of two quasiparticles is described by some unitary matrix, as it shifts between different degenerate ground states [@moore-read]. Recently, this property has triggered the interest in topological phases, as quantum gates operating with non-Abelian anyons might pave the way to fault-tolerant quantum computers [@nayak]. For this purpose, the possibility of controlling or even designing quantum many-body states seems to be advantageous if not inevitable. A very important tool to do so are cold atoms which can, in various ways, be manipulated through interactions with light [@mlbook]. Despite being electroneutral, the atoms may even couple to external gauge potentials, which, instead of being true electromagnetic vector potentials, have to be synthesized. Artificial magnetic fields have been achieved experimentally by rotating the atomic cloud [@schweikhard] (see also [@cooper-aip] for a review), though instabilities at high rotation frequencies have so far prevented from reaching the highly correlated regime. A promising alternative pursues the generation of gauge fields by a laser dressing of the atoms [@dalibard; @*brunoPRA; @*brunoNJP]. This method has already been proven experimentally [@lin; @*spielmanPRL]. It is readily generalized to non-Abelian gauge fields [@fleischhauer], and has recently allowed for synthesizing a spin-orbit coupling [@spielman-sobec].
The strongly correlated states occurring in such systems have been numerically investigated for the case of spinless or spin-polarized bosons [@cooper-wilkin-gunn], in which the incompressible phases are well described by the Read-Rezayi (RR) series of parafermionic states [@read-rezayi], including the Laughlin wave function [@laughlin], and the Moore-Read (MR) wave function [@moore-read]. Although also bosons with a pseudo-spin-1/2 degree of freedom have already been condensed [@ketterle-2comp; @*wieman-2comp], and their behavior under rotation has been studied both theoretically [@mueller-ho-vortices; @*reiman-1vortex; @*ueda-vortices] and experimentally [@cornell-vortices; @*cornell-vortex-lattice], few attention has yet been paid to the strongly correlated regime in such systems. For the electronic FQH effect, however, a variety of states have been proposed for systems with spin, which despite the strong magnetic field might be in a singlet phase due to the small gyromagnetic ratio of electrons in many solids. Within a conformal field theory (CFT) framework, a generalization of the RR series to the so-called non-Abelian spin singlet (NASS) series [@ardonne-schoutens; @*nass-nucl; @*estienne-bernevig] has been formulated. Here the term “non-Abelian” refers to the nature of the excitations. For bosons, this series of topological states occurs at fillings $\nu=2k/3$, and includes the Halperin state [@halperin] with Abelian excitations at $k=1$. Like the RR states, the NASS states are constructed as the zero-energy eigenstates of a repulsive $(k+1)$-body contact interaction.
In this paper we show, using exact diagonalization, that also two-body contact (2BC) interaction leads to the formation of incompressible states precisely at the NASS filling factors in a 2D Bose gas exposed to an Abelian gauge field. For $k=1$, this is trivially the Halperin state, but also for $k=2$ significant overlap with the corresponding NASS state and a similar spectral structure indicate that the system can be brought into the NASS phase within this realistic setup.
This finding is in close analogy to the spin-polarized case with RR-like states [@cooper-wilkin-gunn] at filling $\nu=k/2$. However, as it is experimentally difficult to get into the regime of low filling, the two-component system benefits from supporting analogs of the RR states at higher filling. Furthermore, its possibility of tuning independently inter-spin and intra-spin interactions provides additional control. For pseudo-spin-independent repulsive 2BC interactions, a mechanism is discussed which favors pseudo-spin singlet states. We show that, counterintuitively, states with zero interspin interaction energy are obtained by increasing the interspin interaction strength. Finally, effects stemming from a spin-orbit (SO) coupling can be integrated in the framework of two-component gases by generalizing to a non-Abelian gauge field. We investigate this scenario, and find a transition from the single-component RR states [@trombettoni; @*trombettoniPRA; @palmer-pachos; @*komineas-cooper] to the NASS states at SO coupling strengths corresponding to a Landau level (LL) degeneracy.
We consider a two-component Bose gas in 2D, which on the single-particle level is described by the Hamiltonian $H_{\mathrm{sp}}=(\vec{p}-\vec{A})^2/(2m)$. The vector potential $\vec{A}$ is given by $$\begin{aligned}
\label{A}
\vec{A} = B (0,x,0) \mathbb{1} + q (\sigma_x,\sigma_y,0).\end{aligned}$$ The first term, acting equally on both pseudo-spin components, describes a magnetic field of strength $B$ in the Landau gauge. The second term, with $\sigma_i$ being the Pauli matrices, accounts for a Rashba SO coupling controlled by the coupling strength $q$, which we set to zero in the first part of this work. A detailed description how to implement this Hamiltonian for electroneutral atoms can be found in the supplementary material of Ref. [@trombettoni]. As the boundary conditions we choose a periodic square of length $a$. This eliminates edge effects from our study, and, moreover, generates a non-trivial toroidal geometry, on which topological phases can be classified by the number of degenerate ground states [@wen-niu]. Solving the single-particle Hamiltonian yields, for $q=0$, the usual LL structure [@yoshioka-halperin-lee], where every state ${\left|n,j,s\right\rangle}$ within a fully degenerate Landau level $n$ is characterized by a momentum quantum number $j$ and an additional two-fold degenerate spin quantum number $s=\uparrow,\downarrow$. Since an energy gap $2B$ separates the different LLs, we restrict ourselves to the lowest LL, $n=0$. Fixing the number of fluxes per unit cell, $N_{\Phi}$, restricts $j$ to values running from 1 to $N_{\Phi}$, and the many-body Hamiltonian is, up to a constant amounting for the kinetic energy, given by $$\begin{aligned}
\label{H}
{\ensuremath{\Hat{\mathrm{H}}}} = \sum_{\{j,s\}} V_{\{j,s\}} {\ensuremath{\Hat{\mathrm{a}}}}_{j_1s_1}^{\dagger} {\ensuremath{\Hat{\mathrm{a}}}}_{j_2s_2}^{\dagger} {\ensuremath{\Hat{\mathrm{a}}}}_{j_3s_3}{\ensuremath{\Hat{\mathrm{a}}}}_{j_4s_4},\end{aligned}$$ where ${\ensuremath{\Hat{\mathrm{a}}}}_{js}$ annihilates a particle in state ${\left|0,j,s\right\rangle}$, and $\{j,s\}$ denotes the set of quantum numbers $j_1,\dots,j_4$ and $s_1,\dots,s_4$. The interaction conserves total momentum $J=\sum_i j_i \ \mathrm{mod} \ N_{\Phi}$, and the spin of the particles, so the coefficients read $$\begin{aligned}
\label{V}
V_{\{j,s\}} &=& \delta'_{j_1+j_2,j_3+j_4} \delta_{s_1,s_3} \delta_{s_2,s_4} \times \\ \nonumber && {\left\langlej_1,s_1\right|}\otimes{\left\langlej_2,s_2\right|} g_{s_1s_2} \delta(z_1-z_2) {\left|j_3,s_3\right\rangle}\otimes{\left|j_4,s_4\right\rangle},\end{aligned}$$ with $g_{s_1s_2}$ the spin-dependent contact interaction strength of two particles, and $\delta'$ a Kronecker delta modulo $N_{\Phi}$. The Hamiltonian is diagonalized in blocks with a fixed particle number $N$, defining the filling factor $\nu=N/N_{\Phi}$, fixed Haldane momentum $\vec{K}$ [@haldane], and a fixed spin polarization $S=N_{\uparrow}-N_{\downarrow}$, where $N_{\uparrow}$ ($N_{\downarrow}$) denotes the number of spin-up (spin-down) particles.
First, we show that for spin-independent interactions, i.e. $g_{s_1s_2}=g$, the singlet state is energetically favored. This configuration is a natural choice, as many experiments [@ketterle-2comp; @*wieman-2comp] work with the ${\left|F=1, m_F=0,1\right\rangle}$ states of $^{23}$Na, or the ${\left|F=1,m_F=-1\right\rangle}$ and ${\left|F=2,m_F=1\right\rangle}$ states of $^{87}$Rb with almost equal s-wave scattering lengths within and between the components. We start by contrasting the singlet to the fully polarized situation, $S=N$, in which we recover the results from Ref. [@cooper-wilkin-gunn], with incompressible phases at $\nu=k/2$, forming the RR series. The fully polarized zero-energy state with highest filling has $\nu=1/2$, and is the Laughlin state. In the opposite limit of a fully unpolarized system, zero-energy states occur up to $\nu=2/3$, where the 221-Halperin state is the exact unique ground state of the 2BC interaction. The analytic expression of this state on a disc reads $$\begin{aligned}
\Psi_{\mathrm{H}} = \prod_{i<j}^{N_{\uparrow}} (z_{i \uparrow}-z_{j \uparrow})^2 \prod_{k<l}^{N_{\downarrow}} (z_{k \downarrow}-z_{l \downarrow})^2 \prod_{i,k} (z_{i \uparrow} - z_{k \downarrow}),\end{aligned}$$ where we omit the irrelevant exponential term $\exp\left(-\sum
|z_i|^2/4\right)$. This wave function vanishes whenever two particles are at the same position, and it is symmetric under exchange of two spin-up or two spin-down particles. However, it is antisymmetric under exchange of particles with different spin, and thus cannot be used for describing spinless bosons. For this case, also the terms in the last product had to be squared, which would result in a less dense Laughlin-like wave function. Generalizing this observation, we state that any spatial wave function which solves the fully polarized problem must also be a solution for $|S|<N$, while the opposite is not true. Thus, for a spin-independent interaction, it strictly follows for the ground state energies $E(S_i)$ with $|S_1|<|S_2|$ that $E(S_1) \leq
E(S_2)$. Our numerical data, partially shown in the left side of Fig. \[filling\], supports this finding. In fact, in all cases we have investigated, the ground state of the $S=2$ subspace occurs in the spectrum of $S=0$ as a low-lying excitation, except for $N=6$ and $N_{\Phi}=8$, where we have a degeneracy, $E(S=0)=E(S=2)$. In most cases, thus, the ground state is a singlet, which is understood by noting that interactions of a $\uparrow\downarrow$-pair can be fully avoided by a single zero of the form $z_{i \uparrow} - z_{k \downarrow}$ in the wave function, while at least two zeros must be spent if a pair of particles with equal spin is to be anticorrelated in a similar way.
![\[filling\] (Color online) Left: Ground state energies, not showing possible degeneracies, at $N_{\Phi}=6$ for different $N$ and $S$ subspaces with $g_{s_1s_2}=g$. Even (odd) values of $S$ correspond to even (odd) $N$. Right: $\Delta$ vs. $\nu$ for $N_{\Phi}=6$ and $g_{s_1s_2}=g$.](fig1.eps)
Feshbach resonances allow to tune the spin-dependency of interactions (cf. [@mlbook]). Increasing $g_{\uparrow\downarrow}$ while leaving $g_{\uparrow\uparrow}=g_{\downarrow\downarrow}=g$ should favor spin polarizations with less $\uparrow \downarrow$-pairs. However, this does not necessarily drive the system into the fully polarized configuration: As shown in Fig. \[eud\] for $N=6$ at $\nu=1$, the energy of even the fully unpolarized system saturates for large $g_{\uparrow\downarrow}$. In this limit, the real ground state has $S=2$ and $E=0.449$ (in units of $gB$), close to the $S=0$ and $S=4$ ground states with $E=0.466$ and $E=0.475$. All of them are well separated from the fully-polarized ground state with $E=0.581$. In contrast to this, the energy of spin-unpolarized systems will never saturate at larger filling, as is shown in Fig. \[eud\] for $\nu=6/5$. This different behavior is understood by noting that for $\nu=1$ the number of available zeros in the wave function is sufficient to completely suppress interactions between pairs of different spins, $\langle
E_{\uparrow\downarrow} \rangle \rightarrow 0$, while it is not for larger $\nu$. Jumps in the curve of $\langle E_{\uparrow\downarrow} \rangle$ as a function of $g_{\uparrow\downarrow}$ show that the states with $\langle
E_{\uparrow\downarrow} \rangle = 0$ are reached by several abrupt re-organizations of the ground states. We note that once the state with $\langle E_{\uparrow\downarrow}\rangle=0$ is obtained, this state must be an eigenstate for arbitrary $g_{\uparrow\downarrow}$. Thus, by abruptly switching off the interspin interaction, one could produce binary mixtures of highly entangled, non-interacting systems.
![\[eud\] (Color online) Interaction energy for $N=6$ and $S=0,2$ as a function of $g_{\uparrow\downarrow}$. Left: $N_{\Phi}=6$. Right: $N_{\Phi}=5$.](fig2.eps){width="9cm"}
Coming back to spin-independent interactions, we ask now the question whether, in analogy to the RR states for spin polarized systems, incompressible states can be found amongst the spin-unpolarized ground states. Since incompressibility is connected to a discontinuity in the chemical potential, we define the particle-hole excitation gap as $\Delta(N) = N [ E(N+1)/(N+1) +
E(N-1)/(N-1) - 2 E(N)/N ]$, which is plotted for $N_{\Phi}=6$ in Fig. \[filling\] (right). Upwards peaks correspond to downward cusps in the energy as a function of $\nu$, and signal incompressible phases. They occur at fillings $\nu=2k/3$ with $k\in \mathbb{N}$. For these fillings, a series of singlet states being exact zero-energy eigenstates of a $(k+1)$-body contact interaction is known [@nass-nucl]. For $k=1$, this is the 221-Halperin state, which has Abelian excitations, while the states for $k>1$ are predicted to have non-Abelian excitations, which established the name NASS series.
![(Color online) Energy spectra of singlet states for $N=12$ particles at filling $\nu=4/3$. Black crosses (with the left energy scale) show the results for an SU(2)-symmetric 2BC interaction, while red triangles (with the right energy scale) are the NASS spectrum for any three-body interaction with $g_{s_1s_2s_3}>0$. We do not show the three-fold center-of-mass degeneracy, but visualize further degeneracies by slight shifts in $\vec{K}$. \[2spectra\]](fig3.eps)
We first construct these states for $k=2$ and 3 by diagonalizing the three- and four-body contact interaction for up to $N=12$ particles: As predicted by CFT, these states are characterized by a $(k+1)(k+2)/2$-fold topological degeneracy on the torus. A subtlety occurs at $k=3$, where the CFT construction [@nass-nucl], being restricted to $N=2kp$ with $p \in \mathbb{N}$, is not able to predict zero-energy ground states for all $N$ possible at $\nu=2$. In fact, we find zero-energy states also for $N=8$ and 10, but, in contrast to the states with $N=6$ and 12 which have the predicted ten-fold degeneracy, they are non-degenerate, and thus belong to a different topological phase.
The overlaps of these states with the ground states of Eq. (\[H\]), given in Table \[ov\], turn out to be significant: For $k=2$, the states are exactly equal for $N=4$, and even for a relatively large-sized system with $N=12$ the overlaps are still around 0.8. Such a decrease of the overlap with large $N$ also occurs in the case of the RR states in single-component systems with 2BC interaction [@cooper-aip], suggesting the use of ground state degeneracies to characterize the topological phase in the thermodynamic limit [@wen-niu]. Topological degeneracies are usually lifted in finite systems, as also happens here. In the case of $\nu=4/3$ we may consider the fact that the six lowest eigenstates of ${\ensuremath{\Hat{\mathrm{H}}}}$ agree in their Haldane momenta $\vec{K}$ with the ones of the sixfold degenerate three-body eigenstates as a signal for belonging to the same topological phase. Both spectra are shown for $N=12$ in Fig. \[2spectra\]. We note that both the degeneracy splitting $\Delta_{\mathrm{GS}}$ and the gap to the excited band $\Delta_{\mathrm{EX}}$ behave non-monotonic with $N$. The situation is different for $\nu=2$, where the candidates for the degenerate ground state manifold mix up with excited states.
Our results therefore suggests that only the $\nu=4/3$-NASS phase can be produced with 2BC interaction. For the experimental feasibility of this state to be high, it should also be robust with respect to deviations from the SU(2)-symmetric interaction. In the case of a $(k+1)$-body contact interaction at $\nu=2k/3$, the numerical values of each interaction parameter do not influence the ground states. Though this is different for 2BC interaction at $\nu=4/3$, the overlaps with the NASS states remain high within a wide range of $g_{\uparrow\downarrow}/g$, and have a maximum in the SU(2)-symmetric configuration. For $N=8$, all overlaps are above 0.8 if $g_{\uparrow\downarrow}/g \in [0.8,1.6].$
-------------- ----------- ------------------------------------------------------- ------------------- -- -------------- ----------- ------------------------------------------------------- -------------------
$N,N_{\Phi}$ $\vec{K}$ $|{\left< \mathrm{2b} \vphantom{\mathrm{3b}} \right| $E_{2\mathrm{b}}$ $N,N_{\Phi}$ $\vec{K}$ $|{\left< \mathrm{2b} \vphantom{\mathrm{4b}} \right| $E_{2\mathrm{b}}$
\left. \mathrm{3b} \vphantom{\mathrm{2b}} \right>}|$ \left. \mathrm{4b} \vphantom{\mathrm{2b}} \right>}|$
4,3 (0,0) 1 0.435 12,6 (0,0) 0.334 2.604
1 0.520 0.828 2.677
8,6 (0,0) 0.906 0.901 (0,2) 0.775 2.708
0.920 0.908 (0,4) 0.775 2.708
12,9 (0,0) 0.729 1.289 (2,0) 0.775 2.708
0.885 1.333 (2,2) 0.646 2.685
(2,4) 0.646 2.285
(4,0) 0.775 2.708
(4,2) 0.646 2.685
(4,4) 0.646 2.685
-------------- ----------- ------------------------------------------------------- ------------------- -- -------------- ----------- ------------------------------------------------------- -------------------
: Overlaps between zero-energy eigenstates of $(k+1)$-body interaction with corresponding eigenstates of the 2BC interaction at energy $E_{2\mathrm{b}}$ (in units of $gB$) and Haldane momentum $\vec{K}$ (in units of $2\pi \hbar/a$). All states at filling $\nu=4/3$ have an additional three-fold center-of-mass degeneracy. \[ov\]
Finally, we consider an additional SO coupling with $q>0$ in Eq. (\[A\]). This coupling mixes internal and external degrees of freedom on the single-particle level, such that the LLs become superpositions of the form ${\left|n,j,\pm\right\rangle} \equiv \alpha_{n}^{\pm}
{\left|n-1,j,\uparrow\right\rangle} + \beta_{n}^{\pm} {\left|n,j,\downarrow\right\rangle}$ where $\alpha_{n}^{\pm}$ and $\beta_{n}^{\pm}$ are functions of $q$ [@trombettoni]. For general $q$, this lifts the previous spin-degeneracy, as the single-particle energies now read $E_{n}^{\pm} = 2Bn +2q^2 \pm
\sqrt{B^2+8Bq^2n}$. As has been pointed out in Refs. [@trombettoni; @palmer-pachos], the system then behaves basically like a single-component system, and the incompressible states are described by the RR series projected into the lowest LL of the SO coupled system. However, for $q^2/B = 2n+1$, a degeneracy between $E_{n}^-$ and $E_{n+1}^-$ occurs as the single-particle ground state switches from ${\left|n,j,-\right\rangle}$ to ${\left|n+1,j,-\right\rangle}$. The Hamiltonian from Eq. (\[H\]) can then be adapted to the system with SO coupling by identifying the $s$ quantum number with the two-fold degeneracy between $n$ and $n+1$, and by re-defining $V_{\{js\}}$ accordingly. Note that in this case $S$ does not correspond to the spin polarization, but quantifies the population imbalance between ${\left|n,j,-\right\rangle}$ and ${\left|n+1,j,-\right\rangle}$. As observed in Ref. [@palmer-pachos] within a lowest LL approximation, the projected RR states do not describe the system close to these degeneracy points. For $q^2/B = 3$, the system becomes gapless at both the Laughlin and the MR filling $\nu=1/2$ and 1. However, we find a clear gap in the spectrum at $\nu=2/3$ with $N=6$, and a small gap at $\nu=4/3$ with $N=8$. Projecting this $\nu=2/3$ state into the lowest LL of the system without SO coupling, and comparing with the 221-Halperin state, we find an overlap of only 0.19. It should be noted that, in contrast to the previous case of a pure spin degeneracy, interactions now can induce flips within the two-fold degenerate manifold, and therefore $S$ is not conserved. For a complete description of the state, we use generalizations of the Halperin state, obtained by diagonalizing the 2BC interaction at $\nu=2/3$ in an Abelian field ($q=0$) within different spin subspaces. Then the highest overlap of 0.60 is obtained with the state having $S=-6$, which corresponds to all particles being in ${\left|0,j,-\right\rangle}$. This is also the most populated LL, showing that, despite the degeneracy on the single-particle level, interactions favor the LL with lower $n$. The total fidelity for $-6
\leq S \leq 6$ is 0.82. For $\nu=4/3$ with $N=8$, but now comparing the non-degenerate ground state of the SO coupled system with the ground states of a three-body contact interaction at $q=0$ for different $S$, we still obtain a total fidelity of 0.52, summing from $-8 \leq S \leq 8$.
We have investigated two-component Bose gases in artificial gauge fields. The possibility of tuning the interaction constants allows to control and manipulate the strongly correlated nature of the eigenstates. Our main finding is the formation of incompressible phases described by the NASS series within the experimentally feasible setup of particles interacting via two-body contact interaction. The internal degree of freedom permits also to realize systems with spin-orbit coupling.
This work has been supported by EU (NAMEQUAM, AQUTE), ERC (QUAGATUA), Spanish MINCIN (FIS2008-00784, FIS2010-16185, FIS2008-01661, 2009 SGR1289), Alexander von Humboldt Stiftung, and AAII-Hubbard. B. J.-D. is supported by the Ramón y Cajal program. M. L. acknowledges Hamburg Theory Award.
*Note added:* After submission of this article, a preprint by S. Furukawa and M. Ueda appeared presenting additional numerical evidences for the NASS phase [@furukawa-ueda].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Mid-infrared observations of the Andromeda galaxy, M31, obtained with the Infrared Array Camera on board the Spitzer Space Telescope are presented. The image mosaics cover areas of approximately $3\fdg7 \times 1\fdg6$ and include the satellite galaxies M32 and NGC 205. The appearance of M31 varies dramatically in the different mid-infrared bands, from the smooth bulge and disk of the old stellar population seen at 3.6[ $\mu$m ]{}to the well-known ‘10 kpc ring’ dominating the 8[ $\mu$m ]{}image. The similarity of the 3.6[ $\mu$m ]{}and optical isophotes and nearly constant optical-mid-infrared color over the inner 400 confirms that there is no significant extinction at optical wavelengths in M31’s bulge. The nuclear colors indicate the presence of dust but not an infrared-bright active galactic nucleus. The integrated 8[ $\mu$m ]{}non-stellar luminosity implies a star formation rate of $0.4 M_{\sun}$ yr$^{-1}$, consistent with other indicators that show M31 to be a quiescent galaxy.'
author:
- 'P. Barmby, M.L.N. Ashby, L. Bianchi, C.W. Engelbracht, R.D. Gehrz, K.D. Gordon, J.L. Hinz, J.P. Huchra, R.M. Humphreys, M.A. Pahre, P.G. Pérez-González, E.F. Polomski, G.H. Rieke, D.A. Thilker, S.P. Willner, C.E. Woodward'
title: 'Dusty waves on a starry sea: the mid-infrared view of M31'
---
Introduction
============
The proximity of the Andromeda galaxy, M31, has long made it a prime target for studies of galaxy structure and stellar populations. One technique has been to isolate particular populations: for example, studies of luminous stars [@hmf90] showed that M31 has a lower massive star formation rate than M33 and the LMC. Global studies of particular components have also been valuable: H I maps provide estimates of the mass distribution [@bt04] and have recently revealed a population of high-velocity clouds [@thil04]. [*ISOPHOT*]{} [@haas98], [*MSX*]{} [@kraemer02], and [*IRAS*]{} [@habing84] observations showed a bright ring of infrared emission with radius 10 kpc coincident with many of the H II regions.
Mapping the entire disk of M31 at mid-infrared wavelengths allows [*both*]{} local and global studies of the galaxy. Observations with the Infrared Array Camera [IRAC; @irac] on the [*Spitzer Space Telescope*]{} simultaneously trace the dust in the spiral arms at 8[ $\mu$m ]{}and the oldest stars in the disk and bulge at 3.6 and 4.5[ $\mu$m ]{}without the complicating extinction or distance effects that make such studies in the Milky Way difficult. IRAC observations of M31 are complemented by deep data now available at many other wavelengths. They also complement [*Spitzer*]{} studies of other nearby galaxies.
This paper presents an initial look at the IRAC observations of M31, focusing on the surface brightness profiles and extended emission. Companion papers discuss longer-wavelength MIPS observations of M31 [@gordon06], IRAC and MIPS observations of the M31 satellite galaxy NGC 205 [@marleau06], and the implications of the galaxy’s morphology as seen in 8[ $\mu$m ]{}non-stellar emission [@block06]. A distance to M31 of 783 kpc [@sg98] is assumed throughout. All magnitudes are on the Vega system, using the calibration given by @reach05.
Observations and Data Reduction
===============================
The IRAC observations of M31 were taken as part of [*Spitzer*]{} General Observer program 3126 in 2005 January and August.[^1] Fifteen Astronomical Observation Requests (AORs) were used to map a region approximately $3\fdg7 \times 1\fdg6$ (chosen to match the [*Spitzer*]{}/MIPS observations made as part of program ID 99), with an extension to the NW to include NGC 205. The central $1\fdg6 \times 0\fdg4$ was covered by three AORs, each having two 12-second frames per position. The outer regions were covered by two AORs each with two dithered 30-second frames per position. This mapping strategy ensured that observations of each position in the galaxy were separated by at least 2.5 hours, allowing efficient asteroid rejection in data processing. The complete dataset consists of 3000 individual images in each of the IRAC channels.
Data reduction began with the Basic Calibrated Data (BCD) produced by versions 11 (for the January data) or 12 (the August data) of the [*Spitzer*]{} Science Center (SSC) Pipeline. A ‘delta dark’ offset correction was applied to the 12-second 5.8[ $\mu$m ]{}frames to correct for the first-frame effect, followed by use of the ‘artifact corrector’ software developed by S. Carey, which attempts to remove the electronic effects caused by bright stars. The remaining processing steps were carried out using the “IRAC\_proc”[^2] software with the 2005 September 5 version of the SSC MOPEX package. The ‘overlap correction’ necessary to make the sky background consistent between frames was computed and applied. Median-combined mosaics of the full dataset were created, then projected into the reference frame of each input BCD image to create ‘source maps’. These source maps were used as the input ‘uncertainty images’ for creating the final photometric mosaics — this step prevents the outlier rejection used by MOPEX from rejecting pixels in bright sources and thereby biasing the photometry. The final mosaics were created with a pixel scale of 086, sub-sampling the native IRAC pixel scale by a factor of $\sqrt{2}$ to better sample the point spread function (FWHM $\sim1\farcs9$). The sky background was removed using the [iraf]{} task [imsurfit]{} to model the background as a polynomial surface sampled by ‘blank sky’ regions near the edge of the mosaics. After subtraction, the mean image counts in small areas within these regions were computed, and the standard deviation of these means used to estimate the photometric uncertainties due to the background.
The final step in mosaic production was photometric calibration. Although calibration of point source photometry varies slightly over the IRAC arrays’ field of view [@reach05], we did not correct for this since doing so would have corrupted the extended emission. The calibration of IRAC is known to differ between point and extended sources, particularly at 5.8 and 8[ $\mu$m ]{}where a significant fraction of point-source light is scattered on scales comparable to the array size. We used the results of recent work by T. Jarrett (priv. comm.) to correct from point to extended source calibration in the M31 mosaics by multiplying the images by (0.91, 0.94, 0.66, 0.74) in the four IRAC bands, respectively \[these values differ slightly from those given in @reach05\]. The standard deviations of the background levels are about 0.04[ MJy sr$^{-1}$]{} in all bands except 5.8[ $\mu$m]{}, where the background subtraction was more difficult, and the noise was 0.1[ MJy sr$^{-1}$]{}(1$\sigma$). These correspond to limiting surface brightness levels of $21.2, 20.9, 19.0$, and 19.6 mag arcsec$^{-2}$ in the four IRAC bands.
The final of mosaics are shown in Figure \[m31\_bw\]. As with other recent works on IRAC observations of nearby galaxies we assumed that the 3.6[ $\mu$m ]{}emission is purely due to stars, and used this to construct maps of the ‘non-stellar’ emission at 5.8 and 8[ $\mu$m]{}. The stellar emission at the two wavelengths is assumed to correspond to 0.399 and 0.232 $\times$ the 3.6[ $\mu$m ]{}emission [@helou04]. The 8[ $\mu$m ]{}non-stellar image is shown in Figure \[m31\_bw\].
Morphology and surface brightness profiles
==========================================
M31’s appearance changes dramatically over the IRAC bands, in a similar way to other spiral galaxies such as M81 [@irac_m81] and NGC 300 [@helou04]. The 3.6[ $\mu$m ]{}view is similar to the $K_s$-band image presented in @2mass_lga: dominated by the bulge, with the spiral arms and disk relatively faint.[^3] At 8[ $\mu$m]{}, M31’s bulge is much less prominent, and the spiral arms and the 10 kpc ring are the dominant features. The outer (14 kpc) ring detected at far-infrared wavelengths [@haas98; @gordon06] and tentatively detected with [*MSX*]{} [@kraemer02] is also seen in the 8[ $\mu$m ]{}image. The 8[ $\mu$m ]{}image — particularly the non-stellar version — looks extremely similar to the 24[ $\mu$m ]{}image presented by @gordon06, showing the same split on the southwest side of the 10 kpc ring that @gordon06 attribute to interactions with M32. The ‘hot spots’ seen near the nucleus at 160[ $\mu$m ]{}are also prominent in the 8[ $\mu$m ]{}non-stellar image, where they coincide with the ends of spiral arms.
Disk images $3\fdg3\times0\fdg9$ were extracted from the mosaics and used to derive IRAC surface brightness profiles of M31. Peaks of bright stars, a 1-radius region around M32, and image regions with no coverage were all masked from the profile measurements. Profiles were measured using two methods: making 20-wide cuts along the disk major and minor axes using the [iraf]{} task [pvector]{}, and fitting elliptical isophotes to the images using the [ellipse]{} task. Because [ellipse]{} can be unstable at low surface brightnesses, the isophote shape parameters (ellipticity, position angle, center) were constrained to be constant beyond the edge of the visible disk at semi-major axes $>0\fdg88$. The isophote fit at this distance has an ellipticity ($\epsilon=1-b/a$) of 0.73 and position angle of 38. Profiles in the three longer-wavelength bands were measured on the ellipses fit to the 3.6[ $\mu$m ]{}image, so that colors could be measured at the same spatial locations.
The relative prominence of the M31 disk and bulge in the four IRAC bands is reflected in the major and minor axis profiles, shown in Figure \[fig\_axes\]. Out to a semi-major axis of $\sim400$ (approximately the edge of M31’s bulge), all four bands have very similar profiles. Beyond 400, the 5.8[ $\mu$m ]{}and 8.0[ $\mu$m]{}-band profiles decline more slowly and show more variability due to the clumpy nature of the dust emission. The 10 kpc ring is visible in all of the major-axis profiles; the 8.0[ $\mu$m ]{}profile in particular appears highly asymmetric because of the split in the ring. The minor axis profiles are smoother because they sample mainly the bulge. Both major and minor axis profiles are quite similar in shape to the optical profiles presented by @wk87, indicating that dust extinction does not have a major effect on the optical profiles. The nucleus of M31 is smaller than the IRAC PSF, and does not separate from the bulge emission in the radial profile.
The color profiles of M31 shown in Figure \[fig\_colors\] also depict the disk/bulge separation. The median isophote colors within 400 are $[3.6]-[4.5] = -0.14$, $[3.6]-[5.8]=0.03$ and $[3.6]-[8.0]=0.13$. The $[3.6]-[4.5]$ color is consistent with theoretical expectations for M0 giant stars [see @pahre04]; the $[3.6]-[5.8]$ and $[3.6]-[8.0]$ colors are slightly redder, presumably because of dust emission. At larger distances, the disk dominates the light and the colors become redder, particularly at the location of the 10 kpc ring. Comparing the 3.6[ $\mu$m ]{}profile to the $r$-band optical profile of @kent87 shows that the $r-[3.6]$ color is nearly constant over the central 400, with a luminosity-weighted mean of $r-[3.6]=3.41$ mag. This corresponds to $r-K\approx3.1$, reasonably consistent with the predictions of @mar05. The colors of the M31 nucleus (measured in a 24-radius aperture, with the IRAC point source calibrations and aperture corrections applied) are $[3.6]-[4.5] = -0.15$, $[3.6]-[5.8]=0.19$ and $[3.6]-[8.0]=0.51$. The red colors at the longer wavelengths suggest that the nucleus has more dust than the bulge, but are not the very red colors characteristic of an AGN [e.g. @stern05].
Table \[tab\_mag\] gives integrated flux densities and magnitudes of M31 as measured in the largest elliptical isophote ($a=$18). The uncertainties in these values include the effects of background variation and a 10% uncertainty in the absolute extended source calibration. The IRAC flux densities are consistent with the COBE/DIRBE measurements at 3.5 and 4.9[ $\mu$m ]{}[245 and 128 Jy with 20% uncertainties; @ons98] and the [*MSX*]{} A-band (8.3[ $\mu$m]{}) value [$159\pm32$ Jy; @gordon06]. Near-to-mid-infrared colors are another way to check our photometry: @amh80 measured $H=1.42$ in a 1554 diameter circular aperture centered on M31. In the same aperture the 3.6[ $\mu$m ]{}magnitude is 0.84, giving $H-[3.6]=0.58$. Using M81’s $K_s-[3.6]\approx0.3$ [@irac_m81] and the mean $H-K_s= 0.27$ measured for early-type (S0/Sa/Sb) spirals by @2mass_lga, one would predict $H-[3.6]=0.57$, in very good agreement with our measurement.
A model light distribution consisting of an $r^{1/4}$ bulge and exponential disk were fit to the 3.6[ $\mu$m ]{}semi-major axis surface brightness (SB) profile. The structural parameters fitted to the disk are sensitive to inclusion of the ring at semi-major axis lengths $2000 \leq a \leq 3500$ arcsec, while the bulge effective radius is sensitive to the inner fitting cutoff due to deviations from the idealized $r^{1/4}$ model at $a < 100$ arcsec [see @kent83]. The sky value for the image was determined at isophotes roughly corresponding to $3.5$ disk scale lengths. As there is still some galaxy emission at this radius, our sky value is likely over-estimated and is probably the dominant cause of the significant reduction in the SB at $a > 4000$ arcsec (smaller, but still significant, errors will be present in the profile at $a < 4000$ arcsec). We chose the fitting regions of $100 \leq a \leq 2000$ and $3500 \leq a \leq 4000$ arcsec to minimize these effects, and for consistency with previous work.
The resulting model has (bulge,disk) half-light semi-major axis lengths of $(10.3\pm 0.4,44.7\pm 0.6)$ arcmin, average ellipticities near to those scale lengths of $(0.48\pm 0.02,0.68\pm 0.04)$, total magnitudes $(0.84\pm 0.13,0.58\pm 0.06)$, and RMS of $0.02$ mag. The corresponding circularized effective radii are $(7.5\pm 0.3,25.2\pm 0.7)$ arcmin. This disk major axis scale length of $6.08\pm 0.09$ kpc compares well with the longest wavelength ($R$-band) value of 5.9 kpc measured by @wk88. The ratio of the bulge effective radius of $1.70\pm 0.07$ kpc to disk scale length place M31 on the upper envelope of local spiral galaxies [@dej96]. The fitted bulge and disk models have a flux ratio $B/D = 0.78 \pm 0.11$. This value is biased high relative to the true ratio, however, because the bulge model is too bright at $a < 100$ arcsec and the disk model is too faint at the location of the 10 kpc ring. @wk88 calculate a similar value in the $R$-band.
The model-fitting allows an estimate of M31’s total magnitude at 3.6[ $\mu$m ]{}through extrapolation of the disk model to infinite radius. The magnitude within $a < 4000$ mag is 0.19 mag, the extrapolated disk from 2.5 scale lengths to infinity adds an additional 0.38 mag, and the sky subtraction errors on the SB profile within this isophote result in an additional 0.15 mag. The implied total magnitude of M31 is therefore $-0.34$ mag. We have chosen not to quote this model-dependent value in Table 1 for consistency with previous work and our measurements at other wavelengths, but it is consistent with the measurement in the 18 isophote given there when similar corrections are applied. The errors on the total magnitude of M31 are dominated by the systematic error of sky subtraction on this large galaxy, and less so on the extrapolation of the disk emission.
The 3.6[ $\mu$m ]{}luminosity of M31 can be used to estimate the galaxy’s stellar mass. Using a typical value of $B-R=1.5$ from @wk87 and the color-dependent mass-to-light ratios of @bdj01, an estimated $M/L_K$ for M31 is about 0.8 in solar units. Using $K-[3.6]=0.3$ and multiplying by $M/L_K$, the resulting stellar mass is $1.1\times10^{11} M_{\sun}$. Recent dynamical mass estimates of the M31 disk+bulge [e.g., @wps03; @gee06] are $\sim10^{11}M_{\sun}$, in reasonable agreement with this derived stellar mass. Asymptotic giant branch (AGB) stars can be important contributors to the total near-infrared luminosities of galaxies [@mar05]. However, the most luminous AGB stars are predicted by @gro06 to have much redder colors ($0.5<[3.6]-[4.5]<1$) than observed for M31, so we conclude that they do not dominate M31’s 3.6[ $\mu$m ]{}luminosity.
The 8[ $\mu$m ]{}non-stellar emission is dominated by the 7.7[ $\mu$m ]{}polycyclic aromatic hydrocarbon (PAH) band, which is a useful though imperfect star formation rate (SFR) indicator. It is therefore instructive to compare the M31 8[ $\mu$m ]{}luminosity with other star formation indicators, such as H$\alpha$, radio, and total infrared luminosities and with these values for other galaxies. @wu05 derived a calibration for SFR as a function of 8[ $\mu$m ]{}non-stellar luminosity $\nu L_{\nu}[8\mu{\rm m(dust)}]$ (hereafter $L_8$) using correlations between $L_8$ and $L({\rm H}\alpha)$ and $L(1.4 {\rm GHz})$. The IRAC 8[ $\mu$m ]{}non-stellar flux density measured for M31 corresponds to a luminosity of $\log(L_8/L_{\sun})={8.8}$. The @wu05 calibration yields an SFR of $0.4 M_{\sun}$ yr$^{-1}$.
It is possible to compare M31’s 8[ $\mu$m]{}-derived SFR with SFRs from other indicators, but since all such indicators have calibration uncertainties \[e.g., the @wu05 8[ $\mu$m]{}/SFR calibration is based on a small number of galaxies with a limited range of properties\], we instead compare the observed properties directly. To estimate the $H\alpha$ luminosity of M31, we convert the value given by @dev94 to the 780 kpc distance, multiply by 0.8 to account for \[NII\] contamination, and multiply by 3.4 [as done by @wb94] to correct for extinction. The resulting $L({\rm H}\alpha)=2.6\times 10^7 L_{\sun}$, which predicts $\log(L_8/L_{\sun})=9.1$. The 1.4 GHz radio flux density measured by @bbh98, $3.34\times 10^{20}$ W Hz$^{-1}$, yields a predicted $\log(L_8/L_{\sun})=8.4$. Figure 12 of @dale05 gives the ratio of $\nu f_{\nu}[8\mu{\rm m(dust)}]/f(3-1100\mu{\rm m})$ as a function of $f_{\nu}(70 \mu{\rm m})/f_{\nu}(160 \mu{\rm m})$ for SINGS galaxies. From @gordon06, M31 has $f(70)/f(160)=0.12$, with the corresponding $f(3-1100\mu {\rm m})=2.57\times10^{-10}$ W m$^{-2}$, yielding a predicted $\log(L_8/L_{\sun})=9.0$. The H$\alpha$ and far-infrared predictions are in reasonably good agreement, and are consistent with the observed 8[ $\mu$m ]{}luminosity given the scatter in the relationships. The 8[ $\mu$m ]{}luminosity predicted from the radio flux is lower than observed, perhaps indicating that some extended emission was resolved out of the radio maps. All of the ‘star-formation’ luminosities are at the low end of the galaxy distribution. Despite its prominent star forming ring, M31 is a predominantly quiescent galaxy.
We thank the referee for helpful suggestions. This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech.
Facilities:
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[crr]{} 3.6 & $259\pm32$ &$ 0.09\pm0.13$\
4.5 & $144\pm20$ &$ 0.24\pm0.15$\
5.8 & $190\pm35$ &$ -0.55\pm0.20$\
8.0 & $151\pm21$ &$ -0.93\pm0.15$\
8.0 (non-stellar) & $91\pm21$&
[^1]: A large solar proton event made most of the January data unusable; re-observations were made in August.
[^2]: IRAC\_proc was developed by M. Schuster, M. Marengo and B. Patten at the Smithsonian Astrophysical Observatory.
[^3]: The 2MASS Large Galaxy Atlas image of M31 is suspected of having an overly-faint disk due to problems with background subtraction (T. Jarrett, pers. communication). We do not compare the IRAC quantitative measurements to those given by @2mass_lga, instead estimating M31’s $K-[3.6]$ color by M81’s value of 0.3 [@irac_m81].
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'R. Mathieu, S. A. Ivanov, P. Nordblad, and M. Weil'
date: 'Received: date / Revised version: date'
title: 'Enhancement of antiferromagnetic interaction and transition temperature in $M_3$TeO$_6$ systems ($M$ = Mn, Co, Ni, Cu)'
---
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Introduction
============
There is currently a great interest in designing and finding new materials with multiferroic properties, or in a larger sense magnetic and dielectric ordering near room-temperature. It has been found that complex crystal structures with several independent sites for magnetic cations as well as complex magnetic structures favor concomitant coupled magnetic and dielectric states[@mf].
The structural, magnetic and dielectric properties of several members of the corundum-related class of complex oxides with structural formula $A_3B$O$_6$ ($A$ cation is divalent or trivalent, $B$ cation is pentavalent or hexavalent) have been investigated, as for example the $M_3$Te$^{6+}$O$_6$ systems with $M$ = Mg$^{2+}$[@blasse], Cd$^{2+}$[@kosse], Mn$^{2+}$, Co$^{2+}$, Ni$^{2+}$, Cu$^{2+}$[@MTO; @CTO; @NTO; @CuTO], or ($M_2A$)Sb$^{5+}$O$_6$ compounds with $M$ = Mn$^{2+}$, Ni$^{2+}$ and $A$ = In$^{3+}$, Sc$^{3+}$[@MTO-InSc; @NTO-InSc].
The $M_3$TeO$_6$ oxides with $M$ = Mn, Co, Ni, Cu order antiferromagnetically at low temperatures. Ni$_3$TeO$_6$ crystallizes in a non-centrosymmetric rhombohedral $R3$ structure, and displays a relatively simple collinear magnetic structure, consisting of ferromagnetic $ab$-planes stacked antiferromagnetically along the $c$-axis [@NTO]. On the contrary, the other compounds crystallize in centrosymmetric structures, with complex magnetic structures: monoclinic $C2/c$ Co$_3$TeO$_6$ displays an incommensurate, multi-propagation-vector and temperature-dependent, magnetic structure[@CTO]; rhombohedral $R\bar{3}$ Mn$_3$TeO$_6$ displays a two-orbit incommensurate antiferromagnetic structure[@MTO]; and cubic $Ia\bar{3}$ Cu$_3$TeO$_6$ orders in a “three-dimensional spin web” of hexagonal arrangements of magnetic moments[@CuTO].
We here report structural and magnetic properties of $M_3$TeO$_6$ single-crystals with $M$ = Mn, Co, Ni, Cu and Mn-doped $M_3$TeO$_6$ ($M$ = Co, Ni) ceramics. The antiferromagnetic interaction and antiferromagnetic transition temperature increase when Mn replaces the magnetic cation.\
---------------------- --------------- ------------- ------- ------- ------- -- ------ --------- ------ ----------- ------------ ----------- -------
$M$-O $M$-$M$
$M^{2+}$ $<$ $r_A$ $>$ $s. g.$ min. max. mean min. max. mean $V_A$ $\omega_A$ $x_A$ $T_N$
Mn 0.83 $R\bar{3}$ 2.091 2.373 2.207 3.21 3.83 3.56 12.7 0.103 0.077 23
Mn$_{0.2}$Co$_{0.8}$ 0.762 $R\bar{3}$ 1.976 2.408 2.159 3.10 3.81 3.46 11.81 0.099 0.071 40
$^{a,b}$Co 0.745 $C2/c$ 1.856 2.333 2.064 2.71 3.93 3.38 3.5-13.0 0.02-0.06 0.06-0.24 26
$^a$Ni 0.69 $R3$ 2.008 2.148 2.079 2.78 4.02 3.40 11.6-11.9 0.01-0.03 0.14-0.24 52
Cu 0.73 $Ia\bar{3}$ 1.949 2.369 2.116 3.18 3.60 3.39 10.99 0.105 0.17 61
---------------------- --------------- ------------- ------- ------- ------- -- ------ --------- ------ ----------- ------------ ----------- -------
\[table1\]
Results and discussions
=======================
$M_3$TeO$_6$ single crystals ($M$ = Mn, Co, Ni, Cu) and ceramics ($M$ = (Mn,Co), (Mn,Ni) were synthesized by chemical transport (see Refs. [@MTO; @CTO; @NTO; @CuTO] and [@MTO-MW]) and solid state reactions[@MTO], respectively. The magnetic properties were investigated by magnetization and heat capacity measurements (MPMS squid magnetometer and PPMS setup from Quantum Design Inc.). Crystal and magnetic structures of the samples with $M$ = Mn, Co, and (Mn,Co) have been studied by x-ray (XRD) and neutron (NPD) powder diffraction[@MTO; @CTO; @MCTO]. Corresponding data for the samples with $M$ = Ni and Cu was obtained from literature[@NTO; @CuTO]. The magnetic properties of ceramic samples with $M$ = Mn, Co, Ni, Cu were found to be similar to those of corresponding single crystals.
Magnetic susceptibility and heat capacity experiments indicate low temperature antiferromagnetic states for all $M_3$TeO$_6$ single crystals. The magnetic transitions are evident from the heat capacity results plotted in the main frame of Fig. \[HC\]. The obtained antiferromagnetic transition temperatures ($T_N$), listed in Table \[table1\], are in agreement with the temperatures for which $ \partial $($MT/H$)$/ \partial T$ curves is maximum. Linear fits of $H/M$($T$) data recorded up to higher temperatures were performed, yielding estimates of Curie-Weiss temperatures $\theta_{CW}$ and the magnetic frustration parameter $f$=$-\theta_{CW}/T_N$. The variation of $f$ with $M$ is illustrated in the inset of Fig. \[HC\]. The incommensurate magnetic structure of Mn$_3$TeO$_6$ is frustrated. For this compound, $\theta_{CW}$ $\sim$ -120 K, i.e. about 5 times $T_N$ ($f$ $\sim$ 5), to compare to $\theta_{CW}$ $\sim$ -49 K ($f$ $\sim$ 1) and $\theta_{CW}$ $\sim$ -54 K ($f$ $\sim$ 2) for Ni$_3$TeO$_6$ and Co$_3$TeO$_6$ respectively. Similarly we find $\theta_{CW}$ $\sim$ -223 K ($f$ $\sim$ 3.6) for Cu$_3$TeO$_6$. Note that in that case only the higher-temperature data ($T$ $\ge$ 200 K) closely follows a Curie-Weiss behavior (see also [@CuTO]).
As seen from the heat capacity and magnetization measurements shown in Fig. \[HCMT\], the antiferromagnetic ordering temperature increases in Mn-doped Co$_3$TeO$_6$ and Ni$_3$TeO$_6$ ceramics. In the case of (Mn$_{0.2}$Ni$_{0.8}$)$_3$TeO$_6$, the onset of antiferromagnetic ordering takes place near 67 K, i.e. above the $T_N$ of Cu$_3$TeO$_6$ (which has the highest reported undoped $T_N$ (61 K)). The $f$ parameter is nearly unaffected by Mn-doping, as shown in the inset of Fig. \[HC\], reflecting the co-variation of $T_N$ and $\theta_{CW}$, and thus the increase in antiferromagnetic interaction strength.
We have also tried to dope Mn in Cu$_3$TeO$_6$; however, the temperature-dependent magnetization curves of e.g. (Mn$_{0.2}$Cu$_{0.8}$)$_3$TeO$_6$ show two features, near 50 K and 21 K, suggesting phase separation. Similar two phase features were obtained when doping Co in Ni$_3$TeO$_6$. For comparison, also $B$-site substitutions were attempted, e.g. Ni$_3$(Te,Mo)O$_6$, again resulting in phase separation.
The structural and magnetic properties of the pure $M_3$TeO$_6$ systems and the 20% Mn-doped Co$_3$TeO$_6$ compound are summarized in Table \[table1\] and Fig. \[DIAG\]. The compounds contain different magnetic cations, and adopt different crystal structures with sometimes non-equivalent crystallographic and magnetic sites. It is thus difficult to determine an exact correlation between structural and magnetic properties. We have attempted to extract more structural parameters from the refinements, such as $M$-O-$M$ bond angles, but those angles were found to vary greatly (e.g. between 29$^o$ and 161$^o$ for $M$ = Mn). Nevertheless, as illustrated in Table \[table1\] and Fig. \[DIAG\], while Mn-doped Ni$_3$TeO$_6$ remains rhombohedral $R3$, Mn-doped Co$_3$TeO$_6$ surprisingly adopts the rhombohedral $R\bar{3}$ structure of Mn$_3$TeO$_6$ (even with only 10% Mn, see also Ref. [@MCTO]). As seen in the table, all the obtained structural and polyhedral parameters for Mn$_3$TeO$_6$ and (Mn$_{0.2}$Co$_{0.8}$)$_3$TeO$_6$ are relatively similar. Interestingly, by extrapolating the data presented in Fig. \[DIAG\], one can predict that if rhombohedral $R\bar{3}$ Co$_3$TeO$_6$ could be stabilized (e.g. by synthesis under pressure), it would have a $T_N$ of about 45 K, i.e. nearly twice that of monoclinic Co$_3$TeO$_6$. Similarly, assuming that Mn-doped Ni$_3$TeO$_6$ remains in the rhombohedral $R3$ structure for large amounts of Mn, such an Mn-rich (Ni,Mn)$_3$TeO$_6$ or pure Mn$_3$TeO$_6$ could order antiferromagnetically at temperatures around 100 K.
Dielectric and ferroelectric properties of some $M_3$TeO$_6$ ($M$= Cd, Mg) have been investigated in earlier studies[@blasse; @kosse]. More recently, it was observed that Co$_3$TeO$_6$ could acquire an electronic polarization at low temperatures[@CTO-POL], and that Ni$_3$TeO$_6$ is ferroelectric below 1000 K[@NTO-InSc]. Since the non-centrosymmetric structure of Ni$_3$TeO$_6$ is preserved upon doping with Mn, our results suggest that rhombohedral $R3$ Mn-rich (Ni,Mn)$_3$TeO$_6$ or pure Mn$_3$TeO$_6$ would display enhanced magnetic properties (higher $T_N$) while retaining ferroelectric properties.
Conclusions
===========
We have observed significant changes in the structural and magnetic properties of $M_3$TeO$_6$ solid solutions. For example, the replacement of one fifth of the magnetic cation with Mn in Ni$_3$TeO$_6$ and Co$_3$TeO$_6$ yields an increase in antiferromagnetic interaction and transition temperatures. We predict that new $(M',M'')_3$TeO$_6$ multiferroics with higher magnetic ordering temperature, such as rhombohedral $R3$ Mn-rich (Ni,Mn)$_3$TeO$_6$ could be designed and synthesized.
acknowledgement
===============
We thank the Swedish Research Council (VR), the Göran Gustafsson Foundation, and the Russian Foundation for Basic Research for financial support.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Our recent discovery of magnetic fields in a small number of Herbig Ae/Be stars has required that we survey a much larger sample of stars. From our FORS1 and ESPaDOnS surveys, we have acquired about 125 observations of some $70$ stars in which no magnetic fields are detected. Using a Monte Carlo approach, we have performed statistical comparisons of the observed longitudinal fields and LSD Stokes $V$ profiles of these apparently non-magnetic stars with a variety of field models. This has allowed us to derive general upper limits on the presence of dipolar fields in the sample, and to place realistic upper limits on undetected dipole fields which may be present in individual stars. This paper briefly reports the results of the statistical modeling, as well as field upper limits for individual stars of particular interest.'
author:
- 'G.A. Wade'
- 'E. Alecian'
- 'C. Catala'
- 'S. Bagnulo'
- 'J.D. Landstreet'
- 'J. Flood'
- 'T. Böhm'
- 'J.-C. Bouret'
- 'J.-F. Donati'
- 'C.P. Folsom'
- 'J. Grunhut'
- 'J. Silvester'
date: 'March 8, 2003'
title: |
How non-magnetic are\
“non-magnetic” Herbig Ae/Be stars?
---
Introduction {#intr}
============
Observations of magnetic fields in pre-main sequence Herbig Ae/Be (HAeBe) stars can serve to address several important astrophysical problems: (1) The role of magnetic fields in mediating accretion, and the validity of models which propose that HAeBe stars are simply higher-mass analogues of the T Tau stars. (2) The origin of the magnetic fields of main sequence A and B type stars. (3) The development and evolution of chemical peculiarities and chemical abundance structures in the atmospheres of A and B type stars. (4) The loss of rotational angular momentum which leads to the slow rotation observed in some main sequence A and B type stars.
Since 2004, we have been engaged in a systematic assay of the magnetic properties of bright ($m_{\rm V}\ltsim 12$) HAeBe stars using the FORS1 spectropolarimeter at the ESO-VLT (Wade et al. 2007), and the ESPaDOnS spectropolarimeter at the CFHT (Wade et al., in preparation). We have acquired about 130 Stokes $V$ (circular polarisation) spectra of over 75 HAeBe stars, with the aim of measuring the longitudinal Zeeman effect in their spectra.
The ESPaDOnS observations are of high resolving power ($R\sim 65000$), and provide the capability to resolve the complex line profiles presented by many HAeBe stars. The longitudinal magnetic field was measured from each observation using the standard first-moment method applied to Least-Squares Deconvolved (LSD; Donati et al. 1997) profiles. The dependence of the field diagnosis on the LSD masks was explored in detail for each star, and “clean” photospheric masks were constructed by excluding lines in the spectrum that exhibited clear contamination by emission, or other significant departures from the predictions of an LTE synthetic spectrum. In most cases we found that global departures of the metallic line spectrum were relatively small, and the improvement achieved using tuned masks was minor. The magnetic field diagnosis obtained from the ESPaDOnS data is very sensitive to the projected rotational velocity $v\sin i$, and the formal uncertainties achieved consequently span a large range of precision, from very good (for most stars with $v\sin i\ltsim 80$ ) to essentially useless (for some stars with $v\sin i\gtsim 150$ ).
The FORS1 observations, on the other hand, are of low resolving power ($R\sim 1000-1500$). Although such spectra fail to resolve the complex profiles of most lines, they are relatively insensitive to rotational broadening. Consequently, the longitudinal magnetic fields derived from FORS1 spectra (using the linear regression method developed by Bagnulo et al. 2000) provide a relatively uniform diagnosis over a large range of $v\sin i$.
From these surveys magnetic fields have been detected in 6 stars[^1]. As the detailed properties of these magnetic HAeBe stars are discussed by Alecian et al. (these proceedings) and Folsom et al. (these proceedings), we will only review the general results here. The two stars modeled in great detail (HD 200775, HD 72106) show stable, oblique, dipolar magnetic fields with polar intensities of about 1 kG, low $v\sin i$, and rotation periods of several days. The two stars modeled in moderate detail (HD 190073, V380 Ori) show stable, organised magnetic fields, low $v\sin i$, and rotation periods of days, possibly years. Finally, the two stars with a small number of observations (HD 101412, HD 104237) show strong longitudinal fields and simple Stokes $V$ profiles, suggesting organised magnetic fields. Both of these stars have low $v\sin i$.
Based on these results, it is clear that some Herbig Ae/Be stars host strong, organised magnetic fields, qualitatively identical to those of the main sequence Ap/Bp stars. Here, we turn our eye to our much larger collection of $\sim 125$ observations of $\sim 70$ undetected HAeBe stars, in order to examine the extent to which our observations can constrain their magnetic properties. In other words, how non-magnetic are “non-magnetic” HAeBe stars?
Modeling and Results
====================
To explore the properties of the undetected sample of HAeBe stars, we have followed a Monte Carlo approach similar to that employed by Wade et al. (2007). We developed synthetic populations of magnetic stars where each star was characterised by the inclination of its rotation axis $i$, the obliquity of its dipolar magnetic field $\beta$, the rotation phase at which it was observed $\phi$, and the intensity of its dipolar magnetic field at the magnetic pole, $B_{\rm d}$. For the purposes of our simultations, the parameters $i$ and $\phi$ were selected randomly for each star ($i$ with a $\sin i$ PDF, $\phi$ with a uniform PDF), $\beta$ was randomly set equal to either $0\degr$ or $90\degr$ (with equal probability), while the magnetic intensity $B_{\rm d}$ was fixed for all stars in a given population (and therefore defined the characteristic field strength of that population). We then created synthetic distributions of longitudinal field measurements from each of these populations, assuming the same uncertainties characterising the real observations of undetected HAeBe stars. This procedure was repeated 100 times, using different realisations of the randomly-selected variables.
Fig. 1 compares the observed histogram of uncertainty-normalised longitudinal field measurements $z=\langle B_z\rangle/\sigma_B$ (including both FORS1 and ESPaDOnS data), with synthetic histograms compiled from the Monte Carlo simultations.
To quantitatively test whether the observed and computed distributions are representative of the same population (with the practical goal of testing if the observations imply fields which are weaker than those which characterise the models), we have performed a one-sided Kolmogorov-Smirnov (K-S) test (e.g. Conover 1971) on the cumulative distributions of longitudinal fields. The test statistic $D$ used in the K-S test is the maximum fractional difference betwen two cumulative distributions (i.e. the observed distribution and that compiled from a model). In this case, we find differences $D=0.058, 0.143, 0.228$ and 0.297 for models corresponding to dipole fields of polar intensity 0, 300, 450 and 600 G, respectively. For a sample size $N\sim 125$, a model distribution can be rejected at the 99% level if $D\geq 0.135$. Therefore the longitudinal field measurements allow us to rule out uniform populations of stars with fields above about 300 G. On the other hand, the observations are consistent with a uniform population of stars with fields of about 300 G or smaller[^2].
------------- -----
Model \#
$B_{\rm d}$ det
100 G 1
300 G 2
450 G 6
600 G 6
1000 G 6
2000 G 13
------------- -----
: Results of LSD profile modeling. See text for details.[]{data-label="t1"}
----------- ------ ----------------------- ------ ----- ------------
Target Spec $B_{\rm d}^{\rm max}$ P(%) \# $\sigma_B$
Type (G) obs (G)
HD 17081 B8 100 95 2 9
HD 139614 A6 300 98 3 14
HD 36112 A4 450 96 4 30
HD 142666 A6 450 98 6 35
HD 169142 A8 600 92 3 24
HD 31648 A3 2000 95 2 52
HD 144432 A9 2000 93 2 30
BF Ori A5 2000 86 1 32
----------- ------ ----------------------- ------ ----- ------------
: Results of LSD profile modeling. See text for details.[]{data-label="t1"}
We then set out to provide a clearer evaluation of the upper limits on dipole fields for individual stars. Unfortunately, such upper limits are nearly impossible to derive using small numbers of longitudinal field measurements because the longitudinal field for most stars becomes null at some point during the rotation cycle, even in the case of a strong surface field. We have therefore employed the individual LSD profiles obtained for those stars observed using ESPaDOnS. The velocity-resolved LSD profiles allow the detection of the magnetic field even when the longitudinal field is null, thanks to the spectral separation of polarised contributions from different parts of the stellar disc due to rotational Doppler effect. However, the interpretation of an LSD profile is more complicated than a longitudinal field measurement, requiring that we create synthetic LSD profiles corresponding to each observation (reproducing its associated LSD profile depth, $v\sin i$ and signal-to-noise ratio).
To model the LSD profiles, we first fit each LSD Stokes $I$ profile with a rotationally-broadened model, to determine $v\sin i$, line depth and radial velocity. We then used the model populations of magnetic stars to create synthetic Stokes V LSD profiles corresponding to each of our observations (using the profile synthesis procedure described by Alecian et al. 2007), and introduced synthetic Gaussian noise corresponding to the noise level in the real LSD $V$ profile. Finally, for each synthetic LSD profile we evaluated the probability that a Stokes $V$ signature was detected, using the same criteria that are applied in the real LSD procedure (see Donati et al. 1997). Again, this procedure was repeated 100 times for each observation and for each model, using different realisations of the randomly-selected variables. Examples of observed and synthetic LSD profiles are shown in Fig. 2.
The results of this procedure were twofold: first, a global comparison of the predictions of each of the population models with the observations, and secondly a quantitative evaluation of the compatibility of [*each*]{} LSD profile with the predictions of dipole surface field models of different intensities.
Table 1 summarises the results of this analysis. On the left, we show the number of detections of individual stars we would expect in the case of each population. Even for models as weak as 300 G, we obtain detections of small numbers of stars in over 90% of model realisations. This result is consistent with that derived from the longitudinal field measurements, and demonstrates that the LSD profiles strongly constrain models which propose the presence of weak, organised magnetic fields in all HAeBe stars. On the other hand, there may still exist a small number of magnetic stars, with magnetic properties similar to the detected magnetic HAeBe stars, present in our “non-magnetic” sample (but which remain undetected with the current observations). We note however that the dipole intensities derived by Alecian et al. and Folsom et al. (these proceedings) for the detected magnetic HAeBe stars, when evolved to the main sequence (Table 1 of Alecian et al.), are typical of those of the majority of Ap/Bp stars (Power et al., these proceedings). This indicates that the Herbig stars in which fields have been detected do not have unusually strong fields, and therefore that most of the magnetic stars in the sample are probably already detected.
On the right-hand side of Table 1, we show the inferred upper limits for dipole fields $B_{\rm d}^{\rm max}$ in individual sample stars obtained from fitting the LSD profiles. We also report the fraction of model realisations which generate a detection P(%) when $B_{\rm d}=B_{\rm d}^{\rm max}$, the number of observations, and the derived longitudinal field error bar $\sigma_B$. Of particular interest are the lack of detections for stars in which marginal magnetic field detections had previously been claimed (HD 139614, HD 144432, HD 31648, HD 36112 and BF Ori; Hubrig et al. 2004, 2006a; Wade et al. 2007) based on observations obtained with FORS1.
EA is funded by the Marie-Curie FP6 programme, while JDL and GAW acknowledge support from NSERC and DND-ARP (Canada).
Alecian E., Catala C., Wade G.A., Donati J.-F., et al., 2007, MNRAS, in press Bagnulo S., Landstreet J. D., Lo Curto G., Szeifert T., Wade G. A., 2003, A&A 403, 635 Conover W.J., 1971, [*Practical nonparametric statistics*]{}, 1st ed., p. 400, Wiley (New York) Donati, J.-F., Semel, M., Carter, B. D., Rees, D. E., Cameron, A. C., 1997, MNRAS 291, 658 Hubrig, S., Sch$\ddot{\rm{o}}$ller, M., Yudin, R. V., 2004, A&A 428, L1 Hubrig, S.; Yudin, R. V.; Schšller, M.; Pogodin, M. A., 2006, A&A 446, 1089 Wade G.A., Bagnulo S., Drouin D., Landstreet J.D. and Monin D., 2007, MNRAS 376, 1145
[^1]: One magnetic star (HD 101412) discovered using FORS1, 4 magnetic stars (V380 Ori, HD 72106, HD 190073, HD 200775) discovered using ESPaDOnS, and 1 magnetic star (HD 104237) discovered previously by Donati et al. (1997). We do not discuss here results from the survey of HAeBe stars in young open clusters, briefly introduced by Alecian et al. (these proceedings).
[^2]: We underscore that these field intensities refer to the dipole field polar strength at the stellar surface, and not to the mean longitudinal field.
| {
"pile_set_name": "ArXiv"
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author:
- Richard S Ellis
title: Conference Summary
---
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Introduction
============
Clusters of galaxies are of wide interest in the astronomical community both as laboratories where baryons and dark matter can be conveniently studied and, through their statistical properties at various look-backtimes, as tracers of cosmic structure and evolution. The Local Organising Committee wisely narrowed the topics of this meeting to four broad themes and so I will organise my summary remarks under these headings.
First I think I should address the plea “Why bother to study clusters?”, raised provocatively by Colin Norman at one stage during the proceedings[^1]. We have learnt enough about clusters and the growth of structure to know that two of the traditional motivations require much more careful consideration.
For years on time allocation committees, I read observers repeat a [*mantra*]{} in the first sentence of their proposals: [*“Clusters of galaxies represent the largest bound structures in the Universe.."*]{} in justification of their role as tracers of large scale structure. A number of speakers at this meeting reminded us that clusters no longer uniquely occupy this role. We have other tracers of large scale structure and, moreover, it seems we need to be much clearer of what exactly we mean by a cluster before we can convincingly use them as cosmological probes.
A second, traditional, motive for studying clusters has been observational convenience, e.g. in studying constituent populations such as galaxies at various look-back times. At first sight, it’s an attractive proposition for an observer to study a few rich clusters at various redshifts each containing hundreds of accessibly-luminous galaxies and then to “join the results" in a timeline to make some evolutionary claim. But, at most redshifts of interest, we can expect to find a wide range of overdensities represented (groups, merging systems, virialised clusters). Put simply, rich cluster A at $z$=1.2 is unlikely to evolve into rich cluster B at $z$=0.5 and neither may necessarily become a present-day Coma.
Both these worries indicate a high degree of rigour is needed in using clusters. We need very large samples spanning ranges of mass at any redshift of interest and perhaps it would be foolish to adopt one single selection criterion for their study. A similar “panchromatic" theme has emerged over the past decade in understanding how to address the question of evolution using samples of galaxies.[^2]
The motivation for studying clusters that emerged at this meeting focused broadly along the following themes:
- Testing gravitational instability by measuring the number density of massive clusters at high redshift, viz. $\Phi(M_{cluster},z)$.
- Breaking degeneracies in estimates of the the cosmological parameters by examining the local population of clusters as a probe of the mean mass density and the normalisation of the mass power spectrum, $\Omega_M$ and $\sigma_8$.
- Verifying hierarchical structure and the nature of the dark matter and keeping those theorists in check who predict a universal mass profile with central cusps, $\rho_M(r) \propto r^{-1.5}$.
- Determining the origin of the heating of the intracluster medium, its enrichment history and examining whether non-gravitational processes are involved.
- Examining the history of spheroidal galaxies and role of the environment, for example in understanding the origin of the morphology-density relation and the destiny of the infalling component of gas-rich field galaxies.
Let me make a disclaimer in what follows. I am not an expert in any of the areas I was asked to summarise and so I submit these concluding remarks only as someone who tried to listen carefully to most talks, dutifully avoiding the lunchtime mountain hikes to make sense of what I heard[^3]. With $\simeq$70 talks and $\simeq$80 posters, inevitably I have had to be very selective in discussing results.
If one could summarise the meeting in two paragraphs, I would say the following:
- There is an explosion of terrific data, from large surveys (X-ray and ground-based) which offer qualitatively new ways in which we find and do statistical studies of clusters, and also in [*resolved*]{} data within clusters which opens up new opportunities for understanding the detailed astrophysics of dense environments.
- The subject is moving from exploratory surveying to detailed astrophysics. As part of this “growing up" there is a need to admit defeat on some of the old methods and to embrace new ones, particularly on the statistical questions. For many years cluster workers had something of a monopoly in the study of early galaxies and large scale structure, but it is time to take advantage of other datasets being delivered and view cluster astrophysics as only one part of a larger body of information.
Searching for Distant Clusters
==============================
During the first part of our meeting we listened to progress made in finding clusters, mostly (but not exclusively) at high redshift. We heard about a bewildering number of deep surveys (MACS, REFLEX, WARPS, LCDCS, EIS, RDCS to name but a few..Figure 1) each requiring significant follow-up with the world’s most powerful facilities. Excellent reviews of this active field have been given by Borgani (2001) and Borgani & Guzzo (2001).
The motivation is a sound one: testing gravitational instability in the context of a known cosmological model, or alternatively, by assuming a structure formation model such as CDM, constraining the cosmological parameters. The fraction of collapsed large massive structures ($M\simeq10^{14}M_{\odot}$) at, say, $z\simeq$1 for a given cosmology is one of the most robust predictions of the dark matter models (Figure 2). The necessary ingredients are a large well-defined sample and robust estimates of the cluster masses.
An additional motivation for locating high $z$ clusters, is to undertake evolutionary studies. For galaxies, this is less appealing now we can find statistically-complete field samples at similar redshifts by independent photometric means (Adelberger 2000, Daddy et al 2001, McCarthy et al 2001), many of which are extensive enough for locating clustered objects in a self-consistent way. Marc Postman stressed the dangers here by contrasting differences found between conclusions drawn from galaxies studied in the X-ray selected CNOC and optically-selected Morphs samples. If clusters are found by a manner which relates to the properties of their constituent galaxies (e.g. by only searching for ones with prominent red sequences as discussed by Mike Gladders, or finding ones associated with powerful radio galaxies as reviewed by Philip Best), one can get different views of what is going on in dense regions.
The quest for high $z$ clusters has a chequered history. Most of the early work was motivated by a quest for the deceleration parameter based on brightest cluster galaxies (Gunn & Oke 1975, Kristian et al 1976). Denis Zaritsky gave us a new twist to this story with his evidence for varying accretion onto brightest cluster galaxies in the Las Campanas Distant Cluster Survey (see also Aragón-Salamanca et al 1998). Claims for number evolution in cluster counts with redshift have also swung to and fro over the years. Until recently only a handful of optical or X-ray selected clusters were available[^4]. The most exciting aspect about this meeting is the enormous advance in the numbers of distant clusters.
So how best to find distant clusters? Marc Postman nicely summarised the various search techniques, their advantages and biases. The most useful techniques are those that minimise projection effects. Faint X-ray surveying in the 0.1-10 keV range is cerainly an expensive method in telescope time (but good things don’t come cheap): the background is low, we expect $L_x \propto
n_e^2$ and thus, so long as massive clusters have hot intracluster media, this is the method of choice. Kathy Romer emphasized how, even with the higher EPIC background, XMM promises to extend the depth of the ROSAT surveys to mean redshifts well beyond $z\simeq$1.
Whilst we have yet to reap the benefits of SZ effect as a survey tool, James Bartlett showed us that this method has the great advantage of being distance independent (modulo evolution) with a signal $\propto n_e$. Instruments such as Planck Surveyor, and the proposed Arcminute Microkelvin Imager which Rüdiger Kneissl described, should also readily probe beyond $z\simeq$1.
In a similar category to the SZ effect is weak lensing. As this technique wasn’t explicitly reviewed at the meeting, I reproduce the nice discovery of a $z$=0.28 cluster by Wittman et al (2001) in Figure 3. Although lensing offers the most direct route to the total mass, it still suffers from projection effects, the effect of the mass sheet degeneracy[^5] and, as a high density of background galaxies is needed, the technique will most likely be limited to the study of clusters with 0.1$<z<$0.8.
Concerning the optical searches, Marc showed us, from the analysis of Goto et al (in preparation) how, even locally within the SDSS, changing the search algorithms can deliver very different samples. This is worrying to me as I imagine this kind of uncertainty can only be worse in faint surveys. While smart ideas such as “matched filters" designed to minimise projection and maximise contrast, are of course to be welcomed there is a natural concern that, like Maximum Entropy restoration techniques, such methods work reliably only when we have a clear idea of what we are searching for. Do we?
The Red Cluster Sequence method described by Mike Gladders seems the most promising of the optical methods. Indeed the authors claim it to be “comparable or superior to X-ray methods" (Gladders & Yee 2000) - a bold assertion! Certainly panoramic detectors sensitive in $R$ and $z'$ make this a much more efficient way to locate red colour-magnitude sequences to $z\simeq$1 and, at CFHT, there is the prospect of coupling lower $z$ detections with weak lensing constraints to directly get masses. Simulations suggest this search technique should be complete to $z\simeq$1, even allowing for some dispersion and bluing in the color-magnitude relation. Unless there is a (perverse) population of massive clusters devoid of a uniform population of spheroidals, the technique seems a sound one.
But the next thorny issue is how to get a [*reliable mass*]{}, the second important requirement to test Figure 2 and one worth thinking about before embarking on any time-consuming survey! The mass of a distant cluster has traditionally been inferred from the X-ray luminosity (assuming a $L_x-T_x$ relation), an optical richness (very crude) or a velocity dispersion based on a few members. Clearly none of these is really adequate. This is, I think, where the X-ray surveys win hands down even if “smart” algorithms rescue the optical searches. We can fully expect, with adequate investment, X-ray temperatures for large numbers of distant clusters in the coming years. Brian Holden showed new Chandra data that indicates little evolution in the $L_x-T_x$ relation to $z\simeq$0.8. Whilst assumptions are necessary to link $T_x$ and virial mass (Voit 2000), the prospects look good, particularly with cross-checks from the SZ effect. Lauro Grego showed us the promise of deriving masses independently from her interferometric SZ techniques.
The major problem in my opinion is that the optical searchers have no real route to cluster mass (except by recourse to independent X-ray techniques or perhaps weak lensing over a limited $z$ range); projection also remains an issue. Multiple systems, often separated by only modest velocity differences, would be very hard to detect without a huge spectroscopic investment. A good example in showing the pitfalls is the cluster surrounding the $z$=1.206 galaxy 3C324 (Smail & Dickinson 1995, Figure 4) now known from spectroscopic evidence to consist of two components separated by only $\delta z\simeq$0.06. Although radio-selected, not only would its richness have been overestimated without careful spectroscopy, it seems unlikely the Red Sequence method [*applied to a similar system at any redshift*]{} would be able to photometrically-separate the two systems.
In the case of the local X-ray cluster data essential as a basis for all high $z$ comparisons, Luigi Guzzo and Hans Böhringer demonstrated the remarkably precise cluster luminosity function obtained from the REFLEX survey. Few would disagree that this is a major achievement in the subject not just in statistical terms but also because of the considerable care taken to ensure homogeneity in the survey.
Turning then to the cosmological results and necessarily being somewhat selective, Harald Ebeling (MACS) and Piero Rosati (RDCS) discussed a fairly modest decline in the abundance of massive clusters to $z\simeq$1 (Figure 5); this is a fairly convincing conclusion for $z<$0.5 but beyond I suspect there is still some room for manoever. Qualitatively, this is similar to the 4.7$\sigma$ decline claimed by Gioia (2001) in her NEP sample (not discussed at this meeting). Stefano Borgani has modelled the decline with redshift in the RDCS-3 sample with available $L_x-T_x$ data and claims the data is consistent with popular choices of $\Omega_M$ and $\sigma_8$ (see $\S$3).
Can we say there is “concordance" between the observers? Not yet but good progress is being made. Most of any agreement refers to evolution in luminous, presumed massive, clusters. A number of niggly issues remain including the effects of incompleteness, cosmic variance for the smaller field surveys, AGN/cooling flow contamination, and how to compare surveys with minimal $L_x,z$ overlap. For example, at the lower luminosities probed by the WARPS survey, the situation is quite unclear with some arguing for no evolution at all. Much of the dispersion in the inferred evolution arises from uncertainties in the number of [*local*]{} massive clusters against which comparisons are needed. WARPS and RDCS appear to disagree on the “local" abundance to an extent that seriously affects their respective analyses. However, the prospects for extending these tests to higher redshift and clarifying the temperatures with Chandra appear very promising, even before the new XMM surveys are underway.
As an outsider to the subject of distant X-ray cluster surveys, I was struck in both Harald and Marc Postman’s reviews of the large number of competing surveys underway. If the labour involved in constructing these surveys was not enough, remember that the time required to fully exploit them is even larger. For example, the construction of Ebeling’s ROSAT-based MACS sample of 840 clusters involved two colour CCD imaging and spectroscopic verification of a significant sample. To exploit all 109 clusters beyond $z>$0.3 with multi-slit spectrographs, a weak lensing deep imaging study, Sunyaev-Zel‘dovich (SZ) detections and other applications would require almost dedicated access to a huge range of facilities (a tall order even by Harald’s standards!). When one realizes this is just [*one*]{} of the many ambitious surveys we heard about with more in the pipeline, I worry has the planning of all these surveys seriously taken into account the necessary investment for a full exploitation?
Mapping Large Scale Structure with Clusters
===========================================
Bob Nichol gave us a very balanced review of the changing role of clusters as new probes of large scale structure became available. He also posed some controversial questions, e.g: “in the era of galaxy and lensing surveys, who needs clusters?" He also stressed many of the complications that arise from the frequent merging of clusters on our assumption of relaxed systems in virial equilibrium.
I think it fair to say the traditional role of using clusters selected in various way (optically from the Abell/ACO catalogues or from X-rays) as the most efficient way to get to the mass power spectrum $P(k)$ on large scales is being overtaken by the large redshift surveys. In the era of 2dF and SDSS, huge volumes populated by over 100,000 galaxies are publically available[^6] and delineate the power spectrum more reliably than the earlier Abell/ACO cluster samples whose integrity is still being debated. In the case of the X-ray cluster samples, Luigi Guzzo’s analysis of the REFLEX survey is comparable in importance to 2dF and SDSS at the time of writing. A particularly impressive achievement in that sample is the detection of infall in the $\xi_v(\sigma,\pi)$ plane convincingly demonstrating the quality of the sample. However, the redshift surveys are continuing apace and from them one can create far more reliable spectroscopically-based cluster samples. Of particular interest is the volume to $z\simeq$0.55 probed by SDSS Luminous Red Galaxies survey. The message is clear: wholesale galaxy mapping has arrived..use the data!
Bob drew attention to possible “baryon wiggles" in both the power spectrum of Abell/ACO clusters seen by Miller et al (2001) and referred to one tentatively in the 2dF data (Percival et al 2001). It must be remembered that, in such plots, the redshift-space power spectrum is significantly affected by aliasing introduced by the window function of the survey volume. Without careful simulation of this effect it is very hard to be sure whether bumps are artefacts or genuine baryon oscillations. Neither the REFLEX not the 2dF surveys claim to have detected these features. Indeed, the 2dF team estimates unless the baryon density is much higher than expected, they will remain undetected in the completed SDSS survey.
Elena Pierpaoli described a second aspect of local cluster statistics, namely constraining the [*amplitude*]{} of the mass power spectrum. The abundance of clusters of known mass provides a joint constraint on the variance, $\sigma_8$ on 8$h^{-1}$ Mpc scales and the mean mass density, $\Omega_M$ in a form dependent on $\sigma_8\,\Omega_M^{0.5}$. By adding X-ray temperature measurements and improving the mass-$T_x$ connection, an improved constraints has been determined. In the past year, the same constraint has also been probed in a completely independent way via weak lensing studies of randomly-chosen fields (so called “cosmic shear” surveys) (eg. van Waerbeke et al 2001, Bacon et al 2000, 2001 in prep.). Reassuringly both methods get fairly similar answers with $\sigma_8\,\Omega_M^{0.6}$=0.40-0.50 although it is important to remember that cosmic variance and the redshift distribution of the background faint galaxies may affect uncertainties in the lensing estimates[^7]
The weak lensing results are independent of any assumed cluster population and do not rely on Gaussian fluctuations in the mass spectrum. Accordingly, comparisons of lensing and cluster-based methods are an important way to verify the assumptions we may take for granted, not only in the cosmological framework but also in cluster physics.
Physical Processes Within Clusters
==================================
We had an interesting session on the internal mass distributions within cluster where Andrea Biviano gave a very comprehensive review of years of effort invested in estimating orbital anisotropies and mass distributions in clusters. The number of clusters which have been comprehensively surveyed spectroscopically is increasing rapidly and the new field surveys (SDSS, 2dF) will generate lots more. Cluster galaxy dynamics will remain a critical tool in determining the orbital anistropy of cluster galaxies of different types. As Andrea showed, there is convincing evidence that spheroidal galaxies retain an isotropic velocity field, consistent with their long-standing membership, whereas late type systems show a detectable radial anisotropy reflecting their continuous infall.
The peripheries of clusters at moderate redshifts are largely unstudied regions important in linking cluster-based environmental evolution to that in field galaxies. Tommaso Treu and Taddy Kodama described complementary dynamical/HST and photometric imaging programs to address this question. Kodama finds a potentially important result whereby the colours of field galaxies undergo a sharp transition from blue-to-red at some galaxian surface density, presumably reflecting some process associated with their arrival at the cluster. Treu is embarking on a longer term project to understand the mechanistic details of infall in a single well-studied cluster. Both routes are important ones to pursue.
Andrea was cautious about deriving mass profiles from galaxy velocities. This has been a long standing problem in individual clusters, even Coma, because of the inevitable degeneracies arising from unknown orbital anisotropies as a function of radius. If a functional form is adopted for the mass profile (e.g. NFW), solutions can be found but the present motivation is surely to determine these forms directly, as is possible in a more competitive way with X-ray data in the inner regions and weak lensing to the periphery. Steve Allen showed how much progress has been made in reconciling strong lensing and X-ray based mass estimates following improved Chandra data on 7 clusters and found no significant deviations from the universal mass profile seen in numerical simulations.
In an effort to overcome some limitations, it has become common practice to co-add data from many clusters in the hope of improving signal to noise and erase asymmetries. Peter Katgert and Roland van der Marel showed us results from coadded samples based on the ENACS and CNOC surveys respectively, and concluded mass traces light to a reasonable approximation. However, is a composite cluster really a physical entity? A number of critics were unconvinced because of the dangerous effect one or two “complex" systems might have on the final conclusion. Even with a substantial number of redshifts, it seems one has to be cautious in interpreting dynamical data in terms of a simple gravitational potential.
A salutory lesson can be learnt from the comprehensive survey of the regular cluster Cl0024+16 (see Treu et al, Metevier et al, these proceedings) by Czoske et al (2001). For many years this was regarded as a classic regular virialised system at $z$=0.4 and its mass profile even formed the basis for promoting self-interacting dark matter as discussed by Oleg Gnedin. However, when 650 redshifts were gathered (of which 295 are members), Czoske et al resolved a foreground system (whose mass was initially thought to be quite modest but now is claimed to be significant) undergoing an end-on merger with the main body of the cluster (Figure 6). This discovery may explain its low X-ray luminosity c.f. the original high velocity dispersion. With less than 100 member velocities, Cl0024+16 would surely have been considered a representative cluster for coaddition but now we know it is a more complex beast. Bootstrap-style experiments might be helpful to clarify the robustness of the coaddition procedure.
Soon after the first X-ray detections of hot gas from the intracluster medium (ICM) were detected, it was realised that most of the baryons in clusters are in gaseous form. Assuming an isothermal distribution, the surface brightness profiles offer a valuable probe of cluster mass disribution, and the X-ray morphology indicates the evolutionary state. Clusters are expected to continuously assemble in popular hierarchical models and we can expect to directly witness this growth via shock heating events.
XMM and Chandra are transforming this field and this conference has provided one of the first opportunities for us to discuss the assumptions made in analysing earlier data. Monique Arnaud illustrated some of this progress in her excellent review. Although merging substructures produce shock-heating Chandra reveals unforeseen complexities, for example in the “cold fronts" seen in some clusters which suggest the associated timescales may have been underestimated. Monique demonstrated several lines of evidence for the continuing assembly of clusters (X-ray substructure, anisotropic accretion etc). However, even in the well-studied case of the outskirts of the Coma cluster, Uli Briel demonstrated significant uncertainties in understanding the merger timescales involved. Such data are thus [*illustrative*]{} of hierarchical assembly but may not give us the needed [*quantitative*]{} verification of mass assembly.
The most striking aspect of the new high quality data is the growing evidence for non-gravitational processes in understanding the physics of the ICM. Whilst temperature profiles obtained with ASCA and BeppoSAX were often discrepant (as discussed by De Grandi), the XMM data gives reasonable support to the assumption of isothermal cores. However, the expected self-similar “scaling laws" between the gas mass, virial radius and X-ray temperature, e.g. in $L_x \propto T^2$ are not obeyed. The steep $L_x-T$ relation implies non-gravitational processes, almost certainly associated with additional heating mechanisms.
Theoretical explanations for the steep $L_x-T$ relation were the subject of almost an entire afternoon session. There was a bewildering number of ideas. Fabio Governato reviewed for us how feedback heating can be incorporated in numerical simulations and showed that a steep relation cannot be easily explained via conventional heating sources such as supernovae. Paolo Tozzi demonstrated that, with an energy of 1-2 kev/baryon, AGN are a promising possibility but it seems there is no obvious evidence for heating from these sources. Peter Thomas from the VIRGO consortium can predict the slope but only at the expense of a strong evolution in the relationship. Uros Seljak gave us an analytical model based on a universal gas profile where departure from self-similarity arises from a temperature-dependent hot gas fraction.
Perhaps that glass of wine at lunchtime was not a good idea but this was a confusing session for me. Unlike observers who compare ASCA/BeppoSAX/XMM temperature profiles, worrying about sensitivity, psf and background differences, when it comes to numerical simulations, I am puzzled that there appears to be no necessity to inter-compare results. Surely this is a first step in convincing anyone of the believability of a particular simulation?
Additional physical complexities in the inner cool cores of clusters were reviewed by Andy Fabian. In the physical environment of a dense cluster core, hot gas is able to cool radiatively in a timescale of less than $10^{9-10}$ years and a stable flow of cooling gas to the cluster centre is expected. Almost 50% of all X-ray clusters show some signs of the associated temperature gradient implying mass inflow and deposition rates of 10-100 $M_{\odot}$ yr$^{-1}$. However, XMM spectra show no sign of the expected Fe XX lines of radiative cooling in the 13-18 Å range. Hans Böhringer also discussed additional temperature diagnostics unaffected by absorption effects. Either the gas is not cooling radiatively or there is perhaps is a balancing effects from a hitherto-unidentified heating mechanism. And there are serious duty cycle issues concerned with preventing cooling beyond 2-3 kev.
As usual, Andy had no shortage of ideas for resolving the puzzle, the most intriguing from my perspective being the hypothesis of a starkly bimodal metallicity distribution which has the desired effect of suppressing the amount of cooling witnessed through the diagnostic iron lines. Like most conference participants I suspect, I have little idea where this additional complexity in the ICM physics is taking us. The Chandra images of the Perseus cluster discussed by Schmidt (with such enthusiasm!) show holes in the X-ray emission where the gas appears hotter and more metal rich. Dynamical complexities were introduced by Ettori in the wealth of data he presented for Abell 1795. A number of speakers introduced magnetic fields associated with radio polarisation (in the case of Abell 2255 discussed by Govoni), radio sources (although as Fabian remarked the gas is not obviously hotter in these regions) or in explaining the survival of cold clouds with associated fronts (Arnaud).
I think we can be optimistic that there is a lot to learn which will assist in the entire interplay between gas cooling, star formation and enrichment. Many years ago at a conference I attended in Cambridge, Andy claimed confidently that cooling flows provided a key mechanism for the formation of giant elliptical galaxies. Although cooling flows remain controversial, Andy can still assert (as Richard Bower emphasised independently) that the physics of gas cooling in cluster cores will tell us much about feedback and star formation: still key ingredients in galaxy formation.
Evolution of Galaxy Populations within Dense Environments
=========================================================
On the last day, Pieter van Dokkum surveyed the literature on the role that clusters continue to play in our understanding of galaxy formation and evolution. There is a subtle shift of emphasis however. Rather than being “laboratories of convenience" [^8], galaxies in clusters are now being studied alongside equivalent field populations and as likely successors to well-studied Lyman break galaxies at $z>$2.
Most of the discussion, by van Dokkum, Rosati & Kelson, concentrated on the role of very distant ($z>$0.5) clusters, largely in terms of differentiating stellar ages and the ages of mass assembly of giant ellipticals. Our large telescopes are being flexed to their limits to secure impressive fundamental plane data (to $z=$1.2 in 12 hour Keck exposures!) which continues to support the notion that old stellar populations in at least some fraction of the data. That the stars in ellipticals may be older than the assemblies in which they now reside is illustrated in the red mergers seen in the well-studied X-ray luminous cluster MS1054 ($z$=0.83). Pieter posed the question of how common such a system might be and gave some new examples of red mergers in other clusters. It is good to see some progress in separating the ages of stars from those of the accumulated mass that makes up giant ellipticals.
The biases introduced not only in selecting galaxies within clusters (by HST morphology, by colour or infrared magnitude) but also by how the clusters themselves were selected (X-ray, optical..) continue to worry me. D. Fadda also reminded us how gas rich and dusty systems may be common even in dense clusters. As remarked earlier, it is hard to know what errors are made by connecting data at different redshifts to delineate an evolutionary picture. One suspects we are reliant either on theoretical modelling (heaven forbid!) or comparative field samples. Beyond $z>$0.5 this will be some time coming although a number of speakers alluded to the upcoming Keck and VLT spectroscopic surveys.
It is now 20 years since Alan Dressler published his quantitative study of the morphology-density relation in a nearby sample of 55 rich clusters (Dressler 1980). Pieter and Bianca Poggianti urged us to go back and improve the local samples now we have panoramic CCD cameras and multi-object spectrographs so we can be sure of the local fractions (Figure 7). A key issue here is the origin of S0s; the Morphs team (Dressler et al 1997) proposed their recent demise from gas stripping and tidal effects and presented a strong claim for an evolving E/S0 fraction with lookback time. Pieter urged us to be much more cautious in differentiating Es and S0s; with resolved spectroscopy of $z>$0.5 galaxies feasible on many telescopes, this appears to be a profitable route in conjunction with HST data.
Parting Thoughts
================
This has been a fascinating meeting and we have a lot to be thankful for. Foremost we have amazing observational facilities capable of finding clusters and studying them to great depths in complementary ways. We now can resolve clusters at the arcsec level in X-rays and locate their concentrated masses with S-Z and weak lensing techniques. We can also locate local clusters as byproducts in the comprehensive redshift surveys being undertaken. Instead of viewing these as competing techniques we should exploit them all as complementary probes to check our physical assumptions.
We also have the opportunity to learn new physics in the cluster cores from the paradoxes emerging from the steeper $L_x-T$ relations and the absence of lines of radiative cooling in those with temperature gradients. It should perhaps be no surprise that simple gravitational physics is not enough to explain what we see. The fall-out in our understanding of galaxy and cluster formation from resolving this paradox could be a big one.
As the concluding speaker I want to thank Drs Mardirossian and Mezzetti for all the necessary arrangements here at beautiful Sesto Pusteria and to the Organising Committee for their hard work. We should in particular thank Stefano Borgani for his obvious insight in constructing a thoughtful scientific program and amazing energy he and his colleagues invested in making this a great meeting!
Adelberger, K. 2000 in [*Clustering at High Redshift*]{}, eds. Mazure, A. et al, ASP Conference Series, 200, p13. Aragón-Salamanca, A. et al 1998 MNRAS, 297, 427. Bacon, D.J. et al 2000 MNRAS 318, 625. Beijersbergen, M. et al 2001 MNRAS in press (astro-ph/0106354). Borgani, S. 2001 in [*Galaxy Clusters and the High Redshift Universe Observed in X-rays*]{}, eds. Neumann, E. et al in press (astro-ph/0106471). Borgani, S. & Guzzo, L. 2001 Nature 409, 39. Czoske O., et al. 2001, A&A, in press(astro-ph/0111118) Daddy, E. et al 2001 A&A 376, 825. Dressler A. 1980, , 236, 351. Dressler A. et al. 1997, , 490, 577. Fukugita, M. 2000, in Proc. 30th ICHEP, eds. Lim, C.S. & Yamanata, T., World Sci. p299. Gioia, I. 2001 in [*Chemical Enrichment of the Intracluster Medium*]{} in press (astro-ph/0107452). Gladders, M. & Yee, H.C. 2001 in [*Cosmic Evolution & Galaxy Formation*]{}, ed. Franco, J. et al, ASP Conf. Series 215, p233 Gunn, J.E. & Oke, B. 1975 , 195, 255. Kodama T., et al. 2001, , 562, L9 Kristian, J. et al 1976 , 205, 688 McCarthy, P. et al 2001 , 560, L31. Miller et al 2001 , 555, 68. Percival, W. et al 2001 MNRAS 327, 1279. Smail, I. & Dickinson, M.E. 1995 , 455, L99. van Waerbeke, L. et al 2001 A&A 374, 757. Voit, G.M. 2000 , 543, 112. Wittman, D. 2001 , 557, L89.
[^1]: Fortunately, I have forgotten to which speaker he made this remark!
[^2]: Witness the controversy surrounding the use of UV/optical and far-infrared probes of the cosmic star formation history. No single technique wins: several are required.
[^3]: I apologise if I missed some talks but I tried to secure powerpoint files or transparencies for most of those.
[^4]: On more than one occasion, challenges to a standard paradigm have been made on the basis of the existence of a single high $z$, assumed massive, cluster.
[^5]: Mass estimates based on weak shear of a background population are insensitive to an additional unclustered component of dark matter.
[^6]: See http://www.mso.anu.edu.au/2dFGRS/ & http://archive.stsci.edu/sdss/edr\_main.html.
[^7]: Since the conference, estimates about 20% lower for the combination of $\sigma_8$ and $\Omega_M$ have been published, both from cluster studies and large scale structure results. Lahav et al (astro-ph/0112162 gives a good summary)
[^8]: Perhaps there is a better terminology here!
| {
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[**Pulsatile lipid vesicles under osmotic stress**]{}\
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[ABSTRACT ]{} The response of lipid bilayers to osmotic stress is an important part of cellular function. Recent experimental studies showed that when cell-sized giant unilamellar vesicles (GUVs) are exposed to hypotonic media, they respond to the osmotic assault by undergoing a cyclical sequence of swelling and bursting events, coupled to the membrane’s compositional degrees of freedom. Here, we establish a fundamental and quantitative understanding of the essential pulsatile behavior of GUVs under hypotonic conditions by advancing a comprehensive theoretical model of vesicle dynamics. The model quantitatively captures the experimentally measured swell-burst parameters for single-component GUVs, and reveals that thermal fluctuations enable rate-dependent pore nucleation, driving the dynamics of the swell-burst cycles. We further extract constitutional scaling relationships between the pulsatile dynamics and GUV properties over multiple time scales. Our findings provide a fundamental framework that has the potential to guide future investigations on the non-equilibrium dynamics of vesicles under osmotic stress.
INTRODUCTION {#introduction .unnumbered}
============
In their constant struggle with the environment, living cells of contemporary organisms employ a variety of highly sophisticated molecular mechanisms to deal with sudden changes in their surroundings. One often encountered environmental assault on cells is osmotic stress, where the amount of dissolved molecules in the extracellular environment drops suddenly [@christensen1987; @hoffmann2009]. If left unchecked, this perturbation will result in a rapid flow of water into the cell through osmosis, causing it to swell, rupture, and die. To avoid this catastrophic outcome, even bacteria have evolved complex molecular machineries, such as mechanosensitive channel proteins, which allow them to release excess water from their interior [@berrier1996; @blount1997; @levina1999; @wood1999]. This then raises an intriguing question of how might primitive cells, or cell-like artificial constructs, that lack the sophisticated protein machinery for osmosensing and osmoregulation, respond to such environmental insults and preserve their structural integrity.
Using rudimentary cell-sized giant unilamellar vesicles (GUVs) devoid of proteins and consisting of amphiphilic lipids and cholesterol as models for simple protocells, we showed previously that vesicular compartments respond to osmotic assault created by the exposure to hypotonic media by undergoing a cyclical sequence of swelling and poration [@oglecka2014]. In each cycle, osmotic influx of water through the semi-permeable boundary swells the vesicles and renders the bounding membrane tense, which in turn, opens a microscopic transient pore, releasing some of the internal solutes before resealing. This swell-burst process, [depicted in Fig. \[fig1\](A), ]{}repeats multiple times producing a pulsating pattern in the size of the vesicle undergoing osmotic relaxation. From a dynamical point of view, this autonomous osmotic response results from an initial, far-from-equilibrium, thermodynamically unstable state generated by the sudden application of osmotic stress. The subsequent evolution of the system, characterized by the swell-burst sequences described above, occurs in the presence of a global constraint, namely constant membrane area, during a dissipation-dominated process [@peterlin2008; @ho2016].
The study of osmotic response of lipid vesicles has a rich history in theoretical biophysics, beginning with the pioneering work by Koslov & Markin [@koslov1984], who provided some of the early theoretical foundations of osmotic swelling of lipid vesicles. In this work, they predicted that the response of a sub-micrometer sized vesicles to osmotic stress is likely pulsatile and due to the formation of successive transient pores (see Fig. 9, in [@koslov1984], for a schematic for the volume change of the vesicle over time). They further approximated the characteristic quantities of swell-burst cycles (e.g. swelling time, critical volumes), based on the probability of the membrane overcoming the nucleation energy barrier to form a pore. Independently, the dynamics of a single transient pore in a tense membrane were first theorized by Litster [@litster1975], and later investigated theoretically and experimentally by Brochard-Wyart and coworkers [@sandre1999; @brochard-wyart2000]. Idiart & Levin [@idiart2004] combined the osmotic swelling theory and pore dynamics, and calculated the dynamics of a pulsatile behavior assuming a constant lytic tension. These modeling efforts made great strides in our understanding of some of the essential physics underlying vesicle responses to osmotic stress.
Previously, we used these ideas to provide a qualitative interpretation of pulsatile behavior of GUVs (see schematics in [@oglecka2014] Fig. 7h,i). However a general framework that quantitatively describes the response of pulsatile vesicles to osmotic stress at all relevant time scales is still missing. The success of such a model must rely on (a) the integration of vesicle dynamics, pore dynamics with nucleation, and long-time solute concentration dynamics within a unified framework, and (b) the assessment of the model predictions with respect to experimental measurements, in order to establish the physical relevance of the essential parameters that govern the system dynamics. Here, we build on the findings and theories reported previously [@litster1975; @koslov1984; @brochard-wyart2000; @idiart2004; @evans2003; @ryham2011] to develop such a quantitative model for the dynamics of swell-burst cycles in giant lipid vesicles subject to osmotic stress.
In analyzing the pulsatile dynamics of GUVs, a number of general questions naturally arise: (i) Is the observed condition for membrane poration deterministic or stochastic? (ii) Is poration controlled by a unique value of membrane tension (*i.e.* lytic tension) introduced by the area-volume changes, which occur during osmotic influx, or does it involve coupling of the membrane response to thermal fluctuations? (iii) Does the critical lytic tension depend on the strain rate, and thus the strength of the osmotic gradient? Such questions arise beyond the present context of vesicle osmoregulation in other important scenarios where the coupling between the dissipation of osmotic energy and cellular compartmentalization has important biological ramifications [@rand2004; @diz-munoz2013; @stroka2014; @porta2015].
Motivated by these considerations, we carried out a combined theoretical-experimental study integrating membrane elasticity, continuum transport, and statistical thermodynamics. We gathered quantitative experimental data to address the questions above, and developed a general model that recapitulates the essential qualitative features of the experimental observations, emphasizes the importance of dynamics, and places the heretofore neglected contribution of thermal fluctuations in driving osmotic response of stressed vesicular compartments.
MATERIALS AND METHODS {#materials-and-methods .unnumbered}
=====================
The detailed materials and methods used in this work are available in Supporting Materials and Methods in the Supporting Material. The experimental configuration is similar to that already described [@angelova1992; @oglecka2014]. Briefly, we prepared GUVs consisting essentially of a single amphiphile, namely 1-palmitoyl-2-oleoyl-sn-1-glycero-3-phosphocholine (POPC), doped with a small concentration (1 mol$\%$) of a fluorescently labeled phospholipid (1,2-dipalmitoyl -sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl)) or Rho-DPPE using standard electroformation technique [@angelova1992]. The GUVs thus obtained were typically between 7 and 20 $\mu$m in radius, encapsulated 200 mM sucrose, and were suspended in the isotonic glucose solution of identical osmolarity. Diluting the extra-vesicular dispersion medium with deionized water produces a hypotonic bath depleted in osmolytes, subjecting the GUVs to osmotic stress. Shortly ($\sim$1 min delay) after subjecting the GUVs to the osmotic differential, GUVs were monitored using time-lapse epifluorescence microscopy at a rate of 1 image per 150 ms, and images were analyzed using a customized MATLAB code to extract the evolution of the GUV radii with time, with a precision of about 0.1 $\mu$m.
We developed a mathematical model predicting the pulsatile behavior of GUVs in hypotonic environment. Essentially, the model couples pore nucleation by thermal fluctuations, osmotic swelling, and solute transport. These aspects are represented by Eqs. \[eq:sde\_r\], \[eq:ode\_R\] and \[eq:ode\_c\] respectively, and discussed below. Details regarding the theory and its numerical implementation are reported in Model Development and Simulations in the Supporting Materials.
RESULTS {#results .unnumbered}
=======
Homogeneous GUVs display swell-burst cycles in hypotonic conditions {#homogeneous-guvs-display-swell-burst-cycles-in-hypotonic-conditions .unnumbered}
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A selection of snapshots, revealing different morphological states, and a detailed trace showing [the time-dependence of the vesicle radius $R$ and corresponding area strain ($\epsilon_\text{exp}=(R^2-R_0^2)/R_0^2$, where and $R_0$ is the resting initial vesicle radius) are shown]{} in Fig. \[fig1\](B, C, and D), for a representative GUV. Swelling phases are characterized by a quasi-linear increase of the GUV radius, while pore openings cause a sudden decrease of the vesicle radius.
We outline here three key observations about the dynamics of swell-burst cycles from these experiments.
1. The period between two consecutive bursting events increases with each cycle, starting from a few tenths of a second for the early cycles, to several hundreds of seconds after the tenth cycle.
2. The maximum radius and therefore the maximum strain at which a pore opens decreases with cycle number, suggesting that lytic tension is a dynamic property of the membrane.
3. The observed transient pores are short lived, stay open for about a hundred milliseconds, and reach a maximum radius of up to 60 $\%$ of the GUV radius.
We seek to explain these observations through a quantitative understanding of the pulsatile GUVs in hypotonic conditions. To do so, we first investigate the mechanics of pore nucleation and its relationship to the GUV swell-burst dynamics.
Thermal fluctuations drive the dynamics of pore nucleation {#thermal-fluctuations-drive-the-dynamics-of-pore-nucleation .unnumbered}
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[ In the framework of classical nucleation theory [@litster1975], the energy potential $V(r,\epsilon)$ of a pore of radius $r$ in a lipid membrane under surface tension $\sigma$, is the balance of two competitive terms: $V_s(\epsilon)$, the strain energy, and $V_p(r)$, the pore energy. The strain energy tends to favor the opening and enlargement of the pore while the pore closure is driven by the pore line tension $\gamma$. Accordingly, the energy potential reads]{}
$$\begin{aligned}
V(r,\epsilon)=&V_s(\epsilon) + V_p(r) \nonumber\\
=&\dfrac{1}{2}\kappa_\text{eff} A_0 \epsilon^2 + 2\pi r \gamma \;.
\end{aligned}$$
[The area strain is defined as $\epsilon=(A-A_0)/A_0$, where $A=4\pi R^2-\pi r^2$ is the surface of the membrane, and $A_0=4\pi R_0^2$ is the resting vesicle area. Here $V_s(\epsilon)$ is assumed to have a Hookean form, where $\kappa_\text{eff}$ is the effective stretching modulus, which relates the surface tension to the strain as $\sigma=\kappa_\text{eff}\epsilon$ (see next section for a discussion on $\kappa_\text{eff}$). These two energetic terms oppose each other, resulting in an energy barrier that the system has to overcome in order for a pore to nucleate. The competition between the strain and pore energy is expressed by the ratio $r_b=\gamma/\sigma$, which is the critical radius associated with the crossing of the energy barrier. That is, if a pore in a tensed membrane has a radius $r<r_b$, the pore energy $V_p(r)$ dominates and the pore closes. On the contrary, for $r>r_b$, the strain energy $V_s(\epsilon)$ prevails and the pore grows. The energy required to open a pore of radius $r$ in a tensed GUV is given by $\Delta V(r,\epsilon) = V(r,\epsilon)-V(0,\epsilon)$ and is represented in Fig. \[fig2\](A). The corresponding critical radius of the the energy barrier $r_b$ is shown as a function of the strain $\epsilon$ in Fig. \[fig2\](B). The height of the energy barrier and its critical radius are dependent on the membrane strain; the more the membrane is stretched, the lower the energy barrier is, and the smaller the amount of energy required to nucleate a pore. ]{}
The amplitude of this energy barrier is strictly positive for finite strain values, making pore nucleation impossible without the addition of external energy. This issue has been often resolved by assuming a predetermined and *constant* lytic [strain ($\epsilon^*$)]{} corresponding to a critical energy barrier under which the pore opens (Fig. \[fig2\](C and E)). However, this approach is in contradiction with our experimental observations that the lytic strain in the membrane varies with each swell-burst cycle (Fig. \[fig1\](D)), due to a dependence on the strain rate [@evans2003]. In order to account for this variation, we included thermal fluctuations associated with the pore nucleation barrier in our analysis [@ting2011; @bicout2012]. In this scenario, increasing the membrane tension of the vesicle reduces the minimum pore radius $r_b$ at which a pore opens (Fig. \[fig2\](A and B)), lowering the energy barrier down to the range of thermal fluctuations, eventually letting the free energy of the system to overcome the nucleation barrier (Fig. \[fig2\](D and F)). The stochastic nature of the fluctuations can then explain a distribution of pore opening tensions, eliminating the need to assume constant lytic tension.
A direct consequence of the fluctuation-mediated pore nucleation is that the membrane rupture properties become dynamic. Indeed, fluctuations naturally cause the strain at which the membrane ruptures to be dependent on the *strain rate*, as illustrated in Fig. \[fig2\](D). In order to understand this dynamic nucleation process, consider stretching the membrane at different strain rates $\dot{\epsilon}$. Doing so decreases the radius of the nucleation barrier at corresponding speeds, as shown in Fig. \[fig2\](F). For slow strain rates, as $r_b$ tends to zero, it spends more time in the accessible range of the thermal pore fluctuations, increasing the probability that a fluctuation will overcome the energy barrier. On the other hand, at faster strain rates, $r_b$ decreases quickly, reaching small values in less time, lowering the probability for above average fluctuations to occur during this shorter time.
We use a Langevin equation to capture the stochastic nature of pore nucleation and the subsequent pore dynamics. This equation includes membrane viscous dissipation, a conservative force arising from the membrane potential, friction with water, and thermal fluctuations for pore nucleation (see Model Development and Simulations in the Supporting Material for detailed derivation). This yields the stochastic differential equation for the pore radius $r$ $$\label{eq:sde_r}
\overbrace{\left(h \eta_m + C \eta_s r\right)}^{\mathclap{\text{viscous drag}}}
\underbrace{\frac{d}{dt}(2\pi r)}_{\mathclap{\text{change of pore radius}}} =
\overbrace{2\pi \left(\sigma r-\gamma\right)}^{\mathclap{\substack{\text{surface and}\\\text{line tension}}}} +
\underbrace{\xi(t)}_{\mathclap{\substack{\text{thermal}\\\text{pore fluctuations}}}} \;,$$ where the noise source $\xi(t)$ has zero mean and satisfies, $\langle \xi(t)\xi(t^\prime) \rangle = 2\left(h \eta_m + C \eta_s r\right) k_B T \delta(t-t^\prime)$ according to the fluctuation dissipation theorem [@kubo1966]. Here, $\eta_m$ and $\eta_s$ are the membrane and solute viscosities respectively, $h$ is the membrane thickness, $C$ is a geometric coefficient [@ryham2011; @aubin2016], $k_B$ is the Boltzmann constant and $T$ is the temperature. [We assume here that the pore nucleation probability is independent on the total membrane surface area.]{} The values of the different parameters used in the model are given in Table S\[tab\_parameters\] in the Supporting Material.
Model captures experimentally observed pulsatile GUV behavior {#model-captures-experimentally-observed-pulsatile-guv-behavior .unnumbered}
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![Dynamics of swell-burst cycles from the model for a GUV of radius 14 $\mu$m in 200 mM hypotonic stress. (A and C) GUV radius and (B and D) pore radius as a function of time. The model captures the dynamics of multiple swell-burst cycles, in particular the decrease of maximum GUV radius and increase of cycle period with cycle number (A). Looking closely at a single pore opening event corresponding to the grey region, the model predicts three stage pore dynamics (C and D), namely opening, closing, and resealing, with a characteristic time of a few hundred milliseconds. Numerical reconstruction of the GUV is shown in Movies S3 and S4 in the Supporting Material. Results for $R_0=8$ and 20 $\mu$m are shown in Fig. S\[figS\_R\] in the Supporting Material.[]{data-label="fig3"}](fig_cycles_num)
In addition to pore dynamics (Eq. \[eq:sde\_r\]), we need to consider mass conservation of the solute and the solvent. We assume that the GUV remains spherical at all times and neglect spatial effects. The GUV volume changes because of osmotic influx through the semi-permeable membrane and the leak-out of the solvent through the pore. The osmotic influx is the result of two competitive pressures, the osmotic pressure driven by the solute differential ($\Delta p_{osm} = k_B T N_A \Delta c$), and the Laplace pressure, arising from the membrane tension ($\Delta p_L = 2\sigma / R$), resulting in the following equation for the GUV radius $R$: $$\label{eq:ode_R}
\underbrace{\frac{d}{dt}\left( \frac{4}{3}\pi R^3 \right)}_{\mathclap{\text{change of GUV volume}}}
= \overbrace{\frac{P \nu_s }{k_B T N_A} \left( \Delta p_{osm} - \Delta p_L \right) A}^{\substack{\text{influx of solvent}\\\text{ through the membrane} }}
- \underbrace{ v_L \pi r^2}_{\mathclap{\substack{\text{leak-out}\\\text{through the pore}}}} \;.$$ Here $A=4\pi R^2$ is the membrane area, $P$ is the membrane permeability to the solvent, $\nu_s$ is the solvent molar volume, and $N_A$ is the Avogadro number. Assuming low Reynolds number regime, the leak-out velocity is given by $ v_L = \Delta p_L r / (3\pi\eta_s)$ [@happel1983; @aubin2016].
Mass conservation of solute in the GUV is governed by the diffusion of sucrose and convection of the solution through the pore, which gives the governing equation for the solute concentration differential $\Delta c$: $$\label{eq:ode_c}
\underbrace{\frac{d}{dt}\left(\frac{4}{3}\pi R^3 \Delta c \right)}_{\mathclap{\substack{\text{molar differential}\\\text{of solute}}}}
= -\pi r^2 \bigg( \overbrace{D \frac{\Delta c}{R}}^{\mathclap{\substack{\text{diffusion through}\\\text{the pore}}}}
+ \underbrace{v_L\Delta c}_{\mathclap{\substack{\text{convection}\\\text{through the pore}}}} \bigg) \;,$$ where $D$ is the solute diffusion coefficient. These three coupled equations (Eqs. \[eq:sde\_r\] to \[eq:ode\_c\]) constitute the mathematical model.
In order to completely define the system, we need to specify the relationship between the membrane surface tension $\sigma$ and the area strain of the GUV. We note that the GUV has irregular contours during the pore opening event and for a short time afterwards, when “nodules" are observed at the opposite end from the pore, indicating accumulation of excess membrane generated by pore formation (Fig. \[fig1\](B) middle and right panels). In the low tension regime, GUVs swell by unfolding these membrane nodules, and the stretching is controlled by the membrane bending modulus $\kappa_b$ and thermal energy, yielding an effective “unfolding modulus" $\kappa_u = 48\pi\kappa_b^2 / (R_0^2k_BT)$ of the order of 10$^{-5}$ N/m [@brochard1976]. In contrast, in the high tension regime, elastic stretching is dominant, and the elastic area expansion modulus $\kappa_e$ is roughly equal to 0.2 N/m [@evans1990]. Since the maximum area strain plotted in Fig. \[fig1\](D), is about 15 $\%$, significantly larger than the expected 4 $\%$ for a purely elastic membrane deformation, the experimental data suggests the occurrence of two stretching regimes: an unfolding driven stretching, and an elasticity driven stretching [@ertel1993; @hallett1993; @karatekin2003a]. [Therefore, for simplicity, we assume an effective stretching modulus $\kappa_\text{eff}$, which takes into account both unfoldoing and elastic regimes [@evans1990; @bloom1991] through a linear dependence between the membrane tension and the strain ($\sigma = \kappa_\text{eff} \epsilon$). Note that $\kappa_\text{eff}$ is the only adjustable parameter of the model.]{}
We solved the three coupled equations (Eqs. \[eq:sde\_r\] to \[eq:ode\_c\]) for an initial inner solute concentration of $c_0=200$ mM, and different GUV radii of $R_0=8$, 14 and 20 $\mu$m. All the results presented here are obtained for $\kappa_\text{eff}=2 \times10^{-3}$N/m, the value that best fits the experimental observations (see Supplemental Fig. \[figS\_kappa\] for the effect of this parameter on the GUV dynamics). Dynamics of the GUV radius and the pore radius are shown in Fig. \[fig3\] for a typical simulation with $R_0=14$ $\mu$m (see Supplemental Fig. \[figS\_R\] for simulations with different values of $R_0$). Our model qualitatively reproduces the dynamics of the GUV radius during the swell burst cycle (compare Figs. \[fig1\](C) and \[fig3\](A)). Importantly, we recover the key features of the swell-burst cycle – namely an increase of the cycle period with each bursting event (point 1), and a decrease of the maximum radius with time (point 2). The stochastic nature of the thermodynamic fluctuations leads to variations and irregularities in the pore opening events, and therefore, the cycle period and maximum strain. The dynamics of a single cycle is shown in Fig. \[fig3\](C and D). Our numerical results show an abrupt drop in the GUV radius, followed by a slower decrease, suggesting a sequence of two leak-out regimes: a fast-burst releasing most of the membrane tension, and a low tension leak-out. This two-step tension release is confirmed by the pore radius dynamics, which after suddenly opening (release of membrane tension), reseals quasi-linearly due to dominance of line tension compared to membrane tension in Eq. \[eq:sde\_r\]. Furthermore, the computed pore amplitude and lifetime are in agreement with experimental observations (point 3). Overall, our model is able to reproduce the quantitative features of GUV response to hypotonic stress over multiple time scales.
If thermal fluctuations are ignored, the strain to rupture needs to be adjusted to roughly 15$\%$ in order to match the range of maximum GUV radius observed experimentally (Fig. S\[figS\_det\] in the Supporting Material). However such a deterministic model does not capture the pulsatile dynamics as well as the stochastic model in terms of cycle period and strain rate (Fig. S\[figS\_comparison\] in the Supporting Material), and fails to reproduce a strain rate dependent maximum stress (Fig. S\[figS\_det\]).
Solute diffusion is dominant during the low tension regime of pore resealing {#solute-diffusion-is-dominant-during-the-low-tension-regime-of-pore-resealing .unnumbered}
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![[Diffusion of sucrose through the transient pore produces a step-wise decrease of the inner solute concentration. (A) In hypotonic conditions, the model predicts a step-wise decrease of solute concentration differential with time (blue line), which is solely due to diffusion of solute through the transient pore. In comparison, when diffusion is neglected in the model (grey line), the solute concentration differential decreases smoothly (also see Fig. S\[figS\_D\] in the Supporting Material for further analysis on the effect of diffusion). In isotonic conditions (dashed line), the solute concentration differential is constant with time. (Here $t_0=$ 40 s).]{} (B) Time evolution of the normalized fluorescence intensity of a GUV in hypotonic condition, encapsulating fluorescent glucose analog. $\Delta I$ is the difference in mean intensity between the inside of the GUV and the background. In hypotonic conditions (solid lines) the normalized intensity decreases with time due to the constant influx of water through the membrane, and shows sudden drops in intensity at each pore opening (indicated by arrows), due to diffusion of sucrose through the pore (see Movie S5 in the Supporting Material). In comparison, GUVs in an isotonic environment (dashed lines) exhibit a rather constant fluorescence intensity (see Movie S6 in the Supporting Material). (C) Micrographs of a GUV in hypotonic condition, encapsulating fluorescent glucose analog, just prior to bursting (left panel), with an open pore (middle panel), and just after pore resealing (right panel). The leak-out of fluorescent dye is observed in the middle frame, coinciding with a drop of the GUV radius. Frames extracted from Movie S7 in the Supporting Material. (D) Same as panel (C), with the images processed to increase contrast and attenuate noise. The blue, red, and white lines are the isocontours of the 90, 75, and 60 grey scale values respectively, highlighting the leak-out of fluorescent dye.[]{data-label="fig4"}](fig_efflux)
The concentration differential of sucrose decreases exponentially and drops from 200 mM to about 10 mM in about 1000 seconds (Fig. S\[figS\_R\] in the Supporting Material). Even after 2000 s when the concentration differential is as low as 10 mM, the osmotic influx is still large enough to maintain the dynamics of swell-burst cycles (Fig. \[fig1\](C), Fig. S\[figS\_R\]). We further observe that every pore opening event produces a sudden drop in inner solute concentration (Fig. \[fig4\](A), blue line). This suggests that diffusion of sucrose plays an important role in governing the dynamics of solute. In the absence of diffusive effects, the model does not show the abrupt drops in concentration but a rather smooth exponential decay (Fig. \[fig4\](A), grey line).
To experimentally verify the model predictions of sucrose dynamics, we quantified the evolution of fluorescence intensity in GUVs encapsulating 200 mM sucrose plus 58.4 $\mu$M 2-NBDG, a fluorescent glucose analog (see Supporting Material and Methods). Fig. \[fig4\](B) presents the evolution of fluorescent intensity of sucrose in time. GUVs in isotonic conditions (dashed lines) do not show a significant change in fluorescence intensity. GUVs in hypotonic conditions (solid lines) exhibit an overall decrease of intensity due to permeation of water through the membrane. Strikingly, consecutive drops of fluorescence intensity are observed coinciding with the pore opening events (Fig. \[fig4\](C and D) middle panels), and point out the importance of sucrose diffusion through the pore. While the quantitative dynamics of sucrose depends on the value of the diffusion constant (Fig. S\[figS\_D\]), the qualitative effect of diffusion on the dynamics remains unchanged. On the other hand, leak-out induced convection does not influence the inner concentration of sucrose, as both solvent and solute are convected, conserving their relative amounts. These observations are in agreement with the existence of the low tension pore closure regime discussed above, where Laplace pressure produces negligible convective transport compared to solute diffusion though the pore.
Cycle period and strain rate are explicit functions of the cycle number and GUV properties {#cycle-period-and-strain-rate-are-explicit-functions-of-the-cycle-number-and-guv-properties .unnumbered}
------------------------------------------------------------------------------------------
Given that lytic tension is a dynamic quantity, we asked how cycle period and strain rate evolve along with the cycles. We analyzed the simulated dynamics of GUVs with resting radii of 8, 14 and 20$\mu m$, each data point representing the mean and the standard deviation of 10 simulations with identical parameters (the variations being due to the stochastic nature of the model). [The details of this burst cycle analysis is reported in the Supporting Material]{}. Cycle periods and strain rates show a dependence on the GUV radius, as depicted in Fig. \[fig5\_1\] where larger GUVs have slower dynamics, resulting in smaller strain rates and longer cycle periods (Fig. S\[figS\_R\]). To verify this experimentally, a total of eight GUVs were similarly analyzed with resting radii ranging from 7.02 to 18.76 $\mu$m (Fig. S\[figS\_exp\] in the Supporting Material). The measured cycle period and strain rate as a function of the cycle number (corrected for the lag between the application of the hypotonic stress and the beginning of the observations) are shown in Fig. \[fig5\_1\](A) and (B), respectively. Experimental and model results quantitatively agree, and show a exponential dependence of the cycle period and strain rate on cycle number (Insets Fig. \[fig5\_1\](A and B)).
Two further questions arise: How can we relate the cycle number to the driving force of the process, namely the osmotic differential? And, is there a scaling law that governs the GUV swell-burst dynamics? To answer these questions we computed the cycle solute concentration (defined as the solute concentration at the beginning of each cycle) as a function of the cycle number (Fig. S\[figS\_cn\] in the Supporting Material). We found that the solute concentration follows an exponential decay function of the cycle number, and is independent of the GUV radius. Additionally, plotting the cycle period and strain rate against the cycle solute concentration (Fig. \[fig5\_1\](C and D)), we observe that the cycle period increases as $\Delta c$ decreases, while the strain rate is a linear function of $\Delta c$. The data presented in Fig. \[fig5\_1\] suggest that the dynamics of GUVs swell-burst cycle can be scaled to their size. From the non-dimensional form of Eq. \[eq:ode\_R\], we extracted a characteristic time associated with swelling, defined by $\tau = R_0 / (P \nu_s c_0)$, and scaled the cycle period and strain rates with this quantity. As shown in Fig. \[fig5\_2\], all the scaled experimental and model data collapse onto the same curve, within the range of the standard deviations. The scaled relationships can be justified analytically, by estimating the cycle period and strain rates as [ $$\label{eq:cycle_period}
\frac{ T_n}{\tau} \simeq \frac{\epsilon^*}{\left( 2\sqrt{\epsilon^*+1} \Delta c/c_0 \right)}
\qquad \text{and} \qquad
\tau \dot{\epsilon} \simeq \frac{2\sqrt{\epsilon^*+1} \Delta c}{c_0}$$ ]{} respectively (see Supporting Material for full derivation). These analytical expressions are plotted in Figs. \[fig5\_1\](C, D) and \[fig5\_2\](C, D) for a characteristic lytic strain of $\varepsilon^*=0.15$, showing good agreement with the numerical data. Taken together, these results suggest that the GUV pulsatile dynamics is governed by the radius, the membrane permeability, the solute concentration, and importantly the stochastic pore nucleation mechanism which determines the strain to rupture.
DISCUSSION {#discussion .unnumbered}
==========
Explaining how membrane-enclosed compartments regulate osmotic stress is a first step towards understanding how cells control volume homeostasis in response to environmental stressors. In this work, we have used a combination of theory, computation, and experiments in a simple model system to study how swell-burst cycles control the dynamics of GUV response to osmotic stress. Using this system, we show that the pulsatile dynamics of GUVs under osmotic stress is controlled through thermal fluctuations that govern pore nucleation and lytic tension.
The central feature of a GUV’s osmotic response is the nucleation of a pore. Even though Evans and coworkers [@evans2003; @evans2011] identified that rupture tension was not governed by an intrinsic critical stress, but rather by the load rate, the idea of a constant lytic tension has persisted in the literature [@idiart2004; @popescu2008; @peterlin2008]. By coupling fluctuations to pore energy, we have now reconciled the dynamics of the GUV over several swell-burst cycles with pore nucleation and dependence on strain rate. Our model is not only able to capture the experimentally observed pulsatile dynamics of GUV radius and solute concentration (Figs. \[fig3\] and \[fig4\]), but also predicts pore formation events and pore dynamics (Fig. \[fig3\](B and D)). We also found that during the pore opening event, a low-tension regime enables a diffusion dominated transport of solute through the pore (Fig. \[fig4\]), a feature that has been until now neglected in the literature.
Specifically, we have identified a scaling relationship between (a) the cycle period and cycle number and (b) the strain rate and the cycle number, highlighting that swell-burst cycles of the GUVs in response to hypotonic stress is a dynamic response (Fig. \[fig5\_2\]). One of the key features of the model is that we relate the cycle number, an experimentally observable quantity, to the concentration difference of the solute, a quantity that is hard to measure in experiments (Fig. S\[figS\_cn\]). This allows to interpret the scaling relationships described above in terms of solute concentration differential. The cycle period increases as the solute concentration difference decreases, while the strain rate is a linear function of the concentration difference. Both relationships are derived theoretically in the Supplemental material. These features indicate long time scale relationships of pulsatile vesicles in osmotic stress.
Thermal fluctuations and stochasticity are known to play diverse roles in cell biology. Well-recognized examples include Brownian motors and pumps [@julicher1997; @oster2002], noisy gene expression [@elowitz2002], and red blood cell flickering [@turlier2016]. The pulsatile vesicles presented here provide yet another example of how fluctuations can be utilized by simple systems to produce dynamical adaptive behavior. Given the universality of fluctuations in biological processes, it appears entirely reasonable that simple mechanisms similar to these pulsatile vesicles may have been exploited by early cells, conferring them with a thermodynamic advantage against environmental osmotic assaults. On the other hand, if such swell-burst mechanisms were at play, the chronic leak-out of inner content could have led protocells to evolve active transport mechanisms to compensate for volume loss, and endure osmotic stress without a high energetic cost.
In this study, we experimentally measure the dynamics of swell-burst cycles in GUVs, and provide for the first time a model that captures quantitatively the pulsatile behavior of GUVs under hypotonic conditions for long time scales. In order to do so, we developed a general framework which integrated parts of existing models [@koslov1984; @brochard-wyart2000; @ryham2011], with novel key elements: (a) the explicit inclusion of thermal pore fluctuations, which enables dynamic pore nucleation; (b) the definition of an effective stretching modulus, which combines membrane unfolding and elastic stretching; (c) the incorporation of solute diffusion through the pore, which results in a non-trivial contribution to the evolution of the osmotic differential. The coupling of these key features results in a unified model that is valid in all regimes of the vesicle, pore, and solute dynamics.
While we have been able to explain many fundamental features of the pulsatile GUVs in response to osmotic stress, our approach has some limitations and there is a need for further experiments. We have assumed a linear relationship between stress and strain. Although this assumption is reasonable and appears to work well for the present experimental conditions, a more general expression should be considered to include both membrane (un)folding and elastic deformation [@helfrich1984]. Another important aspect of biological relevance is membrane composition, where the abundance of proteins and heterogeneous composition leading to in-plane order and asymmetry across leaflets influence the membrane mechanics [@alberts2014; @rangamani2014]. We have previously found experimentally that the dynamics of swell-burst cycles is related to the compositional degrees of freedom of the membrane [@oglecka2014]. Future efforts will be oriented toward the development of theoretical framework and quantitative experimental measures that provide insight into how the membrane’s compositional degrees of freedom influence the pulsatile dynamics of cell-size vesicles. [In addition to osmotic response and membrane composition, we will focus on how membrane components such as aquaporins and ion channels may couple thermal fluctuations with membrane tension to regulate their functions. Additionally, we are also investigating how the properties of the encapsulated bulk fluid phase may affect the response of the GUV in response to osmotic shock. The current work is a first and critical step in these directions. ]{}
AUTHOR CONTRIBUTIONS {#author-contributions .unnumbered}
====================
J.C.S.H. and A.N.P. designed the experiments; J.C.S.H. performed the experiments; M.C. and P.R. derived the model; M.C. performed the simulations; M.C. and J.C.S.H. analyzed the data; all authors discussed and interpreted results; all authors wrote and agreed on the manuscript.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are grateful to Prof. Wouter-Jan Rappel and Prof. Alex Mogilner for insightful comments on the manuscript. We also thank Prof. Daniel Tartakovsky for enriching discussions. This work was supported in part by the FISP 3030 for the year 2015-2016 to M.C., NTU provost office to J.C.S.H., AFOSR FA9550-15-1-0124 award to P.R., and NSF PHY-1505017 award to P.R. and A.N.P.
SUPPORTING CITATIONS {#supporting-citations .unnumbered}
====================
References (42-51) appear in the Supporting Material.
Table of Content {#table-of-content .unnumbered}
================
****
---------------------------------------------------------------------------------------------------------- ---
Supporting Materials and Methods 1
Model Development and Simulations 2
Derivation of the analytical relations between cycle period, strain rate, and concentration differential 4
Supporting References 5
Supporting Figures and Movies 6
---------------------------------------------------------------------------------------------------------- ---
Supporting Materials and Methods {#supporting-materials-and-methods .unnumbered}
================================
**Swell-burst cycle experiments.** The experimental methods for the GUVs preparation has been described in [@oglecka2014; @angelova1992]. Briefly, GUVs (100$\%$ POPC + 1mol$\%$ Rho-DPPE) containing 200 mM of sucrose were prepared by electroformation, yielding vesicles with radii ranging from 7 to 20 $\mu$m. GUVs were then placed in a bath of deionized water at room temperature, inducing hypotonic stress proportional to the inner sucrose concentration. The kinetics of eight GUVs were recorded by time-lapse microscopy at 1/150 images/ms. In order to allow for the sedimentation of GUVs to the bottom of the well, observations were started about one minute after the GUVs were subject to hypotonic conditions.
For each frame, the GUV radius was measured using a customized MATLAB (Mathworks, Natick, MA) code to streamline the image analysis. This code uses a Circular Hough Transform method based on a phase-coding algorithm to detect circles , and measure their radii and centers. For our data, this custom code gives the evolution of the GUV radius in time with a precision of about 0.1 $\mu$m. Due to slow movement of the GUVs, in some cases the observation fields had to be adjusted to follow the GUVs, and the recording was paused. These are indicated by black dashed lines in Fig. S\[figS\_exp\]. In order to define a systematic experimental initial GUV radius, $R_0$ was determined for each GUV as 0.995 times the first measured local minimum GUV radius, in accordance with our numerical results. Furthermore, burst events were identified by drops of GUV radius larger than 0.2 $\mu$m within a 1.5 s interval, and are plotted as solid red triangle. Bursting events that were likely to happen during the video gaps were indicated by plain red triangle (these “likely" bursting events were not taken into account in the data processing for Figs. \[fig5\_1\] and \[fig5\_2\]).
**Leak-out quantification.** To quantify the leak-out amount when a membrane pore is formed, giant unilamellar vesicles (GUVs) were electroformed in 200 mM sucrose, supplemented with 58.4 $\mu$M 2-NBDG (2-(N- (7-Nitrobenz-2-oxa-1,3-diazol-4-yl) Amino) -2-Deoxyglucose), a fluorescent glucose analog that has an almost identical molecular weight as sucrose. Fluorescence imaging was performed on a deconvolution microscope, equipped with a FITC filter. Time-lapse imaging of the vesicles was performed approximately one minute after exposing the vesicles to either deionized water (hypotonic conditions $n=3$) or glucose (isotonic conditions $n=3$) environment to ensure sedimentation of GUVs to the bottom of the well. All acquisitions were performed using identical settings to facilitate comparison of vesicles submerged in water or equi-osmotic glucose environment.
For Fig. \[fig4\](C), the GUVs were detected with a MATLAB (Mathworks, Natick, MA) code adapted from the one described above, where the mean gray intensity inside and outside of the GUV are measured. For every time frame, the difference between the inner and outer mean intensity $\Delta I(t)$ was computed, and normalized by the intensity difference of the first frame $\Delta I(t_0)$. Bursting events were identified by visual inspection of the videos, and reported by arrows on Fig. \[fig4\](B).
In order to highlight the efflux of fluorescent dyes during a GUV bursting event, three frames (before, during and after the event) were extracted form the video of a GUV containing 200 mM sucrose + 58.4 $\mu$M 2-NBDG in hypotonic conditions (Fig. \[fig4\](C)). These images were further processed with ImageJ software to plot Fig. \[fig4\](D). Briefly, the noise was attenuated by successively applying ImageJ built-in routines (background suppression, contrast enhancing, median filter), and ploting the isovalues of gray at 90, 75 and 60 with the pluggin Contour Plotter (http://rsb.info.nih.gov/ij/plugins/contour-plotter.html).
**Burst cycle analysis.** [ The following analysis has been applied for both experimental and numerical data in order to produce Figs. \[fig5\_1\], \[fig5\_2\] and S\[figS\_comparison\]. For a given GUV radius dynamics, a swell-burst cycle was defined between two successive minimum GUV radii that immediately followed a bursting event. Cycle periods were computed as the time between two consecutive minima in vesicular radii. For experimental data, if there was a video gap between two consecutive radius minima, the cycle was not taken into account for the analysis, that is to say, only cycles between two successive solid triangles in Figs. \[fig1\](C) and S\[figS\_exp\] were taken into account. The strain rate was computed as the difference between the maximum and minimum radii within these cycles, divided by the time between these two events. For experimental data, because of the lag between the beginning of the experiments and the beginning of the video recordings, the initial observed cycle number was adjusted between $n=1$ and $n=4$, depending on $R_0$.]{}
Model Development and Simulations {#model-development-and-simulations .unnumbered}
=================================
Here we derive a theoretical model to describe the swell-burst cycle of a GUV under hypotonic conditions. In line with previous work [@koslov1984; @idiart2004; @popescu2008], the model has three conservation equations, governing the dynamics of the solvent, solute, and membrane pore.
**Mass conservation of solvent.** Mass conservation of the solvent (water) within the vesicle is governed by the flux through the membrane ($j_w$), and the leak-out through the pore. For a spherical GUV, the general form of the mass conservation equation for the solvent is $$\label{eq:sol_mass_cons1}
\frac{d}{dt}\left( \frac{4}{3}\pi R^3 \rho_s \right) = j_w - \pi r^2 \rho_s v_L \;,$$ where $R$ and $r$ are the radius of the vesicle and the pore respectively, $\rho_s$ is the mass density of the solvent, and $v_L$ is the leak-out velocity of the solvent. The osmotic flux is influenced by the permeability of the membrane to the solvent ($P$), the osmotic pressure ($\Delta p_{osm}$), and the Laplace pressure ($\Delta p_L$). A phenomenological expression for the osmotic flux is [@koslov1984; @popescu2008] $$\label{eq:jw}
j_w = \frac{P \nu_s \rho_s}{k_B T N_A} A \left( \Delta p_{osm} - \Delta p_L \right) \;,$$ where $\nu_s$ is the solvent molar volume, and the membrane area is defined as $A = 4\pi R^2 - \pi r^2$. The two pressures involved in Eq. \[eq:jw\] are defined as $$\left\{
\begin{array}{l}
\Delta p_{osm} = k_B T N_A \Delta c \\
\Delta p_L = \dfrac{2\sigma}{R}
\end{array}\right. \;.$$ The Laplace pressure originates from the surface tension in the membrane $\sigma$, which we assume to be proportional to the membrane strain $$\label{eq:sigma_linear}
\sigma = \kappa_\text{eff} \frac{A-A_0}{A_0} = \kappa_\text{eff} \epsilon \;.$$ Here $\kappa_\text{eff}$ is the effective area extension modulus (combining the effects of membrane unfolding and elastic deformation), and $A_0 = 4\pi R_0^2$ is the surface of the vesicle in its unstretched state. The leak-out velocity $v_L$ can be analytically approximated at low Reynolds number in order to relate it to the Laplace pressure [@happel1983] $$\label{eq:vl}
v_L = \frac{\Delta p_L r}{3\pi\eta_s} \;.$$
Substituting these definitions into Eq. \[eq:sol\_mass\_cons1\], the mass conservation equation for the solvent takes the form of an ordinary differential equation (ODE) for the GUV radius $$\label{eq:sol_mass_cons2}
4\pi R^2\frac{dR}{dt}= \frac{P \nu_s }{k_B T N_A} A \left( k_B T N_A \Delta c - \frac{2\sigma}{R} \right) - \frac{2\sigma }{3\eta_s R} r^3 \;.$$
**Mass conservation of solute.** The permeability of lipid membranes to water is several orders of magnitude larger than for most solutes . Consequently the lipid bilayer is supposed to be semi-permeable, neglecting sucrose transport through the membrane. Thus, variation of solute in the vesicle is exclusively limited to diffusive and convective transport through the pore, such that $$\label{eq:solute_mass_cons1}
\frac{d}{dt}\left(\frac{4}{3}\pi R^3 \Delta c \right) = \pi r^2 \left( - D \frac{\Delta c}{R} - v_L\Delta c \right) \;.$$ While the diffusive flux through the pore is usually neglected over the convective efflux of solute, theoretical analysis of long lived pores indicates that the Laplace pressure decreases rapidly after the pore opening, and stays low for most of the pore life time [@brochard-wyart2000]. This suggests that the convective efflux directed by the leak-out velocity may not always be the dominant solute transport mechanism, as confirmed by our numerical and experimental results (see main text Fig. \[fig4\]). Expanding Eq. \[eq:solute\_mass\_cons1\] we obtain an ODE for the concentration difference in solute $$\label{eq:solute_mass_cons2}
\frac{4}{3}\pi R^3 \frac{d \Delta c }{dt} = - D\pi r^2 \frac{\Delta c}{R} - \frac{2\sigma }{3\eta_s R} r^3\Delta c - 4\pi R^2\Delta c \frac{d R}{dt} \;.$$
**Pore force balance.** The pore in the lipid bilayer is modeled as an overdamped system, where the pore radius is governed by the following Langevin equation $$\label{eq:pore_energy1}
\zeta \frac{d}{dt}\left( 2\pi r \right) = F(r,t) + \xi(t) \;,$$ where $\zeta$ is the membrane drag coefficient (inverse of the mobility), $F(r,t)$ is a conservative force, and $\xi$ is a noise term accounting for independent thermally-induced pore fluctuations. The drag coefficient includes two in-plane contributions $\zeta = \zeta_m + \zeta_s$: one from membrane dissipation, proportional to the membrane viscosity and thickness $\zeta_m=\eta_m h$ [@brochard-wyart2000], and a second from the friction of the solvent with the moving pore – proportional to the solvent viscosity $\zeta_s = C \eta_s r $, where $C=2\pi$ is a geometric coefficient [@ryham2011; @aubin2016]. The conservative force $F(r,t)=-\partial V(r,t)/\partial r$ arises from the membrane potential $V(r,t)$, which is equal to the sum of the strain energy $V_s$, and the pore energy $V_p$. We assume the membrane strain energy to take a Hookean form $V_s = \kappa_\text{eff} \left(A-A_0\right)^2 /(2 A_0)$, where $\kappa_\text{eff}$ is an effective stretching modulus approximating the combined contributions of membrane unfolding and elastic stretching. The pore energy depends on the edge energy and length as $V_p = 2\pi r \gamma$, where $\gamma$ is the pore line tension, here assumed independent of the pore radius. Using the definition $\sigma = \partial V_s/\partial A$, we can therefore express the force as $$F(r,t) =2\pi\sigma r - 2\pi\gamma \;.$$ The fluctuation term has a zero mean, and a correlation function given by $$\langle \xi(t)\xi(t^\prime) \rangle = 2\zeta k_B T \delta(t-t^\prime) \;,$$ following the dissipation-fluctuation theorem, where $\delta$ is the Dirac delta function.
Rearranging Eq. \[eq:pore\_energy1\] with these definitions, we obtain a stochastic differential equation for the pore radius $$\label{eq:pore_energy2}
\left(\eta_m h + C \eta_s r\right)\frac{d}{dt}(2\pi r) = 2\pi(\sigma r - \gamma) + \xi(t) \;,$$ with $r\ge0$. The last term in Eq. \[eq:pore\_energy2\] is responsible for thermally driven pore nucleation.
**Deterministic model.** In the absence of thermal fluctuations, a critical value for the membrane tension (or strain) has to be defined, and an initial pore has to be set artificially in order for a large pore to open. In that case, Eq. \[eq:pore\_energy2\] is simply $$\label{eq:pore_energy_det}
\left(\eta_m h + C \eta_s r\right)\frac{dr}{dt} = \sigma r - \gamma \;.$$ When the pore is closed ($r=0$) and the strain overcomes the predetermined critical value ($\epsilon\ge\epsilon^*$), an initial pore large enough to overcome the nucleation barrier ($r=\gamma/\sigma$) is artificially created.
**Numerical implementation.** All numerical computations have been carried out using a custom code in MATLAB (Mathworks, Natick, MA). The stochastic model, composed of Eqs. \[eq:sde\_r\], \[eq:ode\_R\] and \[eq:ode\_c\] was solved using an order-1 Runge-Kutta scheme. Because a pore nucleation event occurs due to a single fluctuation overcoming the energy barrier, the numerical implementation of the noise requires the definition of a fluctuation frequency $f_T$ [(number of fluctuation “kicks” per seconds)]{} that is independent of the time step. For comparison a deterministic model (Eqs. \[eq:sol\_mass\_cons2\], \[eq:solute\_mass\_cons2\], and \[eq:pore\_energy\_det\]) was solved using Euler method. All parameters are shown in Table S\[tab\_parameters\]. All time steps were taken as 0.1 ms, (smaller time steps did not improve the accuracy of the results significantly). For the cycle analysis of the stochastic model, Figs. \[fig5\_1\] and \[fig5\_2\], shows the average and standard deviations of 10 runs with same parameters.
Parameter Typical value References
--------------------- ----------------------------------------- --------------
$R_0$ 8-20 $\mu$m this work
$c_0$ 200 mM this work
$d$ 3.5 nm
$\rho_s$ 1000 kg m$^{-3}$
$\nu_s$ 18.04$\times$10$^{-6}$ m$^3$ mol$^{-1}$
$P$ 20 $\mu$m/s
$T$ 294 K
$\gamma$ 5 pN
$\kappa_\text{eff}$ 2$\times$10$^{-3}$ N/m this work
$\eta_m$ 5 Pa s
$\eta_s$ 0.001 Pa s
$D$ 5$\times$10$^{-10}$ m$^2$/s
$C$ $2\pi$ [@aubin2016]
$f_T$ 150 Hz this work
Derivation of the analytical relations between cycle period, strain rate, and concentration differential {#derivation-of-the-analytical-relations-between-cycle-period-strain-rate-and-concentration-differential .unnumbered}
========================================================================================================
First, we derive the linear dependence of the strain rate on the concentration difference shown in Fig. \[fig5\_2\](D). For a closed vesicle ($r=0$), the membrane area is $A=4\pi R^2$, and the strain rate is $$\dot{\epsilon} = \frac{d}{dt} \left( \frac{A - A_0}{A_0} \right) = \frac{2R}{R_0}\frac{dR}{dt} \;.$$ This allows us to write Eq. \[eq:sol\_mass\_cons2\] in terms of the strain rate as $$\dot{\epsilon} = \frac{P \nu_s }{k_B T N_A} \frac{A}{2\pi R R_0^2} \left( k_B T N_A \Delta c - \frac{2\sigma}{R} \right)\;.$$ When the osmotic pressure is the dominant process influencing GUV swelling, we can neglect the Laplace pressure and obtain $$\label{eq:eps_dot}
\tau \dot{\epsilon} \simeq \frac{2R}{R_0}\frac{\Delta c}{c_0} \;,$$ where $\tau=R_0/(P \nu_s c_0)$. At maximum GUV radius amplitude, $R/R_0$ can be expressed in term of the lytic strain as $R_\text{max}/R_0 = \sqrt{\epsilon^*+1}$, allowing to write Eq. \[eq:eps\_dot\] as $$\label{eq:eps_dot_s}
\tau \dot{\epsilon} \simeq 2\sqrt{\epsilon^*+1}\frac{\Delta c}{c_0} \;.$$ Plotting this relationship in Figs. \[fig5\_1\](D) and \[fig5\_2\](D) for a typical lytic strain $\epsilon^*=0.15$, we get a good agreement with the numerical results from the stochastic model.
We now derive an approximate relation between the cycle period and the strain rate. During a cycle of period $T_n$, the lytic strain can be written $$\epsilon^* \simeq T_n \dot{\epsilon} \;.$$ Introducing Eq. \[eq:eps\_dot\_s\], we get $$\frac{T_n}{\tau} \simeq \frac{\epsilon^*}{2\sqrt{\epsilon^*+1}} \left( \frac{\Delta c}{c_0}\right)^{-1} \;.$$ Taking $\epsilon^*=0.15$, this relationship fits well the simulation results, as shown in Figs. \[fig5\_1\](C) and \[fig5\_2\](C).
It should be noted that, because the Laplace pressure is neglected in the derivation of Eq. \[eq:eps\_dot\_s\], the analytical expression slightly overestimates the strain rate as shown in Figs. \[fig5\_1\](D) and \[fig5\_2\](D). Moreover the cycle period is also overestimated for low solute concentrations due to the constant lytic strain assumed in the analytical expression (Figs. \[fig5\_1\](C) and \[fig5\_2\](C)).
Supporting Figures and Movies {#supporting-figures-and-movies .unnumbered}
=============================
#### Supporting Movie 1
Membrane nodules appearance after membrane pore reseals. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 2 fps; total duration, 17 s; image size, 82.43 $\mu$m $\times$ 82.43 $\mu$m; scale bar, 10 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM).
#### Supporting Movie 2
Multiple swell-burst cycles of GUVs subject to hypotonic stress. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 77 s; image size, 82.43 $\mu$m $\times$ 82.43 $\mu$m; scale bar, 10 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM).
#### Supporting Movie 3
Model results showing multiple swell-burst cycles of a GUV subject to hypotonic stress. GUV radius (top-left panel), pore radius (middle-left panel), and solute differential (bottom-left panel) as a function of time. Right panel is a representation of the numerical GUV in time, where the grey intensity is proportional to the inner sucrose concentration. GUV initial radius is $R_0=14\;\mu$m, initial solute concentration is $c_0=200$ mM. All parameters are shown in Supporting Table S\[tab\_parameters\].
#### Supporting Movie 4
Model results showing a single pore opening dynamics of a GUV subject to hypotonic stress. GUV radius (top-left panel), pore radius (middle-left panel), and solute differential (bottom-left panel) as a function of time. Right panel is a representation of the numerical GUV in time, where the grey intensity is proportional to the inner sucrose concentration. GUV initial radius is $R_0=14\;\mu$m, initial solute concentration is $c_0=200$ mM. All parameters are shown in Supporting Table \[tab\_parameters\].
#### Supporting Movie 5
Solute leakage of a GUV in multiple swell-burst cycles under hypotonic condition. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 11 s; image size, 119.14 $\mu$m $\times$ 125.58 $\mu$m; scale bar, 20 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM).
#### Supporting Movie 6
GUV under isotonic condition. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 8 s; image size, 101.11 $\mu$m $\times$ 101.11 $\mu$m; scale bar, 10 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a isotonic solution (no osmotic differential).
#### Supporting Movie 7
Solute efflux from GUV during one swell-burst cycle. Movie assembled from time-lapse fluorescence microscopy images (frame rate, 12 fps; total duration, 8 s; image size, 164.86 $\mu$m $\times$ 164.86 $\mu$m; scale bar, 20 $\mu$m) obtained for a population of electroformed GUVs consisting of POPC doped with 1$\%$ Rhodamine-B labeled DPPE membrane in a hypotonic solution (Osmotic differential of 200 mM).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the spectrum of a second-order elliptic operator in divergence form with periodic coefficients, which is known to be completely described by Bloch eigenvalues. We show that under small perturbations of the coefficients, a multiple Bloch eigenvalue can be made simple. The Bloch wave method of homogenization relies on the regularity of spectral edge. The spectral tools that we develop, allow us to obtain simplicity of an internal spectral edge through perturbation of the coefficients. As a consequence, we are able to establish Bloch wave homogenization at an internal edge in the presence of multiplicity by employing the perturbed Bloch eigenvalues. We show that all the crossing Bloch modes contribute to the homogenization at the internal edge and that higher and lower modes do not contribute to the homogenization process.'
address: 'Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India.'
author:
- Sivaji Ganesh Sista
- Vivek Tewary
bibliography:
- 'mylit.bib'
title: Generic Simplicity of Spectral Edges and Applications to Homogenization
---
Bloch eigenvalues ,Genericity ,Periodic Operators ,Homogenization 47A55 ,35J15 ,35B27
Introduction
============
The goal of the paper is to study regularity properties of spectral edges of a periodic second-order elliptic operator in divergence form, given by $$\mathcal{A}u:=-\frac{\partial}{\partial y_k}\left(a_{kl}(y)\frac{\partial u}{\partial y_l}\right),\label{eq1:operator}$$ where summation over repeated indices is assumed. We make the following assumptions on the coefficients of the operator : The coefficients $a_{kl}(y)$ are measurable bounded real-valued periodic functions defined on $\mathbb{R}^d$. Let $Y=[0,2\pi)^d$ be a basic cell for its lattice of periods in the $d$-dimensional euclidean space $\mathbb{R}^d$. The space of measurable bounded periodic real-valued functions in $Y$ is denoted by $L^\infty_\sharp(Y,\mathbb{R})$. Hence, $a_{kl}\in L^\infty_\sharp(Y,\mathbb{R})$. In many instances, we will identify $Y$ with a torus $\mathbb{T}^d$ and the space $L^\infty_\sharp(Y,\mathbb{R})$ with $L^\infty(\mathbb{T}^d,\mathbb{R})$, in the standard way. The matrix $A=(a_{kl})$ is symmetric, i.e., $a_{kl}(y)=a_{lk}(y)$. Further, the matrix $A$ is [*coercive*]{}, i.e., there exists an $\alpha>0$ such that $$\label{coercivity}
\forall\, v\in\mathbb{R}^d\mbox{ and } a.e.\, y\in\mathbb{R}^d,\langle A(y)v, v\rangle\geq \alpha||v||^2.$$
Let $Y^{'}=\left[-\dfrac{1}{2},\dfrac{1}{2}\right)^d$ be a basic cell for the dual lattice in $\mathbb{R}^d$. Then, the spectrum of $\mathcal{A}$ can be studied by evaluating, for $\eta\in Y'$, the spectrum of the shifted operator $$\begin{aligned}
\mathcal{A}(\eta)=e^{-i\eta\cdot y}\mathcal{A}e^{i\eta\cdot y}=-\left(\frac{\partial}{\partial y_k}+i\eta_k\right)a_{kl}(y)\left(\frac{\partial}{\partial y_l}+i\eta_l\right).\label{shiftedoperator}\end{aligned}$$
This is an unbounded operator in $L^2_\sharp(Y)$, the space of all $L^2_{loc}(\mathbb{R}^d)$ functions that are $Y$-periodic. The operator $\mathcal{A}$ in $L^2(\mathbb{R}^d)$ is unitarily equivalent to the fibered operator $$\int^{\bigoplus}_{Y^{'}}\mathcal{A}(\eta)d\eta$$ in the Bochner space $L^2(Y^{'},L^2_{\sharp}(Y))$. As a consequence of this fact, the spectrum of $\mathcal{A}$ is the union of the spectra of $\mathcal{A}(\eta)$ in $L^2_{\sharp}(Y)$ as $\eta$ varies in $Y^{'}$. For a proof, see [@Reed1978 p. 284]. Let $(\lambda_n(\eta))_{n=1}^\infty$ denote the sequence of increasing eigenvalues for $\mathcal{A}(\eta)$, counting multiplicity. The functions $\eta\mapsto\lambda_n(\eta)$ are known as the Bloch eigenvalues of the operator $\mathcal{A}$. Let $\sigma_n^-=\displaystyle\min_{\eta\in Y^{'}}\lambda_n(\eta)$ and $\sigma_n^+=\displaystyle\max_{\eta\in Y^{'}}\lambda_n(\eta)$, then, the spectrum of the operator $\mathcal{A}$ is given by $\bigcup_{n\in\mathbb{N}}[\sigma_n^-,\sigma_n^+]$. Therefore, it is a union of closed intervals, which may overlap. However, it may also be written as $[0,\infty)\setminus\sqcup_{j=1}^N (\mu_j^-,\mu_j^+)$, where $N$ takes values in $\mathbb{N}\cup\{\infty\}$. The pairwise disjoint intervals $(\mu_j^-,\mu_j^+)$ are known as spectral gaps and $(\mu_j^{\pm})_{j=1}^N$ are known as spectral edges. As depicted in Fig. \[figure1\], $\sigma_n^\pm$ may not be spectral edges, even though the corresponding Bloch eigenvalue is simple.
In the first part of the paper, we will study regularity of Bloch eigenvalues near the points where the spectral edge is attained in the dual parameter space. Regularity properties of the Bloch eigenvalues in the parameter are important in applications in the theory of effective mass [@AllairePiatnitski2005] and Bloch wave method in homogenization [@Conca1997]. For periodic Schrödinger operators, Wilcox [@Wilcox78] proved that outside a set of measure zero in the dual parameter space, the Bloch eigenvalues are analytic and the Bloch eigenfunctions may be chosen to be analytic. However, this measure zero set might intersect the spectral edge, which would limit applicability of such a result. For the linear elasticity operator, the Bloch eigenvalues are not analytic near the bottom spectral edge, which poses a major difficulty in the passage to limit in the Bloch wave method of homogenization [@SivajiGanesh2005].
Study of parametrized eigenvalue problems is an active area of research, even in finite dimensions [@michor98], [@Rainer2014]. Broadly speaking, regularity results for parametrized eigenvalues of selfadjoint operators are available in two cases: (i) for one parameter eigenvalue problems [@Rellich1969], [@Kato1995]. (ii) for simple eigenvalues, regardless of the number of parameters. Multiple parameters are unavoidable in most applications of interest. Examples include propagation of singularities for hyperbolic systems of equations with multiple characteristics leading to novel phenomena such as conical refraction [@lax1982], [@dencker1988], stability of hyperbolic initial-boundary-value problems [@Zumbrun2005] and Bloch waves for elasticity system [@SivajiGanesh2005]. Hence, an assumption of simplicity is useful in applications [@AllaireVanni2004], [@AllaireRauch2011], [@AllaireRauch2013].
In the literature, it has been shown that under perturbations of some relevant parameters like domain shape, coefficients, potentials etc, a multiple eigenvalue can be made simple. In a well-known paper [@Albert1975], Albert proves that, for a compact manifold $M$, the set of all smooth potentials $V\in C^{\infty}(M)$ for which the operator $-\Delta+V$ has only simple eigenvalues is a residual set in the space of all smooth admissible potentials. Similar results were proved by Uhlenbeck [@Uhlenbeck1976] using topological methods. Generic simplicity of the spectrum with respect to domain has been established and applied in proving stabilizability and controllability results for the plate equation [@Zuazua2000] and the Stokes system in two dimensions [@Zuazua2001] by Ortega and Zuazua.
We intend to generalize Albert’s method to the spectrum of periodic operators. Albert’s result is applicable to operators with discrete spectrum, whereas a periodic operator typically has no eigenvalues. The symmetries of the periodic operator allow us to write it as a direct integral of operators with compact resolvent. Hence, the method of Albert may be applied in a fiberwise manner. However, the fiber is an operator with complex-valued coefficients. Further, the perturbation is sought in the second-order term as opposed to the zeroth-order term in [@Albert1975]. In this paper, we overcome these difficulties and prove that the Bloch eigenvalues can be made simple locally in the parameter through a perturbation in the coefficients of the operator $\mathcal{A}$. Further, by applying fiberwise perturbation on the fibered operator $\int^{\bigoplus}_{Y^{'}}\mathcal{A}(\eta)d\eta$, we can make sure that the corresponding eigenvalue of interest is simple for all parameter values.
![Bloch eigenvalues $\lambda_4$ and $\lambda_5$ are simple, but have no spectral gap between them.[]{data-label="figure1"}](1.pdf)
The latter part of the paper is concerned with the theory of Bloch wave homogenization. In homogenization, one studies the limits of solutions to equations with highly oscillatory coefficients, such as $$\begin{aligned}
\label{anotherequation}
-\nabla\cdot\left(A\left(\frac{x}{\epsilon}\right)\nabla u^{\epsilon}\right)+\varkappa^2u^{\epsilon}=f~~\mbox{ in }\mathbb{R}^d,\end{aligned}$$
for $f\in L^2(\mathbb{R}^d)$ and $\varkappa>0$.
Suppose that $u^\epsilon$ converges weakly in $H^1(\mathbb{R}^d)$ to $u^{*}$. Then, the theory of homogenization [@Tartar2009], [@Bensoussan2011] shows that the limit $u^{*}$ solves an equation of the same type and identifies the matrix $A^*$: $$\begin{aligned}
-\nabla\cdot\left(A^*\nabla u^{*}\right)+\varkappa^2u^{*}=f~~\mbox{ in }\mathbb{R}^d.\end{aligned}$$ Bloch wave method of homogenization achieves this characterization through regularity properties of Bloch eigenvalues at the bottom of the spectrum. In particular, the homogenized matrix $A^{*}$ is characterized by the Hessian of the lowest Bloch eigenvalue at $0\in Y^{'}$ [@Conca1997]. Similarly, in the theory of internal edge homogenization [@Birman2006], the following regularity properties of Bloch eigenvalues near the spectral edge play an important role in obtaining operator error estimates:
1. The spectral edge must be simple, i.e., it is attained by a single Bloch eigenvalue.
2. The spectral edge must be attained at finitely many points by a Bloch eigenvalue.
3. The spectral edge must be non-degenerate, i.e., for some $m,r\in\mathbb{N}$, if the Bloch eigenvalue $\lambda_m(\eta)$ attains the spectral edge $\lambda_0$ at the points $\{\eta_j\}_{j=1}^r$, then the Bloch eigenvalue must satisfy, for $j=1,2,\ldots,r$, $$\begin{aligned}
\lambda_{m}(\eta)-\lambda_0=(\eta-\eta_j)^T B_j (\eta-\eta_j)+{O}(|\eta-\eta_j|^3), \mbox{ for } \eta \mbox{ near } \eta_j,
\end{aligned}$$ where $B_j$ are positive definite matrices.
While these features are readily available for the lowest Bloch eigenvalue corresponding to the divergence-type scalar elliptic operator, these properties may not be available for other spectral gaps of the same operator [@Kuchment]. However, the following results are available regarding these properties: Klopp and Ralston [@KloppRalston00] proved the simplicity of a spectral edge of Schrödinger operator $-\Delta+V$ under perturbation of the potential term. In two dimensions, spectral edges are known to be isolated [@Filonov15]. Also, in two dimensions, a degenerate spectral edge can be made non-degenerate through a perturbation with a potential having a larger period [@ParShteren17].
The validity of hypotheses (A), (B), (C) is usually assumed in the literature [@Kuchment]; for example, in establishing Green’s function asymptotics [@KuchmentRaich2012], [@KhaKuchmentRaich2017], for internal edge homogenization [@Birman2006] and to establish localization for random Schrödinger operators [@Veselic2002]. Local simplicity of Bloch eigenvalues is assumed in the study of diffractive geometric optics [@AllaireRauch2011], [@AllaireRauch2013] and homogenization of periodic systems [@AllaireVanni2004]. Following Klopp and Ralston [@KloppRalston00], we apply a perturbation to the coefficients of the operator $\mathcal{A}$ so that a multiple spectral edge becomes simple, under the condition that the coefficients are in $W^{1,\infty}$. However, if the coefficients of $\mathcal{A}$ are in $L^\infty$, a multiple spectral edge can be made simple through a small perturbation of the coefficients with the added assumption that the spectral edge is attained at only finitely many points. Thus, our results suggest a possible interplay between the validity of these assumptions and the regularity of the coefficients. Further, these spectral tools also allow us to achieve homogenization at an internal edge in the presence of multiplicty.
More details on the spectrum of elliptic periodic operators may be found in Reed and Simon [@Reed1978] and for state of the art on periodic differential operators, see the review by Kuchment [@Kuchment].
Main Results
------------
Let $S\!ym(d)$ denote the space of all real symmetric matrices, i.e., if $A=(a_{kl})\in S\!ym(d)$, then $a_{kl}=a_{lk}$. Let $$M_B^{>}=\{A:\mathbb{R}^d\to{S\!ym}(d):a_{kl}\in L^{\infty}_\sharp(Y,\mathbb{R}) \mbox{ and $A$ is coercive }\}.$$
$M_B^{>}$ may be identified as a subset of the space of $d(d+1)/2$-tuples of $L^{\infty}_\sharp$ functions and we shall use the norm-topology on this space in our further discussion. A Baire space is a topological space in which the countable intersection of dense open sets is dense. Note that $M_B^{>}$ is an open subset of the space of all symmetric matrices with $L^\infty_\sharp(Y,\mathbb{R})$ entries, which forms a complete metric space, and hence $M_B^{>}$ is a Baire Space. We shall call a property [*generic*]{} in a topological space $X$, if it holds on a set whose complement is of first category in $X$. In particular, a property that is generic on a Baire space holds on a dense set.
The rest of the subsection will be devoted to the statements of the main results.
\[theorem:1\] Let $\eta_0\in Y^{'}$. The eigenvalues of the shifted operator $\mathcal{A}(\eta_0)$ are generically simple with respect to the coefficients $A=\left(a_{kl}\right)_{k,l=1}^d$ in $M_B^{>}$.
Theorem \[theorem:1\] is an extension of the theorem of Albert [@Albert1975] which proves that the eigenvalues of $-\Delta+V$ are generically simple with respect to $V\in C^{\infty}(M)$ for a compact manifold $M$. The potential $V$ is the quantity of interest for Schrödinger operator, $-\Delta+V$. For the applications that we have in mind, for example, the theory of homogenization, the periodic matrix $A$ in the divergence type elliptic operator $-\nabla\cdot(A\nabla)$ is of physical importance. The spectrum of such operators is not discrete, and is analyzed through Bloch eigenvalues, which introduces an extra parameter $\eta\in Y^{'}$ to the problem. The determination of real-valued perturbation for the shifted operator $\mathcal{A}(\eta)$, which has complex-valued coefficients, poses additional difficulties, when coupled with the lack of regularity of the coefficients which the applications demand.
\[theorem:2\] Let $m\in\mathbb{N}$, then for the Bloch eigenvalue $\lambda_m(\eta)$ of the periodic operator $\mathcal{A}=-\nabla\cdot(A\nabla)$, where $A\in M_B^>$, there exists a perturbation of $\mathcal{A}$ such that the perturbed eigenvalue $\tilde{\lambda}_m(\eta)$ is simple for all $\eta\in Y^{'}$.
A spectral edge $\lambda_0$ is said to be [*simple*]{} if the set $\displaystyle\{m\in\mathbb{N}:\exists\,\eta\in Y^{'}\mbox{ such that }\lambda_m(\eta)=\lambda_0\}$ is a singleton. A spectral edge is said to be multiple if it is not simple.
\[theorem:3\] Let $A\in M_B^>$. Further, suppose that its entries $A=\left(a_{kl}\right)_{k,l=1}^d$ belong to the class $W^{1,\infty}_\sharp(Y,\mathbb{R})$. Then, a multiple spectral edge of the operator $\mathcal{A}=\displaystyle -\nabla\cdot (A\nabla)$ can be made simple by a small perturbation in the coefficients.
\[theorem:4\] Let $A\in M_B^>$. Further, suppose that its entries $A=\left(a_{kl}\right)_{k,l=1}^d$ belong to the class $L^{\infty}_\sharp(Y,\mathbb{R})$. Let $\lambda_0$ correspond to the upper edge of a spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$. Assume that the spectral edge is attained by $\lambda_m(\eta)$ at finitely many points. Then, there exists a matrix $B=(b_{kl})_{k,l=1}^d$ with $L^\infty_\sharp(Y,\mathbb{R})$-entries and $t_0>0$ such that for every $t\in(0,t_0]$, a spectral edge is achieved by the Bloch eigenvalue $\lambda_m(\eta;A+tB)$ of the operator $\mathcal{A}=\displaystyle -\nabla\cdot (A+tB)\nabla$ and the spectral edge is simple.
1. While Theorem \[theorem:2\] achieves global simplicity for a Bloch eigenvalue, the perturbed operator is no longer a differential operator, i.e., it is non-local. In the theory of homogenization, non-local terms usually appear as limits of non-uniformly bounded operators [@Briane2002], [@BrianeCalc2002]. In the presence of crossing modes, non-locality appears in the theory of effective mass [@Chabu2018].
2. Theorem \[theorem:3\] is an adaptation of the theorem of Klopp and Ralston [@KloppRalston00] to divergence-type operators. Their proof relies heavily on the Hölder regularity for weak solutions of divergence-type operators. In our proof, we require Hölder continuity of the solutions as well as their derivatives. Hence, we have to impose $W^{1,\infty}$ condition on the coefficients.
3. In Theorem \[theorem:4\], we weaken the $W^{1,\infty}$ requirement on the coefficients under assumption of finiteness on the number of points at which the spectral edge is attained. This is essential for the applications that we have in mind, in the theory of homogenization, where only $L^\infty$ regularity is available on the coefficients.
We shall also prove a theorem on internal edge homogenization, whose complete statement is deferred to Section \[homogen\]. Let $A\in M_B^>$. Birman and Suslina [@Birman2006] propose an effective operator and prove operator error estimates with respect to the operator norm in $L^2(\mathbb{R}^d)$ for the limit as $\epsilon\to 0$ of the operator $\mathcal{A}^\epsilon\coloneqq-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$, at a non-zero spectral edge $\lambda_0$ under the regularity hypotheses $(A), (B), (C)$.
Under appropriate modifications of the regularity hypotheses on the spectral edge, an effective operator is proposed as an approximation of the operator $\mathcal{A}^\epsilon$ in the limit $\epsilon\to 0$ at a multiple spectral edge, and operator error estimates with respect to the operator norm in $L^2(\mathbb{R}^d)$ are proved.
Multiplicity of Bloch eigenvalues is a crucial difficulty in Bloch wave homogenization. The internal edge homogenization result is an attempt at circumventing this issue. Previously, this was handled by use of directional analyticity of Bloch eigenvalues for the linear elasticity operator whose lowest Bloch eigenvalue has multiplicty $3$ [@SivajiGanesh2005].
1. Bloch wave method belongs to the family of multiplier techniques in partial differential equations. In particular, exponential type multipliers, $e^{\tau\phi}$, with real exponents, are used in obtaining Carleman estimates for elliptic operators [@rousseau12].
2. Any operator of the form $-\nabla\cdot A\nabla$ in $L^2(\mathbb{R}^d)$ may be written in direct integral form, provided $A$ is periodic. A satisfactory spectral theory for such operators is available for real symmetric $A$. However, non-selfadjoint operators are becoming increasingly important in physics [@Sjoestrand09]. For non-symmetric $A$, the eigenvalues of the fibers $\mathcal{A}(\eta)$ may no longer be real and the eigenfunctions may not form a complete set. These difficulties were surmounted in proving the Bloch wave homogenization theorem for non-selfadjoint operators in [@Sivaji2004]. Nevertheless, the generalized eigenfunctions form a complete set for a large class of elliptic operators of even order [@agmon62]. However, we are not aware of physical interpretations of complex-valued Bloch-type eigenvalues.
3. Most of the results of this paper would have similar analogues for internal edges of an elliptic system of equations, for example, the elasticity system. It would be interesting to consider these problems for the spectrum of non-elliptic operators such as the Maxwell operator.
The plan of this paper is as follows; in Section \[local\_simplicity\], we prove Theorem \[theorem:1\] on generic simplicity of Bloch eigenvalues at a point. In Section \[global\_simplicity\], we prove Theorem \[theorem:2\] and in subsequent sections \[simplicity\_edge\_1\] and \[simplicity\_edge\_2\], we prove Theorems \[theorem:3\] and \[theorem:4\] concerning generic simplicity of spectral edges. In the final section \[homogen\], we give a short introduction to internal edge homogenization and furnish an application of perturbation theory to Bloch wave homogenization by proving Theorem \[theorem:5\].
Local Simplicity of Bloch eigenvalues {#local_simplicity}
=====================================
Let $\eta_0\in Y^{'}$. Let $P$ be the set defined by $$P\coloneqq\{A\in M_B^{>}\mbox{ : the eigenvalues of $\mathcal{A}(\eta_0)$ are simple} \}.$$
We can write the set $P$ as an intersection of countably many sets as follows: Let $P_0:=M_B^{>}$, and $$\begin{aligned}
P_n&:=\{A\in M_B^{>}: \mbox{ the first $n$ eigenvalues of $\mathcal{A}(\eta_0)$ } \mbox{are simple} \}.\\
&=\{A\in M_B^>: \lambda_1(\eta_0)<\ldots<\lambda_n(\eta_0)<\lambda_{n+1}(\eta_0)\leq\lambda_{n+2}(\eta_0)\leq\ldots\}.\end{aligned}$$
Note that, $$\begin{aligned}
P \subseteq \ldots \subseteq P_n\subseteq P_{n-1}\subseteq\ldots\subseteq P_1\subseteq P_0
&\qquad\mbox{and}\qquad P=\bigcap_{n=0}^{\infty}P_n.\end{aligned}$$ We shall require the following two lemmas.
\[lemma:11\] $P_n$ is open in $M_B^{>}$ for all $n\in\mathbb{N}\cup\{0\}$.
\[lemma:22\] $P_{n+1}$ is dense in $P_n$, for all $n\in \mathbb{N}\cup\{0\}$.
(of Theorem \[theorem:1\]) We recall that a property is said to be generic in a topological space $X$, if it holds on a set whose complement is of first category in $X$. We can write $P$ as the countable intersection $P=\displaystyle\bigcap_{n=0}^\infty P_n$, where $P_n$ is an open and dense set in $M_B^>$ for all $n\in\mathbb{N}\cup\{0\}$. Hence, the complement of $P$ is a set of first category. Therefore, the simplicity of eigenvalues of $\mathcal{A}(\eta_0)$ is a generic property in $M_B^>$.
The rest of this section is devoted to the proofs of Lemmas \[lemma:11\] and \[lemma:22\].
Proof of Lemma \[lemma:11\]
---------------------------
In this subsection, we begin by proving continuous dependence of the eigenvalues of the shifted operator $\mathcal{A}(\eta)$ on its coefficients. The main tool in this proof is Courant-Fischer min-max principle, which states that
$$\begin{aligned}
\lambda_m(\eta_0)=\min_{\dim F =m}\max_{v\in F}\frac{\int_{Y} A(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dx}{\int_{Y} v^2~dx},\end{aligned}$$
where $F$ ranges over all subspaces of $H^1_\sharp(Y)$ of dimension $m$.
Let $A_1,A_2\in M_B^{>}$ and let $\eta\mapsto\lambda_n^1(\eta), \eta\mapsto\lambda^2_n(\eta)$ be the n-th Bloch eigenvalues of the operators $\mathcal{A}_1$ and $\mathcal{A}_2$ respectively. Then $$\begin{aligned}
|\lambda_n^1(\eta_0)-\lambda^2_n(\eta_0)|\leq d c_n(\eta_0) ||A_1-A_2||_{L^\infty},\end{aligned}$$ where $c_n(\eta_0)$ is the $n^{th}$ eigenvalue of the shifted Laplacian $-(\nabla+i\eta_0)^2$ on $Y$ with periodic boundary conditions.
Let $a_1(v)=\int_{Y}A_1(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dy$ and $a_2(v)=\int_{Y}A_2(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dy$ be the quadratic forms that appear in the min-max principle.
$$\begin{aligned}
|a_1(v)-a_2(v)|&=\left|\int_{Y}(A_1-A_2)(\nabla+i\eta_0) v.\overline{(\nabla+i\eta_0) v}~dy\right|\\
&\leq d||A_1-A_2||_{L^\infty}\int_{Y}|(\nabla+i\eta_0) v|^2~dy,
\end{aligned}$$
Therefore, $$\begin{aligned}
a_1(v)\leq a_2(v)+d||A_1-A_2||_{L^\infty}\int_{Y}|(\nabla+i\eta_0) v|^2~dy. \end{aligned}$$
Now, divide both sides by $\int_Y |v|^2~dy$, the $L^2_\sharp(Y)$ inner product of $v$ with itself and apply the appropriate min-max to obtain $$\begin{aligned}
\lambda^1_m(\eta_0)\leq \lambda^2_m(\eta_0)+d c_m(\eta_0)||A_1-A_2||_{L^\infty}.\end{aligned}$$
Notice that the constant $c_m(\eta_0)$ is precisely the $m^{th}$ eigenvalue of the shifted Laplacian $-(\nabla+i\eta_0)^2$ on $Y$ with periodic boundary conditions. By interchanging the role of $A_1$ and $A_2$, the inequality $$\begin{aligned}
\lambda^2_m(\eta_0)\leq \lambda^1_m(\eta_0)+d c_m(\eta_0)||A_2-A_1||_{L^\infty},\end{aligned}$$ is obtained, which completes the proof of this proposition.
In [@Conca1997], the Bloch eigenvalues have been proved to be Lipschitz continuous in $\eta\in Y^{'}$. Indeed, one may prove that the Bloch eigenvalues are jointly continuous in $\eta\in Y^{'}$ and the coefficients of the operator.
Let $A\in P_n$ and $$\begin{aligned}
\delta=\min\{\lambda_{j+1}(\eta_0)-\lambda_{j}(\eta_0):j=1,2,\ldots,n\}.\end{aligned}$$
Let $\displaystyle c=\max_{1\leq j\leq n}d c_j(\eta_0)$, where $c_j(\eta_0)$ is the $j^{th}$ eigenvalue of the shifted Laplacian $-(\nabla+i\eta_0)^2$ on $Y$ with periodic boundary conditions.
Let $$\begin{aligned}
U=\left\{A'\in M_B^>:||A-A'||_{L^\infty}< \frac{\delta}{4c}\right\}.\end{aligned}$$
$U$ is an open set in $M_B^>$ containing $A$. We shall show that $U$ is a subset of $P_n$. Let $A'\in
U$. Let $\{\lambda'_j(\eta),~j=1,2,\ldots\}$ be the Bloch eigenvalues of operator $\mathcal{A}'$ associated to $A'$. For $j=1,2,\ldots,n$, we have: $$\begin{aligned}
|\lambda'_j(\eta_0)-\lambda_j(\eta_0)| & \leq dc_j(\eta_0)||A-A'||_{L^\infty} \leq dc_j(\eta_0)\frac{\delta}{4c}\leq\frac{\delta}{4}.
\end{aligned}$$
Hence, $$\begin{aligned}
\delta & \leq \lambda_{j+1}(\eta_0)-\lambda_j(\eta_0)\\
& \leq |\lambda'_{j+1}(\eta_0)-\lambda_{j+1}(\eta_0)|+|\lambda'_j(\eta_0)-\lambda'_{j+1}(\eta_0)|+|\lambda'_j(\eta_0)-\lambda_j(\eta_0)|\\
& \leq \frac{\delta}{4}+ |\lambda'_j(\eta_0)-\lambda'_{j+1}(\eta_0)|+\frac{\delta}{4}\\
& = \frac{\delta}{2}+\lambda'_{j+1}(\eta_0)-\lambda'_j(\eta_0).
\end{aligned}$$
Therefore, $\lambda'_{j+1}(\eta_0)-\lambda'_j(\eta_0)\geq\frac{\delta}{2}>0$ for $j=1,2,\ldots,n.$ Therefore, the first $n$ Bloch eigenvalues of $\mathcal{A}'$ are simple at $\eta_0$, as required.
Proof of Lemma \[lemma:22\]
---------------------------
In this section, we shall use perturbation theory of selfadjoint operators to prove Lemma \[lemma:22\]. Let $A\in M_B^>$ and $B$ be a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries. For $|\tau|<\sigma_0\coloneqq\frac{\alpha}{2d||B||_{L^\infty}}$, $A+\tau B\in M^>_B$, where $\alpha$ is a coercivity constant for $A$ as in . Consider the operator $\mathcal{A}(\eta_0)+\tau\mathcal{B}(\eta_0)$ in $L^2_\sharp(Y)$. We shall prove in Appendix \[PerturbationTheory\], that the operator family $\mathcal{F}(\tau)=\mathcal{A}(\eta_0)+\tau\mathcal{B}(\eta_0)$ is a selfadjoint holomorphic family of type $(B)$ for $|\tau|<\sigma_0$. For its definition and related notions, see Kato [@Kato1995].
We shall make use of the following theorem which asserts the existence of a sequence of eigenpairs associated with a selfadjoint holomorphic family of type $(B)$, analytic in $\tau\in(-\sigma_0,\sigma_0)$. The proof of this theorem dates back to Rellich, hence we shall call these eigenvalue branches as Rellich branches.
(Kato-Rellich)\[katorellich\] Let $\mathcal{A}(\eta_0)(\tau)$ be a selfadjoint holomorphic family of type $(B)$, defined for $\tau\in R$ where $R=\{z\in\mathbb{C}:|\operatorname{Re}({z})|<\sigma_0,|\operatorname{Im}({z})|<\sigma_0\}$ and $\sigma_0\coloneqq \frac{\alpha}{2d||B||_{L^\infty}}$. Let $\mathcal{A}(\eta_0)(\tau)+C_*I$ have compact resolvent for some $C_*\in \mathbb{R}$. Then, there exists a sequence of scalar-valued functions $(\lambda_j(\tau;\eta_0))_{j=1}^\infty$ and $L^2_\sharp(Y)$-valued functions $(u_j(\tau;\eta_0))_{j=1}^\infty$ defined on $I=(-\sigma_0,\sigma_0)$, such that
1. For each fixed $\tau\in I$, the sequence $(\lambda_j(\tau;\eta_0))_{j=1}^\infty$ represents all the eigenvalues of $\mathcal{A}(\eta_0)(\tau)$ counting multiplicities and the functions $(u_j(\tau;\eta_0))_{j=1}^\infty$ represent the corresponding eigenvectors.
2. For each $j\in\mathbb{N}$, the functions $(\lambda_j(\tau;\eta_0))_{j=1}^\infty$ and $(u_j(\tau;\eta_0))_{j=1}^\infty$ are analytic on $I$ with values in $\mathbb{R}$ and $L^2_\sharp(Y)$ respectively.
3. The sequence $(u_j(\tau;\eta_0))_{j=1}^\infty$ is orthonormal in $L^2_\sharp(Y).$
4. Suppose that the $m^{th}$ eigenvalue of $\mathcal{A}(\eta_0)(\tau)$ at $\tau=0$ has multiplicity $p$, i.e., $$\begin{aligned}
\lambda_m(0;\eta_0)=\lambda_{m+1}(0;\eta_0)=\ldots=\lambda_{p+m-2}(0;\eta_0)=\lambda_{p+m-1}(0;\eta_0).\end{aligned}$$ For each interval $K\subset\mathbb{R}$ with $\overline{K}$ containing the eigenvalue $\lambda_m(0;\eta_0)$ and no other eigenvalue, $\lambda_m(\tau;\eta_0), $ $\lambda_{m+1}(\tau;\eta_0),\ldots, \lambda_{p+m-1}(\tau;\eta_0)$ are the only eigenvalues of $\mathcal{A}(\eta_0)(\tau)$, counting multiplicities, lying in the interval $K$.
By Kato-Rellich Theorem, an eigenvalue $\lambda(\eta_0)$ of $\mathcal{F}(0)$ of multiplicity $h$, splits into $h$ analytic functions $(\lambda_m(\tau;\eta_0))_{m=1}^h$. Further, the corresponding eigenfunctions $(u_m(\tau;\eta_0))_{m=1}^h$ are also analytic. Let the $h$ eigenvalues and eigenvectors of $\mathcal{F}(\tau)$ have the following power series expansions at $\tau=0$ for $m=1,2,\ldots,h$: $$\begin{aligned}
\lambda_m(\tau;\eta_0)=\lambda(\eta_0)+\tau a_m(\eta_0)+\tau^2\beta_m(\tau,\eta_0)\notag\\
u_{m}(\tau;\eta_0)=u_m(\eta_0)+\tau v_m(\eta_0)+\tau^2 w_m(\tau,\eta_0).\end{aligned}$$
The proof of Lemma \[lemma:22\] will rely on the fact that we may choose $B$ in such a way that $a_m(\eta_0)\neq a_n(\eta_0)$ for some $m,n\in\{1,2,\ldots,h\}$. Then, for sufficiently small $\tau$, $\lambda_m(\tau;\eta_0)\neq\lambda_n(\tau;\eta_0)$. In that case, the multiplicity of the perturbed Bloch eigenvalue at $\eta_0$ will be less than $h$.
The eigenpairs satisfy the following equation: $$\begin{aligned}
\left(-(\nabla+i\eta_0)\cdot (A+\tau B)(\nabla+i\eta_0)-\lambda_m(\tau,\eta_0)\right)u_m(\tau,\eta_0)=0.\end{aligned}$$
Differentiating the above with respect to $\tau$ and setting $\tau$ to $0$, we obtain: $$\begin{aligned}
-(\nabla+i\eta_0)\cdot A(\nabla+i\eta_0)v_m(\eta_0)-(\nabla+i\eta_0)\cdot B(\nabla+i\eta_0)u_m(\eta_0)-\lambda(\eta_0)v_m(\eta_0)\notag-a_m(\eta_0)u_m(\eta_0)=0\end{aligned}$$
Finally, multiply by $u_n(\eta_0)$ and integrate over $Y$ to conclude that $$\begin{aligned}
\label{hellmanfeynman}
\int_Y B(\nabla+i\eta_0)u_m(\eta_0)\cdot(\nabla-i\eta_0)\overline{u_n(\eta_0)}~dy=a_m(\eta_0)\delta_{mn}.\end{aligned}$$
Equation suggests the following construction. Given a perturbation $B$ and a basis $F=\{f_1,f_2,\ldots,f_h\}$ for the unperturbed eigenspace $N(\eta_0)\coloneqq ker(\mathcal{A}(\eta_0)-\lambda(\eta_0)I)$, we can define a selfadjoint operator $G_B$ on $N(\eta_0)$ whose matrix in the basis $F$ is given by $$\begin{aligned}
\left([G_B]_F\right)_{m,n}\coloneqq\int_Y B(\nabla+i\eta_0)f_m\cdot(\nabla-i\eta_0)\overline{f_n}~dy.\end{aligned}$$
In particular, it follows from equation that in the basis of unperturbed eigenfunctions $E=\{u_1(\eta_0), u_2(\eta_0),\ldots,u_h(\eta_0)\}$, $[G_B]_E$ is a diagonal matrix, $$\begin{aligned}
_E=diag(a_1(\eta_0),a_2(\eta_0),\ldots,a_h(\eta_0)).\end{aligned}$$ If $[G_B]_E$ is a scalar matrix, then the operator $G_B$ is a scalar multiple of identity operator. However, if we can find a basis $F$ for the eigenspace and a matrix $B$, corresponding to which, the matrix $[G_B]_F$ has a non-zero off-diagonal entry, then for that choice of $B$, $[G_B]_E$ will not be a scalar matrix, and hence, $a_m(\eta_0)\neq a_n(\eta_0)$ for some $m,n\in\{1,2,\ldots,h\}$.
\[find B\] There exists a symmetric matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$-entries such that the operator $G_B$ is not a scalar multiple of identity.
As noted earlier, the proposition will be proved if we can find a basis $F$ and a matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$ entries, such that the matrix $[G_B]_F$ has a non-zero off-diagonal entry.
Let $F=\{f_1,f_2,\ldots,f_h\}$ be any basis of $ker(\mathcal{A}(\eta_0)-\lambda(\eta_0)I)$. Suppose that for some $j\in\{1,2,\ldots,d\}$, $$\begin{aligned}
\label{alternative1}
(\partial_j+i\eta_{0,j})f_1(\partial_j-i\eta_{0,j})\overline{f_2}\not\equiv 0,
\end{aligned}$$ where $\eta_0=(\eta_{0,1},\eta_{0,2},\ldots,\eta_{0,d})$. Since, $f_i\in H_\sharp^1(Y)$, $g\coloneqq(\partial_j+i\eta_{0,j})f_1(\partial_j-i\eta_{0,j})\overline{f_2}\in L^1_\sharp(Y)$. Hence, by Hahn-Banach Theorem, there is a continuous linear functional $\kappa\in(L^1_\sharp(Y))^*$, such that $\kappa(g)=||g||\neq 0.$ However, by duality, there exists a $\beta\in L^\infty_\sharp(Y)$, such that $\kappa(g)=\int_Y \beta\,gdy=||g||\neq 0.$
Now, either $\int_Y \operatorname{Re}(\beta)g\neq 0$ or $\int_Y \operatorname{Im}(\beta)g\neq 0$. Suppose, without loss of generality that $\int_Y \operatorname{Re}(\beta)g\neq 0$ and define $$\begin{aligned}
B=diag(0,0,\ldots,0,\operatorname{Re}(\beta),0,\ldots,0)\end{aligned}$$ with $\operatorname{Re}(\beta)$ in the $j^{th}$ place, then $$\begin{aligned}
([G_B]_F)_{1,2}&=\int_Y B(\nabla+i\eta_0)f_1\cdot(\nabla-i\eta_0)\overline{f_2}~dy\\
&=\int_Y\operatorname{Re}(\beta)(\partial_j+i\eta_{0,j})f_1(\partial_j-i\eta_{0,j})\overline{f_2}~dy\\
&=\int_Y\operatorname{Re}(\beta) g~dy\neq 0.
\end{aligned}$$ Alternatively, if $(\nabla+i\eta_0)f_1\cdot(\nabla-i\eta_0)\overline{f_2}\equiv 0$, then there exists $j\in\{1,2,\ldots,d\}$, such that $$\begin{aligned}
\label{alternative2}
|(\partial_j+i\eta_{0,j})f_1|^2-|(\partial_j+i\eta_{0,j})f_2|^2\not\equiv 0.
\end{aligned}$$ It is easy to see that if and do not hold, then $f_1$ and $f_2$ are both a scalar multiple of $\exp(i\eta_0\cdot y)$, which contradicts the fact that they are distinct elements of basis of $N=ker(\mathcal{A}(\eta_0)-\lambda(\eta_0)I)$.
Since, for all $m=1,2,\ldots,h$, $f_m\in H_\sharp^1(Y)$, $g{'}\coloneqq|(\partial_j+i\eta_{0,j})f_1|^2-|(\partial_j+i\eta_{0,j})f_2|^2\in L^1_\sharp(Y,\mathbb{R})$. Hence, by Hahn-Banach Theorem, there is a continuous linear functional $\kappa{'}\in(L^1_\sharp(Y,\mathbb{R}))^*$, such that $\kappa{'}(g{'})=||g{'}||\neq 0.$ However, by duality, there exists a $\beta{'}\in L^\infty_\sharp(Y,\mathbb{R})$, such that $\kappa{'}(g{'})=\int_Y \beta{'}g{'}=||g{'}||\neq 0.$
Define $$\begin{aligned}
B=diag(0,0,\ldots,0,\beta{'},0,\ldots,0)\end{aligned}$$ with $\beta{'}$ in the $j^{th}$ place, then in the new basis $F^{'}=\{f_1+f_2,f_1-f_2,f_3,\ldots,f_h\}$, the $(1,2)^{th}$ entry of $[G_B]_{F^{'}}$ is given by $$\begin{aligned}
\int_Y B(\nabla+i\eta_0)(f_1+f_2)\cdot(\nabla-i\eta_0)(\overline{f_1}-\overline{f_2})~dy = \int_Y \beta{'}|(\partial_j+i\eta_{0,j})f_1|^2-|(\partial_j+i\eta_{0,j})f_2|^2~dy\neq 0.
\end{aligned}$$ Thus, either way, we have found a basis in which an off-diagonal entry of $[G_B]_F$ is non-zero. Hence, the operator $G_B$ is not a scalar multiple of identity. In particular, the matrix $[G_B]_E$ cannot be a scalar matrix.
Let $A\in P_n$. Given $\epsilon>0$, we want to find $A'\in P_{n+1}$ such that $||A-A'||_{L^\infty}<\epsilon$. We shall construct $A'$ in the form $A'=A+\tau B$, where $B$ is a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries and $\tau\in\mathbb{R}$. By Lemma \[lemma:11\], we can choose $\tau_0$ so that $A+\tau B\in P_n$ for $|\tau|<\tau_0$. Hence, the first $n$ eigenvalues of the operator $-(\nabla+i\eta_0)\cdot(A+\tau B)(\nabla+i\eta_0)$ are simple for $|\tau|<\tau_0$. Subsequently, we must choose $\tau$ such that $|\tau|<\sigma_0=\frac{\alpha}{2d||B||_{L^\infty}}$, in order to apply the Kato-Rellich Theorem. Now, suppose that the $(n+1)^{th}$ eigenvalue of $\mathcal{A}(\eta_0)$ has multiplicity $h$. By Kato-Rellich Theorem (Theorem \[katorellich\]), the $h$ eigenvalue branches of the perturbed operator $\mathcal{A}(\eta_0)+\tau\mathcal{B}(\eta_0)$ are given by the following power series at $\tau=0$, for $r=1,2,\ldots,h$: $$\begin{aligned}
\lambda_r(\tau;\eta_0)=\lambda(\eta_0)+\tau a_r(\eta_0)+\tau^2\beta_r(\tau;\eta_0).
\end{aligned}$$ If there are $m,n\in\{1,2,\ldots,h\}$ such that $a_m(\eta_0)\neq a_n(\eta_0)$, then there is a $\tau_1$ such that, $\lambda_m(\tau;\eta_0)\neq\lambda_n(\tau;\eta_0)$ for $|\tau|<\tau_1$. Since two of the $h$ eigenvalue branches are distinct for small $\tau$, the multiplicity of the perturbed eigenvalue, which can only go down for small $\tau$, must be less than or equal to $h-1$. This can be achieved through an application of Proposition \[find B\] which gives us a matrix $B_1$ such that at least two of $(a_r(\eta_0))_{r=1}^h$ are distinct. Now, starting from the matrix $A+\tau_1 B_1$, we repeat the procedure above so that the multiplicity of the $(n+1)^{th}$ eigenvalue is further reduced. The perturbed matrix is now labelled $A+\tau_1B_1+\tau_2B_2$. Finally, after a finite number of such steps, we can reduce the multiplicity of the $(n+1)^{th}$ eigenvalue to $1$. At the end of this procedure, we obtain a matrix of the form $A'=A+\sum_{j=1}^N\tau_jB_j$, for some $N\in\mathbb{N}$. Each perturbation must be chosen so that $\sum_{j=1}^{N}\tau_j||B_j||_{L^\infty}<\epsilon$.
\[simple\_in\_neighborhood\] Theorem \[theorem:1\] proves that an eigenvalue $\lambda(\eta_0)$ of the shifted operator $\mathcal{A}(\eta_0)$ can be made simple by a perturbation of the matrix $A\in M_B^>$. However, since the Bloch eigenvalues are Lipschitz continuous functions of the parameter $\eta\in Y^{'}$ [@Conca1997], the perturbed eigenvalue $\tilde{\lambda}(\eta)$ will continue to remain simple in some neighborhood of $\eta_0$.
1. The perturbation formula may be thought of as a variation of the Hellmann-Feynman theorem in the physics literature. The coefficients of the differential operator are real-valued functions, in as much as they are related to properties of materials. The presence of complex-valued coefficients in the perturbation formula complicates the choice of the real-valued perturbation $B$.
2. In the theory of homogenization, the coefficients of the second order divergence-type periodic elliptic operator are usually only measurable and bounded. By regularity theory [@Ladyzhenskaya68], the eigenfunctions of the shifted operator $\mathcal{A}(\eta)$ are known to be Hölder continuous. However, derivatives of eigenfunctions, which may not be bounded, appear in the perturbation formula . Therefore, the perturbation $B$ is chosen using the Hahn-Banach Theorem.
Global Simplicity {#global_simplicity}
=================
In the previous section, we have proved that a given Bloch eigenvalue $\lambda_m(\eta)$ of the operator $\mathcal{A}$ can be made simple locally in $Y^{'}$ through a small perturbation in the coefficients. In this section, we shall perform perturbation on the operator $\mathcal{A}$ in such a way that its spectrum still retains the fibered character, i.e., $\sigma(\tilde{A})=\cup_{\eta\in Y^{'}}\sigma(\tilde{A}(\eta))$ and the $m^{th}$ eigenvalue function $\eta\mapsto\tilde{\lambda}_m(\eta)$ is simple for all $\eta\in Y^{'}$. However, the perturbed operator $\tilde{\mathcal{A}}$ may no longer be a differential operator.
The operator has a direct integral decomposition $\mathcal{A}=\int_{\eta\in\mathbb{T}^d}^{\bigoplus}\mathcal{A}(\eta)d\eta$ where $\mathcal{A}(\eta)=-(\nabla+i\eta)\cdot A(\nabla+i\eta)$ is an unbounded operator in $L^2_\sharp(Y)$. We would like to point out that $Y^{'}$ is understood to parametrize the torus, $\mathbb{T}^d$. Consider the $m^{th}$ Bloch eigenvalue $\lambda_m(\eta)$ of $\mathcal{A}$. By Lemma \[lemma:22\], at any point $\eta_0\in Y^{'}$, we can find a perturbation of the coefficients $A=(a_{kl})$ of $\mathcal{A}(\eta_0)$ so that the perturbed eigenvalue $\tilde{\lambda}_m(\eta_0)$ is simple. By Remark \[simple\_in\_neighborhood\], there is a neighborhood of $\eta_0$, $\mathcal{G}_{\eta_0}$ in which the perturbed eigenvalue $\tilde{\lambda}_m(\eta)$ of the perturbed shifted operator $\tilde{\mathcal{A}}(\eta)$ is simple. In this manner, for each $\xi\in \mathbb{T}^d$, we obtain a perturbation $B_{\xi}$ and a neighborhood, $\mathcal{G}_{\xi}$ in which the eigenvalue of the perturbed operator $\tilde{\mathcal{A}}(\eta)=-(\nabla+i\eta)\cdot (A+B_{\xi})(\nabla+i\eta)$ is simple. These sets form an open cover of the torus. By compactness of $\mathbb{T}^d$, there is a finite subcover having the property that in each member $\mathcal{G}_{\xi}$ of the subcover, the corresponding perturbation $B_{\xi}$ causes the perturbed eigenvalue $\tilde{{\lambda}}_m(\eta)$ to be simple in $\mathcal{G}_{\xi}$.
Let $\{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_n\}$ be the finite subcover of the torus obtained above. Define $\mathcal{O}_1=\mathcal{G}_1$. For $r\geq 1$, define $\mathcal{O}_{r+1}=\displaystyle\mathcal{G}_{r+1}\setminus\bigcup_{j=1}^r\mathcal{G}_j$. Suppose that $B_j$ is the perturbation corresponding to the set $\mathcal{O}_j$.
Now, define the parametrized operator $$\begin{aligned}
\tilde{\mathcal{A}}(\eta)=-(\nabla+i\eta)\cdot (A+\sum_{j=1}^n B_{j}\,\chi_{\mathcal{O}_j})(\nabla+i\eta)\end{aligned}$$ which depends measurably on $\eta\in\mathbb{T}^d$. Finally, define the direct integral $\tilde{\mathcal{A}}=\int_{\eta\in\mathbb{T}^d}^{\bigoplus}\tilde{\mathcal{A}}(\eta)d\eta$, where each of the fibers is a differential operator in $L^2_\sharp(Y)$. Then, it is known [@Reed1978 p.284] that, $$\sigma(\tilde{A})=\bigcup_{\eta\in Y^{'}}\sigma(\tilde{A}(\eta)).$$
Hence, we may define an $m^{th}$ eigenvalue function $\eta\mapsto\tilde{\lambda}_m(\eta)$ with the property that $$\begin{aligned}
|\lambda_m(\eta)-\tilde{\lambda}_m(\eta)|\leq C\max_{1\leq j\leq n}||B_j||_{L^{\infty}},\end{aligned}$$ where $\lambda_m(\eta)$ is the $m^{th}$ Bloch eigenvalue of $\mathcal{A}$.
1. Although the $m^{th}$ eigenvalue of the perturbed operator is simple for all parameter values, $\tilde{\lambda}_m(\eta)$ may only be measurable in $\eta\in Y^{'}$. However, $\tilde{\lambda}_m(\eta)$ is analytic in each $\mathcal{O}_j\subset \mathbb{T}^d$.
2. The perturbed operator $\tilde{\mathcal{A}}$ is no longer a differential operator, even though each fiber $\tilde{\mathcal{A}}(\eta)$ is a differential operator. In fact. we shall prove in Theorem \[notdiffop\] that $\tilde{\mathcal{A}}$ is a differential operator if and only if $B_1=B_2=\ldots=B_n.$
3. A rigorous account of direct integral decomposition of operators, such as the one employed above for periodic operators, may be found in [@schmudgen90] and [@maurin68].
\[notdifferential\] Let $B$ be a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries. Define $\mathcal{B}(\eta)=-(\nabla+i\eta)\cdot B(\nabla+i\eta)$. Let $\mathcal{O}\subset Y^{'}$ be a proper subset of $Y^{'}$. Then, the direct integral defined by $\mathcal{B}=\int^{\bigoplus}_{\eta\in\mathbb{T}^d}\mathcal{B}(\eta)\chi_{\mathcal{O}}$ is not a differential operator.
By Peetre’s Theorem [@peetre60], [@duistermaatkolk2010 p. 236], a linear operator $\mathcal{B}:\mathcal{D}(\mathbb{R}^d)\to\mathcal{D}^{'}(\mathbb{R}^d)$ is a differential operator if and only if $supp(Pu)\subset supp(u)$ for all $u\in\mathcal{D}(\mathbb{R}^d)$. Here, $\mathcal{D}(\mathbb{R}^d)$ denotes the space of compactly supported smooth functions on $\mathbb{R}^d$ with the topology of test functions. Also, let $\mathcal{S}(\mathbb{R}^d)$ denote the Schwartz class of rapidly decreasing smooth functions on $\mathbb{R}^d$. In order to show that $\mathcal{B}$ is not a differential operator, we will show that it does not preserve supports.
Given $g\in\mathcal{D}(\mathbb{R}^d)$, we define its Gelfand transform as $$\begin{aligned}
g_\sharp(y,\eta)=\sum_{p\in\mathbb{Z}^d}g(y+2\pi p)e^{-i(y+2\pi p)\cdot\eta}.\end{aligned}$$ This is a function in $L^2(Y^{'},L^2_\sharp(Y))$. The map from $g\mapsto g_\sharp$ is an isometry on $\mathcal{D}(\mathbb{R}^d)$ in the $L^2$-inner product and hence it may be extended to a unitary isomorphism from $L^2(\mathbb{R}^d)$ to $L^2(Y^{'},L^2_\sharp(Y))$. We shall show that $\mathcal{B}(g)$ is not compactly supported. $\mathcal{B}(g)$ is a tempered distribution defined as: $$(\mathcal{B}(g),\phi)=\int_{\mathcal{O}}\int_Y B(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\phi}_\sharp(y,\eta)~dyd\eta.$$
We may define the Fourier transform of $\mathcal{B}(g)$ in $\mathcal{S}^{'}(\mathbb{R}^d)$ as $$\begin{aligned}
(\widehat{\mathcal{B}(g)},\phi)=(\mathcal{B}(g),\mathcal{F}^{-1}(\phi)),\end{aligned}$$ where $\mathcal{F}^{-1}(\phi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\phi(\eta)e^{iy\cdot\eta}~d\eta$ is the inverse Fourier transform of $\phi$. Since $\phi\in\mathcal{S}(\mathbb{R}^d)$, there exists a $\psi\in\mathcal{S}(\mathbb{R}^d)$ such that $\phi=\widehat{\psi}$. Therefore, $$\begin{aligned}
(\widehat{\mathcal{B}(g)},\phi)=(\mathcal{B}(g),\mathcal{F}^{-1}(\phi))=(\mathcal{B}(g),\psi).\end{aligned}$$ By Poisson Summation Formula [@grafakosclassical2008 p. 171], we conclude that $$\begin{aligned}
\label{poissonsum}
\psi_\sharp(y,\eta)&=\sum_{p\in\mathbb{Z}^d}\psi(y+2\pi p)e^{-i(y+2\pi p)\cdot\eta}=\frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}\widehat{\psi}(\eta+q)e^{iq\cdot y}\notag\\&\quad=\frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}{\phi}(\eta+q)e^{iq\cdot y}.
\end{aligned}$$
Now, suppose that $\phi\in\mathcal{S}(\mathbb{R}^d)$ vanishes on $\bigcup_{q\in\mathbb{Z}^d}(\mathcal{O}+q)$, then $\psi_\sharp$, as obtained in , vanishes on $\mathcal{O}$. Hence, $$\begin{aligned}
(\widehat{\mathcal{B}(g)},\phi)&=(\mathcal{B}(g),\psi)=\int_Y\int_{\mathcal{O}}B(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\psi}_\sharp(y,\eta)~d\eta\,dy=0.
\end{aligned}$$ Therefore, $\widehat{\mathcal{B}(g)}$ vanishes on the open set $\bigcup_{q\in\mathbb{Z}^d}(\mathcal{O}+q)$. By Schwartz-Paley-Wiener Theorem [@rudinfunctional1991 p. 191], $\widehat{\mathcal{B}(g)}$ cannot be the Fourier transform of a compactly supported distribution, i.e., $\mathcal{B}(g)$ is not compactly supported.
Let $\{\mathcal{O}_1,\mathcal{O}_2,\ldots,\mathcal{O}_n\}$ be a partition of $Y^{'}$ up to a set of measure zero, i.e., $Y^{'}\setminus\bigcup_{j=1}^n\mathcal{O}_j$ is a set of measure zero. Define $\mathcal{B}:\mathcal{D}(\mathbb{R}^d)\to\mathcal{D}^{'}(\mathbb{R}^d)$ by $\mathcal{B}(g)=\sum_{j=1}^n\int^{\bigoplus}_{\eta\in\mathcal{O}_j}\mathcal{B}_j(\eta)g_\sharp(y,\eta)$ where $\mathcal{B}_j(\eta)=-(\nabla+i\eta)\cdot B_j(\nabla+i\eta)$ where for all $j\in\{1,2,\ldots,n\}$, $B_j$ are matrices with $L^\infty_\sharp(Y,\mathbb{R})$-entries, then $\mathcal{B}$ is a differential operator if and only if $B_1=B_2=\ldots=B_n$.\[notdiffop\]
If $B\coloneqq B_1=B_2=\ldots=B_n$, then $\mathcal{B}(g)=-\nabla\cdot B\nabla(g)$ which is a differential operator.
Conversely, without loss of generality, assume that $B_1\neq B_2$ and suppose that $\mathcal{B}$ is a differential operator. Then, $$\begin{aligned}
\mathcal{B}(g)=&\int^{\bigoplus}_{\eta\in Y^{'}}\mathcal{B}_1(\eta)d\eta+\int^{\bigoplus}_{\eta\in\mathcal{O}_2}(\mathcal{B}_2-\mathcal{B}_1)(\eta)d\eta+\int^{\bigoplus}_{\eta\in\mathcal{O}_3}(\mathcal{B}_3-\mathcal{B}_1)(\eta)d\eta+\ldots\\&\qquad
+\int^{\bigoplus}_{\eta\in\mathcal{O}_n}(\mathcal{B}_n-\mathcal{B}_1)(\eta)d\eta.
\end{aligned}$$ Hence, $$\begin{aligned}
\mathcal{B}(g)-\int^{\bigoplus}_{\eta\in Y^{'}}\mathcal{B}_1(\eta)d\eta=\sum_{j=2}^{n}\int^{\bigoplus}_{\eta\in\mathcal{O}_j}(\mathcal{B}_j-\mathcal{B}_1)(\eta)d\eta
\end{aligned}$$ The left hand side of the above equation is a differential operator. We will show that the right hand side is not a differential operator to obtain a contradiction.
We proceed as in Lemma \[notdifferential\].
Define $\mathcal{C}:\mathcal{D}(\mathbb{R}^d)\to\mathcal{D}^{'}(\mathbb{R}^d)$ by $$(\mathcal{C}(g),\phi)=\sum_{j=2}^{n}\int_{\mathcal{O}_j}\int_Y (B_j-B_1)(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\phi}_\sharp(y,\eta)~dy\,d\eta$$
It is easy to see that $\mathcal{C}(g)\in\mathcal{S}^{'}(\mathbb{R}^d)$.
Therefore, we may define its Fourier transform by $$\begin{aligned}
(\widehat{\mathcal{C}(g)},\phi)=(\mathcal{C}(g),\mathcal{F}^{-1}(\phi)),\end{aligned}$$ where $\mathcal{F}^{-1}(\phi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}\phi(\eta)e^{iy\cdot\eta}~d\eta$ is the inverse Fourier transform of $\phi$. Since $\phi\in\mathcal{S}(\mathbb{R}^d)$, there exists $\psi\in\mathcal{S}(\mathbb{R}^d)$ such that $\phi=\widehat{\psi}$. Therefore, $$\begin{aligned}
(\widehat{\mathcal{C}(g)},\phi)=(\mathcal{C}(g),\mathcal{F}^{-1}(\phi))=(\mathcal{C}(g),\psi).\end{aligned}$$ By Poisson Summation Formula [@grafakosclassical2008 p. 171], we conclude that $$\begin{aligned}
\label{poissonsum2}
\psi_\sharp(y,\eta)&=\sum_{p\in\mathbb{Z}^d}\psi(y+2\pi p)e^{-i(y+2\pi p)\cdot\eta}= \frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}\widehat{\psi}(\eta+q)e^{iq\cdot y}\notag\\
&=\frac{1}{(2\pi)^{d/2}}\sum_{q\in\mathbb{Z}^d}{\phi}(\eta+q)e^{iq\cdot y}.
\end{aligned}$$
Now, suppose that $\phi\in\mathcal{S}(\mathbb{R}^d)$ vanishes on $\bigcup_{q\in\mathbb{Z}^d}(\bigcup_{j=2}^n\mathcal{O}_j+q)$, then $\psi_\sharp$, as obtained in , vanishes on $\bigcup_{j=2}^n\mathcal{O}_j$. Hence, $$\begin{aligned}
(\widehat{\mathcal{C}(g)},\phi)&=(\mathcal{C}(g),\psi)\\
&=\sum_{j=2}^{n}\int_Y\int_{\mathcal{O}_j}(B_j-B_1)(\nabla+i\eta)g_\sharp(y,\eta)\cdot(\nabla-i\eta)\overline{\psi}_\sharp(y,\eta)~d\eta\,dy=0.
\end{aligned}$$ Therefore, $\widehat{\mathcal{C}(g)}$ vanishes on the open set $\bigcup_{q\in\mathbb{Z}^d}(\bigcup_{j=2}^n\mathcal{O}_j+q)$. By Schwartz-Paley-Wiener Theorem [@rudinfunctional1991 p. 191], $\widehat{\mathcal{C}(g)}$ cannot be the Fourier transform of a compactly supported distribution, i.e., $\mathcal{C}(g)$ is not compactly supported. Therefore, $\mathcal{C}$ is not a differential operator.
Proof of Theorem \[theorem:3\] {#simplicity_edge_1}
==============================
In this section, we prove that a spectral edge of a periodic elliptic differential operator can be made simple through a perturbation in the coefficients. The proof essentially follows Klopp and Ralston [@KloppRalston00], with the straightforward modification that the coefficients must come from $W^{1,\infty}_\sharp(Y,\mathbb{R})$. This condition is required to ensure that the eigenfunctions and their derivatives are Hölder continuous functions. We produce the proof here for completeness.
Suppose that the coefficients of the operator , $a_{kl}\in W^{1,\infty}_\sharp(Y)$. Note that the Bloch eigenvalues which are defined for $\eta\in Y^{'}$ are Lipschitz continuous in $\eta$ and may be extended as periodic functions to $\mathbb{R}^d$. In the sequel, we shall treat the Bloch eigenvalues as functions on $\mathbb{T}^d$, which is identified with $Y^{'}$ in a standard way. Also, we shall write $\lambda_j(\eta,A)$ to specify that a Bloch eigenvalue corresponds to a particular matrix $A$, appearing in the operator $\mathcal{A}$. We shall prove the theorem for an upper endpoint of a spectral gap. The proof for a lower endpoint is identical. We shall require the following lemma.
Consider the operator $\mathcal{A}$ as in , with $A\in M_B^>$. Let $\lambda_0$ correspond to the upper edge of a spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$, then
1. There exist numbers $a,b\in\mathbb{R}$ such that $\lambda_{m-1}(\eta)<a<\lambda_0<\lambda_m(\eta)<b$ for all $\eta\in Y^{'}$. Further, there exists $M\in\mathbb{N}$ such that $M>m$ and the Bloch eigenvalue $\lambda_M$ satisfies $\lambda_M(\eta)>b$ for all $\eta\in Y^{'}$.\[lemma.1\]
2. Let $B$ be a symmetric matrix with $L^\infty_\sharp(Y,\mathbb{R})$-entries. There is a finite open cover of $Y^{'}$, $\{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_n\}$ such that for each $\mathcal{G}_j$, we have an orthonormal set in $L^2_\sharp(Y)$ of functions analytic for $\eta\in\mathcal{G}_j$ and for sufficiently small $t$, $$\label{orthonormal set}
\{ \phi_m^{(j)}(\eta,A-tB), \phi_{m+1}^{(j)}(\eta,A-tB),\ldots, \phi_{R_j}^{(j)}(\eta,A-tB) \}.$$ Further, for each fixed $t$, the linear subspace generated by the functions in contains the eigenspaces corresponding to eigenvalues of $-\nabla\cdot(A-tB)\nabla$ between $a$ and $b$.\[lemma.2\]
3. The functions in may be chosen such that the following equation is satisfied $$\left\langle\frac{d{\phi}^{(j)}_r}{dt},\phi^{(j)}_s\right\rangle=0,\label{orthogonality1}$$ where $\langle\cdot,\cdot\rangle$ denotes the $L^2_\sharp(Y)$ inner product. \[lemma.5\]
**Proof of \[lemma.1\]** As noted in Remark \[simple\_in\_neighborhood\], the Bloch eigenvalues are Lipschitz continuous functions on a compact set $\mathbb{T}^d$. Hence, the function $\eta\mapsto\lambda_m(\eta)$ is bounded above, say by $b$. Since, $\lambda_0$ is a spectral edge, $\displaystyle\delta\coloneqq\min_{\eta\in Y^{'}}\lambda_m(\eta)-\max_{\eta\in Y^{'}}\lambda_{m-1}(\eta)$ is positive. Choose $a=\lambda_0-\frac{\delta}{2}$. These choices of $a$ and $b$ satisfy our requirements.
By Weyl’s law [@Reed1978], the eigenvalues of the periodic Laplacian on $Y$ satisfy the following inequality, for some $s>0$ and $C_1>0$, for large $M$, $$\begin{aligned}
\label{BlochBoundedBelow1}
\lambda_M(0,I)\geq\lambda_M^{N}\geq C_1 M^s,
\end{aligned}$$ where $\lambda_M^{N}$ denotes the $M^{th}$ eigenvalue of the Neumann Laplacian on $Y$.
By Lipschitz continuity of Bloch eigenvalues in the dual parameter, we have $$\begin{aligned}
|\lambda_M(\eta,I)-\lambda_M(0,I)|\leq C|\eta|\leq C_2.\end{aligned}$$ Therefore, for all $\eta\in Y^{'}$, $$\label{BlochBoundedBelow2}\lambda_M(\eta,I)\geq \lambda_M(0,I)-C_2.$$ On combining and , for all $\eta\in Y^{'}$, we obtain $$\begin{aligned}
\lambda_M(\eta,I)\geq C_1M^s-C_2.\end{aligned}$$
It follows from a standard argument involving min-max principle, that $\lambda_M(\eta,I)\leq C_3||A^{-1}||_{L^\infty}\lambda_M(\eta,A)$.
Therefore, for all $\eta\in Y^{'}$, $$\begin{aligned}
\lambda_M(\eta,A)&\geq \dfrac{1}{C_3||A^{-1}||_{L^\infty}}\lambda_M(\eta,I)\notag\\
&\quad\geq \dfrac{C_1M^s}{C_3||A^{-1}||_{L^\infty}}-\dfrac{C_2}{C_3||A^{-1}||_{L^\infty}}.
\end{aligned}$$ Finally to prove \[lemma.2\], choose $M$ large enough so that $$\begin{aligned}
\dfrac{C_1M^s}{C_3||A^{-1}||_{L^\infty}}-\dfrac{C_2}{C_3||A^{-1}||_{L^\infty}}>b.\end{aligned}$$
**Proof of \[lemma.2\]** For each $\xi\in\mathbb{T}^d$, there is a circle $\Gamma_\xi$ in the complex plane containing the eigenvalues of $\mathcal{A}(\xi)$ between $a$ and $b$. Let $B$ be a $d\times d$ real symmetric matrix with $W^{1,\infty}_\sharp(Y)$ entries. Observe that the operator $P_\xi$ defined by $$\label{ProjectionOperator}
P_\xi(\eta;A-tB)\coloneqq-\frac{1}{2\pi i}\int_{\Gamma_\xi} \left(\mathcal{A}(\eta;A-tB)-zI\right)^{-1}~dz$$ is real-analytic in a neighborhood $R_\xi$ of $\xi$ and for small $t$, where $$\begin{aligned}
\mathcal{A}(\eta;A-tB)\coloneqq -(\nabla+i\eta)\cdot(A-tB)(\nabla+i\eta).\end{aligned}$$ The operator $P_\xi$ is an orthogonal projection onto the eigenspace of $\mathcal{A}(\eta;A-tB)$ corresponding to the eigenvalues between $a$ and $b$. The analyticity of the projection operator follows from the analyticity of the integrand, which is a consequence of the operator family $\mathcal{A}(\eta;A-tB)$ being a holomorphic family of type $(B)$. A proof of this fact is available in [@Sivaji2004] for perturbation in $\eta$. For a perturbation in $t$, a proof is given in Appendix \[PerturbationTheory\].
Therefore, in a neighborhood of $\eta=\xi, t=0$, we obtain an orthonormal basis for the range of $P_{\xi}(\eta,A-tB)$. In this manner, we obtain an open cover of $\mathbb{T}^d$. By compactness of $\mathbb{T}^d$, the open cover has a finite subcover $\{\mathcal{G}_1,\mathcal{G}_2,\ldots,\mathcal{G}_n\}$ with the following properties.
1. For each $\mathcal{G}_j$, we have an orthonormal set in $L^2_\sharp(Y)$ $$\begin{aligned}
\{\phi^{(j)}_{m}(\eta,A-tB),\ldots,\phi^{(j)}_{R_j}(\eta,A-tB)\}\end{aligned}$$ whose elements are analytic for $\eta\in\mathcal{G}_j$ and $|t|<\delta$.
2. The linear subspace generated by $$\begin{aligned}
\{\phi^{(j)}_{m}(\eta,A-tB),\ldots,\phi^{(j)}_{R_j}(\eta,A-tB)\}\end{aligned}$$ contains the eigenspaces corresponding to eigenvalues of $\mathcal{A}(\eta;A-tB)$ that lie between $a$ and $b$.
**Proof of \[lemma.5\]** Let $\tilde\phi_r=\sum_{p=m} u_{rp}\phi_p$, then
$$\begin{aligned}
\left\langle\frac{d{\tilde{\phi_r}}}{dt},\tilde\phi_s\right\rangle=\sum_p\frac{d{u}_{rp}}{dt}\overline{u_{sp}}+\sum_{p,q}u_{rp}\overline{u_{sq}}\left\langle\frac{d{\phi}_p}{dt},\phi_q\right\rangle.
\end{aligned}$$
If we set $U$ to be the matrix with entries $u_{rs}$ and $A$ to be the matrix with entries $-\left\langle\phi_r,\frac{d{\phi}_s}{dt}\right\rangle$, will hold if $$\begin{aligned}
\frac{dU}{dt}=UA.
\end{aligned}$$ This is solved with the initial condition $U(0)=I$. The matrix $A$ is skew-symmetric, therefore, $U(t)$ is unitary and analytic for $\eta\in\mathcal{G}$. Replace $\phi_r$ with $\tilde{\phi}_r$ to complete the proof of \[lemma.5\].
Consider the sesquilinear form $$\begin{aligned}
a(\eta,t)(u,v)\coloneqq\int_Y(A-tB)(\nabla+i\eta)u\cdot(\nabla-i\eta)\overline{v}.\end{aligned}$$ For the functions constructed in \[lemma.2\], $\langle\frac{d\phi^{(j)}_r}{dt},\phi^{(j)}_s\rangle=0$ for all $r,s$. Thus, $$\begin{aligned}
\frac{d}{dt}\left(a(\eta,t)(\phi_r^{(j)}(\eta,t),\phi_s^{(j)}(\eta,t))\right)=-\int_Y B(\nabla+i\eta)\phi_r^{(j)}(\eta,t)\cdot(\nabla-i\eta)\overline{\phi_s^{(j)}(\eta,t)}.
\end{aligned}$$
A function $f$ defined on $\mathbb{R}^d$ is said to be $(\eta,Y)$-periodic if for all $p\in\mathbb{Z}^d$, $y\in\mathbb{R}^d$, $u(y+2\pi p)=e^{2\pi i p\cdot \eta}u(y)$. The eigenfunctions of $\mathcal{A}(\eta,A)$ with periodic boundary conditions, when multiplied by $\exp(-i\eta\cdot y)$, become eigenfunctions of $\mathcal{A}\coloneqq-\nabla\cdot A\nabla$ with $(\eta,Y)$-periodic boundary conditions, i.e., there are $\lambda$ and $u$ such that $-\nabla\cdot(A\nabla)u=\lambda u$, where $u$ is $(\eta,Y)$-periodic. Since $u$ is a complex-valued function, the regularity theorem [@Ladyzhenskaya68 Chapter 3, Section 15], cannot be applied directly. However, since the operator is linear, we may write $u=v+iw$ and express the eigenvalue equation for $u$ as two equations for the real-valued functions $v$ and $w$. In particular, $v$ and $w$ satisfy $-\nabla\cdot(A\nabla)v=\lambda v$ and $-\nabla\cdot(A\nabla)w=\lambda w$ in the interior of $Y$. Hence, by the regularity theory for elliptic equations with $W^{1,\infty}$ coefficients, $v$ and $w$ and their first-order derivatives are Hölder continuous in the interior of $Y$. Further, the Hölder estimates in the interior of $Y$ are independent of $\eta\in Y^{'}$. Consequently, $u$ and its derivatives are Hölder continuous in the interior of $Y$.
Choose $\widehat{\eta}$ and $\phi_0$ such that $\mathcal{A}(\widehat{\eta},A)\phi_0=\lambda_0\phi_0.$ Choose $\phi_0\neq \exp(-i\hat{\eta}\cdot y)$. This can be achieved because the multiplicity of the Bloch eigenvalue at $\hat{\eta}$ is greater than one. Therefore, $(\nabla+i\widehat{\eta})\phi_0$ is non-zero. Consequently, there exist a $y_0$ in the interior of $Y$, an $l$ with $1\leq l\leq d$, and a $\theta>0$ such that $\left|\left(\frac{\partial}{\partial x_l}+i\widehat{\eta}_l\right)\phi_0(y_0,\widehat{\eta})\right|^2\geq\theta$. Since $\phi_0$ and its derivatives are Hölder continuous in the interior of $Y$, there is a small $\epsilon_0>0$ such that, $$\begin{aligned}
\label{continuityofderivatives1}\mbox{for }|y-y_0|<\epsilon_0,\qquad\left|\left(\frac{\partial}{\partial x_l}+i{\widehat{\eta}_l}\right)\phi_0(y,\widehat{\eta})\right|^2>\frac{2\theta}{3}.\end{aligned}$$
Additionaly, since $\phi_r^{(j)}$ obtained earlier in \[lemma.2\] are linear combinations of eigenfunctions, by the Hölder continuity of the eigenfunctions and their derivatives, an $\epsilon_0$ may be chosen so that $$\begin{aligned}
\label{continuityofderivatives2}
\sum_{p=m}^{R_j} \left|\left(\frac{\partial}{\partial x_l}+i{\eta}_l\right)\phi_p^{(j)}(y,\eta,A)-\left(\frac{\partial}{\partial x_l}+i{\eta}_l\right)\phi_p^{(j)}(y_0,\eta,A)\right|^2<\frac{\theta}{3},
\end{aligned}$$ for $\eta\in\mathcal{G}_j$ and $|y-y_0|<\epsilon_0$. Define the matrix $B=diag(0,\ldots,0,b_l,0,\ldots,0)$ all of whose diagonal entries are zero other than $b_{l}$ which is chosen as a function $b_{l}\in C_0^{\infty}(|y-y_0|<\epsilon_0)$ such that $b_{l}\geq 0$ and $\int_Y b_{l}=1$. Extend $B$ periodically to $\mathbb{R}^d$.
There is an index $q$ such that $\widehat{\eta}\in\mathcal{G}_q$. Therefore, $\displaystyle\phi_0(y,\widehat{\eta})=\sum_{r=m}^{R_q} c_r\phi_r^{(q)}(y,\widehat{\eta},A)$.
Define $\phi_0(y,\widehat{\eta},t)=\displaystyle\sum_{r=m}^{R_q} c_r\phi_r^{(q)}(y,\widehat{\eta},A-tB)$. Then, by , $$\begin{aligned}
\frac{d}{dt}\left(a(\widehat{\eta},t)(\phi_0(\cdot,\widehat{\eta},t),\phi_0(\cdot,\widehat{\eta},t))\right)|_{t=0}&=-\int_Y b_l\left(\frac{\partial}{\partial y_l}+i\widehat{\eta_l}\right)\phi_0(y,\widehat{\eta})\left(\frac{\partial}{\partial y_l}-i\widehat{\eta_l}\right)\overline{\phi_0(y,\widehat{\eta})}~dy\notag\\
&\qquad\leq\frac{-2\theta}{3}.
\end{aligned}$$
Hence, $$\begin{aligned}
\label{eq:var3}
a(\widehat{\eta},t)(\phi_0(\cdot,\widehat{\eta},t),\phi_0(\cdot,\widehat{\eta},t))\leq\lambda_0-\frac{2\theta}{3}t+t^2\beta(t).
\end{aligned}$$
For each $\eta\in\mathcal{G}_j$, we define the function $$\begin{aligned}
\label{stareq}
\phi_{*}^{(j)}(y,\eta,t)=\sum_{r=m}^{R_j}\overline{\left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y_0,\eta,A)}\phi_r^{(j)}(y,\eta,t).
\end{aligned}$$ For $\phi(\cdot,\eta,t)=\displaystyle\sum_{k=r}^{R_j} a_r\phi_r^{(j)}(\cdot,\eta,A-tB)$, $\phi(\cdot,\eta,t)$ is perpendicular to $\phi_*^{(j)}(\cdot,\eta,t)$ if and only if $$\begin{aligned}
\label{eq:perp1}
\sum_{r=1}^{R_j} a_r \left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y_0,\eta,A) = 0.
\end{aligned}$$ For $\phi(\cdot,\eta,t)$ satisfying and $||\phi||_{L^2_\sharp(Y)}=1$, the following holds for $\eta\in\mathcal{G}_j$, $$\begin{aligned}
{}&\frac{d}{dt}\left(a(\eta,t)(\phi(\cdot,\eta,t),\phi(\cdot,\eta,t))\right)|_{t=0}\\
&\quad=-\int_Y b_{l}\left(\frac{\partial}{\partial y_l}+i{\eta_l}\right)\phi(y,\eta,0)\left(\frac{\partial}{\partial y_l}-i{\eta_l}\right)\overline{\phi(y,\eta,0)}~dy\notag\\
&\quad=-\int_{B_{\epsilon_0}(y_0)}\left|\sum_{r=m}^{R_j} a_r\left(\left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y,\eta,A)- \left({\partial_l}+i{\eta}_l\right)\phi_r^{(j)}(y_0,\eta,A)\right)\right|^2b_{l}~dy\notag\\
&\quad\geq-\frac{\theta}{3},
\end{aligned}$$where the last inequality follows from . Therefore, the following holds true, uniformly for $\eta\in\mathcal{G}_j$ and $||\phi||_{L^2_\sharp(Y)}=1$, $$\begin{aligned}
\label{eq:var4}
a(\eta,t)(\phi(\cdot,\eta,t),\phi(\cdot,\eta,t))\geq\lambda_0-\frac{\theta}{3}t+t^2\gamma(t).
\end{aligned}$$
To find an upper bound for $\lambda(\widehat{\eta},t)$, we apply the following variational characterization of the eigenvalues of $\mathcal{A}(\eta,t)$ to . If $\phi_1,\phi_2,\ldots,\phi_{m-1}$ are the first $m-1$ eigenfunctions corresponding to the selfadjoint operator $\mathcal{A}(\eta,t)$, then the $m^{th}$ eigenvalue of $\mathcal{A}(\eta,t)$ is given by the formula $$\begin{aligned}
\lambda_{m}(\eta,t)=\min_{\phi\perp\{\phi_1,\phi_2,\ldots,\phi_{m-1}\},~||\phi||_{L^2_\sharp(Y)}=1}~a(\eta,t)(\phi,\phi).\end{aligned}$$ Therefore, $$\begin{aligned}
\label{estimate1}
\lambda_{m}(\widehat{\eta},t)<\lambda_0-\frac{7\theta}{12}t,
\end{aligned}$$ for $t$ sufficiently small. To find a lower bound for $\lambda_{m+1}(\eta,t)$, we apply another variational characterization for the eigenvalues to , viz., $$\begin{aligned}
\lambda_{m+1}(\eta,t)=\max_{{\dim V = m}}~\min_{{\phi\perp V,~||\phi||_{L^2_\sharp(Y)}=1}}~a(\eta,t)(\phi,\phi),
\end{aligned}$$
where $V$ varies over $m$-dimensional subspaces of $H^1_\sharp(Y)$.
For each fixed $\eta$ and $t$, take the $m$-dimensional subspace $V$ spanned by the first $m-1$ eigenfunctions of $\mathcal{A}(\eta,t)$ and $\phi_*^{(j)}$ as defined in , i.e., $$\begin{aligned}
V=\{\phi_1(\eta,t), \phi_2(\eta,t),\ldots,\phi_{m-1}(\eta,t),\phi_*^{(j)}(\eta,t)\}.\end{aligned}$$ Then, $\phi(\eta,t)$ satisfying the equation is perpendicular to $V$ and allows us to conclude that $$\begin{aligned}
\label{estimate2}
\lambda_{m+1}(\eta,t)>\lambda_0-\frac{5\theta}{12}t,
\end{aligned}$$ for small $t$. The two estimates obtained above and together imply that the perturbed spectral edge is attained by a single Bloch eigenvalue.
The proof of Theorem \[theorem:3\] depends crucially on the interior Hölder continuity of the Bloch eigenfunctions and their derivatives. This requires the coefficients of the elliptic operator to have $W^{1,\infty}_\sharp(Y)$ entries. We attempt to reduce this regularity requirement to $L^\infty$ in Section \[simplicity\_edge\_2\].
Proof of Theorem \[theorem:4\] {#simplicity_edge_2}
==============================
We shall prove Theorem \[theorem:4\] for an upper endpoint of a spectral gap. The proof for a lower endpoint is identical. Let $\lambda_0$ be the upper endpoint of a spectral gap of $\mathcal{A}\coloneqq -\nabla\cdot(A\nabla)$, which is achieved by the Bloch eigenvalue $\lambda_m(\eta)$ at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$ in $Y^{'}$. The proof uses ideas from Parnovski and Shterenberg [@ParShteren17] and is divided into the following steps:
1. By Proposition \[corollarymultiple\], there is a single perturbation $B$ of the coefficients so that the Bloch eigenvalue $\lambda_m(\eta;A+tB)$ is simple at the points $\eta_1,\eta_2,\ldots,\eta_N$.
2. However, the perturbation creates new points at which the new spectral edge has been attained. We shall prove that given $\delta>0$, we can find perturbation parameter $t$ such that all the points at which the spectral edge is attained are within $\delta$-distance of the old spectral edge (Lemma \[continuity\_spectral\_edge\]).
3. We prove that these new spectral edges are not multiple.
We shall require the following preliminaries.
The multiplicity of a Bloch eigenvalue can be reduced at a finite number of points in the dual parameter by application of the same perturbation. This will be the content of the next proposition.
\[corollarymultiple\] Fix $m\in\mathbb{N}$. Let $S=\{\eta_1,\eta_2,\ldots,\eta_N\}$ be a finite collection of points in $Y^{'}$. Then, there exists a matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$-entries and a $t_0$ positive such that for all $t\in(0,t_0]$, the Bloch eigenvalue $\lambda_m(t,\eta)$ of the operator $\mathcal{A}+t\mathcal{B}=-\nabla\cdot(A+tB)\nabla$ is simple for all $\eta_n\in S$, $1\leq n\leq N$.
To this end, we require the following lemma.
\[hahnbanach\] Let $N\in\mathbb{N}$. Let $X$ be a normed linear space over $\mathbb{K}~(\mathbb{R} \mbox{ or } \mathbb{C})$ and let $x_1,x_2,\ldots,x_N$ be non-zero elements of $X$. Then there exists an $x^*\in X^*$ such that $\forall$ $n=1,2,\ldots,N$, $\langle x^*,x_n\rangle\neq 0$
Consider the finite dimensional subspace $F$ of $X$ spanned by $x_1,x_2,\ldots,x_N$. For each $n=1,2,\ldots,N$, let $F_n^*$ denote the subspace of $F^*$ containing $x^*\in F^*$ such that $\langle x^*,x_n \rangle=0$. Then, $F^*\neq \cup_{n=1}^N F^*_n$ since a vector space cannot be written as a finite union of its proper subspaces. Hence, there exists an $x^*\in F^*$ such that $x^*\not\in\cup_{n=1}^N F^*_n$. Hence, for all $n=1,2,\ldots,N$, $\langle x^*,x_n\rangle\neq 0$. Finally, extend $x^*$ to $X^*$ using the Hahn-Banach Theorem.
As a part of the proof of Lemma \[lemma:22\], we prove that for a given $m\in\mathbb{N}$ and $\eta_0\in Y^{'}$, there exists a $t_0$ positive such that for all $t\in(0,t_0]$, the Bloch eigenvalue $\lambda(t,\eta)$ of the perturbed operator $\mathcal{A}+t\mathcal{B}$ is simple at $\eta_0$. In the present proposition, we shall make a Bloch eigenvalue $\lambda_m(\eta)$ of the operator $\mathcal{A}$ simple at a finite number of points in $Y^{'}$ through a perturbation in the coefficients.
As in the proof of Proposition \[find B\], the perturbation at any $\eta_n \in S$ gives rise to a selfadjoint holomorphic family of type $(B)$, analytic in $\tau\in(-\sigma_0,\sigma_0)$, where $\sigma_0=\frac{\alpha}{2d||B||_{L^\infty}}$. Suppose that the eigenvalue $\lambda_m(\eta_n)$ of the operator $\mathcal{A}(\eta_n)$ has multiplicity $h_n$. For the perturbed operator $-\nabla\cdot(A+\tau B)\nabla$, the eigenvalue $\lambda_m(\eta_n)$ splits into $h_n$ branches. Suppose that the $h_n$ eigenvalues and eigenvectors are given as follows. For $n=1,2,\ldots,N$ and $r=1,2,\ldots,h_n$: $$\begin{aligned}
\lambda_m^{r}(\tau;\eta_n)=\lambda_m(\eta_n)+\tau a_m^r(\eta_n)+\tau^2\beta_m^r(\tau,\eta_n)\\
u_m^{r}(\tau;\eta_n)=u_m^r(\eta_n)+\tau v_m^r(\eta_n)+\tau^2 w_m^r(\tau,\eta_n).\end{aligned}$$
As before, the following system of equations holds true for $n=1,2,\ldots,N$ and $r=1,2,\ldots,h_n$: $$\int_{Y}B(\nabla+i\eta_n) u_m^r(\eta_n)\cdot\overline{(\nabla+i\eta_n) u_m^s(\eta_n)}~dy=a_m^r(\eta_n)\delta_{rs}.$$
The above equations define operators that act on the unperturbed eigenspaces at each $\eta_n$. The multiplicity would go down if we find $B$ and bases for the unperturbed eigenspaces in which some off-diagonal entry, in particular, the $(1,2)$-entry is non-zero. To achieve this, we proceed as in the proof of Proposition \[find B\]. For any choice of basis of the unperturbed eigenspace at $\eta_n$, we find that either or holds. However, we cannot use this idea anymore, since, different $\eta_n$ would have different matrices $B$. To remedy this, we notice that, at each $\eta_n=(\eta_{n,1},\eta_{n,2},\ldots,\eta_{n,d})$, for a basis given by $\{f_n^1,f_n^2,\ldots,f_n^{h_n}\}$ either $$\label{alternative3}
\sum_{l=1}^d(\partial_l+i\eta_{n,l}){f_n^1}(\partial_l-i\eta_{n,l})\overline{f_n^2}\not\equiv0,$$
or, if the above sum is zero, then in the modified basis $\{f_n^1,f_n^1+f_n^2,f_n^3,\ldots,f_n^{h_n}\}$, $$\label{alternative4}
\sum_{l=1}^d(\partial_l+i\eta_{n,l}){f_n^1}(\partial_l-i\eta_{n,l})\overline{(f_n^1+f_n^2)}=\sum_{l=1}^d|(\partial_l+{i}\eta_{n,l}){f_n^1}|^2\not\equiv0,$$ provided that $f_n^1\neq \exp(-i\eta_n\cdot y)$.
We can always choose $f_n^1$ to be a function different from $\exp(-i\eta_n\cdot y)$ since at any of the $\eta_n$, we have an eigenspace of dimension greater than $1$. For each $\eta_n$, call the non-zero sum between and as $p_n$. Further, take either $\Re(p_n)$ or $\Im(p_n)$ depending on whichever is non-zero. If both are non-zero, we may take either one. This will make sure that we have a collection of only real-valued functions.
By the above procedure, we have $N$ elements of $L^1_\sharp(Y,\mathbb{R})$, again labelled as $\{p_1,p_2,\ldots,p_N\}$. By Lemma \[hahnbanach\], there is an $\alpha\in (L^1_\sharp(Y,\mathbb{R}))^*$ such that $\alpha(p_n)\neq 0$ for all $n=1,2,\ldots,N$. By duality, there exists a $\beta\in L^\infty_\sharp(Y,\mathbb{R})$ such that $\alpha(p_n)=\int_Y \beta p_n~dy\neq 0$.
Define $B=diag(\beta,\beta,\ldots,\beta)$, then either, $$\Re{\int_{Y}B(\nabla+i\eta_n) f_n^1\cdot\overline{(\nabla+i\eta_n) f_n^2}~dy}\neq 0,$$ or $$\Re{\int_{Y}B(\nabla+i\eta_n) f_n^1\cdot(\nabla-i\eta_n)(\overline{f_n^1}+\overline{f_n^2})~dy}\neq 0,$$ depending on $\eta_n$.
At the end of this step, the multiplicity of $\lambda_m(\eta)$ at each of the points $\eta_n$ will reduce at least by $1$. We repeat the procedure with the points among $\{\eta_1,\eta_2,\ldots,\eta_N\}$ where the eigenvalue is still multiple. Finally, we require at most $M$ steps to make the Bloch eigenvalue simple at each of these points, where $M=\displaystyle\max_{1\leq n\leq N}{h_n}$.
In the next lemma, we shall prove that a spectral edge does not move very far for small perturbations in the coefficients of the periodic operator $\mathcal{A}$. We shall denote the operator $-\nabla\cdot(A+tB)\nabla$ as $\mathcal{A}+t\mathcal{B}$, where $\mathcal{A}=-\nabla\cdot(A\nabla)$ and $\mathcal{B}=-\nabla\cdot(B\nabla)$. Let $S_t$ denote the set of points at which the new spectral edge is attained, i.e., $$\begin{aligned}
S_t \coloneqq \{\eta\in Y^{'} : \mbox{ The Bloch eigenvalue } \lambda_{m}(\eta;A+tB)\mbox{ attains the spectral edge at } \eta.\}\end{aligned}$$
\[continuity\_spectral\_edge\] Let $N\in\mathbb{N}$. Let $A\in M_B^>$ and let $B$ be a real symmetric matrix with $L^\infty_\sharp(Y)$-entries. Let $\mathcal{A}=-\nabla\cdot(A\nabla)$ be a periodic elliptic differential operator. Let $\lambda_0$ be the upper endpoint of a spectral gap, which is attained by the Bloch eigenvalue $\lambda_m(\eta)$ at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$ in $Y^{'}$. Given a $ \delta$ belonging to the open interval $(0,1)$, there is a $t_0$ such that $$\begin{aligned}
\mbox{for }t\in(0,t_0],\qquad S_t\subset \bigcup_{j=1}^{N}B(\eta_j,\delta).\end{aligned}$$
We prove this lemma by contradiction. Assume that there is a $\delta\in(0,1)$ and sequences $(t_n)$ and $(\xi_n)$ such that $t_n\to 0$ and $\xi_n\in S_{t_n}$ such that $$\forall~1\leq j\leq N,\qquad|\xi_n-\eta_j|\geq \delta.\label{farpoint}$$
Let $\lambda_0(A+tB)$ denote the spectral edge associated to the operator $\mathcal{A}+t\mathcal{B}$. The perturbed spectral edge satisfies the following inequality. $$\begin{aligned}
|\lambda_0(A)-\lambda_0(A+t_nB)|& = |\min_{\eta\in Y^{'}}\lambda_m(\eta;A)-\min_{\eta\in Y^{'}}\lambda_m(\eta;A+t_nB)|\notag\\
&\quad = |-\max_{\eta\in Y^{'}}(-\lambda_m(\eta;A))+\max_{\eta\in Y^{'}}(-\lambda_m(\eta;A+t_nB))|\notag\\
&\quad\leq\max_{\eta\in Y^{'}}|\lambda_m(\eta;A)-\lambda_m(\eta;A+t_nB)|\notag\\
&\quad\leq Ct_n.\label{edgeconvergence}
\end{aligned}$$
Since $(\xi_n)$ is a bounded sequence in $Y^{'}$, a subsequence of $(\xi_n)$ converges to $\hat{\xi}$, which we continue to denote by $(\xi_n)$.
We shall prove that $$\begin{aligned}
\label{diagconvergence}
\lambda_m(\xi_n;A+t_nB)\to\lambda_m(\hat{\xi};A).
\end{aligned}$$
Observe that, $$\begin{aligned}
{}&\left|\int (A+t_n B)(\nabla+i\xi_n)u(\nabla-i\xi_n)\bar{u}dy-\int A(\nabla+i\hat{\xi})u(\nabla-i\hat{\xi})\bar{u}dy\right|\leq\\
&\quad\left|\int A(\nabla+i\xi_n)u(\nabla-i\xi_n)\bar{u}dy-\int A(\nabla+i\hat{\xi})u(\nabla-i\hat{\xi})\bar{u}dy\right|+t_n||B||_{L^\infty}\int (\nabla+i\xi_n)u(\nabla-i\xi_n)\bar{u}dy
\end{aligned}$$
Divide throughout by $||u||^2_{L^2(Y)}$ and apply the min-max principle to obtain the following inequality. $$\begin{aligned}
\label{triangle1}
|\lambda_m(\xi_n;A+t_nB)-\lambda_m(\hat{\xi};A)|\leq|\lambda_m(\xi_n)-\lambda_m(\hat{\xi})|+t_n||B||_{L^\infty}|\lambda_m(\xi_n;I)|.
\end{aligned}$$
In order to establish , notice that the first and second part of converge to $0$ by the Lipschitz continuity of $\lambda_m(\cdot)$ and the boundedness of $\lambda_m(\xi_n;I)$, respectively.
It follows from and that $\lambda_0(A)=\lambda_m(\hat{\xi};A)$ and hence, $\hat{\xi}$ is also a spectral edge. By , this contradicts the initial assumption that there are only $N$ points at which the spectral edge is attained.
The spectral edge of the operator $\mathcal{A}$ is attained at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$ in $Y^{'}$. Choose among $\eta_1,\eta_2,\ldots,\eta_N$ the points where the Bloch eigenvalue $\lambda_m(\eta)$ is not a simple eigenvalue. Now, apply Proposition \[corollarymultiple\] to these points, so that for the perturbed operator $\mathcal{A}+t\mathcal{B}$, the corresponding Bloch eigenvalue becomes simple at these points. The points which were simple to begin with, will remain simple for sufficiently small $t$.
There is a neighborhood $\mathcal{O}_j$ of each of the points $(\eta_j)_{j=1}^N$ in which the Bloch eigenvalue is simple for a range of $t$. Each of these neighborhoods contain a ball, $B(\eta_j,\delta_j)$ of radius $\delta_j$ centered at $\eta_j$. Let $\delta\coloneqq\displaystyle\min_{1\leq j\leq N}\delta_j$, then by Lemma \[continuity\_spectral\_edge\], there exists a $t_0$ positive such that for all $t\in(0,t_0]$, the spectral edge of the perturbed operator $\mathcal{A}+t\mathcal{B}$ is contained in the union of the balls $\displaystyle\bigcup_{j=1}^N B(\eta_j,\delta)$.
Hence, we have obtained a perturbation of the operator $\mathcal{A}$ such that its spectral edge is simple.
An Application to the Theory of Homogenization {#homogen}
==============================================
Birman and Suslina [@BirmanSuslina2003] have described homogenization as a [*spectral threshold effect*]{}. Their analysis focuses on finding norm resolvent estimates of different orders. For the operator $\mathcal{A}$, it is known that $\inf \sigma(\mathcal{A})=0$. This corresponds to the bottom edge of its spectrum. A non-zero spectral edge is called an internal edge. The notion of homogenization has been extended to internal edges in [@Birman2004], [@Birman2006].
Internal Edge Homogenization {#internal1}
----------------------------
In this subsection, we review the internal edge homogenization theorem of Birman and Suslina [@Birman2006]. Consider the equation corresponding to the operator $\mathcal{A}$ . Let $\lambda_0$ denote an internal edge, corresponding to the upper endpoint of a spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$, then $$\begin{aligned}
\lambda_0=\min_{\eta\in Y^{'}}\lambda_{m}(\eta).\end{aligned}$$
Birman and Suslina [@Birman2006] make the following regularity assumptions on $\lambda_0$. These are exactly the properties of spectral edge that are required in order to define effective mass in the theory of motion of electrons in solids [@Filonov15].
1. \[assumptions1:1\] $\lambda_0$ is attained by the $m^{th}$ Bloch eigenvalue $\lambda_{m}(\eta)$ at finitely many points $\eta_1,\eta_2,\ldots,\eta_N$.
2. \[assumptions1:2\]For $j=1,2,\ldots,N$, $\lambda_{m}(\eta)$ is simple in a neighborhood of $\eta_j$, therefore, $\lambda_{m}(\eta)$ is analytic in $\eta$ near $\eta_j$.
3. \[assumptions1:3\] For $j=1,2,\ldots,N$, $\lambda_{m}(\eta)$ is non-degenerate at $\eta_j$, i.e., $$\begin{aligned}
\lambda_{m}(\eta)-\lambda_0=(\eta-\eta_j)^T B_j (\eta-\eta_j)+{O}(|\eta-\eta_j|^3),\mbox{ for } \eta \mbox{ near } \eta_j,
\end{aligned}$$ where $B_j$ are positive definite matrices.
Under these assumptions, the internal edge homogenization theorem is proved.
\[BirmanSuslina\] Let $\mathcal{A}$ be the operator in $L^2(\mathbb{R}^d)$ defined by and let $\lambda_0$ be an internal edge of the spectrum of $\mathcal{A}$. Assume conditions \[assumptions1:1\], \[assumptions1:2\], \[assumptions1:3\] and let $\varkappa^2>0$ be small enough so that $\lambda_0-\varkappa^2$ is in the spectral gap. Let $\mathcal{A}^\epsilon$ denote the unbounded operator $-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$ defined in $L^2(\mathbb{R}^d)$. For $1\leq j\leq N$, let $\psi_j(y,\eta_j)\coloneqq\exp(iy\cdot\eta_j)\phi_j(y)$, where $\phi_j$ is the eigenvector corresponding to the eigenvalue $\lambda_{0}=\lambda_{m}(\eta_j)$ of the operator $\mathcal{A}(\eta_j)=-(\nabla+i\eta_j)\cdot A(\nabla+i\eta_j)$. Then, $$\begin{aligned}
||R(\epsilon)-R^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad\epsilon\to 0\quad&\mbox{where},\\
R(\epsilon)=\left(\mathcal{A}^\epsilon-(\epsilon^{-2}\lambda_{0}-\varkappa^2)I\right)^{-1} \mbox{and}\quad R^0(\epsilon)\coloneqq&|Y|\sum_{j=1}^N [\psi_j^\epsilon]\left(B_j\nabla^2+\varkappa^2I\right)^{-1}[\overline{\psi_j^\epsilon}]\end{aligned}$$ are bounded operators on $L^2(\mathbb{R}^d)$ and $||\cdot||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}$ denotes the operator norm. Here, $[f]$ denotes the operation of multiplication by the function $f$.
Internal Edge Homogenization for a multiple spectral edge {#internal2}
---------------------------------------------------------
In this section, we shall prove a theorem corresponding to internal edge homogenization of the operator $\mathcal{A}^\epsilon=-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$ in $L^2(\mathbb{R}^d)$ in the presence of multiplicity. We shall interpret the three assumptions \[assumptions1:1\], \[assumptions1:2\], \[assumptions1:3\] that have been made on the spectral edge as hypotheses on the shape and structure of the spectral edge. Without knowledge of the shape and structure of the spectral edge, it is not possible to obtain any explicit homogenization result.
Starting with a spectral edge which is not simple, we shall appeal to Theorem \[theorem:4\] to modify the spectral edge so that it becomes simple. We shall make the following assumptions on the spectral edge. We assume the finiteness of the number of points at which the spectral edge is attained, however, since the contributions from different points are added up, we may as well assume that the spectral edge is attained at one point. Therefore, suppose that for the operator , a spectral gap exists. Let $\lambda_0$ denote the upper endpoint of this spectral gap of $\mathcal{A}$ and let $m$ be the smallest index such that the Bloch eigenvalue $\lambda_{m}$ attains $\lambda_0$, then $\displaystyle\lambda_0=\min_{\eta\in Y^{'}}\lambda_{m}(\eta)$.
Suppose that the spectral edge is attained at a unique point $\eta_0\in Y^{'}$. Also suppose that the eigenvalue $\lambda_0$ has multiplicity $2$. Therefore, there exists a neighborhood $\mathcal{O}$ of $\eta_0$, on which the Bloch eigenvalue $\lambda_{m}(\eta)$ is simple except at $\eta_0$. Now, a perturbation matrix $B$ with $L^\infty_\sharp(Y,\mathbb{R})$ entries, as in Theorem \[theorem:4\], is applied to the coefficients of operator $\mathcal{A}$, so that the new operator $\tilde{\mathcal{A}}(t)=\mathcal{A}+t\mathcal{B}$, has a simple spectral edge $\tilde{\lambda}_0(t)$ for sufficiently small $t$. However, the perturbed Bloch eigenvalues $\tilde{\lambda}_{m}(\eta,t)$ and $\tilde{\lambda}_{m+1}(\eta,t)$ are simple in the neighborhood $\mathcal{O}$ for small enough $t$. These properties follow from the analyticity of the projection operator , $P(\eta;A+tB)$, which is a consequence of the operator family $\tilde{\mathcal{A}}(t)$ being a holomorphic family of type $(B)$. For more details, see Appendix \[PerturbationTheory\].
For the perturbed spectral edge, we assume the following hypothesis
1. \[assumptions2:1\] $\tilde{\lambda}_{m}(\eta;t)$ attains minimum $\tilde{\lambda}_0(t)$ at a unique point $\eta_0(t)\in\mathcal{O}$ and is non-degenerate on $\mathcal{O}$, i.e., $$\begin{aligned}
\tilde{\lambda}_{m}(\eta;t)-\tilde{\lambda}_0(t)=(\eta-\eta_0(t))^T \tilde{B}_0(t) (\eta-\eta_0(t))+{O}(|\eta-\eta_0(t)|^3),
\end{aligned}$$ for $\eta\in\mathcal{O}$, where $\tilde{B}_0(t)$ is positive definite, i.e., there is $\alpha_0>0$, independent of $t$, such that $\tilde{B}_0(t)>\alpha_0 I$. Further, the order above holds uniformly for sufficiently $t$.
2. \[assumptions2:2\] $\tilde{\lambda}_{m+1}(\eta;t)$ attains minimum $\tilde{\lambda}_1(t)$ at a unique point $\eta_1(t)\in\mathcal{O}$ and is non-degenerate on $\mathcal{O}$, i.e., $$\begin{aligned}
\tilde{\lambda}_{m+1}(\eta;t)-\tilde{\lambda}_1(t)=(\eta-\eta_1(t))^T \tilde{B}_1(t) (\eta-\eta_1(t))+{O}(|\eta-\eta_1(t)|^3),
\end{aligned}$$ for $\eta\in\mathcal{O}$, where $\tilde{B}_1(t)$ is positive definite, i.e., there is $\alpha_1>0$, independent of $t$, such that $\tilde{B}_1(t)>\alpha_1 I$. Further, the order above holds uniformly for sufficiently $t$.
In essence, we are asking for the Bloch eigenvalues to have the shapes before and after the perturbation as in Fig. \[figure2\] and Fig. \[figure3\].
![Spectral Edge before perturbation.[]{data-label="figure2"}](2.pdf)
![Spectral Edge after perturbation.[]{data-label="figure3"}](3.pdf)
We will now set up notation for the internal edge homogenization theorem that we intend to prove. For $j=0,1$, let $\tilde{\psi}_{m+j}(y,\eta_j(t))=\exp(iy\cdot\eta_j(t))\tilde{\phi}_{m+j}(y;t)$, where $\tilde{\phi}_{m+j}$ is a normalized eigenvector corresponding to the eigenvalue $\tilde{\lambda}_{j}(t)=\tilde{\lambda}_{m+j}(\eta_j(t))$ of $\tilde{\mathcal{A}}(\eta_j;t)=-(\nabla+i\eta_j)\cdot (A+tB)(\nabla+i\eta_j)$. In what follows, we shall choose $t={O}(\epsilon^4)$. Define the following operators $$\begin{aligned}
\label{resolvent1}
{R}(\epsilon)\coloneqq\left(\mathcal{A}^\epsilon-(\epsilon^{-2}\lambda_{0}-\varkappa^2)I\right)^{-1},\mbox{ and}\end{aligned}$$ $$\begin{aligned}
\label{resolvent2}
\tilde{R}^0(\epsilon)\coloneqq|Y|[\tilde{\psi}_m^\epsilon]\left(\tilde{B}_0(t)\nabla^2+\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_m^\epsilon}]+|Y|[\tilde{\psi}_{m+1}^\epsilon]\left(\tilde{B}_1(t)\nabla^2+\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_{m+1}^\epsilon}].\end{aligned}$$
We shall require the following two lemmas.
\[resolventlemma1\] Let $$\begin{aligned}
\label{resolvent3}
\tilde{{R}}(\epsilon)\coloneqq\left(\tilde{\mathcal{A}}^\epsilon(t)-(\epsilon^{-2}\tilde{\lambda}_{0}(t)-\varkappa^2)I\right)^{-1},
\end{aligned}$$ where $\tilde{\mathcal{A}}^\epsilon(t)=-\nabla\cdot\left(A(\frac{x}{\epsilon})+tB(\frac{x}{\epsilon})\right)\nabla$ is an unbounded operator in $L^2(\mathbb{R}^d)$, satisfying assumptions \[assumptions2:1\] and \[assumptions2:2\]. Choose $t={O}(\epsilon^4)$. Then, $$||R(\epsilon)-\tilde{R}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad \epsilon\to 0.$$
\[resolventlemma2\] With the same notation as in Lemma \[resolventlemma1\], it holds that $$||\tilde{R}(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad \epsilon\to 0.$$
The proofs of these lemmas will be the content of subsections \[continuity\_of\_resolvents\] and \[internal\_homogenization\_result\]. Now, we state the internal edge homogenization theorem for a multiple spectral edge.
\[theorem:5\] Let $\mathcal{A}$ be the operator defined in $L^2(\mathbb{R}^d)$ as $\mathcal{A}\coloneqq -\nabla\cdot(A\nabla)$. Suppose that the entries of the matrix $A$ belong to $M_B^>$. Let $\lambda_0$ be the upper edge of a spectral gap associated to operator $\mathcal{A}$. Suppose that $\lambda_0$ is attained at one point $\eta_0\in Y^{'}$ and its multiplicity is $2$. Let $\varkappa^2>0$ be small enough so that $\lambda_0-\varkappa^2$ remains in the spectral gap. Let $\mathcal{A}^\epsilon$ be defined as $\mathcal{A}^\epsilon=-\nabla\cdot\left(A(\frac{x}{\epsilon})\nabla\right)$ in $L^2(\mathbb{R}^d)$.
Let $\tilde{\mathcal{A}}(t)=\mathcal{A}+t\mathcal{B}$ be a perturbation of $\mathcal{A}$ such that the perturbed operator has a simple spectral edge at $\tilde{\lambda}_0(t)$. Let $\tilde{\mathcal{A}}^\epsilon(t)=-\nabla\cdot\left(A(\frac{x}{\epsilon})+tB(\frac{x}{\epsilon})\right)\nabla$. Choose $t={O}(\epsilon^4)$. Assume conditions \[assumptions2:1\], \[assumptions2:2\] on the perturbed eigenvalues. Then, $$\begin{aligned}
\label{resolventinequality0}
||R(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(\epsilon)\quad\mbox{as}\quad \epsilon\to 0,
\end{aligned}$$ where $R(\epsilon)$ and $\tilde{R}^0(\epsilon)$ are defined in and , respectively.
Observe that $$\begin{aligned}
\label{intermediateinequality}
{}&||R(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)} \notag\\ &\quad\quad\leq||R(\epsilon)-\tilde{R}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}+||\tilde{R}(\epsilon)-\tilde{R}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}.
\end{aligned}$$ Applying Lemmas \[resolventlemma1\] and \[resolventlemma2\] to , we obtain .
1. Theorem \[theorem:5\] allows the computation of the homogenized coefficients through perturbed Bloch eigenvalues. Both the crossing modes contribute to homogenization, even though the spectral edge is simple after the perturbation.
2. A perturbation of the form $\tilde{\mathcal{A}}(t)$, as mentioned in Theorem \[theorem:5\], exists for sufficiently small $t$ by Theorem \[theorem:4\].
3. If the spectral edge is attained at finitely many points, the contribution to the effective operator from each of those points, are merely added up, as in Theorem \[BirmanSuslina\]. Hence, our assumption that the spectral edge is attained at one point is not restrictive. Further, the assumption that multiplicity of the spectral edge is $2$ can also be relaxed, since our method allows successive reduction of multiplicity of Bloch eigenvalues at multiple points.
Proof of Lemma \[resolventlemma1\] {#continuity_of_resolvents}
----------------------------------
The aim of this section is to prove Lemma \[resolventlemma1\]. We begin by introducing some notation. Define the two resolvents $S(\epsilon)$ and $\tilde{S}(\epsilon)$ by $$\begin{aligned}
\label{scaledresolvents}
S(\epsilon)=\left(\mathcal{A}-(\lambda_{0}-\epsilon^2\varkappa^2)I\right)^{-1}
\quad\mbox{and}\quad
\tilde{S}(\epsilon)=\left(\tilde{\mathcal{A}}(t)-(\tilde{\lambda}_{0}(t)-\epsilon^2\varkappa^2)I\right)^{-1}\end{aligned}$$ Define $$\begin{aligned}
\mathfrak{h}[u]\coloneqq\int_{\mathbb{R}^d}A\nabla u\cdot\nabla \overline{u}~dy-\lambda_0\int_{\mathbb{R}^d}|u|^2~dy.\end{aligned}$$ Then, $\mathfrak{h}$ is a closed sectorial form with domain $H^1(\mathbb{R}^d)$.
Consider another form $\mathfrak{p}(t)$ with domain $H^1(\mathbb{R}^d)$ defined by $$\begin{aligned}
\mathfrak{p}(t)[u]\coloneqq\int_{\mathbb{R}^d}tB\nabla u\cdot\nabla \overline{u}~dy-(\tilde{\lambda}_0-\lambda_0)\int_{\mathbb{R}^d}|u|^2~dy.\end{aligned}$$
To the sectorial forms $\mathfrak{h}$ and $\mathfrak{p}$, we shall apply the following theorem about continuity of resolvents which can be found in [@Kato1995 p. 340].
\[KLMN\] Let $\mathfrak{h}$ be a densely defined, closed sectorial form bounded from below and let $\mathfrak{p}$ be a form relatively bounded with respect to $\mathfrak{h}$, so that $D(\mathfrak{h})\subset D(\mathfrak{p})$ and $$\begin{aligned}
\label{eq:hypothesis1}
|\mathfrak{p}[u]|\leq a||u||^2+b\mathfrak{h}[u],
\end{aligned}$$
where $0\leq b<1$, but $a$ may be positive, negative or zero. Then $\mathfrak{h}+\mathfrak{p}$ is sectorial and closed. Let $H, K$ be the operators associated with $\mathfrak{h}$ and $\mathfrak{h}+\mathfrak{p}$, respectively. If $\zeta$ is not in the spectrum of $H$ and $$\begin{aligned}
\label{eq:hypothesis2}
||(a+bH)R(\zeta,H)||<1,
\end{aligned}$$ then $\zeta$ is not in the spectrum of $K$ and $$\begin{aligned}
\label{inequality5}
||R(\zeta,K)-R(\zeta,H)||\leq \frac{4||(a+bH)R(\zeta,H)||}{(1-||(a+bH)R(\zeta,H)||)^2}||R(\zeta,H)||.
\end{aligned}$$
In order to apply the theorem, we must verify the hypotheses and . We shall prove that $\mathfrak{p}$ is relatively bounded with respect to $\mathfrak{h}$, i.e., there exist $a,b\in\mathbb{R}$, such that: $$\begin{aligned}
|\mathfrak{p}[u]|\leq a||u||^2+b\mathfrak{h}[u],\end{aligned}$$
Observe that
$$\begin{aligned}
\mathfrak{h}[u]\geq \alpha\int_{\mathbb{R}^d}|\nabla u|^2~dy-\lambda_0\int_{\mathbb{R}^d}|u|^2~dy,\end{aligned}$$
and $$\begin{aligned}
\mathfrak{p}[u]&\leq t||B||_{L^\infty}\int_{\mathbb{R}^d}|\nabla u|^2~dy+|\tilde{\lambda}_0-\lambda_0|\int_{\mathbb{R}^d}|u|^2~dy\notag\\
&\quad=\frac{t||B||_{L^\infty}}{\alpha}\left\{\int_{\mathbb{R}^d}\alpha|\nabla u|^2~dy-\lambda\int_{\mathbb{R}^d}|u|^2~dy\right\}+\left\{|\tilde{\lambda}_0-\lambda_0|+\frac{t||B||_{L^\infty}}{\alpha}\lambda_0\right\}\int_{\mathbb{R}^d}|u|^2~dy\notag\\
&\quad= b\mathfrak{h}[u]+a||u||^2,\end{aligned}$$
where $a=\left\{|\tilde{\lambda}_0-\lambda_0|+\frac{t||B||_{L^\infty}}{\alpha}\lambda_0\right\}\approx c_1t$ and $b=\frac{t||B||_{L^\infty}}{\alpha}=c_2t$ for some constants $c_1$ and $c_2$.
Next, observe that for selfadjoint operator $H$, the resolvent $R(\zeta,H)$ is a normal operator, therefore, we have (see [@Kato1995 p. 177]) $$\begin{aligned}
||R(\zeta,H)||\leq\dfrac{1}{dist(\zeta,\sigma(H))}.\end{aligned}$$ Further, $$\begin{aligned}
||(a+bH)R(\zeta,H)||&\leq||aR(\zeta,H)||+||bHR(\zeta,H)||&\notag\\
&\quad\leq\frac{a}{dist(\zeta,\sigma(H))}+||b(I-\zeta R(\zeta,H))||&\notag\\
&\quad\leq\frac{a}{dist(\zeta,\sigma(H))}+b||I||+b||\zeta R(\zeta,H)||&\notag\\
&\quad\leq\frac{a}{dist(\zeta,\sigma(H))}+b+b\frac{|\zeta|}{dist(\zeta,\sigma(H))}.\end{aligned}$$
The operator corresponding to the sectorial form $\mathfrak{h}$ is $H\coloneqq-\nabla\cdot A\nabla-\lambda_0I$, therefore, $0\in\sigma(H)$, so that, for $\zeta=-\epsilon^2\varkappa^2$ $$\begin{aligned}
||(a+bH)R(\zeta,H)||&\leq\frac{a}{\epsilon^2\varkappa^2}+2b.&\end{aligned}$$
Notice that $R(\zeta,H)=S(\epsilon)$ and $R(\zeta,K)=\tilde{S}(\epsilon)$. Let us assume that $t$ is small enough so that Theorem \[KLMN\] can be applied to the resolvents in . In particular, we have $$\begin{aligned}
||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}&=~||R(\zeta,H)-R(\zeta,K)||&\notag\\&\leq\dfrac{4(c_1t+2c_2t\epsilon^2\varkappa^2)}{(\epsilon^2\varkappa^2-c_1t-2c_2t\epsilon^2\varkappa^2)^2}.&\end{aligned}$$
Choose $t$ so that $c_1t=\epsilon^4\varkappa^2$, then, $$\begin{aligned}
||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}\leq\frac{4(1+2c_3\epsilon^2\varkappa^2)}{\varkappa^2(1-\epsilon^2-2c_3\epsilon^4\varkappa^2)^2}.\end{aligned}$$
Further, for $\epsilon^2<1/2$, $$\begin{aligned}
\label{some_inequality_1}
||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}\leq\frac{16(1+c_3\varkappa^2)}{\varkappa^2(1-c_3\varkappa^2)^2}.\end{aligned}$$
Define the scaling transformation $T_\epsilon$ by $$\begin{aligned}
T_\epsilon:u(y)\mapsto \epsilon^{d/2}u(\epsilon y).
\end{aligned}$$ These are unitary operators on $L^2(\mathbb{R}^d)$. For the operators and , it holds that $$\begin{aligned}
R(\epsilon)=\epsilon^2T^*_\epsilon S(\epsilon)T_\epsilon\quad\mbox{and}\quad
\tilde{R}(\epsilon)=\epsilon^2T^*_\epsilon \tilde{S}(\epsilon)T_\epsilon.
\end{aligned}$$
Proving Lemma \[resolventlemma1\] is equivalent to proving that $$\begin{aligned}
||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right).
\end{aligned}$$ In fact, in , we proved $$\begin{aligned}
||S(\epsilon)-\tilde{S}(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(1).
\end{aligned}$$
Proof of Lemma \[resolventlemma2\] {#internal_homogenization_result}
----------------------------------
The aim of this section is to prove Lemma \[resolventlemma2\]. Let $\left(\tilde{\lambda}_l(\eta;t)\right)_{l=1}^\infty$ and $\left(\tilde{\phi}_l(y,\eta;t)\right)_{l=1}^\infty$ be the Bloch eigenvalues and the corresponding orthonormal Bloch eigenvectors for the operator $\tilde{\mathcal{A}}(t)$, defined in Theorem \[theorem:5\]. Let, $\tilde{\psi}_l(y,\eta;t)=e^{iy\cdot\eta}\tilde{\phi}_l(y,\eta;t)$. In the sequel, we shall suppress the dependence on $t$ for notational convenience. The operator $\tilde{\mathcal{A}}$ may be decomposed in terms of the Bloch eigenvalues as in the theorem below, a proof of which may be found in [@Bensoussan2011].
Let $g\in L^2(\mathbb{R}^d)$. Define $l^{\,th}$ Bloch coefficient of $g$ as follows: $$\begin{aligned}
(\tilde{\mathcal{B}}_l g)(\eta)=\int_{\mathbb{R}^d}\overline{\tilde{\psi}_l(y,\eta)}g(y)~dy,~l\in\mathbb{N},\eta\in Y^{'}.
\end{aligned}$$
Then, the following inverse formula holds. $$\begin{aligned}
g(y)=\sum_{l=1}^{\infty}\int_{Y^{'}}(\tilde{\mathcal{B}}_l g)(\eta){\psi}_l(y,\eta)~d\eta=&\sum_{l=1}^{\infty}(\tilde{\mathcal{B}}_l^*)(\tilde{\mathcal{B}}_l g),\quad\mbox{where},\\ (\tilde{\mathcal{B}}_l^*h)(y)=\int_{Y^{'}}h(\eta)\tilde{\psi}_l(y,\eta)\,d\eta&\quad\mbox{for}\quad h\in L^2(Y^{'}).\end{aligned}$$
In particular, the following representation holds for the operator $\tilde{\mathcal{A}}$: $$\begin{aligned}
\tilde{\mathcal{A}}=\sum_{l\in\mathbb{N}}\tilde{\mathcal{B}}_l^*\tilde{\lambda}_l\tilde{\mathcal{B}}_l.
\end{aligned}$$
Also, $$\begin{aligned}
R(\zeta,\tilde{\mathcal{A}})=\left(\tilde{\mathcal{A}}-\zeta I\right)^{-1}=\sum_{l\in\mathbb{N}}\tilde{\mathcal{B}}_l^*(\tilde{\lambda}_l-\zeta)^{-1}\tilde{\mathcal{B}}_l.
\end{aligned}$$
Define the Fourier Transform and the inverse Fourier Transform $$\begin{aligned}
\left({\mathcal{F}} u\right)(\eta)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{-iy\cdot\eta}u(y)~dy,\qquad\left({\mathcal{F}^{-1}} u\right)(\eta)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}e^{iy\cdot\eta}u(y)~dy.\end{aligned}$$
Define the operator $$\begin{aligned}
\tilde{S}^0(\epsilon)\coloneqq|Y|[\tilde{\psi}_m]\left(\tilde{{B}}_0\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_m}]+|Y|[\tilde{\psi}_{m+1}]\left(\tilde{{B}}_1\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}[\overline{\tilde{\psi}_{m+1}}]
\end{aligned}$$ For the operators and , it holds that $$\begin{aligned}
\tilde{R}(\epsilon)=\epsilon^2T^*_\epsilon \tilde{S}(\epsilon)T_\epsilon\quad\mbox{and}\quad
\tilde{R}^0(\epsilon)=\epsilon^2T^*_\epsilon \tilde{S}^0(\epsilon)T_\epsilon.
\end{aligned}$$ Therefore, to prove Lemma \[resolventlemma2\], it is sufficient to prove that $$\begin{aligned}
\label{tobeproved1}
||\tilde{S}(\epsilon)-\tilde{S}^0(\epsilon)||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right).
\end{aligned}$$
By making $\mathcal{O}$ smaller if required (see \[assumptions2:1\] and \[assumptions2:2\]), we may assume that $$\begin{aligned}
2(\tilde{\lambda}_{m}(\eta)-\tilde{\lambda}_0)\geq \tilde{{B}}_0(\eta-\eta_0)^2,~\eta\in\mathcal{O},\quad \mbox{ and }\\
2(\tilde{\lambda}_{m+1}(\eta)-\tilde{\lambda}_1)\geq \tilde{{B}}_1(\eta-\eta_1)^2,~\eta\in\mathcal{O}.
\end{aligned}$$ Let $\chi$ be the characteristic function of $\mathcal{O}$, then the projections $F=\tilde{\mathcal{B}}_m^*\,\chi\,\tilde{\mathcal{B}}_m+\tilde{\mathcal{B}}_{m+1}^*\,\chi\,\tilde{\mathcal{B}}_{m+1}$ and $F^\perp=I-F$ commute with $\tilde{\mathcal{A}}$.
Now, observe that $$\begin{aligned}
||\tilde{S}(\epsilon)-\tilde{S}^0(\epsilon)||_{L^2\to L^2}&=||\tilde{S}(\epsilon)F^{\perp}+\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F-\tilde{S}^0(\epsilon)F^\perp||_{L^2\to L^2}\notag\\
&\leq||\tilde{S}(\epsilon)F^{\perp}||_{L^2\to L^2}+||\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F||_{L^2\to L^2}+||\tilde{S}^0(\epsilon)F^\perp||_{L^2\to L^2}
\end{aligned}$$
Thus, in order to prove , it is sufficient to prove the following: $$\begin{aligned}
||\tilde{S}(\epsilon)F^{\perp}||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(1),\label{tobeproved3}\\
||\tilde{S}^0(\epsilon)F^\perp||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}(1),\label{tobeproved4}
\end{aligned}$$$$\begin{aligned}
||\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right) \label{tobeproved2}.
\end{aligned}$$
[*Proof of *]{}: Notice that the Bloch wave decomposition of $\tilde{S}(\epsilon)$ is given by $$\begin{aligned}
\tilde{S}(\epsilon)=\sum_{l=1}^{\infty}\tilde{\mathcal{B}}_l^* \left(\tilde{\lambda}_l-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\tilde{\mathcal{B}}_l.\end{aligned}$$
We may write, $$\begin{aligned}
\tilde{S}(\epsilon)&=\tilde{S}(\epsilon)F+\tilde{S}(\epsilon)F^\perp,&
\end{aligned}$$ where $$\begin{aligned}
\tilde{S}(\epsilon)F=\tilde{\mathcal{B}}_m^* \left(\tilde{\lambda}_m-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\chi \tilde{\mathcal{B}}_m+\tilde{\mathcal{B}}_{m+1}^* \left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\chi \tilde{\mathcal{B}}_{m+1},
\end{aligned}$$
and $$\begin{aligned}
\label{awayfromedge1}
\tilde{S}(\epsilon)F^\perp&=\sum_{l\neq m,m+1}\tilde{\mathcal{B}}_l^* \left(\tilde{\lambda}_l-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} \tilde{\mathcal{B}}_l+\tilde{\mathcal{B}}_m^*\left(\tilde{\lambda}_m-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\left(1-\chi\right) \tilde{\mathcal{B}}_m\notag\\
&\quad\quad+\tilde{\mathcal{B}}_{m+1}^*\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\left(1-\chi\right) \tilde{\mathcal{B}}_{m+1}.
\end{aligned}$$
To prove , notice that in the first term of , the sum does not include indices $m$ and $m+1$, therefore, the Bloch eigenvalues $\tilde{\lambda}_l$ are bounded away from the spectral edge $\tilde{\lambda}_0$, uniformly in $\epsilon$ and hence, the expression $\left(\tilde{\lambda}_l-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}$ is bounded independent of $\epsilon$, for $l\neq m, m+1$. Due to the non-degeneracy conditions assumed in \[assumptions2:1\] and \[assumptions2:2\], the Bloch eigenvalues $\tilde{\lambda}_m$ and $\tilde{\lambda}_{m+1}$ are bounded away from $\tilde{\lambda}_0$ outside $\mathcal{O}$, independent of $\epsilon$. Hence, the last two terms in are bounded independent of $\epsilon$.
[*Proof of *]{}: Similarly, we may write $$\begin{aligned}
\tilde{S}^0(\epsilon)&=\tilde{S}^0(\epsilon)F+\tilde{S}^0(\epsilon)F^\perp,\quad\mbox{where}\\
\tilde{S}^0(\epsilon)F&=|Y|[\tilde{\psi}_m]\left(\tilde{{B}}_0\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\chi[\overline{\tilde{\psi}_m}]+|Y|[\tilde{\psi}_{m+1}]\left(\tilde{{B}}_1\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\chi[\overline{\tilde{\psi}_{m+1}}],\quad\mbox{and},
\end{aligned}$$ $$\begin{aligned}
\label{awayfromedge2}
\tilde{S}^0(\epsilon)F^\perp&=|Y|[\tilde{\psi}_m]\left(\tilde{{B}}_0\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)[\overline{\tilde{\psi}_m}]&\notag\\
&\quad+|Y|[\tilde{\psi}_{m+1}]\left(\tilde{{B}}_1\nabla^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)[\overline{\tilde{\psi}_{m+1}}].&
\end{aligned}$$ $\tilde{S}^0(\epsilon)F^\perp$ may be further written as $$\begin{aligned}
\tilde{S}^0(\epsilon)F^\perp&=|Y|[\tilde{\phi}_m]\mathcal{F}^{-1}\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)\mathcal{F}[\overline{\tilde{\phi}_m}]\notag\\
&\quad+|Y|[\tilde{\phi}_{m+1}]\mathcal{F}^{-1}\left(\tilde{{B}}_1(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}\left(1-\chi\right)\mathcal{F}[\overline{\tilde{\phi}_{m+1}}].\notag
\end{aligned}$$
The proof of follows from the positive-definiteness of $\tilde{B}_0$ and $\tilde{B}_1$ assumed in \[assumptions2:1\] and \[assumptions2:2\], which makes the operator norm of the terms in independent of $\epsilon$. Now, it only remains to prove .
[*Proof of *]{}: Write $\tilde{S}(\epsilon)F=S_0+S_1$, where, for $j=0,1$, $$\begin{aligned}
\label{res1}
S_j\coloneqq\tilde{\mathcal{B}}_{m+j}^*\left(\tilde{\lambda}_{m+j}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\,\chi\, \tilde{\mathcal{B}}_{m+j}\notag\\
=X_{m+j}^* \left(\tilde{\lambda}_{m+j}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} X_{m+j},
\end{aligned}$$ and, for $j=0,1$, $$\begin{aligned}
\left(X_{m+j} u\right)(\eta)=\int_{\mathbb{R}^d}\chi\overline{\tilde{\psi}_{m+j}}(y,\eta)u(y)~dy\quad\mbox{and}\quad\left(X_{m+j}^* u\right)(y)=\int_{\mathbb{Y}^{'}}\chi\tilde{\psi}_{m+j}(y,\eta)u(y)~d\eta.
\end{aligned}$$
Write $\tilde{S}^0(\epsilon)F=S_0^0+S_1^0$, where, for $j=0,1$, $$\begin{aligned}
\label{res2}
S_j^0&=|Y|[\tilde{\phi}_{m+j}]\mathcal{F}^{-1}\left(\tilde{{B}}_j(\eta-\eta_j)^2+\epsilon^2\varkappa^2I\right)^{-1}\left(\chi\right)\mathcal{F}[\overline{\tilde{\phi}_{m+j}}]\notag\\
&= (X^0_{m+j})^*\left(\tilde{{B}}_j(\eta-\eta_j)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m+j},
\end{aligned}$$ and, for $j=0,1$, $$\begin{aligned}
\left(X^0_{m+j} u\right)(\eta)=\int_{\mathbb{R}^d}\chi e^{-iy\cdot\eta}\overline{\tilde{\phi}_{m+j}}(y,\tilde{\eta}_j)u(y)~dy\quad\mbox{and}\quad\left(X^0_{m+j} u\right)^*(\eta)=\int_{\mathbb{R}^d}\chi e^{-iy\cdot\eta}\tilde{\phi}_{m+j}(y,\tilde{\eta}_j)u(y)~dy.
\end{aligned}$$
Observe that, $$\begin{aligned}
||\tilde{S}(\epsilon)F-\tilde{S}^0(\epsilon)F||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}\leq||S_0-S_0^0||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}+||S_1-S_1^0||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}.
\end{aligned}$$
Therefore, to prove , it remains to prove that for $j=0,1$, $$\begin{aligned}
||{S}_j-{S}_j^0||_{L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)}={O}\left(\dfrac{1}{\epsilon}\right).
\end{aligned}$$
where $S_j, S_j^0$ are defined in , .
Consider, $$\begin{aligned}
\epsilon||{S}_0-{S}_0^0||&=\epsilon||X_{m}^* [\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m}-(X^0_{m})^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m}||.
\end{aligned}$$ Therefore, $$\begin{aligned}
\label{final1}
\epsilon||{S}_0-{S}_0^0||& \leq \epsilon||X_{m}^* [\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m}-X_{m}^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m}||&\notag\\
&\qquad+\epsilon||X_{m}^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m}-(X^0_{m})^*\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m}||.&
\end{aligned}$$
The first of the two terms on the right hand side (RHS) in the inequality is estimated by using the following chain of inequalities. $$\begin{aligned}
&\epsilon|\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} -\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}|&\notag\\
&\qquad\leq c\epsilon|\eta-\eta_0|^3\left(\tilde{\lambda}_{m}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}&\notag\\
&\qquad\leq\left(c|\eta-\eta_0|^2\left(\tilde{{B}}_0(\eta-\eta_0)^2\right)^{-1}\right)\left(2\epsilon|\eta-\eta_0|\left(\tilde{{B}}_0(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}\right)\leq C_1.&
\end{aligned}$$
The proof of the boundedness of the second term on the RHS in inequality hinges on the analyticity of the Bloch eigenfunctions, and may be found in [@Birman2006]. Finally, consider $$\begin{aligned}
\epsilon||{S}_1-{S}_1^0||&=\epsilon||X_{m+1}^* [\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m+1}-(X^0_{m+1})^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m+1}||.
\end{aligned}$$ Therefore, $$\begin{aligned}
\label{final2}
\epsilon||{S}_1-{S}_1^0||&\leq \epsilon||X_{m+1}^* [\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1}] X_{m+1}-X_{m+1}^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m+1}||&\notag\\
&~+\epsilon||X_{m+1}^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X_{m+1}-(X^0_{m+1})^*\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}X^0_{m+1}||.&
\end{aligned}$$
The first of the two terms on RHS in inequality is estimated by using the following chain of inequalities. $$\begin{aligned}
&\epsilon|\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_0+\epsilon^2\varkappa^2\right)^{-1} -\left(\tilde{{B}}_1(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}|&\notag\\
&\qquad\leq\epsilon|\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_1+\epsilon^2\varkappa^2\right)^{-1} -\left(\tilde{{B}}_1(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}|&\notag\\
&\qquad\leq c\epsilon|\eta-\eta_1|^3\left(\tilde{\lambda}_{m+1}-\tilde{\lambda}_1+\epsilon^2\varkappa^2\right)^{-1}\left(\tilde{{B}}_0(\eta-\eta_1)^2+\epsilon^2\varkappa^2I\right)^{-1}&\notag\\
&\qquad\leq\left(c|\eta-\eta_1|^2\left(\tilde{{B}}_1(\eta-\eta_0)^2\right)^{-1}\right)\left(2\epsilon|\eta-\eta_1|\left(\tilde{{B}}_1(\eta-\eta_0)^2+\epsilon^2\varkappa^2I\right)^{-1}\right)\leq C_2.&
\end{aligned}$$
As before, the proof of the boundedness of the second term on RHS in inequality may be found in [@Birman2006].
Perturbation Theory of holomorphic family of type $(B)$ {#PerturbationTheory}
=======================================================
In this section, we show that a perturbation in the coefficients of the operator $\mathcal{A}$ gives rise to a corresponding holomorphic family of sectorial forms of type $(a)$. Further, the selfadjointness of the forms coupled with the compactness of the resolvent for the operator family ensures that it is a selfadjoint holomorphic family of type $(B)$. For definition of these notions, see Kato [@Kato1995].
Let $A\in M_B^{>}$ and $B=(b_{kl})$ be a symmetric matrix with $L^\infty_{\sharp}(Y,\mathbb{R})$ entries. Then, for $\sigma<\frac{\alpha}{d||B||_{L^\infty}}$, $A+\sigma B$ belongs to $M_B^{>}$, where $\alpha$ is a coercivity constant for $A$, as in . For a fixed $\eta_0 \in Y^{'}$ and for $\sigma_0\coloneqq \frac{\alpha}{2d||B||_{L^\infty}}$, let us define the operator family $$\begin{aligned}
\mathcal{A}(\eta_0)(\tau)=-(\nabla+i\eta_0)\cdot(A+\tau B)(\nabla+i\eta_0),\quad\tau\in R,\end{aligned}$$ where $R=\{z\in\mathbb{C}:|\operatorname{Re}(z)|<\sigma_0,|\operatorname{Im}(z)|<\sigma_0\}.$ For real $\tau$, $-\sigma_0<\tau<\sigma_0$, $A+\tau B$ is coercive with a coercivity constant $\alpha/2$. The holomorphic family of sesquilinear forms $\mathfrak{t}(\tau)$ associated to operator $\mathcal{A}+\tau\mathcal{B}$, with the $\tau$-independent domain $\mathfrak{D}(\mathfrak{t}(\tau))=H^1_\sharp(Y)$, is defined as $$\begin{aligned}
\mathfrak{t}(\tau)[u,v] &\coloneqq \int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{v}}{\partial y_k}~dy+i\eta_{0,l}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\frac{\partial \overline{v}}{\partial y_k}~dy\\
&\quad-i\eta_{0,k}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) \overline{v} \frac{\partial u}{\partial y_l}~dy+\eta_{0,l}\eta_{0,k}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\overline{v}~dy,\end{aligned}$$
where $\eta_0\coloneqq(\eta_{0,1},\eta_{0,2},\ldots,\eta_{0,d})$ and summation over repeated indices is assumed.
$\mathfrak{t}(\tau)$ is a holomorphic family of type $(a)$.
The quadratic form associated with $\mathfrak{t}(\tau)$ is as follows: $$\begin{aligned}
\mathfrak{t}(\tau)[u] & \coloneqq \int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+i\eta_{0,l}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy
\\
& \quad-i\eta_{0,k}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) \overline{u} \frac{\partial u}{\partial y_l}~dy+\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\tau b_{kl}(y)\right) u\overline{u}~dy.
\end{aligned}$$
[**$(i)$**]{} $\mathfrak{t}(\tau)$ is sectorial.
Let us write $\tau=\rho+i\gamma$, then the quadratic form $\mathfrak{t}(\tau)$ can be written as the sum of its real and imaginary parts: $$\begin{aligned}
\mathfrak{t}(\tau)= \Re{\mathfrak{t}(\tau)[u]}+ i\Im{\mathfrak{t}(\tau)[u]}
\end{aligned}$$
where the real part is $$\begin{aligned}
\label{realpart}
\Re{\mathfrak{t}(\tau)[u]} & \coloneqq \int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+i\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\notag
\\
&\quad -i\eta_{0,k}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) \overline{u} \frac{\partial u}{\partial y_l}~dy+\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\overline{u}~dy,
\end{aligned}$$
and the imaginary part is $$\begin{aligned}
\label{imaginarypart}
\Im{\mathfrak{t}(\tau)[u]} & \coloneqq \int_Y \left(\gamma b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+2\Im{\left(\eta_{0,l}\int_Y \left(\gamma b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\right)}\notag
\\
&\quad +\eta_{0,k}\eta_{0,l}\int_Y \left(\gamma b_{kl}(y)\right) u\overline{u}~dy.
\end{aligned}$$
The real part of ${\mathfrak{t}(\tau)[u]}$ may also be written as $$\begin{aligned}
\label{realpart2}
\Re{\mathfrak{t}(\tau)[u]} & \coloneqq \int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy+2\Re{\left(i\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\right)}\notag
\\
&\quad +\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\overline{u}~dy.
\end{aligned}$$
The first term in is estimated from below as follows: $$\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right)\frac{\partial u}{\partial y_l}\frac{\partial \overline{u}}{\partial y_k}~dy \geq \frac{\alpha}{2}\int_{Y}|\nabla u|^2~dy.\label{eq:111}$$
The second term in may be bounded from above as follows: $$\begin{aligned}
\Re{\left(i\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\frac{\partial \overline{u}}{\partial y_k}~dy\right)} & \leq C_1||u||_{L^2_{\sharp}(Y)}||\nabla u||_{L^2_{\sharp}(Y)}\notag
\\
& \leq C_1C_*||u||^2_{L^2_{\sharp}(Y)}+\frac{C_1}{C_*}||\nabla u||^2_{L^2_{\sharp}(Y)}\notag
\\
& = C_2||u||^2_{L^2_{\sharp}(Y)}+\frac{\alpha}{4}||\nabla u||^2_{L^2_{\sharp}(Y)},\label{eq:222}
\end{aligned}$$
where $C_*=\frac{4C_1}{\alpha}$ and $C_1, C_2$ are some constants independent of $u$ and $\rho$.
The last term in is estimated as $$\begin{aligned}
\eta_{0,k}\eta_{0,l}\int_Y \left(a_{kl}(y)+\rho b_{kl}(y)\right) u\overline{u}~dy \leq C_3 ||u||^2_{L^2_{\sharp}(Y)},\label{eq:333}
\end{aligned}$$ for some $C_3>0$.
Finally, combining , and , we obtain $$\begin{aligned}
\Re{\mathfrak{t}(\tau)[u]} \geq \frac{\alpha}{4}||u||^2_{H^1_{\sharp}(Y)}-C_4||u||^2_{L^2_{\sharp}(Y)},\label{eq:444}
\end{aligned}$$ for some $C_4>0$.
Estimating the imaginary part from above, we obtain $$\begin{aligned}
|\Im{\mathfrak{t}(\tau)[u]}| \leq C_5||\nabla u||^2_{L^2_\sharp(Y)}+C_6||u||^2_{L^2_\sharp(Y)},\label{eq:999}
\end{aligned}$$ for some positive $C_5, C_6$.
Now, choose a scalar $C_7$ so that $C_7=\frac{4C_5}{\alpha}$.
The inequality may be written as $$\begin{aligned}
\Re{\mathfrak{t}(\tau)[u]} + C_4||u||^2_{L^2_{\sharp}(Y)} + \frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)} \geq \frac{\alpha}{4}||u||^2_{H^1_{\sharp}(Y)}+\frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:555}
\end{aligned}$$
Now, we define a new quadratic form $\tilde{\mathfrak{t}}[u]\coloneqq \mathfrak{t}[u]+(C_4+\frac{C_6}{C_7})||u||^2_{L^2_\sharp{Y}},$ then inequality becomes $$\begin{aligned}
\Re{\tilde{\mathfrak{t}}(\tau)[u]} \geq \frac{\alpha}{4}||u||^2_{H^1_{\sharp}(Y)}+\frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:666}
\end{aligned}$$
This may be further written as $$\begin{aligned}
\Re{\tilde{\mathfrak{t}}(\tau)[u]} - \frac{\alpha}{4}||u||^2_{L^2_\sharp(Y)} \geq \frac{\alpha}{4}||\nabla u||^2_{L^2_{\sharp}(Y)}+\frac{C_6}{C_7}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:777}
\end{aligned}$$
On multiplying throughout by $C_7$, the inequality becomes $$\begin{aligned}
C_7\left\{\Re{\tilde{\mathfrak{t}}(\tau)[u]} - \frac{\alpha}{4}||u||^2_{L^2_\sharp(Y)}\right\} \geq C_5||\nabla u||^2_{L^2_{\sharp}(Y)}+{C_6}||u||^2_{L^2_{\sharp}(Y)}.\label{eq:888}
\end{aligned}$$
Since $\Im{\tilde{\mathfrak{t}}[u]}=\Im{\mathfrak{t}[u]}$, combining the inequalities and , we obtain $$\begin{aligned}
|\Im{\tilde{\mathfrak{t}}(\tau)[u]}| \leq C_7\left\{\Re{\tilde{\mathfrak{t}}(\tau)[u]} - \frac{\alpha}{4}||u||^2_{L^2_\sharp(Y)}\right\}.
\end{aligned}$$
This proves that the form $\tilde{\mathfrak{t}}$ is sectorial. However, the property of sectoriality is invariant under a shift. Therefore, $\mathfrak{t}$ is sectorial, as well.
[**$(ii)$**]{} $\mathfrak{t}(\tau)$ is closed.
This follows from the inequality . If $u_n\xrightarrow{{\mathfrak{t}-convergence}}u$ then $\Re{\mathfrak{t}[u_n-u_m]}\to 0$ as $n,m\to\infty$. By , $(u_n)$ is a Cauchy sequence in $H^1_\sharp(Y)$. By completeness, there is $v\in H^1_\sharp(Y)$ to which the sequence converges. However, $\mathfrak{t}$-convergence implies $L^2$ convergence, and therefore, $u=v$. Clearly, $\mathfrak{t}[u_n-u]\to 0$.
[**$(iii)$**]{} $\mathfrak{t}(\tau)$ is a holomorphic family of type $ (a) $.
We have proved that $\mathfrak{t}(\tau)[u]$ is sectorial and closed. It remains to prove that the form is holomorphic. This is easily done since $\mathfrak{t}(\tau)[u]$ is linear in $\tau$ for each fixed $u\in H^1_\sharp(Y)$.
The first representation theorem of Kato ensures that there exists a unique m-sectorial operator with domain contained in $H^1_\sharp(Y)$ associated with each $\mathfrak{t}(\tau)$. A proof may be found in [@Kato1995 p.322]. The family of such operators associated with a holomorphic family of sesquilinear forms of type $(a)$ is called a holomorphic family of type $( B )$. The aforementioned m-sectorial operator is given by $$\begin{aligned}
\mathcal{A}(\eta_0)(\tau)=-(\nabla+i\eta_0)(A+\tau B)(\nabla+i\eta_0).\end{aligned}$$ It follows from the symmetry of the matrix $A+\tau B$ that the family $\mathcal{A}(\eta_0)(\tau)$ is a selfadjoint holomorphic family of type $(B)$. Moreover, by the compact embedding of $H^1_\sharp(Y)$ in $L^2_\sharp(Y)$, the operator $\mathcal{A}(\eta_0)(\tau)+C^*I$ has compact resolvent for each $\tau\in R$ for some appropriate constant $C^*$, independent of $\tau\in R$.
Hence, by Kato-Rellich Theorem, there exists a complete orthonormal set of eigenvectors associated with the operator family $\mathbb{A}(\eta_0)(\tau)$ which are analytic for the whole interval $-\sigma_0<\tau<\sigma_0$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like and . We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst and are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by and on some benchmark problems.'
author:
- George Katsirelos
- Nina Narodytska
- Toby Walsh
title: 'On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry[^1]'
---
Introduction
============
One challenge in constraint programming is to develop effective search methods to deal with common modelling patterns. One such pattern is row and column symmetry [@ffhkmpwcp2002]: many problems can be modelled by a matrix of decision variables [@matrix] where the rows and columns of the matrix are fully or partially interchangeable. Such symmetry is a source of combinatorial complexity. It is therefore important to develop techniques to deal with this type of symmetry. We study here simple constraints that can be posted to break row and column symmetries, and analyse their effectiveness both theoretically and experimentally. We prove that we can compute in polynomial time the lexicographically smallest representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and thus remove all symmetric solutions. We are therefore able for the first time to see how much symmetry is left by these commonly used symmetry breaking constraints.
Formal background
=================
A constraint satisfaction problem (CSP) consists of a set of variables, each with a domain of values, and a set of constraints specifying allowed values for subsets of variables. When solving a CSP, we often use propagation algorithms to prune the search space by enforcing properties like domain consistency. A constraint is *domain consistent* (*DC*) iff when a variable in the scope of a constraint is assigned any value in its domain, there exist compatible values in the domains of all the other variables in the scope of the constraint. A CSP is domain consistent iff every constraint is domain consistent. An important feature of many CSPs is symmetry. Symmetries can act on variables or values (or both). A *variable symmetry* is a bijection $\sigma$ on the variable indices that preserves solutions. That is, if $\{X_i = a_i \ | \ i \in [1,n]\}$ is a solution then $\{X_{\sigma(i)} = a_i \ | \ i \in [1,n]\}$ is also. A *value symmetry* is a bijection $\theta$ on the values that preserves solutions. That is, if $\{X_i = a_i \ | \ i \in [1,n]\}$ is a solution then $\{X_{i} = \theta(a_i) \ | \ i \in [1,n]\}$ is also. A simple but effective method to deal with symmetry is to add *symmetry breaking constraints* which eliminate symmetric solutions. For example, Crawford [*et al.*]{} proposed the general lex-leader method that posts lexicographical ordering constraints to eliminate all but the lexicographically least solution in each symmetry class [@clgrkr96]. Many problems are naturally modelled by a matrix of decision variables with variable symmetry in which the rows and/or columns are interchangeable [@ffhkmpwcp2002]. We say that a CSP containing a matrix of decision variables has row symmetry iff given a solution, any permutation of the rows is also a solution. Similarly, it has column symmetry iff given a solution, any permutation of the columns is also a solution.
The Equidistant Frequency Permutation Array (EFPA) problem is a challenging problem in coding theory. The goal is to find a set of $v$ code words, each of length $q\lambda$ such that each word contains $\lambda$ copies of the symbols 1 to $q$, and each pair of code words is Hamming distance $d$ apart. For example, for $v=4$, $\lambda=2$, $q=3$, $d=4$, one solution is:
$$\label{epfa}
\begin{array}{cccccc}
0 & 2 & 1 & 2 & 0 & 1 \\
0 & 2 & 2 & 1 & 1 & 0 \\
0 & 1 & 0 & 2 & 1 & 2 \\
0 & 0 & 1 & 1 & 2 & 2
\end{array}$$
This problem has applications in communication theory, and is related to other combinatorial problems like finding orthogonal Latin squares. Huczynska [*et al.*]{} [@hmmncp09] consider a model for this problem with a $v$ by $q\lambda$ array of variables with domains $1$ to $q$. This model has row and column symmetry since we can permute the rows and columns and still have a solution.
Breaking row and column symmetry
================================
To break all row symmetry we can post lexicographical ordering constraints on the rows. Similarly, to break all column symmetry we can post lexicographical ordering constraints on the columns. When we have both row and column symmetry, we can post a ${\mbox{\sc DoubleLex}\xspace}$ constraint that lexicographically orders both the rows and columns [@ffhkmpwcp2002]. This does not eliminate all symmetry since it may not break symmetries which permute both rows and columns. Nevertheless, it is often effective in practice.
Consider again solution . If we order the rows of lexicographically, we get a solution with lexicographically ordered rows and columns:
$$\label{epfa2}
\begin{array}{cccccc}
0 & 2 & 1 & 2 & 0 & 1 \\
0 & 2 & 2 & 1 & 1 & 0 \\
0 & 1 & 0 & 2 & 1 & 2 \\
0 & 0 & 1 & 1 & 2 & 2
\end{array}
\
\begin{array}{c}
\mbox{\rm order} \\
\ \ \ \ \Rightarrow \ \ \ \ \\
\mbox{\rm rows}
\end{array}
\ \
\begin{array}{cccccc}
0 & 0 & 1 & 1 & 2 & 2 \\
0 & 1 & 0 & 2 & 1 & 2 \\
0 & 2 & 1 & 2 & 0 & 1 \\
0 & 2 & 2 & 1 & 1 & 0
\end{array}$$
Similarly if we order the columns of lexicographically, we get a different solution in which both rows and columns are again ordered lexicographically:
$$\label{epfa3}
\begin{array}{cccccc}
0 & 2 & 1 & 2 & 0 & 1 \\
0 & 2 & 2 & 1 & 1 & 0 \\
0 & 1 & 0 & 2 & 1 & 2 \\
0 & 0 & 1 & 1 & 2 & 2
\end{array}
\
\begin{array}{c}
\mbox{\rm order} \\
\ \ \ \ \Rightarrow \ \ \ \ \\
\mbox{\rm cols}
\end{array}
\ \
\begin{array}{cccccc}
0 & 0 & 1 & 1 & 2 & 2 \\
0 & 1 & 0 & 2 & 1 & 2 \\
0 & 1 & 2 & 0 & 2 & 1 \\
0 & 2 & 2 & 1 & 1 & 0
\end{array}$$
All three solutions are thus in the same row and column symmetry class. However, both
and satisfy the constraint. Therefore can leave multiple solutions in each symmetry class.
The lex-leader method breaks all symmetry by ensuring that any solution is the lexicographically smallest in its symmetry class [@clgrkr96]. This requires linearly ordering the matrix. Lexicographically ordering the rows and columns is consistent with a linearization that takes the matrix in row-wise order (i.e. appending rows in order). We therefore consider a complete symmetry breaking constraint which ensures that the row-wise linearization of the matrix is lexicographically smaller than all its row or column permutations, or compositions of row and column permutations.
Consider the symmetric solutions to . If we linearize these solutions row-wise, the first two are lexicographically larger than the third. Hence, the first two solutions are eliminated by the constraint.
breaks all row and column symmetries. Unfortunately, posting such a constraint is problematic since it is NP-hard to check if a complete assignment satisfies [@bhhwaaai2004; @complexity]. We now give our first major result. We prove that if we can bound the number of rows (or columns), then there is a polynomial time method to break all row and column symmetry. For example, in the EFPA problem, the number of columns might equal the fixed word size of our computer.
\[tm:fpt\] For a $n$ by $m$ matrix, we can check if a complete assignment satisfies a constraint in $O(n! nm \log m)$ time.
[[**Proof: **]{}]{}Consider the matrix model $X_{i,j}$. [[ We exploit the fact that with no row symmetry and just column symmetry, lexicographically ordering the columns gives the lex-leader assignment. ]{}]{} [[ Let $Y_{i,j}= X_{\sigma(i),j}$ be a row permutation of $X_{i,j}$. To obtain $Z_{i,j}$, the smallest column permutation of $Y_{i,j}$ we lexicographically sort the $m$ columns of $Y_{i,j}$ in $O(n m \log(m))$ time. Finally, we check that $[X_{1,1},\ldots,X_{1,m},\ldots,X_{n,1},\ldots,X_{n,m}]
\leq_{\rm lex} [Z_{1,1},\ldots,Z_{1,m},\ldots,Z_{n,1},\ldots,Z_{n,m}]$, where $\leq_{\rm lex}$ is the lexicographic comparison of two vectors. This ensures that $X_{i,j}$ is lexicographically smaller than or equal to any column permutation of this row permutation. ]{}]{} If we do this for each of the $n!-1$ non-identity row permutations, then $X_{i,j}$ is lexicographically smaller than or equal to any row permutation. This means that we have the lex-leader assignment. This can be done in time $O(n! nm\log m)$, which for bounded $n$ is polynomial.
This result easily generalizes to when rows and columns are partially interchangeable. In the experimental section, we show that this gives an effective method to break [*all*]{} row and column symmetry.
Double Lex
==========
When the number of both rows and columns is large, breaking [all]{} row and column symmetry is computationally challenging. In this situation, we can post a constraint [@ffhkmpwcp2002]. However, as we saw in the running example, this may not break all symmetry. In fact, it can leave $n!$ symmetric solutions in an $2n\times 2n$ matrix model.
\[tm:expsol\] There exists a class of $2n$ by $2n$ 0/1 matrix models on which leaves $n!$ symmetric solutions, for all $n \geq 2$.
[[**Proof: **]{}]{}Consider a $2n$ by $2n$ matrix model with the constraints that the matrix contains $3n$ non-zero entries, and each row and column contains between one and two non-zero entries. This model has row and column symmetry since row and column permutations leave the constraints unchanged. There exists a class of symmetric solutions to the problem that satisfy a constraint of the form: $$\begin{array}{cc}
0 & I^R \\
I^R & P \end{array}$$ Where $0$ is a $n$ by $n$ matrix of zeroes, $I^R$ is the [[reflection]{}]{} of the identity matrix, and $P$ is any permutation matrix (a matrix with one non-zero entry on each row and column). For example, as there are exactly two possible permutation matrices of order 2, there are two symmetric 4 by 4 solutions with lexicographically ordered rows and columns: $$\begin{aligned}
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 \\
1 & 0 & 0 & 1
\end{array}
& \ \ \ {\rm and} \ \ \ &
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
1 & 0 & 1 & 0
\end{array}\end{aligned}$$ In general, there are $n!$ row and column symmetries of $P$. Hence, leaves $n!$ symmetric solutions.
Having decided to break row and column symmetry with , how do we propagate it? One option is to decompose it into two ${\mbox{\sc LexChain}}$ constraints, one on the rows and the other on the columns. A ${\mbox{\sc LexChain}}$ constraint ensures that a sequence of vectors are lexicographically ordered. Enforcing domain consistency on each ${\mbox{\sc LexChain}}$ constraint takes polynomial time [@lexchain]. However, this decomposition hinders propagation. For example, in the matrix of decision variables with domains: $$\begin{array}{ccc}
0/1 & 0/1 & 1 \\
0/1 & 0 & 1 \\
1 & 1 & 1
\end{array}$$ constraints on the rows and columns ensure the second row is lexicographically larger than the first row and lexicographically smaller than the third, and the second column is lexicographically larger than the first column and lexicographically smaller than the third. Both such constraints are DC. However, the corresponding ${\mbox{\sc DoubleLex}\xspace}$ constraint is not since there is no solution in which the top left variable is set to 1. We might therefore consider a specialized propagator for the constraint. Unfortunately, whilst checking a constraint takes polynomial time, enforcing DC on this constraint is NP-hard. Thus, even when posting just to break row and column symmetry, there are computational limits on our ability to prune symmetric branches from the search tree.
\[t:lex\_double\] Enforcing DC on the ${\mbox{\sc DoubleLex}\xspace}$ constraint is NP-hard.
[[**Proof: **]{}]{}(Outline) We reduce an instance of 1-in-3SAT on positive clauses to a partially instantiated instance of the constraint with 0/1 variables. The constructed ${\mbox{\sc DoubleLex}\xspace}$ constraint has a solution iff the 1-in-3 SAT formula is satisfiable. Hence, it is NP-hard to enforce DC on the ${\mbox{\sc DoubleLex}\xspace}$ constraint [@bhhwaaai2004], even with a bounded number of values. The full proof appears in [@dlexproof].
Special cases
=============
We consider two special cases where we can check a constraint that breaks all row and column symmetry in polynomial time. In both cases, we show that we can do even better than check the constraint in polynomial time. We prove that in these cases we can enforce DC on a constraint that breaks all row and column symmetry in polynomial time. This provides a counterpoint to our result that enforcing DC on is NP-hard in general.
All-different matrices
----------------------
An all-different matrix is a matrix model in which every value is different. It was shown in [@ffhkmpwcp2002] that when an all-different matrix has row and column symmetry, then ${\mbox{\sc RowWiseLexLeader}}$ is equivalent to ensuring that the top left entry is the smallest value, and the first row and column are ordered. Let be such a symmetry breaking constraint.
DC can be enforced on ${\mbox{\sc Order1stRowCol}}$ in polynomial time.
[[**Proof: **]{}]{}Consider the $n$ by $m$ matrix model $X_{i,j}$. We post $O(nm)$ constraints: $X_{1,1} < \ldots < X_{n,1}$, $X_{1,1} < \ldots < X_{1,m}$, $X_{1,1} < X_{1+i,1+j}$ for $1 \leq i < n$ and $1 \leq j < m$. The constraint graph of this decomposition is acyclic. Therefore enforcing DC on the decomposition achieves DC on ${\mbox{\sc Order1stRowCol}}$. Each constraint in the decomposition can be made DC in constant time (assuming we can change bounds in constant time). Hence, DC can be enforced on ${\mbox{\sc Order1stRowCol}}$ in $O(nm)$ time.
Note that, when applied to an all-different matrix with row and column symmetry, the general method for breaking symmetry in all-different problems proposed in [@pijcai2005] will post binary inequalities logically equivalent to ${\mbox{\sc Order1stRowCol}}$.
Matrix models of functions
--------------------------
A matrix model of a function is one in which all entries are 0/1 and each row sum is 1. If a matrix model of a function has row and column symmetry then ${\mbox{\sc RowWiseLexLeader}}$ ensures the rows and columns are lexicographically ordered, the row sums are 1, and the sums of the columns are in decreasing order, as was shown in [@ilya01; @lex2001; @ffhkmpwcp2002]. We denote this symmetry breaking constraint as ${\mbox{\sc DoubleLexColSum}\xspace}$. Enforcing DC on ${\mbox{\sc DoubleLexColSum}\xspace}$ takes polynomial time, in contrast to partial row and column interchangeability in matrix models of functions, which is NP-hard [@wcp07].
DC can be enforced on ${\mbox{\sc DoubleLexColSum}\xspace}$ in polynomial time.
[[**Proof: **]{}]{}[[We will show that ${\mbox{\sc DoubleLexColSum}\xspace}$ can be encoded with a set of ${\mbox{\sc Regular}}$ constraints. Consider the $n$ by $m$ matrix model $X_{i,j}$. For each row $i$ we introduce an extra variable $Y_i$ and a ${\mbox{\sc Regular}}$ constraint on $[X_{i,1}, \ldots,X_{i,m}, \#, Y_i]$ where $\#$ is a delimiter between $X_{i,m}$ and $Y_i$. Each ${\mbox{\sc Regular}}$ constraint ensures that exactly one position in the $i$th row is set to $1$ and the variable $Y_i$ stores this position. The automaton’s states are represented by the 3-tuple $\left\langle s, d, p \right\rangle$ where $s$ is the row sum, $d$ is the current position and $p$ records the position of the 1 on this row. This automaton has $4m$ states and a constant number of transitions from each state, so the total number of transitions is $O(m)$. The complexity of propagating this constraint is $O(m^2)$. ]{}]{} [[ We also post a ${\mbox{\sc Regular}}$ constraint over $Y_1,\ldots,Y_n$ to ensure that they form a decreasing sequence of numbers and the number of occurrences of each value is decreasing. The first condition ensures that rows and columns are lexicographically ordered and the second condition ensures that the sums of the columns are decreasing. The states of this automaton are 3-tuples $\left\langle v, s, r\right\rangle$ where $v$ is the last value, $s$ is the number of occurrences of this value, and $r$ is the number of occurrences of the previous value. This automaton has $O(n^2m)$ states, while the number of transition from each state is bounded. Therefore propagating this constraint requires time $O(n^3m)$. ]{}]{} This decomposition is logically equivalent to the ${\mbox{\sc DoubleLexColSum}\xspace}$ constraint, therefore it is sound. Completeness follows from the fact that the decomposition has a Berge acyclic constraint graph. Therefore, enforcing DC on each ${\mbox{\sc Regular}}$ constraint enforces DC on ${\mbox{\sc DoubleLexColSum}\xspace}$ in $O(m^2n + n^3m)$ time.
Value symmetry
==============
Problems with row and column symmetry also often contain value symmetries. For example, the EFPA problem has row, column and value symmetry. We therefore turn to the problem of breaking row, column and value symmetry.
Consider again the solution . If we interchange the values 1 and 2, we get a symmetric solution:
$$\label{epfa4}
\begin{array}{cccccc}
0 & 2 & 1 & 2 & 0 & 1 \\
0 & 2 & 2 & 1 & 1 & 0 \\
0 & 1 & 0 & 2 & 1 & 2 \\
0 & 0 & 1 & 1 & 2 & 2
\end{array}
\
\begin{array}{c}
\ \ \ \ \ \ \Rightarrow \ \ \ \ \ \ \\
\mbox{\rm (1 2)}
\end{array}
\ \
\begin{array}{cccccc}
0 & 1 & 2 & 1 & 0 & 2 \\
0 & 1 & 1 & 2 & 2 & 0 \\
0 & 2 & 0 & 1 & 2 & 1 \\
0 & 0 & 2 & 2 & 1 & 1
\end{array}$$
In fact, all values in this CSP are interchangeable.
How do we break value symmetry in addition to breaking row and column symmetry? For example, Huczynska [*et al.*]{} write about their first model of the EFPA problem:
> [*“To break [some]{} of the symmetry, we apply lexicographic ordering (lex-ordering) constraints to the rows and columns …These two constraint sets do not explicitly order the symbols. It would be possible to order the symbols by using value symmetry breaking constraints. However we leave this for [future]{} work.”*]{} (page 53 of [@hmmncp09])
We turn to this future work of breaking row, column and value symmetry.
Double Lex
----------
We first note that the interaction of the problem and constraints can in some circumstances break all value symmetry. For instance, in our (and Huczynska [*et al.*]{}’s) model of the EFPA problem, *all* value symmetry is already eliminated. This appears to have been missed by [@hmmncp09].
Consider any solution of the EFPA problem which satisfies
(e.g. or ). By ordering columns lexicographically, ensures that the first row is ordered. In addition, the problem constraints ensure $\lambda$ copies of the symbols 1 to $q$ to appear in the first row. Hence, the first row is forced to be: $$\overbrace{1\ldots 1}^{\lambda}
\overbrace{2\ldots 2}^{\lambda}
\ldots
\overbrace{q\ldots q}^{\lambda}$$ All value symmetry is broken as we cannot permute the occurrences of any of the values.
Puget’s method
--------------
In general, value symmetries may remain after we have broken row and column symmetry. How can we eliminate these value symmetries? Puget has given a general method for breaking any number of value symmetries in polynomial time [@pcp05]. Given a surjection problem in which all values occur at least once,[^2] he introduces variables $Z_j$ to represent the index of the first occurrence of each value: $$\begin{aligned}
X_i = j & \Rightarrow & Z_j \leq i \\
Z_j = i & \Rightarrow & X_i=j\end{aligned}$$ Value symmetry on the $X_i$ is transformed into variable symmetry on the $Z_j$. This variable symmetry is especially easy to break as the $Z_j$ take all different values. We simply need to post appropriate ordering constraints on the $Z_j$. Consider, for example, the inversion symmetry which maps $1$ onto $m$, $2$ onto $m-1$, etc. Puget’s method breaks this symmetry with the single ordering constraint: $Z_1 < Z_m$. Unfortunately Puget’s method for breaking value symmetry is not compatible in general with breaking row and column symmetry using . This corrects Theorem 6 and Corollary 7 in [@pcp05] which claim that, provided we use the same ordering of variables in each method, it is compatible to post lex-leader constraints to break variable symmetry and Puget’s constraints to break value symmetry. There is no ordering of variables in Puget’s method which is compatible with breaking row and column symmetry using the lex-leader method (or any method like based on it).
There exist problems on which posting and applying Puget’s method for breaking value symmetry remove all solutions in a symmetry class irrespective of the ordering on variables used by Puget’s method.
[[**Proof: **]{}]{}Consider a 3 by 3 matrix model with constraints that all values between 0 and 8 occur, and that the average of the non-zero values along every row and column are all different from each other. This problem has row and column symmetry since we can permute any pair of rows or columns without changing the average of the non-zero values. In addition, it has a value symmetry that maps $i$ onto $9-i$ for $i>0$. This maps an average of $a$ onto $9-a$. If the averages were all-different before they remain so after. Consider the following two solutions: $$\begin{aligned}
\begin{array}{ccc}
0 & 2 & 3 \\
4 & 8 & 5 \\
7 & 6 & 1
\end{array}
&
\ \ \ {\rm and} \ \ \
&
\begin{array}{ccc}
0 & 2 & 3 \\
4 & 1 & 5 \\
7 & 6 & 8
\end{array} \end{aligned}$$ Both matrices satisfy as the smallest entry occurs in the top left corner and both the first row and column are ordered. They are therefore both the lex leader members of their symmetry class.
Puget’s method for breaking value symmetry will simply ensure that the first occurrence of 1 in some ordering of the matrix is before that of 8 in the same ordering. However, comparing the two solutions, it cannot be the case that the middle square is both before [*and*]{} after the bottom right square in the given ordering used by Puget’s method. Hence, whichever ordering of variables is used by Puget’s method, one of these solutions will be eliminated. All solutions in this symmetry class are thus eliminated.
We can pinpoint the mistake in Puget’s proof which allows him to conclude incorrectly that his method for value symmetry can be safely combined with variable symmetry breaking methods like . Puget introduces a matrix of 0/1 variables $Y_{ij} \iff X_i = j$ and observes that variable symmetries $\sigma$ on variables $X_i$ correspond to row symmetries on the matrix $Y_{ij}$, while value symmetries $\theta$ of the variables $X_i$ correspond to column symmetries of the matrix. Using the lex-leader method on a column-wise linearisation of the matrix, he derives the value symmetry breaking constraints on the $Z$ variables. Finally, he claims that we can derive the variable symmetry breaking constraints on the $X$ variables with the same method (equation (13) of [@pcp05]). However, this requires a row-wise linearisation of the matrix. Unfortunately, combining symmetry breaking constraints based on row and column-wise linearisations can, as in our example, eliminate all solutions in a symmetry class.
In fact, we can give an even stronger counter-example to Theorem 6 in [@pcp05] which shows that it is incompatible to post together variable and value symmetry breaking constraints [*irrespective*]{} of the orderings of variables used by [*both*]{} the variable and the value symmetry breaking method.
There exist problems on which posting lex-leader constraints to break variable symmetries and applying Puget’s method to break value symmetries remove all solutions in a symmetry class irrespective of the orderings on variables used by both methods.
[[**Proof: **]{}]{}Consider variables $X_1$ to $X_4$ taking values 1 to 4, an all-different constraint over $X_1$ to $X_4$ and a constraint that the neighbouring differences are either all equal or are not an arithmetic sequence. These constraints permit solutions like $X_1,\ldots,X_4=1,2,3,4$ (neighbouring differences are all equal) and $X_1,\ldots,X_4=2,1,4,3$ (neighbouring differences are not an arithmetic sequence). They rule out assignments like $X_1,\ldots,X_4=3,2,4,1$ (neighbouring differences form the arithmetic sequence $1, 2, 3$). This problem has a variable symmetry $\sigma$ which reflects a solution, swapping $X_1$ with $X_4$, and $X_2$ with $X_3$, and a value symmetry $\theta$ that inverts a solution, swapping $1$ with $4$, and $2$ with $3$. Consider $X_1,\ldots,X_4=2,4,1,3$ and $X_1,\ldots,X_4=3,1,4,2$. These two assignments form a symmetry class of solutions.
Suppose we break variable symmetry with a lex-leader constraint on $X_1$ to $X_4$. This will permit the solution $X_1,\ldots,X_4=2,4,1,3$ and eliminate the solution $X_1,\ldots,X_4=3,1,4,2$. Suppose we break the value symmetry using Puget’s method on the same ordering of variables. This will ensure that $1$ first occurs before $4$. But this will eliminate the solution $X_1,\ldots,X_4=2,4,1,3$. Hence, all solutions in this symmetry class are eliminated. In this case, both variable and value symmetry breaking use the same order on variables. However, we can show that all solutions in at least one symmetry class are eliminated whatever the orders used by both the variable and value symmetry breaking.
The proof is by case analysis. In each case, we consider a set of symmetry classes of solutions, and show that the combination of the lex-leader constraints to break variable symmetries and Puget’s method to break value symmetries eliminates all solutions from one symmetry class. In the first case, suppose the variable and value symmetry breaking constraints eliminate $X_1,\ldots,X_4=3,1,4,2$ and permit $X_1,\ldots,X_4=2,4,1,3$. In the second case, suppose they eliminate $X_1,\ldots,X_4=2,4,1,3$ and permit $X_1,\ldots,X_4=3,1,4,2$. This case is symmetric to the first except we need to reverse the names of the variables throughout the proof. We therefore consider just the first case. In this case, the lex-leader constraint breaks the variable symmetry by putting either $X_1$ first in its ordering variables or $X_3$ first.
Suppose $X_1$ goes first in the ordering used by the lex-leader constraint. Puget’s method ensures that the first occurrence of 1 is before that of 4. Puget’s method therefore uses an ordering on variables which puts $X_3$ before $X_2$. Consider now the symmetry class of solutions: $X_1,\ldots,X_4=2,1,4,3$ and $X_1,\ldots,X_4=3,4,1,2$. Puget’s method eliminates the first solution as 4 occurs before 1 in any ordering that put $X_3$ before $X_2$. And the lex-leader constraint eliminates the second solution as $X_1$ is larger than its symmetry $X_4$. Therefore all solutions in this symmetry class are eliminated.
Suppose, on the other hand, $X_3$ goes first in the lex-leader constraint. Consider now the symmetry class of solutions: $X_1,\ldots,X_4=1,2,3,4$ and $X_1,\ldots,X_4=4,3,2,1$. The lex-leader constraint eliminates the first solution as $X_3$ is greater than its symmetry $X_2$. Suppose now that the second solution is not eliminated. Puget’s method ensures the first occurrence of 1 is before that of 4. Puget’s method therefore uses an ordering on variables which puts $X_4$ before $X_1$. Consider now the symmetry class of solutions: $X_1,\ldots,X_4=1,3,2,4$ and $X_1,\ldots,X_4=4,2,3,1$. Puget’s method eliminates the first solution as 4 occurs before 1 in any ordering that put $X_4$ before $X_1$. And the lex-leader constraint eliminates the second solution as $X_3$ is larger than its symmetry $X_2$. Therefore all solutions in this symmetry class are eliminated.
Value precedence
----------------
We end with a special but common case where variable and value symmetry breaking do not conflict. When values partition into interchangeable sets, Puget’s method is equivalent to breaking symmetry by enforcing value precedence [@llcp2004; @wecai2006]. Given any two interchangeable values $i$ and $j$ with $i<j$, a value constraint ensures that if $i$ occurs then the first occurrence of $i$ is before that of $j$. It is safe to break row and column symmetry with and value symmetry with when value precedence considers variables either in a row-wise or in a column-wise order. This is a simple consequence of Theorem 1 in [@llcp2004]. It follows that it is also safe to use to break value symmetry when using constraints like derivable from the lex-leader method.
Snake Lex
=========
A promising alternative to for breaking row and column symmetries is [@snakelex]. This is also derived from the lex leader method, but now applied to a snake-wise unfolding of the matrix. To break column symmetry, ensures that the first column is lexicographically smaller than or equal to both the second and third columns, the reverse of the second column is lexicographically smaller than or equal to the reverse of both the third and fourth columns, and so on up till the penultimate column is compared to the final column. To break row symmetry, ensures that each neighbouring pair of rows, $X_{1,i},\ldots,X_{n,i}$ and $X_{1,i+1},\ldots,X_{n,i+1}$ satisfy the entwined lexicographical ordering: $$\begin{aligned}
\langle X_{1,i}, X_{2,i+1}, X_{3,i}, X_{4,i+1}, \ldots \rangle & \leq_{\rm lex}
\langle X_{1,i+1}, X_{2,i}, X_{3,i+1}, X_{4,i}, \ldots \rangle\end{aligned}$$
Like , is an incomplete symmetry breaking method. In fact, like , it may leave a large number of symmetric solutions.
\[tm:snake-expsol\] There exists a class of $2n$ by $2n+1$ 0/1 matrix models on which leaves $O(4^n/\sqrt{n})$ symmetric solutions, for all $n \geq 2$.
[[**Proof: **]{}]{}Consider the following 4 by 4 matrix: $$\begin{aligned}
&
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0
\end{array}
& \end{aligned}$$ This is a permutation matrix as there is a single 1 on each row and column. It satisfies the constraints. In fact, we can add any 5th column which reading top to bottom is lexicographically larger than or equal to $0010$ and reading bottom to top is lexicographically larger than or equal to $0010$. We shall add a 4 bit column with 2 bits set. That is, reading top to bottom: $1100$, $1010$, $0110$ or $0011$. Note that all 4 of these 4 by 5 matrices are row and column symmetries of each other. For instance, consider the row and column symmetry $\sigma$ that reflects the matrix in the horizontal axis, and swaps the 1st column with the 2nd, and the 3rd with the 4th: $$\begin{aligned}
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 1\\
0 & 0 & 0 & 1 & 1\\
0 & 0 & 1 & 0 & 0\\
1 & 0 & 0 & 0 & 0
\end{array}
&
\begin{array}{c} \\
\ \ \ \ \Leftrightarrow \ \ \ \ \\
\sigma
\end{array}
&
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 1 & 0 & 1\\
1 & 0 & 0 & 0 & 1
\end{array}\end{aligned}$$ In general, we consider the $2n$ by $2n$ permutation matrix: $$\begin{aligned}
&
{\small
\begin{array}{cccccccccc}
0 & 1 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & \ldots & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 1 & \ldots & 0 & 0 & 0\\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & 0 & 0 & \ldots & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 1 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & & \vdots & \vdots & \vdots \\
0 & 0 & 0 & 0 & 1 & 0 & \ldots & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & \ldots & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & \ldots & 0 & 0 & 0
\end{array}
}
& \end{aligned}$$ This satisfies the constraints. We can add any $2n+1$th column which reading top to bottom is lexicographically larger than or equal to the $2n-1$th column and reading bottom to top is lexicographically larger than or equal to the $2n$th column. In fact, we can add any column with eactly $n$ of the $2n$ bits set. This gives us a set of $2n$ by $2n+1$ matrices that are row and column symmetries of each other. There are $(2n)!/(n!)^2$ bit vectors with exactly $n$ of $2n$ bits set. Hence, we have $(2n)!/(n!)^2$ matrices which satisfy that are in the same row and column symmetry class. Using Stirling’s formula, this grows as $O(4^n/\sqrt{n})$.
Experimental results
====================
The proof of Theorem \[tm:fpt\] gives a polynomial method to break all row and column symmetry. This allows us to compare symmetry breaking methods for matrix models like and , not only with respect to each other but for the first time in absolute terms. Our aim is to evaluate: first, whether the worst-case scenarios identified in theorems \[tm:expsol\] and \[tm:snake-expsol\] are indicative of what can be expected in practice; second, how effective these methods are with respect to each other; third, in cases where they differ significantly, how much closer the best of them is to the optimal.
To answer these questions, we experimented with different symmetry breaking constraints: ${\mbox{\sc DoubleLex}\xspace}$, the column-wise (${\mbox{\sc SnakeLex}}_C$) or the row-wise (${\mbox{\sc SnakeLex}}_R$) [@snakelex]. We use ${\mbox{\sc NoSB}}$ to denote no symmetry breaking constraints. For each problem instance we found the total number of solutions left by symmetry breaking constraints ($ \#s$) and computed how many of them were symmetric based on the method outlined in the proof of Theorem \[tm:fpt\]. The number of *non symmetric* solutions is equal to the number of symmetry classes ($\#ns$) if the search space is exhausted. In all instances at least one model exhausted the search space to compute the of symmetry classes, shown in the column ${\mbox{\sc RowWiseLex}}$. We use ‘$-$’ to indicate that the search is not completed within the time limit. As the model typically could not exhaust the search space within the time limit, we use ‘$>$’ to indicate a lower bound on the number of solutions. Finally, we used a variable ordering heuristic that follows the corresponding lex-leader variable ordering in each set of symmetry breaking constraints (i.e. row-wise snake ordering with ${\mbox{\sc SnakeLex}}_R$). We ran experiments in Gecode 3.3.0 on an Intel XEON X5550, 2.66 GHz, 32 GB RAM with $18000$ sec timeout.
*Unconstrained problems.* We first evaluated the effectiveness of symmetry breaking constraints in the absence of problem constraints. This gives the “pure” effect of these constraints at eliminating row and column symmetry. We considered a problem with a matrix $m_{r \times c}$, $r \leq c = [2,6]$, $D(m_{r,c}) = [0,d-1]$, $d=[2,\ldots,5]$ whose rows and columns are interchangeable. Table \[t:t2\] summarizes the results. The first part presents typical results for 0/1 matrices whilst the second part presents results for larger domains. The results support the exponential worst case in Theorems \[tm:expsol\] and \[tm:snake-expsol\], as the ratio of solutions found to symmetry classes increases from 1.25 (3,3,2) to over 6 (6,6,2), approximately doubling with each increase of the matrix size. As we increase the problem size, the number of symmetric solutions left by and grows rapidly. Interestingly, ${\mbox{\sc SnakeLex}}_C$ achieves better pruning on 0/1 matrices, while ${\mbox{\sc DoubleLex}\xspace}$ performs better with larger domains.
$(r, c, d )$ [${\mbox{\sc RowWiseLex}}$]{} [${\mbox{\sc NoSB}}$]{} [${\mbox{\sc DoubleLex}\xspace}$]{} [${\mbox{\sc SnakeLex}}_R$]{} [${\mbox{\sc SnakeLex}}_C$]{}
-------------- ------------------------------- ------------------------- --------------------------------------------------- ------------------------------------------ ----------------------------------------------- --
\#ns \#s \#s / time \#s / time \#s / time
$ (3,3,2) $ $36$ $ 512 $ $ 45 $ / $ \mathbf{0.00} $ $ \mathbf{44} $ / $ \mathbf{0.00} $ $ \mathbf{44} $ / $ \mathbf{0.00} $
$ (4,4,2) $ $317$ $ 65536 $ $ 650 $ / $ \mathbf{0.00} $ $ \mathbf{577} $ / $ \mathbf{0.00} $ $ \mathbf{577} $ / $ \mathbf{0.00} $
$ (5,5,2) $ $5624$ $ 3.36{\cdot} 10^7 $ $ 24520 $ / $ \mathbf{0.05} $ $ \mathbf{18783} $ / $ 0.06 $ $ \mathbf{18783} $ / $ 0.06 $
$ (6,6,2) $ $251610$ $ > 9.4{\cdot} 10^9$ $2.62 \cdot 10^6$ / $ 22.2 $ $ \mathbf{1.71 \cdot 10^6}$ / $ 22.2 $ $ \mathbf{1.71 \cdot 10^6}$ / $\mathbf{18.1}$
$ (3,3,3) $ $738$ $ 19683 $ $ \mathbf{1169} $ / $ \mathbf{0.00} $ $ 1232 $ / $ \mathbf{0.00} $ $ 1232 $ / $ \mathbf{0.00} $
$ (3,3,4) $ $8240$ $ 2.62{\cdot} 10^5 $ $ \mathbf{14178} $ / $ 0.03 $ $ 15172 $ / $ \mathbf{0.02} $ $ 15172 $ / $ 0.05 $
$ (3,3,5) $ $57675$ $ 1.95{\cdot} 10^6 $ $ \mathbf{1.02{\cdot} 10^5} $ / $ 0.19 $ $ 1.09{\cdot} 10^5 $ / $ \mathbf{0.15} $ $ 1.09{\cdot} 10^5 $ / $ 0.21 $
$ (3,3,6) $ $289716$ $ 1.01{\cdot} 10^7 $ $ \mathbf{5.20{\cdot} 10^5} $ / $ \mathbf{2.32} $ $ 5.54{\cdot} 10^5 $ / $ 3.29 $ $ 5.54{\cdot} 10^5 $ / $ 2.83 $
: \[t:t4\] Covering Arrays. Number of solutions found by posting different sets of symmetry breaking constraints. $b$ is the number of vectors, $k$ is the length of a vector, $g$ is the size of the domains, $t$ is the covering strength.
$(q, \lambda, d, v)$ [${\mbox{\sc RowWiseLex}}$]{} [${\mbox{\sc NoSB}}$]{} [${\mbox{\sc DoubleLex}\xspace}$]{} [${\mbox{\sc SnakeLex}}_R$]{} [${\mbox{\sc SnakeLex}}_C$]{}
---------------------- ------------------------------- ------------------------- ------------------------------------- ------------------------------------------------------ ------------------------------------ --
\#ns \#s \#s / time \#s / time \#s / time
$ (3, 3,2,3) $ $6$ $ 1.81{\cdot} 10^5 $ $ \mathbf{6} $ / $ \mathbf{0.00} $ $ \mathbf{6} $ / $ \mathbf{0.00} $ $ \mathbf{6} $ / $ \mathbf{0.00} $
$ (4, 3,3,3) $ $8$ $ > 3.88{\cdot} 10^7 $ $ \mathbf{16} $ / $ \mathbf{0.01} $ $ \mathbf{16} $ / $ 0.01 $ $ \mathbf{16} $ / $ 0.16 $
$ (4, 4,2,3) $ $12$ $ > 5.87{\cdot} 10^7 $ $ \mathbf{12} $ / $ \mathbf{0.00} $ $ \mathbf{12} $ / $ \mathbf{0.00} $ $ \mathbf{12} $ / $ 0.04 $
$ (3, 4,6,4) $ $1427$ $ > 5.57{\cdot} 10^7 $ $ 11215 $ / $ 5.88 $ $ 10760 $ / $ \mathbf{5.36} $ $ \mathbf{8997} $ / $ 493.87 $
$ (4, 3,5,4) $ $8600$ $ > 2.03{\cdot} 10^7 $ $ 61258 $ / $ 69.90 $ $ 58575 $ / $ \mathbf{51.62} $ $ \mathbf{54920} $ / $ 3474.09 $
$ (4, 4,5,4) $ $9696$ $ > 5.45{\cdot} 10^6 $ $ 72251 $ / $ 173.72 $ $ 66952 $ / $ \mathbf{132.46} $ $ \mathbf{66168} $ / $ 14374.82 $
$ (5, 3,3,4) $ $5$ $ > 4.72{\cdot} 10^6 $ $ \mathbf{20} $ / $ 0.36 $ $ \mathbf{20} $ / $ \mathbf{0.25} $ $ \mathbf{20} $ / $ 31.61 $
$ (3, 3,4,5) $ $18$ $ > 2.47{\cdot} 10^7 $ $ 71 $ / $ 0.17 $ $ 71 $ / $ \mathbf{0.13} $ $ \mathbf{63} $ / $ 30.08 $
$ (3, 4,6,5) $ $4978$ $ > 2.08{\cdot} 10^7 $ $ 77535 $ / $ 167.50 $ $ \mathbf{71186} $ / $ \mathbf{137.88} $ $-$
$ (4, 3,4,5) $ $441$ $ > 6.55{\cdot} 10^6 $ $ 2694 $ / $ 19.37 $ $ 2688 $ / $ \mathbf{12.80} $ $ \mathbf{2302} $ / $ 5960.43 $
$ (4, 4,2,5) $ $12$ $ > 6.94{\cdot} 10^6 $ $ \mathbf{12} $ / $ 0.02 $ $ \mathbf{12} $ / $ \mathbf{0.01} $ $ \mathbf{12} $ / $ 1.60 $
$ (4, 4,4,5) $ $717$ $ > 6.27{\cdot} 10^6 $ $ 4604 $ / $ 38.15 $ $ \mathbf{4397} $ / $ \mathbf{24.58} $ $-$
$ (4, 6,4,5) $ $819$ $ > 4.08{\cdot} 10^6 $ $ 5048 $ / $ 69.83 $ $ \mathbf{4736} $ / $ \mathbf{44.83} $ $-$
$ (5, 3,4,5) $ $3067$ $ > 2.39{\cdot} 10^6 $ $ 20831 $ / $ 403.97 $ $ \mathbf{20322} $ / $ \mathbf{216.93} $ $-$
$ (6, 3,4,5) $ $15192$ $ > 2.16{\cdot} 10^6 $ $ 1.11{\cdot} 10^5 $ / $ 4924.41 $ $ \mathbf{1.06{\cdot} 10^5} $ / $ \mathbf{2006.19} $ $-$
: \[t:t4\] Covering Arrays. Number of solutions found by posting different sets of symmetry breaking constraints. $b$ is the number of vectors, $k$ is the length of a vector, $g$ is the size of the domains, $t$ is the covering strength.
$(v, k, \lambda )$ [${\mbox{\sc RowWiseLex}}$]{} [${\mbox{\sc NoSB}}$]{} [${\mbox{\sc DoubleLex}\xspace}$]{} [${\mbox{\sc SnakeLex}}_R$]{} [${\mbox{\sc SnakeLex}}_C$]{}
-------------------- ------------------------------- ------------------------- --------------------------------------- ------------------------------------ -------------------------------- --
\#ns \#s \#s / time \#s / time \#s / time
$ (5,2,7) $ $1$ $ > 0 $ $ \mathbf{1} $ / $ \mathbf{0.01} $ $ \mathbf{1} $ / $ 0.02 $ $ \mathbf{1} $ / $ 73.26 $
$ (5,3,6) $ $1$ $ > 1.51{\cdot} 10^9 $ $ \mathbf{1} $ / $ \mathbf{0.00} $ $ \mathbf{1} $ / $ \mathbf{0.00} $ $ \mathbf{1} $ / $ 0.82 $
$ (6,3,4) $ $4$ $ > 1.29{\cdot} 10^9 $ $ \mathbf{21} $ / $ 0.01 $ $ 25 $ / $ \mathbf{0.00} $ $ \mathbf{21} $ / $ 12.62 $
$ (6,3,6) $ $6$ $ > 1.21{\cdot} 10^9 $ $ \mathbf{134} $ / $ \mathbf{0.04} $ $ 146 $ / $ 0.07 $ $ \mathbf{134} $ / $ 1685.58 $
$ (7,3,4) $ $35$ $ > 1.18{\cdot} 10^9 $ $ \mathbf{3209} $ / $ \mathbf{0.33} $ $ 9191 $ / $ 1.07 $ $ 5270 $ / $ 7241.92 $
$ (7,3,5) $ $109$ $ > 1.09{\cdot} 10^9 $ $ 33304 $ / $ \mathbf{4.15} $ $ 85242 $ / $ 11.90 $ $-$
: \[t:t4\] Covering Arrays. Number of solutions found by posting different sets of symmetry breaking constraints. $b$ is the number of vectors, $k$ is the length of a vector, $g$ is the size of the domains, $t$ is the covering strength.
$(t, k, g, b )$ [${\mbox{\sc RowWiseLex}}$]{} [${\mbox{\sc NoSB}}$]{} [${\mbox{\sc DoubleLex}\xspace}$]{} [${\mbox{\sc SnakeLex}}_R$]{} [${\mbox{\sc SnakeLex}}_C$]{}
----------------- ------------------------------- ------------------------- ------------------------------------------------------ --------------------------------------------- --------------------------------------- --
\#ns \#s \#s / time \#s / time \#s / time
$ (2, 3,2,4) $ $2$ $ 48 $ $ \mathbf{2} $ / $ \mathbf{0.00} $ $ \mathbf{2} $ / $ \mathbf{0.00} $ $ \mathbf{2} $ / $ \mathbf{0.00} $
$ (2, 3,2,5) $ $8$ $ 1440 $ $ \mathbf{15} $ / $ \mathbf{0.00} $ $ \mathbf{15} $ / $ \mathbf{0.00} $ $ \mathbf{15} $ / $ \mathbf{0.00} $
$ (2, 3,3,9) $ $6$ $ 4.35{\cdot} 10^6 $ $ \mathbf{12} $ / $ \mathbf{0.00} $ $ \mathbf{12} $ / $ \mathbf{0.00} $ $ \mathbf{12} $ / $ 1.95 $
$ (2, 3,3,10) $ $104$ $ > 5.08{\cdot} 10^8 $ $ \mathbf{368} $ / $ \mathbf{0.00} $ $ 370 $ / $ 0.03 $ $ 372 $ / $ 7.06 $
$ (2, 3,3,11) $ $1499$ $ > 5.56{\cdot} 10^8 $ $ \mathbf{6824} $ / $ \mathbf{0.23} $ $ 6905 $ / $ 0.24 $ $ 6892 $ / $ 26.29 $
$ (2, 3,4,16) $ $150$ $ > 0 $ $ \mathbf{576} $ / $ 0.72 $ $ \mathbf{576} $ / $ \mathbf{0.70} $ $-$
$ (2, 3,4,17) $ $8236$ $ > 0 $ $ \mathbf{43368} $ / $ \mathbf{12.43} $ $ 43512 $ / $ 12.82 $ $-$
$ (2, 3,5,25) $ $27280$ $ > 0 $ $ \mathbf{1.61{\cdot} 10^5} $ / $ \mathbf{1166.94} $ $ \mathbf{1.61{\cdot} 10^5} $ / $ 1178.14 $ $-$
$ (2, 4,2,5) $ $5$ $ 1920 $ $ \mathbf{10} $ / $ \mathbf{0.00} $ $ \mathbf{10} $ / $ \mathbf{0.00} $ $ \mathbf{10} $ / $ \mathbf{0.00} $
$ (2, 4,2,7) $ $333$ $ 1.60{\cdot} 10^7 $ $ 2285 $ / $ \mathbf{0.04} $ $ 2224 $ / $ 0.07 $ $ \mathbf{1850} $ / $ \mathbf{0.04} $
$ (2, 4,3,9) $ $5$ $ 2.61{\cdot} 10^7 $ $ 36 $ / $ 0.02 $ $ 36 $ / $ \mathbf{0.01} $ $ \mathbf{26} $ / $ 1102.30 $
: \[t:t4\] Covering Arrays. Number of solutions found by posting different sets of symmetry breaking constraints. $b$ is the number of vectors, $k$ is the length of a vector, $g$ is the size of the domains, $t$ is the covering strength.
*Constrained problems.* Our second set of experiments was on three benchmark domains: Equidistant Frequency Permutation Array (EFPA), Balanced Incomplete Block Designs and Covering Array (CA) problems. We used the non-Boolean model of EFPA [@hmmncp09] (Table \[t:t1\]), the Boolean matrix model of BIBD [@ffhkmpwcp2002] (Table \[t:t3\]) and a simple model of CA [@Hnich06] (Table \[t:t4\]). We consider the satisfaction version of the CA problem with a given number of vectors $b$. In all problems instances the ${\mbox{\sc DoubleLex}\xspace}$, ${\mbox{\sc SnakeLex}}_R$ and ${\mbox{\sc SnakeLex}}_C$ constraints show their effectiveness, leaving only a small fraction of symmetric solutions. Note that ${\mbox{\sc SnakeLex}}_C$ often leaves fewer symmetric solutions. However, it is significantly slower compared to ${\mbox{\sc DoubleLex}\xspace}$ and ${\mbox{\sc SnakeLex}}_R$ because it tends to prune later (thereby exploring larger search trees). For example, the number of failures for the $(5,3,3,4)$ EFPA problem is $21766$, $14072$ and $1129085$ for ${\mbox{\sc DoubleLex}\xspace}$, ${\mbox{\sc SnakeLex}}_R$ and ${\mbox{\sc SnakeLex}}_C$ respectively. On EFPA problems, ${\mbox{\sc SnakeLex}}_R$ is about twice as fast as ${\mbox{\sc DoubleLex}\xspace}$ and leaves less solutions. On the CA problems ${\mbox{\sc DoubleLex}\xspace}$ and ${\mbox{\sc SnakeLex}}_R$ show similar results, while ${\mbox{\sc DoubleLex}\xspace}$ performs better on BIBD problems in terms of the number of solution left.
Overall, our results show that ${\mbox{\sc DoubleLex}\xspace}$ and ${\mbox{\sc SnakeLex}}$ prune most of the symmetric solutions. ${\mbox{\sc SnakeLex}}_C$ slightly outperforms ${\mbox{\sc DoubleLex}\xspace}$ and ${\mbox{\sc SnakeLex}}_R$ in terms of the number of solutions left, but it explores larger search trees and is about two orders of magnitude slower. However, there is little difference overall in the amount of symmetry eliminated by the three methods.
Other related work
==================
Lubiw proved that any matrix has a row and column permutation in which rows and columns are lexicographically ordered and gave a nearly linear time algorithm to compute such a matrix [@lubiw]. Shlyakhter and Flener [*et al.*]{} independently proposed eliminating row and column symmetry using [@ilya01; @lex2001; @ffhkmpwcp2002]. To break some of the remaining symmetry, Frisch, Jefferson and Miguel suggested ensuring that the first row is less than or equal to all permutations of all other rows [@fjmcp2003]. As an alternative to ordering both rows and columns lexicographically, Frisch [*et al.*]{} proposed ordering the rows lexicographically but the columns with a multiset ordering [@fhkmwijcai2003]. More recently, Grayland [*et al.*]{} have proposed , an alternative to based on linearizing the matrix in a snake-like way [@snakelex]. An alternative way to break the symmetry of interchangeable values is to convert it into a variable symmetry by channelling into a dual 0/1 viewpoint in which $Y_{ij}=1$ iff $X_i=j$, and using lexicographical ordering constraints on the columns of the 0/1 matrix [@ffhkmpwcp2002]. However, this hinders propagation [@wecai2006]. Finally, dynamic methods like SBDS have been proposed to remove symmetry from the search tree [@sbds]. Unfortunately, dynamic techniques tend not to work well with row and columns symmetries as the number of symmetries is usually too large.
Conclusions
===========
We have provided a number of positive and negative results on dealing with row and column symmetry. To eliminate some (but not all) symmetry we can post static constraints like and . On the positive side, we proposed the first polynomial time method to eliminate *all* row and column symmetry when the number of rows (or columns) is bounded. On the negative side, we argued that and can leave a large number of symmetric solutions. In addition, we proved that propagating completely is NP-hard. Finally, we showed that it is not always safe to combine Puget’s value symmetry breaking constraints with row and column symmetry breaking constraints, correcting a claim made in the literature.
[10]{} =-0.2pt
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Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Matrix Modelling. Technical Report APES-36-2001, APES group (2001) Presented at Formul’01 (Workshop on Modelling and Problem Formulation), CP2001 post-conference workshop.
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Huczynska, S., McKay, P., Miguel, I., Nightingale, P.: Modelling equidistant frequency permutation arrays: An application of constraints to mathematics. In Gent, I., ed.: Principles and Practice of Constraint Programming - CP 2009, 15th International Conference, CP 2009, Lisbon, Portugal, September 20-24, (2009) 50–64
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Bessiere, C., Hebrard, E., Hnich, B., Walsh, T.: The complexity of global constraints. Constraints, **12(2)** (2007) 239-259
Carlsson, M., Beldiceanu, N.: Arc-consistency for a chain of lexicographic ordering constraints. Technical report [T2002-18]{}, Swedish Institute of Computer Science (2002). Katsirelos, G., Narodytska, N., Walsh, T.: Breaking Generator Symmetry In: Proceedings of SymCon’09 - 9th International Workshop on Symmetry and Constraint Satisfaction Problems, colocated with CP2009.
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Shlyakhter, I.: Generating effective symmetry-breaking predicates for search problems. Electronic Notes in Discrete Mathematics **9** (2001) 19–35
Flener, P., Frisch, A., Hnich, B., Kiziltan, Z., Miguel, I., Pearson, J., Walsh, T.: Symmetry in matrix models. Technical Report APES-30-2001, APES group (2001) Presented at SymCon’01 (Symmetry in Constraints), CP2001 post-conference workshop.
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[^1]: Supported by ANR UNLOC project, ANR 08-BLAN-0289-01 and the Australian Government’s Department of Broadband, Communications and the Digital Economy and the ARC.
[^2]: Any problem can be turned into a surjection problem by the addition of suitable new variables.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study cryptography based on operator theory, and propose quantum no-key (QNK) protocols from the perspective of operator theory, then present a framework of QNK protocols. The framework is expressed in two forms: trace-preserving quantum operators and natural presentations. Then we defined the information-theoretical security of QNK protocols and the security of identification keys. Two kinds of QNK protocols are also proposed. The first scheme is constructed based on unitary transformation, and the other is constructed based on two multiplicative commutative sets.'
address:
- 'State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China'
- 'Data Communication Science and Technology Research Institute, Beijing 100191, China'
author:
- Li Yang
- Min Liang
title: 'Cryptography based on operator theory (): quantum no-key protocols'
---
quantum cryptography ,quantum no-key protocol ,man-in-the-middle attack,information-theoretical security
Introduction
============
The earliest group of quantum message oriented protocols is suggested in [@Boykin00; @Ambainis00; @Nayak07], which can be regarded as a quantum version of one-time pad, the sender and the receiver must preshare secretly a classical key. Later, a public-key encryption scheme of quantum message is proposed [@Yang03]. Recently, this kind of public-key cryptosystems has been developed [@Yang10].
Here we consider another technique to securely transmit quantum message, so called quantum no-key (QNK) protocol. No-key protocol was first proposed by Shamir [@Menezes97]. It is a wonderful idea to transmit classical messages secretly in public channel, independent of the idea of public-key cryptosystem and that of secret-key cryptosystem. However, the protocol presented is computationally secure, cannot resists a man-in-the-middle(MIM) attack. [@Yangli02a; @Yangli02] develop a quantum from of no-key protocol based on single-photon rotations, which can be used to transmit classical and quantum messages secretly. It can be seen that the security of the QNK protocol is based on the laws of quantum mechanics, so it is beyond computational hypothesis. Ref. [@Yangli03] proposed a protocol based on quantum computing of Boolean functions. This protocols is constructed with inherent identifications in order to prevent MIM attack. Similar to the idea of QNK protocol, Kanamori et al.[@Kanamori05] proposed a protocol for secure data communication, Kye et al.[@Kye05] proposed a quantum key distribution scheme, and Kak [@Subhash07] proposed a three-stage quantum cryptographic protocol for key agreement. Wu and Yang [@Wu09] presents a practical QNK protocol, and studied a new kind of attack named unbalance-of-information-source (UIS) attack. This kind of attack may also be effective to quantum secure direct communication protocols, such as those in [@Beige01; @Bostrom02; @Deng2003; @Deng2004].
In the paper, the theory of QNK protocols is studied, and a framework of QNK protocols is presented. Then we defined the information-theoretical security of QNK protocols, and the security of identification keys. Finally, two kinds of QNK protocols are presented.
Quantum no-key protocols
========================
Framework
---------
Let us consider a general framework of quantum no-key protocol, in which two ancillary states are used. Suppose Alice will send quantum message $\rho\in H_M$. The ancillary states used by Alice and Bob are $\rho_A$ and $\rho_B$, respectively. The framework of QNK protocol is described as (see Figure \[fig1\]):
![\[fig1\] A general framework of quantum no key protocol. This figure is divided into two part by a dashed line. The part above the dashed line describes Alice’s operations, and the other part describes Bob’s operations. The quantum state $\rho$ is the plain state, and $\rho_1,\rho_2,\rho_3$ represents the three cipher states transmitted between Alice and Bob. $\rho_A,\rho_B$ are two ancillary states generated randomly by Alice and Bob, respectively.](quantum-no-key){width="12cm"}
1. Alice randomly prepare a quantum state $\rho_A$, then performs $U_A$ on the quantum states $\rho_A\otimes\rho$ and gets $U_A(\rho_A\otimes\rho)U_A^\dagger$. Then she sends to Bob the first cipher state $\rho_1$, $$\rho_1=tr_A(U_A(\rho_A\otimes\rho)U_A^\dagger)\triangleq \mathcal{E}_A(\rho).$$ She retains the state $\rho_A'=tr_M(U_A(\rho_A\otimes\rho)U_A^\dagger)$.
2. Bob randomly prepares a quantum state $\rho_B$, then performs $U_B$ on the quantum states $\rho_1\otimes\rho_B$ and gets $U_B(\rho_1\otimes\rho_B)U_B^\dagger$. Then he sends to Alice the second cipher state $\rho_2$, $$\rho_2=tr_B(U_B(\rho_1\otimes\rho_B)U_B^\dagger)\triangleq \mathcal{E}_B(\rho_1).$$ He retains the state $\rho_B'=tr_M(U_B(\rho_1\otimes\rho_B)U_B^\dagger)$.
3. Alice performs $U_A'$ on $\rho_A'\otimes\rho_2$, and sends to Bob the third cipher state $\rho_3$, $$\rho_3=tr_A(U_A'(\rho_A'\otimes\rho_2)U_A'^\dagger)\triangleq\mathcal{E}_A'(\rho_2).$$
4. Bob performs $U_B'$ on $\rho_3\otimes\rho_B'$, and gets the message $\rho\prime$, $$\rho'=e^{i\phi}\rho=tr_B(U_B'(\rho_3\otimes\rho_B')U_B'^\dagger)\triangleq\mathcal{E}_B'(\rho_3).$$
In the above protocols, the four quantum operations $\mathcal{E}_A,\mathcal{E}_B,\mathcal{E}_A',\mathcal{E}_B'$ are all trace-preserving quantum operators. This protocol is correct if and only if the four quantum operators satisfy this condition: $$\label{eqn5}
\mathcal{E}_B'\circ\mathcal{E}_A'\circ\mathcal{E}_B\circ\mathcal{E}_A=e^{i\phi}\mathcal{I}.$$
[**Remark:**]{} As a special case, the unitary transformations $U_A$,$U_B$ can be chosen as bitwise controlled-unitary transformations where the message qubits act as control qubits, and $U_A'=U_A^\dagger$,$U_B'=U_B^\dagger$. In this case, $(I\otimes U_B)(U_A \otimes I)=(U_A \otimes I)(I\otimes U_B)$, and $(I\otimes U_B')(U_A' \otimes I)(I\otimes U_B)(U_A \otimes I)=I$.
Natural Representation
----------------------
Trace-preserving quantum operator $\mathcal{E}$ can be written as the form of operator-sum representation $$\begin{aligned}
\mathcal{E}(\rho)=\sum_i E_{i}\rho E_{i}^\dagger.\end{aligned}$$ Its natural representation [@Watrous08] is $$\begin{aligned}
\label{eqn7}
\overrightarrow{\mathcal{E}}(\rho)= \sum_i (E_{i}\otimes E_{i}^*)\overrightarrow{\rho} \triangleq B\overrightarrow{\rho},\end{aligned}$$ where $\overrightarrow{\rho}$ is a the column vector, and represents the vector form of the density matrix $\rho$.
The quantum operations $\mathcal{E}_1$ and $\mathcal{E}_2$ are trace-preserving quantum operator, and their natural representations are denoted as $A,B$ respectively. Because trace-preserving quantum operator can be realized physically, the operators $A,B$ can be realized physically.
Suppose four operators $A,B,A',B'$ represent the natural representation of the four trace-preserving quantum operators $\mathcal{E}_A,\mathcal{E}_B,\mathcal{E}_A',\mathcal{E}_B'$. Next we deduce the expressions of the natural representation $A,B,A',B'$. Let the ancillary states of Alice and Bob are $\rho_A=\rho_B=|0\rangle\langle 0|$, the orthogonal basis on the ancillary space is the set $\{e_k\}_k$. From the trace-preserving quantum operator $\mathcal{E}_A(\rho)=tr_A(U_A(|0\rangle\langle 0|\otimes\rho)U_A^\dagger)$, it can be inferred that $$\begin{aligned}
\mathcal{E}_A(\rho)&=&\sum_k\langle e_k|U_A(|0\rangle\langle 0|\otimes\rho)U_A^\dagger|e_k\rangle \\
&=&\sum_k E_k\rho E_k^\dagger,\end{aligned}$$ where $E_k\equiv\langle e_k|U_A|0\rangle$ is an operator acting on the message space $H_M$. Thus, according to Eq.(\[eqn7\]), one know that the natural representation of trace-preserving quantum operator $\mathcal{E}_A$ is as follows $$A=\sum_k E_k\otimes E_k^*=\sum_k \langle e_k|U_A|0\rangle\otimes\langle e_k|U_A^*|0\rangle.$$ Similarly, one can present the natural representations of the other three trace-preserving quantum operators, for example $B=\sum_k \langle e_k|U_B|0\rangle\otimes\langle e_k|U_B^*|0\rangle$.
From the above analysis, the four transformations to the quantum message $\rho$ in the quantum no-key protocol can be described in the form of natural representation. $$\begin{aligned}
\overrightarrow{\rho}&\longrightarrow&\overrightarrow{\rho_1}=A\overrightarrow{\rho}\\
&\longrightarrow&\overrightarrow{\rho_2}=B\overrightarrow{\rho_1}\\
&\longrightarrow&\overrightarrow{\rho_3}=A'\overrightarrow{\rho_2}\\
&\longrightarrow&\overrightarrow{\rho}'=B'\overrightarrow{\rho_3}\end{aligned}$$ Quantum no-key protocol can be described in the form of trace-preserving quantum operators or natural representation, and the two forms are equivalent. In the following sections, we use the natural representation, and the operators $A$ and $B$ are both natural representations of trace-preserving quantum operators. $\mathcal{E}_1\circ\mathcal{E}_2=\mathcal{E}_2\circ\mathcal{E}_1$ if and only if $AB=BA$.
The additive commutator $A,B$ satisfy $$AB=BA+K;$$ The multiplicative commutator $A,B$ satisfy $$BA=e^{i\lambda}AB, \lambda\neq 0,$$ or be written as $B^{-1}A^{-1}BA=e^{i\lambda}I, \lambda\neq 0$.
When $K=\lambda=0$, the additive commutator equals to multiplicative commutator. The multiplicative commutator $BA=e^{i\lambda}AB$ is just additive commutator when $\lambda=0$; Multiplicative commutator is different from additive commutator in only a global phase $e^{i\lambda}$ when $\lambda \neq 0$, and they have no difference in physical implementation.
[**Theorem 1**]{}: The multiplicative commutators $A,B$ are unitary matrices. If $BA=e^{i\lambda}AB$, where $\lambda\neq 0$, then the sum of the eigenvalues of the operators $A,B$ are zero, respectively.
[**Proof**]{} It means to prove: if $A=\sum_j e^{i\varphi_j}|a_j\rangle\langle a_j|$, then $\sum_j e^{i\varphi_j}=0$; if $B=\sum_j e^{i\phi_j}|b_j\rangle\langle b_j|$, then $\sum_j e^{i\phi_j}=0$. The proof is as follows.
From $BA=e^{i\phi_{AB}}AB$ where $A,B$ are both unitary, one know that $A=e^{i\phi_{AB}}B^\dagger AB$. Because $A$ is a unitary transformation, it has spectral decomposition $A=\sum_j\lambda_j|j\rangle\langle j|$, and each eigenvalue $\lambda_j$ can be written as $e^{i\phi_j}$. Thus, $\sum_j\lambda_j|j\rangle\langle j|=\sum_je^{i\phi_{AB}}\lambda_jB^\dagger |j\rangle\langle j|B$. Then $$\label{eqn8}
\lambda_k=\sum_je^{i\phi_{AB}}\lambda_j\langle k|B^\dagger|j\rangle\langle j|B|k\rangle=\sum_je^{i\phi_{AB}}\lambda_j|\langle j|B|k\rangle|^2.$$ Let the $j$-th row and $k$-th column of $B$ is $b_{jk}$. The two side of Eq.(\[eqn8\]) is added with the variable $k$, and obtains $\sum_k\lambda_k=e^{i\phi_{AB}}\sum_j\lambda_j\sum_k|b_{jk}|^2=e^{i\phi_{AB}}\sum_j\lambda_j$. Thus, $\sum_j\lambda_j(e^{i\phi_{AB}}-1)=0$. If $\phi_{AB}\neq 0$, then $\sum_j\lambda_j=0$, that means the sum of the eigenvalues of $A$ is 0. Similarly, one can prove that the sum of the eigenvalues of $B$ is 0.$~\hfill{}\Box$
In the next part of this paper, we consider the case that $A'=A^{-1},B'=B^{-1}$. In this case, Eq.(\[eqn5\]) can also be expressed as $B^{-1}A^{-1}BA=e^{i\phi}I$. It can be seen that $A,B$ are multiplicative commutator.
In order to identify personal identification in the protocols, Alice and Bob must preshare a secret key $k,i$, and the multiplicative commutator $A,B$ should satisfy: $$B^{-1}A^{-1}BA=e^{i\phi}N_k(i).$$ Alice and Bob preshare secret identification key $k,i$, where $k\in\mathcal{K}$. From the value of $k$, a operator $N_k$ can be obtained and a set $I(k)$ can be constructed, and the secret key $i\in I(k)$. From $k,i$, we can get a set $L(k,i)$ and a set of operators satisfying $$S_k(i)=\{A_l^{(i)}(k)|l\in L(k,i)\}\cup \{B_l^{(i)}(k)|l\in L(k,i)\}.$$ The set of operators $S_k(i)$ should satisfy the condition: for any two elements $A_{l_1}^{(i)}(k),B_{l_2}^{(i)}(k)\in S_k(i)$, it holds that $$\label{eqn1}
[B_{l_2}^{(i)}(k)]^{-1}[A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k) =e^{i\phi(l_1,l_2)}N_k(i).$$
Alice and Bob communicates according to $k,i$ and the set $S_k(i)$. The process is as follows
1. Alice randomly selects $l_1\in L(k,i)$, and performs an operator $A_{l_1}^{(i)}(k)$ on quantum message $\overrightarrow{\rho}$, then obtains a state $\overrightarrow{\rho_1}$. She sends it to Bob.
2. Bob randomly selects $l_2\in L(k,i)$, and performs an operator $B_{l_2}^{(i)}(k)$ on quantum state $\overrightarrow{\rho_1}$, then obtains a state $\overrightarrow{\rho_2}$. He sends it to Alice.
3. According to $l_1$, Alice performs an operator $[A_{l_1}^{(i)}(k)]^{-1}$ on quantum state $\overrightarrow{\rho_2}$, then obtains a state $\overrightarrow{\rho_3}$. She sends it to Bob.
4. According to $l_2$, Bob performs an operator $[B_{l_2}^{(i)}(k)]^{-1}$ on quantum state $\overrightarrow{\rho_3}$, then obtains a state $\overrightarrow{\rho_4}$.
From the condition satisfied by $S_k(i)$, it is deduced that $$\overrightarrow{\rho_4}=[B_{l_2}^{(i)}(k)]^{-1} [A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k) \overrightarrow{\rho} = e^{i\phi(l_1,l_2)}N_k(i) \overrightarrow{\rho}.$$ Thus, the quantum state obtained by Bob in the end is $N_k(i) \overrightarrow{\rho}$. He can recovery the quantum message $\overrightarrow{\rho}$ by performing the inverse transformation of $N_k(i)$.
Through the value of $l_1$ is unknown by Bob, Bob randomly selects a value of $l_1'\in L(k,i)$, then it can satisfy the relation in Eq.(\[eqn1\])(only a little difference on total phase). Thus, the inverse transformation of $N_k(i)$ can also be replace by $$N_k^{-1}(i)=[A_{l_1'}^{(i)}(k)]^{-1}[B_{l_2}^{(i)}(k)]^{-1}A_{l_1'}^{(i)}(k)B_{l_2}^{(i)}(k), \forall l_1'\in L(k,i).$$ Because the operators $A_l^{(i)}(k)$, $B_l^{(i)}(k)$ are all trace-preserving quantum operators and can be implemented physically, the quantum operator $N_k^{-1}$ is physically implementable. Thus, Bob can recovery the quantum message $\overrightarrow{\rho}$ by performing $N_k^{-1}(i)$ on the state $\overrightarrow{\rho_4}$.
In order to enhance the security of this protocol, we can select a group of similarity transformations $\{T_i|i\in L(k,i)\}$. The set $S_k(i)$ in the above protocol is replaced with another set $$\tilde{S}_k(i)=\{T_iA_l^{(i)}(k)T_i^{-1}|l\in L(k,i)\}\cup \{T_iB_l^{(i)}(k)T_i^{-1}|l\in L(k,i)\}.$$ Similarly, $N_k(i)$ is replaced with $T_iN_k(i)T_i^{-1}$, but it is still denoted as $N_k(i)$ for convenient. Then according to Eq.(\[eqn1\]), we know $$T_i[B_{l_2}^{(i)}(k)]^{-1}[A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k)T_i^{-1}=e^{i\phi(l_1,l_2)}N_k(i).$$ From the following relation $$\begin{aligned}
& T_i[B_{l_2}^{(i)}(k)]^{-1}[A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k)T_i^{-1} \nonumber\\
=& [T_iB_{l_2}^{(i)}(k)T_i^{-1}]^{-1}[T_iA_{l_1}^{(i)}(k)T_i^{-1}]^{-1} T_iB_{l_2}^{(i)}(k)T_i^{-1} T_iA_{l_1}^{(i)}(k)T_i^{-1},\end{aligned}$$ it is inferred that: any two elements in $\tilde{S}_k(i)$ such as $T_iA_{l_1}^{(i)}(k)T_i^{-1},T_iB_{l_2}^{(i)}(k)T_i^{-1}$ satisfy the following relation $$[T_iB_{l_2}^{(i)}(k)T_i^{-1}]^{-1}[T_iA_{l_1}^{(i)}(k)T_i^{-1}]^{-1} T_iB_{l_2}^{(i)}(k)T_i^{-1} T_iA_{l_1}^{(i)}(k)T_i^{-1}=e^{i\phi(l_1,l_2)}N_k(i).$$
Security
--------
In QNK protocol, there are 3 times of transmitting quantum ciphers through public quantum channel. Denote $\rho_i$ as the $i$-th quantum cipher with respect to the attackers, where $i=1,2,3$.
[**Definition 1:**]{} QNK protocol is information-theoretically secure, if the three quantum ciphers $\rho_1,\rho_2,\rho_3$ satisfy the following condition: for any positive polynomial $p(.)$, and all sufficiently big number $n$, it holds that $$D\left(\rho_i,\rho_j\right)<\frac{1}{p(n)},\forall i,j\in\{1,2,3\},$$ where $\rho_1,\rho_2,\rho_3$ are all $n$-qubit ciphers.
In fact, this definition equals to the following definition.
[**Definition 2:**]{} QNK protocol is information-theoretically secure, if the three quantum ciphers $\rho_1,\rho_2,\rho_3$ satisfy the following condition: there exists a quantum state $\tau$, such that for any positive polynomial $p(.)$ and all sufficiently big number $n$, it holds that $$D\left(\rho_i,\tau\right)<\frac{1}{p(n)},\forall i\in\{1,2,3\},$$ where $\rho_1,\rho_2,\rho_3$ are all $n$-qubit ciphers.
In one hand, if QNK protocol satisfies Definition 2, it can be deduced that $D\left(\rho_i,\rho_j\right)<D\left(\rho_i,\tau\right)+D\left(\rho_j,\tau\right)<\frac{2}{p(n)},\forall i,j\in\{1,2,3\}$, then the protocol satisfies Definition 1; In the other hand, if QNK protocol satisfies Definition 1, and let $\tau=\rho_1$, we know that $D\left(\rho_i,\tau\right)=D\left(\rho_i,\rho_1\right)<\frac{1}{p(n)},\forall i\in\{1,2,3\}$, then the protocol satisfies Definition 2. Thus, the two definitions are equivalent to each other.
Here, $\rho_i$ is the $i$-th quantum cipher with respect to the attackers, where $i=1,2,3$. In the QNK protocol, $\overrightarrow{\rho_1}$ is obtained by performing quantum transformation $A_{l_1}^{(i)}(k)$ on quantum state $\overrightarrow{\rho}$. However, with regard to the attackers, the random number $l_1$ and authentication key $i,k$ used by Alice cannot be obtained, so $$\overrightarrow{\rho_1}=\sum_{i,k,l_1}p_i p_k p_{l_1} A_{l_1}^{(i)}(k)\overrightarrow{\rho},$$ where $p_i, p_k$ are the probability of selecting the authentication key $i,k$, and $p_{l_1}$ is the probability of the local number $l_1$ being selected by Alice. Similarly, the attackers cannot obtain the random number $l_2$ and authentication key $i,k$ used by Bob, so $$\overrightarrow{\rho_2}=\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)\overrightarrow{\rho},$$ where $p_{l_2}$ is the probability of the local number $l_2$ being selected by Bob. $$\overrightarrow{\rho_3}=\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)\overrightarrow{\rho}.$$
[**Definition 3:**]{} QNK protocol is information-theoretically secure, if the three operators $\sum_{i,k,l_1}p_i p_k p_{l_1}A_{l_1}^{(i)}(k)$, $\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)$, and\
$\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)$ satisfy the condition: for any positive polynomial $p(.)$, and all sufficiently large number $n$, it holds that $$\begin{aligned}
&&||\sum_{i,k,l_1}p_i p_k p_{l_1} A_{l_1}^{(i)}(k)-\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)},\\
&&||\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}\left(B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)-(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)\right)||_{\diamondsuit}<\frac{1}{p(n)},\\
&&||\sum_{i,k,l_1}p_i p_k p_{l_1}A_{l_1}^{(i)}(k)-\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)}.\end{aligned}$$ where the notation $||*||_{\diamondsuit}$ represents diamond norm.
It is obvious that a sufficient condition for information-theoretical security is as follow: $$\begin{aligned}
&&||I-\sum_{l_2}p_{l_2}B_{l_2}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)},\forall i,k,\\
&&||I-A_{l_1}^{(i)}(k)^{-1}||_{\diamondsuit}<\frac{1}{p(n)}, \forall i,k,l_1, \\
&&||I-\sum_{l_2}p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)}, \forall i,k,l_1.\end{aligned}$$
For the QNK protocol which uses authentication key, when considering its security, besides analyzing the security of quantum message, we also should analyze the security of the authentication key. Here we present a definition of the security of authentication key in QNK protocol.
[**Definition 4:**]{} The authentication key in QNK protocol is secure, if for any positive polynomial $p(.)$, and all sufficiently large number $n$, it holds that $$\begin{aligned}
&&||\sum_{l_1}A_{l_1}^{(i_1)}(k_1)-\sum_{l_1}A_{l_1}^{(i_2)}(k_2)||_{\diamondsuit}<\frac{1}{p(n)}, \forall i_1,i_2,k_1,k_2.\\
&&||\sum_{l_1,l_2}B_{l_2}^{(i_1)}(k_1)A_{l_1}^{(i_1)}(k_1)-\sum_{l_1,l_2}B_{l_2}^{(i_2)}(k_2)A_{l_1}^{(i_2)}(k_2)||_{\diamondsuit}<\frac{1}{p(n)}, \forall i_1,i_2,k_1,k_2.\\
&&||\sum_{l_1,l_2}(A_{l_1}^{(i_1)}(k_1))^{-1}B_{l_2}^{(i_1)}(k_1)A_{l_1}^{(i_1)}(k_1)-\sum_{l_1,l_2}(A_{l_1}^{(i_2)}(k_2))^{-1}B_{l_2}^{(i_2)}(k_2)A_{l_1}^{(i_2)}(k_2)||_{\diamondsuit}\\
&&<\frac{1}{p(n)}, \forall i_1,i_2,k_1,k_2.\end{aligned}$$
Two schemes of quantum no-key protocols
=======================================
First scheme
------------
Unitary transformation is a special kind of trace-preserving quantum operator. Here we assume the quantum operators used by Alice and Bob in QNK protocols are all unitary. Let $N_k(i)=I$. The sets of operators $\{A_l|l\in\{1,2,\ldots,n_A\}\}$ and $\{B_k|k\in\{1,2,\ldots,n_B\}\}$ are natural representations of unitary operator used by Alice and Bob. The two set satisfy the relation: $$\label{eqn4}
B_{l_2}^{-1}A_{l_1}^{-1}B_{l_2}A_{l_1}=e^{i\phi(l_1,l_2)}I.$$ The above formula can also be written as $B_{l_2}A_{l_1}=e^{i\phi(l_1,l_2)}A_{l_1}B_{l_2}$.
$A_l,B_k$ are natural representations of unitary operators. So we can assume $A_l=E_l\otimes E_l^*,B_k=F_k\otimes F_k^*$, where $E_l, F_k$ are unitary transformations, and $l\in\{1,2,\ldots,n_A\}$, $k\in\{1,2,\ldots,n_B\}$. Then $$\begin{aligned}
B_{l_2}^{-1}A_{l_1}^{-1}B_{l_2}A_{l_1}&=&(F_{l_2}\otimes F_{l_2}^*)^{-1}(E_{l_1}\otimes E_{l_1}^*)^{-1}(F_{l_2}\otimes F_{l_2}^*)(E_{l_1}\otimes E_{l_1}^*)\nonumber\\
&=& (F_{l_2}^{-1}E_{l_1}^{-1}F_{l_2}E_{l_1})\otimes(F_{l_2}^{-1}E_{l_1}^{-1}F_{l_2}E_{l_1})^*\nonumber\\
&\triangleq&V_{l_1,l_2}\otimes V_{l_1,l_2}^*,\end{aligned}$$ where $V_{l_1,l_2}=F_{l_2}^{-1}E_{l_1}^{-1}F_{l_2}E_{l_1}$ is unitary transformation. According to Eq.(\[eqn4\]), we have $V_{l_1,l_2}\otimes V_{l_1,l_2}^*=e^{i\phi(l_1,l_2)}I_{2^{2n}}$. From the identity $I_{2^{2n}}=I_{2^n}\otimes I_{2^n}$, it has $V_{l_1,l_2}\otimes V_{l_1,l_2}^*=aI_{2^n}\otimes bI_{2^n}$, where $ab=e^{i\phi(l_1,l_2)}$, and $b=a^*$. So $e^{i\phi(l_1,l_2)}=aa^*=|a|^2$, then $e^{i\phi(l_1,l_2)}=1$. That means it is impossible to produce a global phase in Eq.(\[eqn4\]). Thus the Eq.(\[eqn4\]) is rewritten as follows $$\label{eqn6}
B_{l_2}^{-1}A_{l_1}^{-1}B_{l_2}A_{l_1}=I.$$
Based on this relation, we construct a QNK scheme, which is described as follows.
Firstly, two sets of operators are selected, such as $$S(A)=\{A_l|l\in\{1,2,\ldots,n_A\}\},$$ $$S(B)=\{B_k|k\in\{1,2,\ldots,n_B\}\}.$$
By using the two sets, Alice and Bob communicate following this QNK protocol.
1. Alice randomly selects a number $l_1\in\{1,2,\ldots,n_A\}$, then performs quantum operator $A_{l_1}$ on quantum message $\overrightarrow{\rho}$, and obtains the state $\overrightarrow{\rho_1}$. She sends it to Bob.
2. Bob randomly selects a number $l_2\in\{1,2,\ldots,n_B\}$, then performs quantum operator $B_{l_2}$ on quantum message $\overrightarrow{\rho_1}$, and obtains the state $\overrightarrow{\rho_2}$. He sends it to Alice.
3. According to the value of $l_1$, Alice performs quantum operator $A_{l_1}^{-1}$ (or $A_{l_1}^\dagger$) on quantum message $\overrightarrow{\rho_2}$, and obtains the state $\overrightarrow{\rho_3}$. She sends it to Bob.
4. According to the value of $l_2$, Bob performs quantum operator $B_{l_2}^{-1}$ (or $B_{l_2}^\dagger$) on quantum message $\overrightarrow{\rho_3}$, and obtains the state $\overrightarrow{\rho_4}$.
According to the relation (Eq.(\[eqn6\])) of the two sets $S(A),S(B)$, we know that $$\overrightarrow{\rho_4}=B_{l_2}^{-1} A_{l_1}^{-1} B_{l_2} A_{l_1} \overrightarrow{\rho} = \overrightarrow{\rho}.$$ Thus, the quantum state obtained by Bob’s performing quantum transformation $B_{l_2}^{-1}$ is just the quantum message sent by Alice.
Second scheme
-------------
Denote the operation of two operators $A,B$: $(A,B)=B^{-1}A^{-1}BA$.
Suppose there exists two groups of operators $S(A)=\{A_i|i=1,\cdots,n_A\}$ and $S(B)=\{B_i|i=1,\cdots,n_B\}$, which satisfy the following condition $$\label{eqn2}
(A_i,B_j)=N, \forall i\in\{1,\cdots,n_A\},j\in\{1,\cdots,n_B\},$$ where $N$ is an operator that is independent of $i,j$.
[**Proposition 1:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j$ $$\begin{aligned}
&& (A_i^{-1},B_jB_i^{-1})=I,\\
&& (A_j^{-1},B_jB_i^{-1})=I,\\
&& (B_i,A_jA_i^{-1})=I,\\
&& (B_j,A_jA_i^{-1})=I.\end{aligned}$$ [**Proof:**]{} See the appendix.$\hfill{}\Box$
[**Proposition 2:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j,k,l$ $$(A_iA_j,B_kB_l)=(A_j^2,B_l^2).$$ [**Proof:**]{} See the appendix.$\hfill{}\Box$
According to Proposition 2, we know the relation $(A_{l_1}A_{k_1},B_{l_2}B_{k_2})=(A_{k_1}^2,B_{k_2}^2)$. Based on this relation, a QNK scheme is constructed as follows.
Suppose the set of similarity transformations $\{T_i|i\in I\}$ is selected. Firstly we construct a set of operators $S_k(i)$ as follows: $$\begin{aligned}
S_{k_1||k_2}(i)&=&\{T_iA_lA_{k_1}T_i^{-1}|l\in\{1,\cdots,n_A\}\} \nonumber\\
&&\cup\{T_iB_lB_{k_2}T_i^{-1}|l\in\{1,\cdots,n_B\}\}.\end{aligned}$$
In the set $S_k(i)$, any two elements $A_{l_1}^{(i)}(k_1||k_2)=T_iA_{l_1}A_{k_1}T_i^{-1}$ and $B_{l_2}^{(i)}(k_1||k_2)=T_iB_{l_2}B_{k_2}T_i^{-1}$ satisfy the following relation $$\begin{aligned}
(A_{l_1}^{(i)}(k_1||k_2),B_{l_2}^{(i)}(k_1||k_2))&=& (T_iA_{l_1}A_{k_1}T_i^{-1},T_iB_{l_2}B_{k_2}T_i^{-1}) \nonumber\\
&=& T_i(A_{l_1}A_{k_1},B_{l_2}B_{k_2})T_i^{-1} \nonumber\\
&=& T_i(A_{k_1}^2,B_{k_2}^2)T_i^{-1}.\end{aligned}$$ Denote $N_{k_1||k_2}=(A_{k_1}^2,B_{k_2}^2)$, $N_{k_1||k_2}(i)=T_i N_{k_1||k_2} T_i^{-1}$, then $$(A_{l_1}^{(i)}(k_1||k_2),B_{l_2}^{(i)}(k_1||k_2))=N_{k_1||k_2}(i), ~\forall l_1,l_2.$$
Alice and Bob communicate according to $k_1||k_2,i$ ($k_1\in\{1,\cdots,n_A\},k_2\in\{1,\cdots,n_B\}$, $i\in I$) and the set of operators $S_{k_1||k_2}(i)$. The process is as follows.
1. Alice randomly selects a number $l_1\in\{1,\cdots,n_A\}$, and performs quantum transformation $A_{l_1}^{(i)}(k_1||k_2)$ on quantum message $\overrightarrow{\rho}$, then obtains $\overrightarrow{\rho_1}$. She sends it Bob.
2. Bob randomly selects a number $l_2\in \{1,\cdots,n_B\}$, and performs quantum transformation $B_{l_2}^{(i)}(k_1||k_2)$ on quantum state $\overrightarrow{\rho_1}$, then obtains $\overrightarrow{\rho_2}$. He sends it to Alice.
3. According to $l_1$, Alice performs quantum transformation $[A_{l_1}^{(i)}(k_1||k_2)]^{-1}$ on quantum state $\overrightarrow{\rho_2}$, and obtains $\overrightarrow{\rho_3}$. She sends it to Bob.
4. According to $l_2$, Bob performs quantum transformation $[B_{l_2}^{(i)}(k_1||k_2)]^{-1}$ on quantum state $\overrightarrow{\rho_3}$, and obtains $\overrightarrow{\rho_4}$.
From the condition satisfied by $S_{k_1||k_2}(i)$, it can be deduced that $$\overrightarrow{\rho_4}=N_{k_1||k_2}(i)\overrightarrow{\rho}=T_i (A_{k_1}^2,B_{k_2}^2) T_i^{-1}\overrightarrow{\rho}.$$ Bob performs the inverse transformation of $N_{k_1||k_2}(i)$, which is $[N_{k_1||k_2}(i)]^{-1}=T_i (B_{k_2}^2,A_{k_1}^2) T_i^{-1}$, then the quantum message $\overrightarrow{\rho}$ is recovered.
[**Proposition 3:**]{} $(A_{j_1}A_{j_2}A_{j_3},B_{i_1}B_{i_2}B_{i_3})=(A_{j_3}A_{j_2}A_{j_3},B_{i_3}B_{i_2}B_{i_3})$, $\forall i_1,i_2,i_3$, $\forall j_1,j_2,j_3$.
[**Proof:**]{} See the appendix.$\hfill{}\Box$
According to Proposition 3, we know the relation $(A_{l_1}A_{k_1}A_{k_2},B_{l_2}B_{k_3}B_{k_4})=(A_{k_2}A_{k_1}A_{k_2},B_{k_4}B_{k_3}B_{k_4})$. Suppose the set of similarity transformations $\{T_i|i\in I\}$ is selected. We construct the set of operators $S_k(i)$ as follows: $$\begin{aligned}
S_{k_1||k_2||k_3||k_4}(i)&=&\{T_iA_lA_{k_1}A_{k_2}T_i^{-1}|l=1,\cdots,n_A\} \nonumber\\
&&\cup\{T_iB_lB_{k_3}B_{k_4}T_i^{-1}|l=1,\cdots,n_B\}.\end{aligned}$$ When the set of operators $S_{k_1||k_2||k_3||k_4}(i)$ is used in the QNK communication, $l$ is a random number selected locally, and $k_1,k_2,k_3,k_4,i$ are authentication keys preshared by Alice and Bob. If Alice selects a local random number $l_1$, then she should perform quantum transformation $T_iA_{l_1}A_{k_1}A_{k_2}T_i^{-1}$; if Bob selects a local random number $l_2$, then he should perform quantum transformation $T_iB_{l_2}B_{k_3}B_{k_4}T_i^{-1}$. The detailed process of the three interactive communication is the same as the last scheme. After the fourth step, Bob obtains a quantum state $$\overrightarrow{\rho_4}=T_i (A_{k_2}A_{k_1}A_{k_2},B_{k_4}B_{k_3}B_{k_4}) T_i^{-1}\overrightarrow{\rho}.$$ Then Bob performs quantum transformation $T_i (B_{k_4}B_{k_3}B_{k_4},A_{k_2}A_{k_1}A_{k_2}) T_i^{-1}$ and obtains the quantum message $\overrightarrow{\rho}$ sent by Alice.
In general, we have the following relation $$(A_{l_1}A_{k_1}A_{k_2}\cdots A_{k_n},B_{l_2}B_{k_1'}B_{k_2'}\cdots B_{k_n'})=(A_{k_n}A_{k_1}A_{k_2}\cdots A_{k_n},B_{k_n'}B_{k_1'}B_{k_2'}\cdots B_{k_n'}).$$ According to this relation, a new QNK scheme can be constructed similarly. In the new scheme, more operators can be used by Alice and Bob.
Construction of the sets $S(A)$ and $S(B)$
------------------------------------------
Suppose there exists two groups of operators such as $S(A)=\{A_i|i=1,\cdots,n_A\}$ and $S(B)=\{B_i|i=1,\cdots,n_B\}$, and they satisfy the relation expressed in Eq.(\[eqn2\]), then $A_i^{-1}B_jA_i=B_jN,\forall j$. Thus $A_{i_1}^{-1}B_jA_{i_1}=A_{i_2}^{-1}B_jA_{i_2}$. That is $$\label{eqn3}
(A_{i_2}A_{i_1}^{-1})B_j=B_j(A_{i_2}A_{i_1}^{-1}),$$ or written as $(A_{i_2}A_{i_1}^{-1},B_j)=I$ or $[A_{i_2}A_{i_1}^{-1},B_j]=0$. (Denote $[A,B]=AB-BA$)
According to Eq.(\[eqn2\]) and the similar deduction, it can be inferred that $\forall i,j,k$, $$\begin{aligned}
&& [A_iA_j^{-1},B_k]=0, \\
&& [A_iA_j^{-1},B_k^{-1}]=0, \\
&& [B_iB_j^{-1},A_k]=0, \\
&& [B_iB_j^{-1},A_k^{-1}]=0,\end{aligned}$$
From $[A_iA_j^{-1},B_k]=0,\forall i,j,k$, it can be deduced that $$\label{eqn3}
[A_iA_j^{-1},B_kB_l^{-1}]=[A_iA_j^{-1},B_k]B_l^{-1}+B_k[A_iA_j^{-1},B_l^{-1}]=0+0=0.$$ That means $(A_iA_j^{-1},B_kB_l^{-1})=I, \forall A_i,A_j\in S(A), B_k,B_l\in S(B)$.
Thus, we can extend the two sets of operators $S(A),S(B)$ to two new sets of operators $\tilde{S}(A),\tilde{S}(B)$ as follows: $$\begin{aligned}
& \tilde{S}(A)=\{A_{i||j}=A_iA_j^{-1}|A_i,A_j\in S(A)\}, \\
& \tilde{S}(B)=\{B_{i||j}=B_iB_j^{-1}|B_i,B_j\in S(B)\},\end{aligned}$$ According to Eq.(\[eqn3\]), if we select one element from each of the two new sets $\tilde{S}(A),\tilde{S}(B)$, such as $A_{i||j},B_{k||l}$ (where $i,j\in\{1,\cdots,n_A\}$, $k,l\in\{1,\cdots,n_B\}$), then the two elements satisfy the relation $(A_{i||j},B_{k||l})=I$. Thus, the new sets extended from $S(A),S(B)$ can also be used in QNK protocol.
When $n_A=n_B=2$, the extended sets of operators are $\tilde{S}(A)=\{A_1A_2^{-1}, \\A_2A_1^{-1},A_1A_1^{-1}=I,A_1A_1^{-1}=I\}$ and $\tilde{S}(B)=\{B_1B_2^{-1},B_2B_1^{-1},B_1B_1^{-1}=I,B_1B_1^{-1}=I\}$. Because the identity operator $I$ in the two sets is meaningless, the number of the operators in the two extended sets does not increase through the above method (The number of the operators in each set remains 2). Thus the extension is meaningless. However, the extension is meaningful when $n_A,n_B\geq 3$.
Discussions and Conclusions
===========================
We study the theory of quantum no-key protocols. Firstly, a framework of quantum no-key protocols is presented. The framework is expressed in two forms: trace-preserving quantum operators and natural presentations. Secondly, we defined the information-theoretical security of quantum no-key protocol, and the security of authentication keys. Finally, two kinds of quantum no-key protocols are presented. In the first scheme, the quantum operators used by Alice and Bob are all unitary. In the second scheme, Alice and Bob use trace-preserving quantum operators, and the two sets of operators used by Alice and Bob are constructed based on two multiplicative commutative sets.
In the paper, there exists the following questions.
1. $k,i$ are secret identification key. How to use them to identify the personal identification of Alice, Bob, or attackers. How to identify in each of the three transformations?
2. How to construct operator $N_k$ and a set $I(k)$ from a given number $k$?
3. How to construct a set $L(k,i)$ and a set of operators $S_k(i)$ from the given $k$ and $i\in I(k)$?
4. In the quantum no-key protocols, the numbers $l_1,l_2\in L(k,i)$ are selected randomly. Whether the random selection of $l_1,l_2$ can prevent from the leakage of identification keys $k\in\mathcal{K}$ and $i\in I(k)$ during the communication?
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the National Natural Science Foundation of China (Grant No. 61173157).
Appendix {#appendix .unnumbered}
========
[**Proposition 1:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j$ $$\begin{aligned}
&& (A_i^{-1},B_jB_i^{-1})=I,\\
&& (A_j^{-1},B_jB_i^{-1})=I,\\
&& (B_i,A_jA_i^{-1})=I,\\
&& (B_j,A_jA_i^{-1})=I.\end{aligned}$$ [**Proof:**]{} According to the condition Eq.(\[eqn2\]), $\forall i\neq j$ $$\begin{aligned}
&& (A_i,B_i)=N,\\
&& (A_i,B_j)=N,\\
&& (A_j,B_i)=N,\\
&& (A_j,B_j)=N.\end{aligned}$$ From the identity $(A_i,B_i)=N=(A_i,B_j)$, it can be deduced $B_i^{-1}A_i^{-1}B_iA_i=B_j^{-1}A_i^{-1}B_jA_i$, then $$B_i^{-1}A_i^{-1}B_i=B_j^{-1}A_i^{-1}B_j.$$ Thus, it can be known that $(B_jB_i^{-1})A_i^{-1}=A_i^{-1}(B_jB_i^{-1})$, that is $(A_i^{-1},B_jB_i^{-1})=I$.
The other three relations can be obtained in the same way.$\hfill{}\Box$
[**Proposition 2:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j,k,l$ $$(A_iA_j,B_kB_l)=(A_j^2,B_l^2).$$ [**Proof:**]{} In the deduction of this proof, the four identities such as $B_lA_i=A_iB_lN$ (that is $(A_i,B_l)=N$), $A_i^{-1}B_kA_i=B_kN$ (that is $(A_i,B_k)=N$), $B_k^{-1}A_j^{-1}B_k=B_l^{-1}A_j^{-1}B_l$, and $B_lN=A_j^{-1}B_lA_j$ (that is $(A_j,B_l)=N$) are used in turn. The deduction is as follows. $\forall i,j,k,l$, $$\begin{aligned}
(A_iA_j,B_kB_l)&=& (B_l^{-1}B_k^{-1})(A_j^{-1}A_i^{-1})B_kB_lA_iA_j \\
&=& (B_l^{-1}B_k^{-1})A_j^{-1}A_i^{-1}B_kA_iB_lNA_j \\
&=& B_l^{-1}B_k^{-1}A_j^{-1}B_kNB_lNA_j \\
&=& B_l^{-1}B_l^{-1}A_j^{-1}B_lNB_lNA_j \\
&=& (B_l^{-1})^2 A_j^{-1}A_j^{-1}B_lA_jA_j^{-1}B_lA_jA_j \\
&=& (B_l^{-1})^2 (A_j^{-1})^2B_l^2A_j^2 \\
&=& (A_j^2,B_l^2).\end{aligned}$$ $\hfill{}\Box$
[**Proposition 3:**]{} $(A_{j_1}A_{j_2}A_{j_3},B_{i_1}B_{i_2}B_{i_3})=(A_{j_3}A_{j_2}A_{j_3},B_{i_3}B_{i_2}B_{i_3})$, $\forall i_1,i_2,i_3$, $\forall j_1,j_2,j_3$.
[**Proof:**]{} It can be inferred from the formula $A_i^{-1}B_kA_i=A_j^{-1}B_kA_j$ that $$\begin{aligned}
A_i^{-1}B_{k_1}B_{k_2}B_{k_3}A_i &=& A_i^{-1}B_{k_1}A_iA_i^{-1}B_{k_2}A_iA_i^{-1}B_{k_3}A_i \\
&=& A_j^{-1}B_{k_1}A_jA_j^{-1}B_{k_2}A_jA_j^{-1}B_{k_3}A_j \\
&=& A_j^{-1}B_{k_1}B_{k_2}B_{k_3}A_j.\end{aligned}$$ Similarly, it can be inferred $$\begin{aligned}
B_i^{-1}A_{k_1}^{-1}A_{k_2}^{-1}A_{k_3}^{-1}B_i &=& B_j^{-1}A_{k_1}^{-1}A_{k_2}^{-1}A_{k_3}^{-1}B_j.\end{aligned}$$ Thus, the following result holds $$\begin{aligned}
(A_{j_1}A_{j_2}A_{j_3},B_{i_1}B_{i_2}B_{i_3})&=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_1}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_1}^{-1}B_{i_1}B_{i_2}B_{i_3}A_{j_1}A_{j_2}A_{j_3} \\
&=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_1}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_2}^{-1}B_{i_1}B_{i_2}B_{i_3}A_{j_2}A_{j_2}A_{j_3} \\
&=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_2}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_2}^{-1}B_{i_2}B_{i_2}B_{i_3}A_{j_2}A_{j_2}A_{j_3} \\
&=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_2}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_3}^{-1}B_{i_2}B_{i_2}B_{i_3}A_{j_3}A_{j_2}A_{j_3} \\
&=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_3}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_3}^{-1}B_{i_3}B_{i_2}B_{i_3}A_{j_3}A_{j_2}A_{j_3} \\
&=& (A_{j_3}A_{j_2}A_{j_3},B_{i_3}B_{i_2}B_{i_3}).\end{aligned}$$ $\hfill{}\Box$
[99]{} P. Boykin and V. Roychowdhury, Optimal Encryption of Quantum Bits, arXiv: quant-ph/0003059.
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A. Nayak and P. Sen, Invertible quantum operations and perfect encryption of quantum states, Quantum Information & Computation 7(1-2): 103-110, 2007.
L. Yang, Quantum public-key cryptosystem based on classical NP-complete problem, arXiv: quant-ph/0310076.
L. Yang, et al, Quantum public-key cryptosystems based on induced trapdoor one-way transformations, arXiv: 1012.5249.
A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton,1997
L. Yang, L. A. Wu, Transmit Classical and Quantum Information Secretly. arXiv: quant-ph/0203089.
L. Yang, L. A. Wu, and S. H. Liu, Proc. SPIE, 4917(2002), 106-111.
L. Yang, Quantum no-key protocol for direct and secure transmission of quantum and classical messages. arXiv: quant-ph/0309200, 28 Sep 2003.
L. Yang and L. Hu, Quantum no-key protocol with inherent identification, Proc. SPIE Vol. 6305, pp. 63050J (2006).
Y. Kanamori, S. M. Yoo and Mohammad, A Quantum No-Key Protocol for Secure Data Communication, 43rd ACM SE Conference, March 18-20, 2005
W. H. Kye, C. M. Kim, M. S. Kim and Y. J. Park, Quantum Key Distribution with Blind Polarization Bases, Phys.Rev.Lett. 95 (4), 2005, 040501.
S. Kak, A Three-Stage Quantum Cryptography Protocol, Foundations of Physics Letters, Vol. 19, No. 3, June 2006.
Y. Wu and L. Yang, Practical quantum no-key protocol with identification. IAS 2009: 540-543, IEEE Computer Society.
A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, Secure communication with a publicly known key. Acta physica Polonica. A 101(3): 357-368. (see also arXiv:quant-ph/0101066).
K. Boström and T. Felbinger. Deterministic secure direct communication using entanglement. Physical Review Letters 89(18): 187902. 2002.
F. G. Deng, G. L. Long, and X. S. Liu. Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block. Physical Review A 68(4): 042317. 2003.
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J. Watrous. Quantum Computational Complexity, arXiv:0804.3401.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Dirac oscillator in a homogenous magnetic field exhibits a chirality phase transition at a particular (critical) value of the magnetic field. Recently, this system has also been shown to be exactly solvable in the context of noncommutative quantum mechanics featuring the interesting phenomenon of re-entrant phase transitions. In this work we provide a detailed study of the thermodynamics of such quantum phase transitions (both in the standard and in the noncommutative case) within the Maxwell-Boltzmann statistics pointing out that the magnetization has discontinuities at critical values of the magnetic field even at finite temperatures.'
author:
- '<span style="font-variant:small-caps;">A. M. Frassino</span>'
- '<span style="font-variant:small-caps;">D. Marinelli</span>'
- '<span style="font-variant:small-caps;">O. Panella</span>'
- '<span style="font-variant:small-caps;">P. Roy</span>'
bibliography:
- 'DiracOscillator.bib'
title: Thermodynamics of Quantum Phase Transitions of a Dirac oscillator in a homogenous magnetic field
---
Introduction {#sec:introduction}
============
Quantum Phase Transitions (QPT) [@Sachdev201105] are a class of phase transitions that can take place at zero temperature when the quantum fluctuations, required by the Heisenberg’s uncertainty principle, cause an abrupt change in the phase of the system. The QPTs occur at a critical value of some parameters of the system such as pressure or magnetic field. In a QPT, the change is driven by the modification of particular couplings that characterise the interactions between the microscopic elements of the system and the dynamics of its phase near the quantum critical point.
For a quantum system at finite temperature $T$, both the thermal and the quantum fluctuations are present. The interplay between the quantum and the thermal fluctuations can either smooth out the differences between the phases (namely, no phase transition occurs at finite temperature) or there can be regimes in which some discontinuities hold, and a phase transition appears. Eventually, the thermal fluctuations prevail [@Sachdev201105].
In this paper, we focus on a particular quantum system, namely the Dirac oscillator, at finite temperature $T$ and with an additional constant magnetic field $\textbf{B}$. The Dirac oscillator system including an interaction in the form of a homogeneous magnetic field is an exactly solvable system [@PhysRevLett.99.123602; @2007PhRvA76Delgado; @PhysRevA.77.033832; @Mandal:2009we; @Mandal:2012wp] and shows interesting properties. In particular, at zero temperature, if the magnitude of the magnetic field either exceeds or is less than a critical value $B_{\text{cr}}$ (which depends on the oscillator strength), this combined system shows a *chirality phase transition* [@2008PhRvA77:Delgado; @Quimbay:2013gta]. Because of the phase transition, the energy spectrum is different for $B>B_{\text{cr}}$ and $B<B_{\text{cr}}$ where $B$ is the magnetic field strength.
The $(2+1)$-dimensional Dirac oscillator in the presence of a constant magnetic field has also been studied at zero temperature in the framework of noncommutative space coordinates and momenta [@Panella:2014hga]. It has been shown that in this case $B_{\text{cr}}$ depends not only on the oscillator strength but also on the noncommutative parameters. An interesting consequence of the noncommutative scenario is that, apart from the left- and right-chiral phases of the commutative case [@PhysRevA.77.033832], there is also a third left phase and thus a second quantum phase transition (right-left) [@Panella:2014hga] (see Fig. \[fig:CNNcom\]). The presence of the third phase with left chirality leads to what is called a Re-entrant Phase Transition (RPT) [@2016:PanellaRoyReE] first observed in nicotine/water mixture [@hudson1904gegenseitige] as noted in [@Mann2016]. This phenomenon has been recently found also in gravitational systems [@Altamirano:2013ane; @Frassino:2014pha].
In this work, we analyze the Dirac oscillator in the presence of a constant magnetic field at finite temperature. While in the first part of the paper we consider the standard system at finite $T$, the second part will be dedicated to the noncommutative Dirac oscillator.
To study what kind of QPT the Dirac oscillator at finite $T$ undergoes, we will investigate the system at high temperatures when the statistics can be approximated with the Maxwell-Boltzmann one [@Kubo1990]. We find that in this regime the QPT does not disappear and therefore the quantum fluctuations prevail upon the thermal ones (as can be seen in the noncommutative case in Fig. \[fig:CNNcom\]). Only in the limit of very high temperatures ($\beta \rightarrow0$) the QPT disappears.
The organization of the paper is as follows: in Sec.\[sec:System\] we shall present the system to be analyzed and discuss the spectrum of the two phases, namely, the left and the right phase; in Sec. \[sec:FiniteT\], we present the method to calculate the partition function that characterizes the chiral phases at finite temperature and describes the phase transition as the strength of the magnetic field varies; in Sec. \[sec:magnetiz\] we scrutinize in details the magnetization at finite temperature and in Sec. \[sec:NCoscillator\] we study the thermodynamic of a noncommutative Dirac Oscillator with a constant magnetic field. Finally, Sec. \[sec:conclusions\] is devoted to conclusions.
The Quantum System: Dirac Oscillator with a constant magnetic field {#sec:System}
===================================================================
The non-relativistic version of the $\left(2+1\right)$-dimensional Dirac oscillator with a constant magnetic field and zero temperature can be associated to a chiral harmonic oscillator [@2008PhRvA77:Delgado], that have been studied in [@Banerjee:1998em]. Moreover, in [@HortaBarreira:1992ec; @Dunne:1989hv] a possible connection to topological Chern-Simons gauge theories has been pointed out. The case of the system presented in this paper can be seen as a relativistic extension of the chiral harmonic oscillators [@2008PhRvA77:Delgado; @Quimbay:2013gta], at finite temperature.
A relativistic spin-1/2 fermion constrained in a 2-dimensional plane with mass $m$, charge $e$, Dirac oscillator frequency $\omega$ and subjected to a a constant magnetic field orthogonal to the plane, is described by the Hamiltonian: $$H= c\, \bm{ \sigma} \cdot
(\bm{p}-im\omega\sigma_z \bm{x}+\frac{e}{c}\bm{\,A})
+\sigma_z \,mc^{2}\,, \label{eq:hamiltonian}$$ where $c$ stands for the speed of light and $\bm{\sigma}=\left(\sigma_x,\sigma_y\right)$, $\sigma_z$ denotes the Pauli matrices. As in the usual notation, the $\bm{p}$ and the $\bm{x}$ represent the momentum and the position operators while the vector potential is related to the magnetic field through $$\bm{A}=(-B{y}/2,B{x}/2)\,.$$ This set-up offers an intriguing interplay: while the two-dimensional Dirac oscillator coupling endows the particle with an intrinsic left-handed chirality [@2007PhRvA76Delgado], the magnetic field coupling favors a right-handed chirality [@PhysRevLett.99.123602]. The interplay between opposed chirality interactions culminates in the appearance of a relativistic quantum phase transition, which can be fully characterized [@2008PhRvA77:Delgado].
Energy levels {#sec:energy-levels}
-------------
In this section, we shall recall the spectrum and the degeneracy of the various energy levels of a two-dimensional relativistic Dirac oscillator in the two phases defined by $B>B_{\text{cr}}$ and $B<B_{\text{cr}}$. The spectum of the system is characterized by two quantum numbers: $n_r = 0,1,2,...$, the radial quantum number and $M=0,\pm 1, \pm 2,...$, the 2-dimensional angular momentum quantum number.
In what we call the *left phase*, the energy level corresponding to the zero mode has positive energy $E=\, mc^2$ and infinite degeneracy with respect to the non-negative magnetic quantum number $M\ge 0$ (the negative magnetic numbers are forbidden [@Panella:2014hga]). The excited states are $$\begin{aligned}
E_{N}^{\pm}=\,\pm mc^{2}\sqrt{1+\,\xi_{L}\,N}\,,\quad\\
N=n_{r}+\frac{|M|-M}{2},\quad N=1,2,\,,\dots\,\label{spectrumL1}\end{aligned}$$ where $N$ labels the energy levels, $\xi_L$ is a constant encoding the parameters of the system that for the standard commutative case simply reads $$\begin{gathered}
\xi_L = -4\,X, \quad X := \frac{1}{mc^{2}}\left(\mu\, B - \hbar\omega\right)
\label{eq:xiLcomm}\end{gathered}$$ and $\mu=e \hbar / (2 m\, c)$ denotes the Bohr magneton. The energy levels in are degenerate. In particular, every level has infinite degeneracy, with respect to the non-negative values of $M$, and $D=N+1$ finite degeneracy with respect to the negative values of $M$.\
The *right phase*, on the other hand, has zero mode degeneracy with respect to the non-positive magnetic quantum number $M \le 0$ and the value of the energy is in the negative branch: $E = - m\, c^2$. Similarly, the excited states have an infinite degeneracy with respect to non-positive values of $M$, while the degeneracy is $D=N+1$ with respect to $M>0$ and the energies read $$\begin{gathered}
E_{N}^{\pm}=\,\pm mc^{2}\sqrt{1+\,\xi_{R}\,N}\,,\\
N=n_{r}+\frac{|M|+M}{2}\quad N=1,2,\dots\,,\label{spectrumR}\end{gathered}$$ where the parameter $\xi_{R}$ is related to $\xi_{L}$ by the relation $\xi_{R}=-\xi_{L}$. Note that, when $\xi_{R} = \xi_{L} = 0$, that is when $$B = B_{\text{cr}} =\hbar\, \mu \, \omega, \label{eq:BcrComm}$$ all the energy levels collapse to the energy $E=\pm mc^2$ and the *quantum phase transition* happens [@2008PhRvA77:Delgado; @Panella:2014hga].
Finite Temperature Quantum Phase Transition {#sec:FiniteT}
===========================================
To study the interplay between thermal and quantum fluctuations for this two-dimensional relativistic Dirac oscillator, we focus on the system at high temperatures when the electron-states statistics can be described with the Maxwell-Boltzmann statistics. We will show that in this regime the QPT will not disappear.
Partition Function
------------------
The partition function for the positive branch of the energy levels (\[spectrumL1\]) and (\[spectrumR\]) and the zero mode, has to take into account the degeneracy of each single energy level. Thus, it is defined by the product of the density of the states and the Boltzmann factor $$\begin{gathered}
Z_{L,R} = \sum_{M,n_r}\, \mathtt{g} \, e^{-\beta\, E^{+}_{N}}, \quad N=0,1,\dots\,, \label{eq:Z-LR}\end{gathered}$$ where $E^{+}_{N}$ are the values given by (\[spectrumL1\]) and (\[spectrumR\]), and $\beta$ is the inverse temperature.
[0.25/0.75/0.5,0.75/1.25/1.0,1.55/2.05/1.8]{} //iin
//in in [0,0.05,...,0.5]{} [ ($(\xz,\from)+(0,\k)$) to ($(\xs,\from)+(0,\k)$) ; ($(\xs,\from)+(0,\k)$) to\[out=0,in=180\] (,) ; ]{}
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(0.5,1.28) – (0.5,1.50) ; (3.0,1.3) – (3.0,1.7) ;
(0.5,2.1) – (0.5,2.250) ; (3.0,2.1) – (3.0,2.25) ;
(,0) to node\[above,sloped,font=,pos=0.8\] [$mc^2\sqrt{1+p^2}$]{} node\[above,sloped,font=,pos=0.2\] [$\omega=B=0$]{} (,2.255) ;
iin
To evaluate the density of the states $ \mathtt{g}$, we can split every chiral phase of the system (left and right introduced in the previous section) into two different sub-systems with respect to the sign of the quantum number $M$.
If we consider a free electron with $M>0$ or alternatively $M<0$, in a standard commutative spacetime and confined to a finite area $\mathtt{L}^{2}$, the number of energy levels in the region $dp_{x}dp_{y}$ around $p_{x}$ and $p_{y}$ is $ d\mathtt{g} = \mathtt{L}^{2}/\left( 2 h^{2} \right)dp_{x}dp_{y}$ [@Kubo1990]. Note that it can be proved that the phase-space volume element $dp_{x}\wedge dp_{y}$ in the relativistic case is invariant for Lorentz transformations [@hakim2011introduction]. Moreover, $\mathtt{L}$ is the length in a rest-frame.
#### Left Phase.
For $M\ge0$, the degeneracy of each energy level (\[spectrumL1\]) defined in the region $\mathscr{K}$ of the momenta $(p_x,p_y)$ such that $E_N<E_{\text{free}}(p_x,p_y)<E_{N+1}$ (see Fig.\[fig:spectrum\]), where $E_{\text{free}}$ is the energy of the free electron, is given by $$\begin{aligned}
\mathtt{g}_{M>0}&=&
\frac{1}{2}\left(\frac{\mathtt{L}}{h}\right)^{2}
\iintop_{\mathscr{K}}dp_{x}dp_{y} \nonumber \\ &=&
\pi\left(\frac{\mathtt{L}}{h}\right)^{2}\left[p^{2}\right]_{\frac{p^{2}}{m^{2}c^{2}}=\,\xi_{L}N}^{\frac{p^{2}}{m^{2}c^{2}}=\,\xi_{L}\left(N+1\right)}
= \left(\frac{\mathtt{L}mc}{h}\right)^{2}\,\pi\xi_{L} .\end{aligned}$$ and $N$ depends only on the quantum number $n_{r}$ because the difference in is zero. In the case $M<0$, the number of levels collapsing into the level $N$ is still equal to $\mathtt{g}_{M>0}$. However, one should consider that, when $M<0$, as shown in (\[spectrumL1\]), $N$ depends on both the quantum number $M$ and $n_{r}$, namely $N = n_r + M $. Therefore, one needs to keep into account, for each energy level $N$, the degeneracy $D$, i.e. $ \mathtt{g}_{M<0}= \mathtt{g}_{M>0}/D$. Thus, the number of states $$\mathtt{g}_{M<0}= \left(\frac{\mathtt{L}mc}{h}\right)^{2}\,\frac{\pi\xi_{L}}{\left(n_{r}-M+1\right)},\label{eq:StateDensity}$$ is not constant along the spectrum, but depends on the energy level.
Taking into account the degeneracy, using the spectrum (\[spectrumL1\]) and its zero point energy, the partition function with Boltzmann statistics of the states of a single oscillator[^1] for the left phase is $$\begin{aligned}
Z_L &=&
\left(\frac{\mathtt{L}mc}{h}\right)^{2}\,\pi\xi_{L}
\left[
e^{-\beta mc^{2}}+
\sum_{n_{r}=1}^{\infty}
\ e^{-\beta mc^{2}\sqrt{1+\,\xi_{L}\,n_{r}}}+\right.\nonumber\\
&+&\left.
\sum_{n_{r}=1}^{\infty}
\sum_{M=0}^{-\infty}
\frac{e^{-\beta mc^{2}\sqrt{1+\,\xi_{L}\,\left(n_{r}-M\right)}}}
{\left(n_{r}-M+1\right)}
\right]. \label{eq:ZSingleOscillator}\end{aligned}$$ Notice that the contribution to the partition function of the zero mode energy appears only for $M\ge 0$ [@Panella:2014hga] and, as we will see later in sec. \[sec:CriticalPoint\], this will be the source of the non-analiticty of the partition function at the phase point. The sum in the third term of coincides with the second if we introduce $M'$ and $N'$ such that $M'=-M$ then $n_{r}+M'=N'$, obtaining $$\label{doublesum} \sum_{n_{r}=1}^{\infty}\sum_{N'=n_{r}}^{\infty}
\frac{e^{-\beta mc^{2}\sqrt{1+\,\xi_{L}\,\left(N'\right)}}} {\left(N'+1\right)}
=\sum_{N'=1}^{\infty}e^{-\beta mc^{2}\sqrt{1+\,\xi_{L}\,\left(N'\right)}},$$ so, over all, the two sectors ($M>0$ and $M<0$) equally contribute to the partition function.
#### Right Phase
The partition function for the right phase, defined by the energy levels (\[spectrumR\]), is equal to (\[eq:ZSingleOscillator\]) except for the zero mode term. This is because, as explained in Sec. \[sec:System\], the zero mode contributes only to the negative energies branch.
Zeta-function representation \[sub:mellin\]
-------------------------------------------
A different representation of the defining series in the partition function that will reveal itself to be more effective when the phase switches from left to right, can be obtained using the Cahen-Mellin integral, as suggested for different systems in [@Brevik:2005ys] and reviewed in [@Dariescu:2007; @boumali2015one]. The Cahen-Mellin integral is defined as [@ParisKaminski200109] $$e^{-x}=\frac{1}{2\pi i}\int_{\mathfrak{c}-i\infty}^{\mathfrak{c}+i\infty}\Gamma\left(s\right)\: x^{-s}\ ds \quad \left(\left|\arg x\right|<\frac{1}{2}\pi;\, x\neq0\right)\label{eq:CahenMellinIntegral}$$ where $\mathfrak{c}$ is real and $\mathfrak{c}>0$. The argument of the integral has poles for all $x=-n$, $n\in \mathbb{N}_0$ and the residuals at the negative poles are $\left(-1\right)^{n}/n!$ (see Fig. \[fig:figure1\]).
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Using (\[eq:CahenMellinIntegral\]) to calculate the second term in the partition function (\[eq:ZSingleOscillator\]) and including the sum into the integral, we have that the term $$\xi_{L}\sum_{n_{r}=1}^{\infty}\ e^{-\beta mc^{2}\sqrt{1+\,\xi_{L}\, n_{r}}}$$ can be rewritten as $$\frac{\xi_{L}}{2\pi i}\int_{\mathfrak{c}-i\infty}^{\mathfrak{c}+i\infty} \frac{\Gamma\left(s\right)}{\left(\beta mc^{2}\right)^{s}}\:\sum_{n_{r}=1}^{\infty}\frac{1}{\left(1+\,\xi_{L}\, n_{r}\right)^{s/2}}\ ds. \label{eq:term2}$$ The series in (\[eq:term2\]) can be recognized as the series representation of the Hurwitz-$\zeta$ function $\zeta\left(v,a\right)=\sum_{n=0}^{\infty}\frac{1}{\left(n+a\right)^{v}}$ that converges only if $\ensuremath{\mathfrak{Re}\left(v\right)}>1$. Under this condition the integral (\[eq:term2\]) becomes $$\frac{\xi_{L}}{2\pi i}\int_{\mathfrak{c}-i\infty}^{\mathfrak{c}+i\infty} \frac{\Gamma\left(s\right)}{\left(\beta mc^{2}\xi_{L}^{1/2}\right)^{s}}\:\:\zeta\left(\frac{s}{2},\frac{1}{\xi_{L}}+1\right)\ ds. \label{eq:IntegrandZeta}$$ In contrast to what has been proposed in the recent literature on the topic (see [@boumali2015one] and references therein), we notice that the condition for the convergence of the series $s/2>1$ implies $\mathfrak{c} >2$. The integral can therefore be evaluated with the method of residue once a proper closed path has been identified. Because of the presence of the $\Gamma$-function that diverges as $\Gamma(s)=\left(2\pi\right)^{1/2}e^{-s}s^{s-\frac{1}{2}}\left[ 1-\mathcal{O}\left(s^{-1}\right)\right] $, the integration in can be closed only on the left part of the complex plane, allowing to use the Cauchy’s residue theorem on the poles $\left\{ s=2,0,\mathbb{Z}^{-}\right\} $ represented in Fig. \[fig:figure1\]. Introducing the adimensional variable $\tilde{\beta}=\beta m\, c^2$, the partition functions for both the phases becomes $$\begin{gathered}
Z_{L,R}=2\pi\left(\frac{\mathtt{L}mc}{h}\right)^{2}\times\\
\times\left[\left(\frac{2}{\tilde{\beta}^{2}}-1\right)+\left(\frac{\Theta e^{-\tilde{\beta}}}{2}-\frac{1}{2}\right)\xi_{L,R}+\right.\\
\left.+\sum_{n=1}^{\infty}\frac{\left(-\tilde{\beta}\right)^{n} \xi_{L,R}^{\frac{n}{2}+1}}{n!}\zeta\left(-\frac{n}{2},1+\frac{1}{\xi_{L,R}}\right)\right],
\label{eq:Zzeta}\end{gathered}$$ where now we use the subscript $L,R$ to indicate both the left and right phase and $\Theta$ is a step function that is equal to zero (one) for the right phase (left phase). This series representation can be seen as a series expansion in power of $\beta$, namely a high-temperature power expansion. From now on the quantity in the square brackets of will be called $\tilde{Z}_{L,R}$ $$Z_{L,R}=2\pi\left(\frac{\mathtt{L}mc}{h}\right)^{2} \tilde{Z}_{L,R}.$$ The series representations of the partition function $ \tilde{Z}_{L,R}$ in terms of the $\zeta$-function and the Boltzmann representation introduced in can be both evaluated numerically considering only a finite number of terms in the series. However, the Boltzmann representation encounters obvious numerical issues for small values of $\xi_{L,R}$, i.e. close to the value where the phase transition happens. These issues do not affect the $\zeta$-function representation of the partition function and, as we will see in the next subsection, the partition function can be evaluated analytically for small values of $\xi_{L,R}$. Fig. \[fig:2\] shows $ \tilde{Z}_{L,R}$ in the $\zeta$-function representation keeping $n_{\text{max}}=100$ terms of the series in .
Partition function near the critical point {#sec:CriticalPoint}
------------------------------------------
The partition function can be used to calculate the asymptotic expansion near the critical point, namely when $\xi_{L,R}\rightarrow 0$. This allows to analytically explore the regimes close to the phase transition and study the interplay between thermal and quantum fluctuations. The asymptotic expansion of the Hurwitz-$\zeta$ function [@MR2124202; @magnus2013formulas] is $$\zeta\left(s,a\right)=\frac{a^{1-s}}{s-1}+\frac{1}{2}a^{-s}+\frac{\mathtt{Z}\left(s,a\right)}{\Gamma\left(s\right)}\label{eq:ZetaExp}$$ where the large-$a$ (Poincaré) asymptotic expansion of the function $\mathtt{Z}\left(s,a\right)$ $$\mathtt{Z}\left(s,a\right)\sim\sum_{k=1}^{\infty}\frac{B_{2k}}{\left(2k\right)!}\frac{\Gamma\left(2k+s-1\right)}{a^{2k+s-1}},\qquad\left|a\right|\rightarrow\infty \label{eq:ZetaExp2}$$ is valid in $|\text{arg}\;\; a| < \pi$ and $B_{2k}$ denote the even-order Bernoulli numbers. Writing explicitly the sum in the partition function $\tilde{Z}_{L,R}$ gives $$\begin{gathered}
\tilde{Z}_{L,R}=\frac{2}{\tilde{\beta}^{2}}-1-\left(1-\Theta e^{-\tilde{\beta}}\right)\frac{\xi_{L,R}}{2}+\\
+\sum_{n=1}^{\infty}\frac{\left(-\tilde{\beta}\right)^{n}}{n!}\left\{ \left[\frac{1}{2}\xi_{L,R}-\frac{2\left(\xi_{L,R}+1\right)}{n+2}\right]\left(\xi_{L,R}+1\right)^{\frac{n}{2}}+\right.\\
\left.+\sum_{k=1}^{\infty}\frac{B_{2k}}{\left(2k\right)!}\frac{\Gamma\left(2k-\frac{n}{2}-1\right)\:}{\Gamma\left(-\frac{n}{2}\right)}\frac{\xi_{L,R}^{2k}}{\left(\xi_{L,R}+1\right)^{2k-\frac{n}{2}-1}}\right\} . \label{eq:ZtildPreBinomial}\end{gathered}$$ Using the binomial series and the relation between Euler gamma functions $$\Gamma\left( -\frac{n}{2} \right) \; \Gamma\left( 1 + \frac{n}{2} \right) = \frac{\pi}{\sin\left( - \pi n/2 \right)},$$ one can rewrite the terms in the curly brackets in (\[eq:ZtildPreBinomial\]) as $$\begin{gathered}
\sum_{w=0}^{\infty} \frac{1}{w!} \left\{ \left[\frac{1}{2}\xi_{L,R}-\frac{2\left(\xi_{L,R}+1\right)}{n+2}\right]
\frac{\Gamma \left(1+ \frac{n}{2} \right)}{\Gamma \left(1+ \frac{n}{2} - w \right)} \xi_{L,R}^{w} +
\right.\\
\left.+\sum_{k=1}^{\infty}\frac{B_{2k}}{\left(2k\right)!}\frac{\Gamma\left( 1+ \frac{n}{2} \right)\:}{\Gamma\left( -2k +\frac{n}{2}+2-w \right)} \xi_{L,R}^{2k+w} \right\}\end{gathered}$$ defining in this way the partition function at all orders in $\xi_{L,R}$ near the critical point. At the second order in $\xi_{L,R}$, one can sum the partition function at all orders in $\beta$, so that, near the critical point, it reads $$\begin{gathered}
\tilde{Z}_{L,R} =\frac{2e^{-\tilde{\beta}}}{\tilde{\beta}}\left[1+\frac{1}{\tilde{\beta}}+\left(\Theta-1\right)\frac{\tilde{\beta} \xi_{L,R}}{4}+\frac{\tilde{\beta}^{2}\,\xi_{L,R}^{2}}{48}\right] + \\
+{\cal O}\left(\xi_{L,R}^{3}\right).\label{eq:Z2ndOrder}\end{gathered}$$ This expression is a continuous but non-analytic function in $\xi_{L,R}$ and, in fact, $\Theta$ brings a discontinuity in the first order derivatives. The function to the second order is compared with in Fig. \[fig:2\].
Magnetization at finite temperature {#sec:magnetiz}
===================================
At this point, it is possible to use the derived expression for the partition function to calculate a physical observable quantity like the magnetization [@Yoshioka1992; @Andersen1995]. Indeed, in [@Panella:2014hga] the magnetization has been proposed as the quantity able to distinguish between the phases of the system.
The magnetization for a system at finite temperature, can be defined as [@Kubo1990] $$\begin{gathered}
M=k_{B}T\frac{\partial\log Z}{\partial B}=\frac{1}{\beta\,Z}\frac{\partial Z}{\partial\xi}\frac{\partial\xi}{\partial X}\frac{\partial X}{\partial B}.\label{eq:Mformula}\end{gathered}$$ In order evaluate the derivative $\partial Z/\partial\xi_{L.R}$ one can use in the following identity [@NIST:DLMF] $$\begin{gathered}
\frac{\partial\zeta\left(s,a\right)}{\partial a}=-s\:\zeta\left(s+1,a\right)\ \mbox{for }s\neq0,1\,\mathfrak{Re}a>0. \end{gathered}$$ The magnetization evaluated numerically with $n_\text{max}=60$ is illustrated in Fig. \[fig:MagnComm\]. Interestingly, we find that the magnetization in the left phase, has opposite sign with respect to the single state result obtained at $T=0$ [@Panella:2014hga]. At finite temperature, in $\xi_L =\xi_R = 0$ the magnetization manifests a discontinuity.
Analogously to what has been done for the partition function in the previous sections, we now inspect the magnetization behaviour at values of the magnetic field close to the critical values. To approximate the full partition function in the neighborhood of $\xi_L=\xi_R=0$, we can either follow the steps described in subsection \[sec:CriticalPoint\] for the full partition function resulting of , or we can use in .
At the critical point, the magnetization is not a continuous function of the magnetic field and can be written as a series expansion around $X=0$ at all order in $\beta$ as $$\begin{gathered}
\frac{M_{R,L}}{\mu}= -\frac{\tilde{\beta}}{\tilde{\beta}+1}H\left(X\right)+\\
+\left(\frac{\tilde{\beta}}{\tilde{\beta}+1}\right)^{2}\left[\frac{2}{3}+\tilde{\beta}\left(H\left(-X\right)-\frac{1}{3}\right)\right]\,X +
\mathcal{O}\left(X^{2}\right)
,\end{gathered}$$ where $H(x) $ is the Heaviside step function. The discontinuity with respect to the magnetic field at the critical point $X=0$ is $$|\Delta M_{\text{cr}}| = \mu\, \frac{\, \tilde{\beta} }{ (1+\tilde{\beta} )}.
\label{eq:Mgap}$$ In the limit $\tilde{\beta} \rightarrow 0 $, the thermal fluctuations overcome the quantum ones, the gap disappears and the magnetization is zero for both the chiralities.
Other thermodynamic functions
=============================
Once the partition function has been calculated, the thermodynamics of the system can be fully explored. Using $$\begin{aligned}
F=-\frac{1}{\beta}\ln Z,\qquad U=-\frac{\partial}{\partial\beta}\ln Z,\\
S=\, k_{B}\beta^{2}\frac{\partial F}{\partial\beta},\qquad C=-k_{B}\beta^{2}\frac{\partial U}{\partial\beta},
\end{aligned}\label{eq:ThermQuant}$$ in Fig. \[fig:Free\], we plot the thermodynamic quantities for the commutative system. The specific heat as function of the $\beta$ manifests a non-monotonic behaviour in the left phase, while it becomes monotonic in the right phase.
Noncommutative Dirac Oscillator with a constant magnetic field {#sec:NCoscillator}
==============================================================
In the previous sections we described the Dirac oscillator under a uniform magnetic field and its associated QPT in a $2D$ space-time. Now, we move to the noncommutative system, indeed, studies in quantum gravity and string theory propose that the space-time can be fundamentally or effectively described by a generalization of geometry where, locally, coordinates do not necessarily commute [@Bertolami2005; @Gamboa2002; @Ito1967; @Bellucci2001b; @Duval2000; @Gamboa2001a; @Gamboa2001; @Horvathy:2010wv; @Nair2001; @Smailagic2002; @Smailagic2002a]. Moreover, recent developments in condensed matter physics, suggest the use of the noncommutative geometry framework to encode geometrical properties of topological quantum systems [@2012PhRvB..86c5125N; @Haldane:2011ia; @2012PhRvB..85x1308P; @2012PhRvB..85g5128B].
The study of the Dirac oscillator in the presence of noncommutative coordinates and momenta and a constant magnetic field in a $2$-dimensional space at zero-temperature has been carried out in [@Panella:2014hga]. The system that in the commutative case undergoes a quantum phase transition at finite magnetic field, also exhibits in the noncommutative case another phase transition at a higher magnetic field. The third region appears to have the same chirality of the first one providing a re-entrant quantum phase transition [@2016:PanellaRoyReE].
In this section, we will study the noncommutative system at finite temperature, the parametrization that has been used for the exact calculation of the spectrum in [@Panella:2014hga] will allow us to exploit all the calculation done in the previous sections and easily study the QPTs in the noncommutative space.
Energy levels {#energy-levels}
-------------
The Hamiltonian for the $(2 + 1)$-dimensional Dirac oscillator in the noncommutative plane with a homogeneous magnetic field can be written in a way similar to (\[eq:hamiltonian\]) $$\hat{H}=
c\bm{\sigma}.
(\hat{\bm{p}}-im\omega\sigma_z\hat{\bm{x}}+\frac{e}{c}\,\hat{\bm{A}})
+\sigma_{z}\,mc^{2}\,,\label{h1}$$ where now the *hat* indicates the noncommutative operators. In this framework, the commutation relation between coordinates and momenta are given by [@Bertolami2005] $$[{\hat{x}},{\hat{y}}]=i\theta,
~~~~[{\hat{p}}_{x},{\hat{p}}_{y}]=i\eta,
~~~~[{\hat{x}}_{i},{\hat{p}}_{j}]=i\hbar(1+\frac{\theta\eta}{4\hbar^{2}})\delta_{ij},
\label{NCEQ}$$ where $\theta,\eta\in\mathbb{R}$. The noncommuting coordinates and momenta can be expressed in terms of commuting ones using the Seiberg-Witten map and are given by $$\begin{aligned}{\hat x}=\displaystyle {x-\frac{\theta}{2\hbar}p_y,~~~~{\hat p}_x=p_x+\frac{\eta}{2\hbar}y}\,,\\
{\hat y}=\displaystyle{y+\frac{\theta}{2\hbar}p_x,~~~~{\hat p}_y=p_y-\frac{\eta}{2\hbar}x}\label{rel}\,.
\end{aligned}$$ The Hamiltonian (\[h1\]) can be rewritten as $$H=
c\left(\begin{array}{cc}
mc & {\hat{\Pi}}_{-}\\
{\hat{\Pi}}_{+} & -mc
\end{array}
\right),\label{h2}$$ where $\hat{\Pi}_{\pm}$ are given by $$\begin{gathered}
{\hat\Pi}_{\pm} = \left( 1-
\frac{\tilde{\omega}-\omega}{\omega_\theta}\right)(p_x \pm i\, p_y)\,\pm \\ \pm\, i\, m\, (\tilde{\omega} -\omega -\omega_\eta) (x \pm i y)\end{gathered}$$ after introducing the following frequencies out of the parameters of the system: $$\tilde{\omega}=\frac{eB}{2mc},\,\qquad \omega_\theta = \frac{2\,\hbar}{m\theta},\, \qquad \omega_\eta=\frac{\eta}{2\hbar m}.$$ With this paramatrization it is clear that there are two critical values of the magnetic field $$\begin{gathered}
\label{bc}
B_{\text{cr}}=\frac{2 mc}{e} (\omega +\omega_\eta)=\frac{2c}{e}(m\omega+\frac{2\eta}{\hbar}) \,,\\
\label{bcrit*}
B_{\text{cr}}^*=\frac{2 mc}{e} (\omega +\omega_\theta)=\frac{2c}{e}(m\omega+\frac{\hbar}{\theta}) \, .\end{gathered}$$ such that for both $B=B_{\text{cr}}$ and $B=B_{\text{cr}}^*$, there are no interactions in the model and the Hamiltonian represents a free particle (only kinetic energy). Since the critical value $B_{\text{cr}}$ depends on the momentum noncommutative parameter $\eta$, this shifts the value of the critical field relative to the value in .The space noncommutativity ($\theta \ne 0$) instead introduces the additional critical value for the magnetic field ($B_{\text{cr}}^*$) [@Panella:2014hga].
Interestingly, the parameters used in the noncommutative massive Dirac oscillator can be re-assorbed in such a way that one can study the single phases independently. In fact, the behaviour of the single phases can be described regardless of the parameters encoding the noncommutative physics [@Panella:2014hga]. Therefore, although the noncommutative system has two more parameters, they can recast into the parameter $\xi_{L,R}$ allowing us to use most of the calculations performed in the previous sections for the commutative case. The main difference now is that the magnetic field and the noncommutative parameters are related by $$\begin{aligned}
\xi_{L}&=&-4\frac{mc^{2}}{\omega_{\theta}\hbar}\left(\frac{\omega_{\theta}\hbar}{mc^{2}}-X\right)\left(X-\frac{\omega_{\eta}\hbar}{mc^{2}}\right) \label{eq:xiL} \end{aligned}$$ while it is still valid the relation $\xi_R = -\xi_L$. Phase changes occur at critical values of the magnetic field that makes the parameter $\xi_{L,R}$ null. Note that, differently from the commutative case , in the noncommutative case the equation is quadratic in $B$ and consequently it admits two critical points.
Energy levels and partition function
------------------------------------
In Section \[sec:energy-levels\] we reviewed the spectrum of the system as a function of the parameter $\xi_{L,R}$. As mentioned above, for the noncommutative case, the spectrum remains equal to and as a function of the parameter $\xi_{L,R}$ that now becomes , quadratic in $B$. Since the density of the energy states is also a function of the parameters $\xi_{L,R}$, its contribution to the partition function of each energy level in the noncommutative case coincides with the one in the commutative case . The resulting partition function is illustrated in Fig. \[fig:ZNonComm\]. In the plot is also represented the outcome of the approximation once the constant $\xi_{L,R}$ in is replaced in . The partition function can be used to calculate the other thermodynamic quantities using . In particular, the heat capacity is shown in Fig. \[fig:CNNcom\]. In [@boumali2013thermal] a different noncommutative model has been studied without considering the degeneracy of the states.
Magnetization
-------------
The magnetization for the noncommutative Dirac oscillator case is illustrated in Fig. \[fig:Magn\]. The figure shows the presence of a fixed point at $M=0$ for every $T$. The point where the magnetization changes its sign appears only in the noncommutative case and is a consequence of the term $ \partial \xi / \partial X $ in , that is a non constant function of $X$: $$\begin{gathered}
\frac{\partial\xi_{L,R}}{\partial X}=\pm 4\frac{mc^{2}}{\omega_{\theta}\hbar}\left[2\, X-\frac{\hbar\left(\omega_{\eta}+\omega_{\theta}\right)}{mc^{2}}\right]= \\
=\pm 4\sqrt{\frac{m\,c^{2}}{\hbar\,\omega_{\theta}}
\xi_{L,R} \pm \frac{(\omega_{\eta}-\omega_{\theta})^{2}}{\omega_{\theta}{}^{2}}} \label{eq:Derxix}\end{gathered}$$ where the plus sign is for the $L$ phase and the minus sign is for the $R$ phase. From the first line of , we find that at the point $X_0 = \frac{\hbar (\omega_\eta +\omega_\theta )}{2 mc^2}$ the magnetization change[**s**]{} sign. Therefore, the point where the magnetization smoothly vanishes is the midpoint between the two critical points. This feature of the noncommutative system is independent of the temperature and was also present in the analysis at $T=0$ [@Panella:2014hga].
Near the critical point, we can evaluate the magnetization in $\xi_{L,R}$ using the second line of and in , obtaining: $$\begin{gathered}
\frac{\partial\xi_{L,R}}{\partial X} = \pm\,4\,\left[\left|\frac{(\omega_{\eta}-\omega_{\theta})}{\omega_{\theta}}\right|\pm\frac{1}{2}\frac{m\,c^{2}}{\hbar\,}\frac{\xi_{L,R}}{\left|\omega_{\eta}-\omega_{\theta}\right|}+\mathcal{O}(\xi^{2})\right],\end{gathered}$$ and the other terms in read $$\begin{gathered}
\frac{1}{\tilde{\beta}}\frac{1}{Z}\frac{\partial Z}{\partial\xi} =
\frac{\tilde{\beta}}{4(\tilde{\beta}+1)}\left(\Theta-1\right)+ \\ +
\left[\frac{\tilde{\beta}}{4(\tilde{\beta}+1)}\right]^{2}\left[\frac{2}{3}+\tilde{\beta}\left(\Theta-\frac{1}{3}\right)\right]\xi+
\mathcal{O}\left(\xi^{2}\right).\end{gathered}$$ Then the discontinuity of the magnetization at the two critical points, for the noncommutative case is $$\begin{gathered}
\Delta M=\pm\mu\, \left|\frac{(\omega_{\eta}-\omega_{\theta})}{\omega_{\theta}}\right|\frac{\tilde{\beta}}{(\tilde{\beta}+1)},\end{gathered}$$ and we can notice that if $\eta$, the noncommutative parameter related with the momenta, vanishes then the transitions become independent from $\theta$, the coordinates noncommutative parameter.
Conclusions {#sec:conclusions}
===========
Over the years, the Dirac oscillator has become a valuable tool for various branches of physics, mostly because it is one of the few relativistic systems whose exact solutions are known. The $(2+1)$-dimensional case, namely when an electron is constrained to live in a plane, manifests an intrinsic chiral behavior of its states. An external magnetic field interacts at the quantum level with the chirality properties of the system and, if strong enough, it forces an abrupt switch in the electron chirality. This behavior studied at zero temperature, distinguish two phases of the electron at different regimes, and their change is recognized to be a quantum phase transition [@2008PhRvA77:Delgado]. Magnetization shows a discontinuity at a finite value of the magnetic field [@Panella:2014hga]. The quantum nature of this behavior raises the question on how thermal fluctuations interacts with the system and when the thermal disorder destroys these quantum effects. In this paper, we analyze the interplay between these two physical phenomena. In particular, we quantify the discontinuity of the magnetization of the QPT at finite temperature $\Delta M \propto \hbar \left[ \frac{1}{k_B\,T}+O(\frac{1}{k_B\,T})^2\right]$. This discontinuity is a quantum phenomenon that tends to zero for high temperatures. While the critical values of the system are not affected by changes in temperature, we noticed a crucial difference from the system at $T=0$ studied in [@Panella:2014hga]: the magnetization at finite temperature has regimes with opposite sign with respect to the single state (zero-temperature) case. This opens to the possibility of a second phase transition at low temperatures, where the statistics of the states becomes quantum and cannot be described by Boltzmann statistics. In Fig. \[fig:Free\] we report various thermodynamic quantities that can be evaluated once the partition function is provided. The specific heat shows different behaviors at different phases of the system.
The series representations of the partition function proposed in this paper allow to explore the system in diverse regimes: far, close and at the critical points. The representation in terms of the Hurwitz-zeta function fixes some misunderstandings present in the literature regarding which poles play a role in the evaluation of the partition function.
We also generalize our calculations to the Dirac oscillator in noncommutative momenta and coordinates. The interest for this system comes from the fact that at zero temperature it manifests a re-entrant quantum phase transition, that we find is present also at finite temperature. Interestingly, there are systems in condensed matter physics that in certain regimes appear to have behavior similar and some times mathematically equivalent to systems living in a noncommutative space-time [@2012PhRvB..86c5125N; @Haldane:2011ia; @2012PhRvB..85x1308P; @2012PhRvB..85g5128B]. This opens the avenue of an *analogue noncommutativity* that is the possibility of studying, also experimentally, systems that are mathematically equivalent to those with noncommutating coordinates and/or momenta but are condensed matter systems living in an ordinary commutative space. The “analogue” approach to theoretical models that are too extreme or too weak to be measured directly has already been successfully employed in high-energy physics, and in particular in gravity with the so called analogue gravity [@Barcelo:2005fc]. This program is in constant development and already lead to important successes such as the measures of the analogue Hawking radiation [@Weinfurtner:2010nu; @Belgiorno:2010wn; @Steinhauer:2014dra; @Steinhauer:2015saa], phenomenon that seemed to be only a theoretical model. It is interesting and challenging to find an analogue system that would allow the experimental measure of the re-entrant phase transition described in this paper, and we leave it for future investigation.
Acknowledgements {#acknowledgements .unnumbered}
================
A.M.F. would like to thank the Stiftung Giersch for the generous support during this work. D.M. acknowledges L. Malagò for discussions and support.
[^1]: The classical multi-oscillator states partition function is simply $ (Z_L)^n / n! $ for $n$ electrons [@Kubo1990].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $p$ be a prime and let $x$ be a $p$-adic integer. We provide two supercongruences for truncated series of the form $$\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot
\frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad
\sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot
\frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}.$$
address: 'Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica,00133 Roma, Italy'
author:
- Roberto Tauraso
title: Two supercongruences related to multiple harmonic sums
---
Introduction and main results
=============================
In [@Ta:10 Theorem 1.1] and [@Ta:12 Theorem 7] we showed that for any prime $p\not=2$, $$\sum_{k=1}^{p-1} \frac{\binom{2k}{k}}{k 4^k}
\equiv_{p^3} -H_{(p-1)/2}\quad\text{and}\quad
\sum_{k=1}^{p-1} \frac{\binom{2k}{k}^2}{k 16^k}
\equiv_{p^3} -2H_{(p-1)/2}$$ where $H_n^{(t)}=\sum_{j=1}^n\frac{1}{j^t}$ is the $n$-th harmonic number of order $t\geq 1$. Here we present two extensions of such congruences which involves the (non-strict) multiple harmonic sums $$S_n(t_1,\dots,t_r):=\sum_{1\le j_1\le\cdots\le j_r\le n}\frac{1}{j_1^{t_1}\cdots j_r^{t_r}}$$ with $t_1,t_2,\dots, t_r$ positive integers. For the sake of brevity, if $t_1 = t_2 = \dots = t_r = t$ we write $S_n(\{t\}^r)$.
Let $(x)_n:=x(x+1)\cdots (x+n-1)$ be the Pochhammer symbol, and let $B_n(x)$ be the $n$-th Bernoulli polynomial. For any prime $p$, $\mathbb{Z}_p$ denotes the ring of all $p$-adic integers and $\langle \cdot\rangle_p$ is the least non-negative residue modulo $p$ of the $p$-integral argument.
\[MT\] Let $p$ be a prime, $x\in\mathbb{Z}_p$ and $r\in\mathbb{N}$. Let $s:=(x+\langle-x\rangle_p)/p$.
i\) If $p>r+3$ then $$\begin{aligned}
\label{SI1}
\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot
\frac{S_k(\{1\}^r)}{k}
&\equiv_{p^2}
-H^{(r+1)}_{\langle-x\rangle_p}-(-1)^rsp B_{p-r-2}(x).\end{aligned}$$
ii\) If $p>2r+3$ then $$\begin{aligned}
\label{SI2}
\sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot
\frac{S_k(\{2\}^r)}{k}
&\equiv_{p^3}
-2H^{(2r+1)}_{\langle-x\rangle_p}-2(2r+1)sp H^{(2r+2)}_{\langle-x\rangle_p}
\nonumber \\&\qquad\qquad
+\frac{2s(1+3sr+2sr^2)}{2r+3}\,p^2 B_{p-2r-3}(x).\end{aligned}$$
Note that, when $r=0$, both and have been established by Zhi-Hong Sun in [@Sunzh:15]. Moreover, for the special value $x=1/2$, and yield $$\label{CI1}
\sum_{k=1}^{p-1} \frac{\binom{2k}{k}}{k 4^k}\cdot S_k(\{1\}^r)
\equiv_{p^2}
\begin{cases}
-H_{(p-1)/2}^{(r+1)}
&\text{if $r\equiv_2 0$},\vspace{3mm}\\
\frac{2^{r+2}-1}{2(r+2)}\,p B_{p-r-2}
&\text{if $r\equiv_2 1$,}
\end{cases}$$ and $$\label{CI2}
\sum_{k=1}^{p-1} \frac{\binom{2k}{k}^2}{k 16^k}\cdot
S_k(\{2\}^r)
\equiv_{p^3}
-2H_{(p-1)/2}^{(2r+1)}
-\frac{r(2^{2r+3}-1)}{2}\,p^2B_{p-2r-3}.$$ For $r=1$, the congruence proves the conjecture [@Sunzw:15 Conjecture 5.3].
In the last section we provide $q$-analogs of two binomial identities related to the congruences and .
Proof of in Theorem \[MT\]
==========================
By taking the partial fraction expansion of the rational function $$x\to \frac{(x)_k}{(x)_n}$$ with $0\leq k<n$, we find $$\label{PDF1}
\sum_{k=0}^{n-1} \frac{(x)_k}{(1)_k}\cdot a_k=
(x)_n\sum_{j=0}^{n-1}\frac{(-1)^j T_j}{j!(n-1-j)!}\cdot \frac{1}{x+j}$$ where $T_j$ is the binomial transform of the sequence $a_k$, $$T_j:=\sum_{k=0}^j(-1)^k\binom{j}{k}\cdot a_k.$$ It is easy to see from that if $a_0,\dots,a_{p-1},x\in \mathbb{Z}_p$ then $$\label{A1}
\sum_{k=0}^{p-1} \frac{(x)_k}{(1)_k}\cdot a_k\equiv_{p} T_{{\langle-x\rangle_p}}.$$In order to show we introduce the function $$G_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k}{(1)_k}\cdot S_k(\{1\}^r).$$ We have that $$G_n^{(0)}(x)=\frac{(1+x)_{n}}{(1)_{n}}-1$$ and $S_{k}(\{1\}^r)= S_{k-1}(\{1\}^r)+S_{k}(\{1\}^{r-1})/k$ implies $$\label{G1}
G_n^{(r)}(x)=\frac{(1+x)_{n}}{(1)_{n}}\cdot S_{n}(\{1\}^r)-\frac{G_n^{(r-1)}(x)}{x}.$$ Moreover $$F_n^{(r)}(x+1)-F_n^{(r)}(x)=\frac{G_n^{(r)}(j)}{x}$$ where $$F_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k}{(1)_k}\cdot
\frac{S_k(\{1\}^r)}{k}.$$ Then, for any positive integer $m$, $$\label{F1}
F_n^{(r)}(x+m)-F_n^{(r)}(x)=
\sum_{j=0}^{m-1}\frac{G_n^{(r)}(x+j)}{x+j}.$$ By , for $u=1,\dots,n$ $$G_n^{(r)}(-u)=\frac{G_n^{(r-1)}(-u)}{u}=\cdots=\frac{G_n^{(0)}(-u)}{u^r}=-\frac{1}{u^r}.$$ Hence by letting $x=-n$ and $m=n$ in we obtain the known identity (see [@He:99]) $$\label{He}
\sum_{k=1}^n (-1)^k\binom{n}{k}\frac{S_k(\{1\}^r)}{k}=-H_n^{(r+1)}.$$ Thus, for $a_k=\frac{S_k(\{1\}^r)}{k}$, we have that $T_j=-H_j^{(r+1)}$, and by , we already have the modulo $p$ version of .
Since $sp=x+\langle-x\rangle_p$ it follows that $$G_{p-1}^{(0)}(x) =\frac{(1+x)_{p-1}}{(1)_{p-1}}-1
\equiv_{p^2} \frac{sp}{x}-1.$$ By [@Zh:07 Theorem 1.6], $S_{p-1}(\{1\}^r)\equiv_p 0$ and therefore $$G_{p-1}^{(r)}(x)\equiv_{p^2}
-\frac{G_{p-1}^{(r-1)}(x)}{x}\equiv_{p^2}\cdots
\equiv_{p^2} (-1)^r\frac{G_{p-1}^{(0)}(x)}{x^{r}}
\equiv_{p^2} \frac{(-1)^r sp}{x^{r+1}}-\frac{(-1)^r}{x^{r}}.$$ Moreover $$\begin{aligned}
F_{p-1}^{(r)}(sp)&=
\sum_{k=1}^{p-1} \frac{(sp)_k}{(1)_k}\cdot
\frac{S_k(\{1\}^r)}{k}\equiv_{p^2}
\sum_{k=1}^{p-1} \frac{sp}{k}\cdot
\frac{S_k(\{1\}^r)}{k}\\
&=spS_{p-1}(\{1\}^{r},2))
\equiv_{p^2}
spB_{p-r-2}\end{aligned}$$ where we used $S_{p-1}(\{1\}^r,2)\equiv_p B_{p-r-2}$ (see [@HHT:14 Theorem 4.5]).
Finally, by , $$\begin{aligned}
F_{p-1}^{(r)}(x)&\equiv_{p^2}\sum_{j=0}^{\langle-x\rangle_p-1}
\left(\frac{(-1)^r}{(x+j)^{r+1}}-\frac{(-1)^rsp}{(x+j)^{r+2}}\right)+spB_{p-r-2}\\
&\equiv_{p^2}-\sum_{j=1}^{\langle-x\rangle_p}
\frac{1}{(j-sp)^{r+1}}-sp\sum_{j=1}^{\langle-x\rangle_p}\frac{1}{j^{r+2}}+spB_{p-r-2}\\
&\equiv_{p^2}-H_{\langle-x\rangle_p}^{(r+1)}-
(r+2)spH_{\langle-x\rangle_p}^{(r+2)}+spB_{p-r-2}\\
&\equiv_{p^2}-H_{\langle-x\rangle_p}^{(r+1)}-(-1)^r spB_{p-r-2}\\\end{aligned}$$ where the last step uses the following congruence: for $2\leq t<p-1$ $$\label{Hp}
H_{\langle-x\rangle_p}^{(t)}
\equiv_p \sum_{j=1}^{\langle-x\rangle_p} j^{p-1-t}=
\frac{B_{p-t}(\langle-x\rangle_p+1)-B_{p-t}}{p-t}
\equiv_p
(-1)^t\frac{B_{p-t}(x)-B_{p-t}}{t}$$ which is an immediate consequence of [@Sunzh:00 Lemma 3.2].
Proof of in Theorem \[MT\]
==========================
We follow a similar strategy as outlined in the previous section. We start by considering the partial fraction decomposition of the rational function $$x\to\frac{(x)_k(1-x)_k}{(x)_n(1-x)_n}$$ with $0\leq k<n$. We have that $$\label{PDF2}
\sum_{k=0}^{n-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k=
(x)_n(1-x)_n
\sum_{j=0}^{n-1}\frac{(-1)^j A_j}{(n+j)!(n-1-j)!}\left(\frac{1}{x+j}+\frac{1}{1-x+j}\right)$$ where $$A_j:=\sum_{k=0}^j(-1)^k\binom{j}{k}\binom{j+k}{k}\cdot a_k.$$ For $n\to \infty$, if the series is convergent, the identity becomes $$\sum_{k=0}^{\infty} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k=
\frac{\sin(\pi x)}{\pi}
\sum_{j=0}^{\infty}(-1)^j A_j\left(\frac{1}{x+j}+\frac{1}{1-x+j}\right).$$ In many cases the transformed sequence $A_j$ has a *nice* formula. For example if $a_k=1/(k+z)$ then $$A_j=\frac{(1-z)_j}{(z)_{j+1}}$$ and for $x=z=1/2$ we recover this series representations the Catalan’s constant $G=\sum_{j=0}\frac{(-1)^j}{(2j+1)^2}$: $$\sum_{k=0}^{\infty} \frac{\binom{2k}{k}^2}{(2k+1)16^k}=\frac{1}{2}\sum_{k=0}^{\infty} \frac{(1/2)_k^2}{(1)_k^2(k+1/2)}=
\frac{1}{2\pi}
\sum_{j=0}^{\infty}(-1)^j \frac{4}{(1/2+j)^2}=
\frac{8G}{\pi}.$$
As regards congruences we have the following result.
Let $p$ be a prime with $a_0,\dots,a_{p-1},x\in \mathbb{Z}_p$. Then $$\label{A2}
\sum_{k=0}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k
\equiv_{p^2}
A_{\langle-x\rangle_p}+s(A_{p-1-\langle-x\rangle_p}-A_{\langle-x\rangle_p})$$ For $x=1/2$ and $p>2$ then $$\sum_{k=0}^{p-1} \frac{\binom{2k}{k}^2}{16^k}\cdot a_k
\equiv_{p^2}
A_{(p-1)/2}.$$
Rearranging in a convenient way, we have $$\sum_{k=0}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k=
\frac{(x)_p(1-x)_p}{(1)_p^2}\binom{2p-1}{p-1}^{-1}
\sum_{j=0}^{p-1}(-1)^j\binom{2p-1}{p+j} A_j\left(\frac{p}{x+j}+\frac{p}{1-x+j}\right).$$ If $0\leq k\leq j\leq p-1$ then $A_{p-1-j}\equiv_p A_{j}$ because $$\begin{aligned}
\binom{p-1-j}{k}\binom{p-1-j+k}{k}
&=\frac{(p-1-j)\cdots (p-j-k)(p-1-j+k)\cdots (p-j) }{(k!)^2}\\&\equiv_p
\frac{(j+1)\cdots (j+k)(j-k+1)\cdots j }{(k!)^2}
=\binom{j}{k}\binom{j+k}{k}.\end{aligned}$$ Thus, since $\langle -x\rangle_p+\langle-(1-x)\rangle_p=p-1$, it follows that $$\begin{aligned}
\sum_{j=0}^{p-1}(-1)^j\binom{2p-1}{p+j} \frac{pA_j}{x+j}
&\equiv_{p^2}\sum_{j=0}^{\langle-x\rangle_p-1} \frac{pA_j}{x+j}+(-1)^{\langle-x\rangle_p}\binom{2p-1}{p+{\langle-x\rangle_p}} \frac{A_{\langle-x\rangle_p}}{s}+\sum_{j=\langle-x\rangle_p+1}^{p-1} \frac{pA_j}{x+j}\\
&\equiv_{p^2}\sum_{j=0}^{\langle-x\rangle_p-1} \frac{pA_j}{x+j}+(-1)^{\langle-x\rangle_p}\binom{2p-1}{p+{\langle-x\rangle_p}} \frac{A_{\langle-x\rangle_p}}{s}
-\!\!\!\!\!\!\sum_{j=0}^{\langle-(1-x)\rangle_p-1}\!\!\!
\frac{pA_{p-1-j}}{1-x+j}.\end{aligned}$$ Therefore $$\begin{aligned}
\sum_{j=0}^{p-1}(-1)^j\binom{2p-1}{p+j} A_j\left(\frac{p}{x+j}+\frac{p}{1-x+j}\right)
&\equiv_{p^2}(-1)^{\langle-x\rangle_p}\binom{2p-1}{p+{\langle-x\rangle_p}} \left(\frac{A_{\langle-x\rangle_p}}{s} +\frac{A_{\langle-(1-x)\rangle_p}}{1-s}\right).\end{aligned}$$ Finally, by using $$\begin{aligned}
&\binom{2p-1}{p-1} \equiv_{p^3} 1,\\
&\binom{2p-1}{p+j} \equiv_{p^2} (-1)^{j}\left(1-2p H_{j}\right),\\
&\frac{(x)_p(1-x)_p}{(1)_p^2}\equiv_{p^2} s(1-s)\left(1+2p H_{\langle-x\rangle_p}\right),\end{aligned}$$ we are done. For $x=1/2$ it suffices to note that $$\langle-x\rangle_p=(p-1)/2= p-1-\langle-x\rangle_p.$$
As an application of the previous theorem, we note that when $a_k=1$ then $A_j=(-1)^j$, and, by , it follows that $$\sum_{k=0}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}
\equiv_{p^2} (-1)^{\langle -x\rangle_p}$$ which has been established in [@Sunzh:14 Corollary 2.1]. Another example worth to be mentioned is $a_k=1/k^r$ for $k\geq 1$ (and $a_0=0$). Then by [@Pr:10 Theorem 1] $$A_j=-\sum_{1\cdot k_1+3\cdot k_3+\dots= r}
\frac{2^{k_1+k_3+\dots} (H_j^{(1)})^{k_1}(H_j^{(3)})^{k_3}\cdots}{1^{k_1}3^{k_3}\cdots k_1!k_3!\cdots}.$$
Now we consider the case $a_k=S_k(\{2\}^r)/k$. Let $$G_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k(-x)_k}{(1)_k^2}\cdot S_k(\{2\}^r).$$ We have that $$G_n^{(0)}(x)=\frac{(1+x)_{n}(1-x)_{n}}{(1)_{n}^2}-1,$$ and $S_{k}(\{2\}^r)= S_{k-1}(\{2\}^r)+S_{k}(\{2\}^{r-1})/k^2$ implies $$\label{G2}
G_n^{(r)}(x)=\frac{(1+x)_{n}(1-x)_{n}}{(1)_{n}^2}\cdot S_{n}(\{2\}^r)+\frac{G_n^{(r-1)}(x)}{x^2}.$$ Moreover $$F_n^{(r)}(x+1)-F_n^{(r)}(x)=\frac{2G_n^{(r)}(x)}{x}$$ where $$F_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot
\frac{S_k(\{2\}^r)}{k}.$$ Hence $$\label{F2}
F_n^{(r)}(x+m)-F_n^{(r)}(x)=
2\sum_{j=0}^{m-1}\frac{G_n^{(r)}(x+j)}{x+j}.$$
The next identity is a variation of and it appears to be new.
For any integers $n\geq 1$ and $r\geq 0$, $$\label{Ta}
\sum_{k=1}^{n} (-1)^k\binom{n}{k}\binom{n+k}{k}\frac{S_k(\{2\}^r)}{k}=-2H_{n}^{(2r+1)}.$$
By , for $u=1,\dots,n$, $$G_n^{(r)}(-u)=\frac{G_n^{(r-1)}(-u)}{u^2}=\dots=\frac{G_n^{(0)}(-u)}{u^{2r}}=-\frac{1}{u^{2r}}.$$ Hence by letting $x=-n$ and $m=n$ in $$\begin{aligned}
\sum_{k=1}^{n} (-1)^k\binom{n}{k}\binom{n+k}{k}\frac{S_k(\{2\}^r)}{k}&=F^{(r)}_n(-n)=F_n^{(r)}(0)-2\sum_{j=0}^{n-1}\frac{G_n^{(r)}(-n+j)}{-n+j}\\
&=2\sum_{j=0}^{n-1}\frac{1}{(-n+j)^{2r+1}}=-2H_{n}^{(2r+1)}.\end{aligned}$$
Thus by applying $\eqref{A2}$ we find a modulo $p^2$ version of $\eqref{SI2}$. A more refined reasoning will lead us to the $p^3$ congruence.
Since $sp=x+\langle-x\rangle_p$, $$G_{p-1}^{(0)}(x) =\frac{(1+x)_{p-1}(1-x)_{p-1}}{(1)_{p-1}^2}-1
\equiv_{p^3} -\frac{s(1-s)p^2}{x^2}-1$$ By [@Zh:07 Theorem 1.6], $S_{p-1}(\{2\}^r)\equiv_p 0$ and therefore $$G_{p-1}^{(r)}(x)\equiv_{p^3}
\frac{G_{p-1}^{(r-1)}(x)}{x^{2}}\equiv_{p^3}\cdots
\equiv_{p^3} \frac{G_{p-1}^{(0)}(x)}{x^{2r}}
\equiv_{p^3} -\frac{s(1-s)p^2}{x^{2r+2}}-\frac{1}{x^{2r}}.$$ It follows that $$\begin{aligned}
F_{p-1}^{(r)}(sp)-F_{p-1}^{(r)}(x)
&\equiv_{p^3}
2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{G_{p-1}^{(0)}(x+j)}{(x+j)^{2r+1}}\\
&\equiv_{p^3}
-2s(1-s)p^2\sum_{j=1}^{\langle-x\rangle_p}\frac{1}{j^{2r+3}}-2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{1}{(x+j)^{2r+1}}.\end{aligned}$$ By $$\sum_{j=1}^{\langle-x\rangle_p}\frac{1}{j^{2r+3}}=H_{\langle-x\rangle_p}^{(2r+3)}
\equiv_p
-\frac{B_{p-2r-3}(x)-B_{p-2r-3}}{2r+3}.$$ Moreover $$\begin{aligned}
F_{p-1}^{(r)}(sp)&=
\sum_{k=1}^{p-1} \frac{(sp)_k(1-sp)_k}{(1)_k^2}\cdot
\frac{S_k(\{2\}^r)}{k}\\
&\equiv_{p^3}
\sum_{k=1}^{p-1} \frac{sp(k-sp)}{k^2}\cdot
\frac{S_k(\{2\}^r)}{k}\\
&=sp\sum_{k=1}^{p-1}
\frac{S_k(\{2\}^r)}{k^2}
-p^2s^2\sum_{k=1}^{p-1}
\frac{S_k(\{2\}^r)}{k^3}\\
&=spS_{p-1}(\{2\}^{r+1}))
-p^2s^2S_{p-1}(\{2\}^{r},3))\\
&\equiv_{p^3}
sp\frac{2pB_{p-2r-3}}{2r+3}
+p^2s^22rB_{p-2r-3}\\
&\equiv _{p^3}
\frac{2sp^2(1+sr(2r+3))B_{p-2r-3}}{2r+3}\end{aligned}$$ where we used $$\frac{(sp)_k(1-sp)_k}{(1)_k^2}
=\frac{sp(k-sp)}{k^2}\cdot \frac{(1+sp)_{k-1}(1-sp)_{k-1}}{ (1)_{k-1}^2}\equiv_{p^3}\frac{sp(k-sp)}{k^2}$$ and the congruences $$S_{p-1}(\{2\}^r)\equiv_{p^2} \frac{2pB_{p-2r-1}}{2r+1}\quad\text{and}\quad
S_{p-1}(\{2\}^r,3)\equiv_p -2rB_{p-2r-3}.$$ which have been established in [@Zh:07 Theorem 1.6] in [@HHT:14 Theorem 4.1] respectively. Finally, $$\begin{aligned}
F_{p}^{(r)}(x)
&\equiv_{p^3}
\frac{2sp^2(1+sr(2r+3))B_{p-2r-3}}{2r+3}
-\frac{2s(s-1)p^2(B_{p-2r-3}(x)-B_{p-2r-3})}{2r+3}\\
&\qquad\qquad+2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{1}{(x+j)^{2r+1}}\\
&\equiv_{p^3}
2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{1}{(x+j)^{2r+1}}
+\frac{2s(1-s)}{2r+3}p^2B_{p-2r-3}(x)
\\&\qquad\qquad
+\frac{2s^2(r+1)(2r+1)}{2r+3}p^2 B_{p-2r-3}\end{aligned}$$
We observe that follows by letting $x=1/2$. Then $\langle-x\rangle_p-1=(p-1)/2$, $B_{2n}(1/2)=(2^{1-2n}-1)B_{2n}$ and for $p-4>t>1$ $$H^{(t)}_{(p-1)/2}\equiv
\begin{cases}
\frac{t(2^{t+1}-1)}{2(t+1)}\,p B_{p-t-1} \pmod{p^2}
&\text{if $t\equiv_2 0$},\vspace{3mm}\\
-\frac{(2^{t}-2)}{t}\, B_{p-t} \qquad \pmod{p}
&\text{if $t\equiv_2 1$}.
\end{cases}$$ see [@Sunzh:00 Theorem 5.2].
Final remarks: $q$-analogs of and
==================================
It is interesting to note that identities and have both a $q$-version (the first one appears in [@Pr:00]).
For any integers $n\geq 1$ and $r\geq 0$, $$\label{Heq}
\sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{n}{k}_q}q^{\binom{k}{2}-(n-1)k}\cdot \frac{S_k(\{1\}^r;q)}{1-q^k}
=-\sum_{k=1}^n\frac{q^{rk}}{(1-q^k)^{r+1}}$$ and $$\label{Taq}
\sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{n}{k}_q}{\genfrac{[}{]}{0pt}{}{n+k}{k}_q} q^{\binom{k}{2}-(n-1)k}
\cdot \frac{S_k(\{2\}^r;q)}{1-q^k}
=-\sum_{k=1}^n\frac{(1+q^k)q^{rk}}{(1-q^{k})^{2r+1}}$$ where ${\genfrac{[}{]}{0pt}{}{m}{k}_q}$ is the Gaussian binomial coefficient $${\genfrac{[}{]}{0pt}{}{m}{k}_q}=\left\{
\begin{array}{ll}
\frac{(1-q^m)(1-q^{m-1})\cdots (1-q^{m-k+1})}{(1-q^k)(1-q^{k-1})\cdots (1-q)}
&\mbox{if $0\leq k\leq m$},\\[3pt]
0 &\mbox{otherwise},
\end{array}\right.$$ and $$S_n(t_1,\dots,t_r;q):=\sum_{1\le j_1\le\cdots\le j_r\le n}\frac{q^{j_1+\dots +j_r}}{(1-q^{j_1})^{t_1}\cdots (1-q^{j_r})^{t_r}}.$$
We show and we leave the proof of other one to the interested reader. The procedure is quite similar to the one given for the corresponding ordinary identity . Let $$G_n^{(r)}(u):=\sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{u}{k}_q}{\genfrac{[}{]}{0pt}{}{u+k-1}{k}_q} q^{\binom{k}{2}-(u-1)k}
\cdot S_k(\{2\}^r;q).$$ Then for $u=1,\dots,n$, $G_n^{(0)}(u)=-1$ and $$G_n^{(r)}(u)=\frac{q^u G_n^{(r-1)}}{(1-q^u)^2}=\dots=\frac{q^{ru }G_n^{(0)}(u)}{(1-q^u
)^{2r}}=-\frac{q^{ru }}{(1-q^u)^{2r}}.$$ Moreover $$F_n^{(r)}(u)-F_n^{(r)}(u-1)=\frac{(1+q^u)G_n^{(r)}(u)}{(1-q^u)}
=-\frac{(1+q^u)q^{ru }}{(1-q^u)^{2r+1}}$$ where $$F_n^{(r)}(u):=\sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{u}{k}_q}{\genfrac{[}{]}{0pt}{}{u+k}{k}_q} q^{\binom{k}{2}-(u-1)k}
\cdot S_k(\{2\}^r;q).$$ Thus, since $F_n^{(0)}(n)=0$, $$\begin{aligned}
F_n^{(r)}(n)&=\sum_{u=1}^n\frac{(1+q^u)G_n^{(r)}(u)}{(1-q^u)}+F_n^{(0)}(n)
=-\sum_{u=1}^n\frac{(1+q^u)q^{ru}}{(1-q^{u})^{2r+1}}\end{aligned}$$ and the proof is complete.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'I present the results of first principles calculations of the electronic structure and magnetic interactions for the recently discovered superconductor YFe$_2$Ge$_2$ and use them to identify the nature of superconductivity and quantum criticality in this compound. I find that the Fe $3d$ derived states near the Fermi level show a rich structure with the presence of both linearly dispersive and heavy bands. The Fermi surface exhibits nesting between hole and electron sheets that manifests as a peak in the susceptibility at $(1/2,1/2)$. I propose that the superconductivity in this compound is mediated by antiferromagnetic spin fluctuations associated with this peak resulting in a $s_\pm$ state similar to the previously discovered iron-based superconductors. I also find that various magnetic orderings are almost degenerate in energy, which indicates that the proximity to quantum criticality is due to competing magnetic interactions.'
author:
- Alaska Subedi
title: 'Unconventional sign-changing superconductivity near quantum criticality in YFe$_2$Ge$_2$'
---
Unconventional superconductivity and quantum criticality are two of the most intriguing phenomena observed in physics. The underlying mechanisms and the properties exhibited by the systems in which these two phenomena occur has not been fully elucidated because unconventional superconductors and materials at quantum critical point are so rare. The dearth of realizable examples has also held back the study of the relationship and interplay between unconventional superconductivity and quantum criticality, if there are any.
Therefore, the recent report of non-Fermi liquid behavior and superconductivity in YFe$_2$Ge$_2$ by Zou *et al.* is of great interest despite a low superconducting $T_c$ of $\sim$1.8 K.[@zou13] This material is also interesting because it shares some important features with the previously discovered iron-based high-temperature superconductors. Like the other iron-based superconductors, its structural motif is a square plane of Fe that is tetrahedrally coordinated, in this case, by Ge. This Fe$_2$Ge$_2$ layer is stacked along the $z$ axis with an alternating layer of Y ions. The resulting body-centered tetragonal structure ($I4/mmm$) of this compound is the same as that of the ‘122’ family of the iron-based superconductors. The nearest neighbor Fe–Ge and Fe–Fe distances of 2.393 and 2.801 Å, respectively, in this compound[@vent96] are similar to the Fe–As and Fe–Fe distances of 2.403 and 2.802 Å, respectively, found in BaFe$_2$As$_2$.[@rott08] This raises the possibility that the direct Fe–Fe hopping is important to the physics of this material, which is the case for the previously discovered iron-based superconductors.[@sing08a]
Furthermore, Zou *et al.* report that the superconductivity in this compound exists in the vicinity of a quantum critical point that is possibly associated with antiferromagnetic spin fluctuations.[@zou13] A related isoelectronic compound LuFe$_2$Ge$_2$ that occurs in the same crystal structure exhibits antiferromagnetic spin density wave order below 9 K,[@avil04; @fers06] and the magnetic transition is continuously suppressed in Lu$_{1-x}$Y$_x$Fe$_2$Ge$_2$ series as Y content is increased, with the quantum critical point lying near the composition Lu$_{0.81}$Y$_{0.19}$Fe$_2$Ge$_2$.[@ran11] The proximity of YFe$_2$Ge$_2$ to quantum criticality is observed in the non-Fermi liquid behavior of the specific-heat capacity and resistivity. Zou *et al.* find that the unusually high Sommerfeld coefficient with a value of $C/T \simeq 90$ mJ/mol K$^2$ at 2 K further increases to a value of $\sim$100 mJ/mol K$^2$ as the temperature is lowered, although the experimental data is not detailed enough to distinguish between a logarithmic and a square root increase. They also find that the resistivity shows a behavior $\rho \propto T^{3/2}$ up to a temperature of 10 K.
In this paper, I use the results of first principles calculations to discuss the interplay between superconductivity and quantum criticality in YFe$_2$Ge$_2$ in terms of its electronic structure and competing magnetic interactions. I find that the fermiology in this compound is dominated by Fe $3d$ states with the presence of both heavy and linearly dispersive bands near the Fermi level. The Fermi surface consists of five sheets. There is an open tetragonal electron cylinder around $X = (1/2, 1/2, 0)$. A large three dimensional closed sheet that is shaped like a shell of a clam is situated around $Z =
(0, 0, 1/2) = (1, 0, 0)$. This sheet encloses a cylindrical and two almost spherical hole sheets. The tetragonal cylinder sheet around $X$ nests with the spherical and the cylindrical sheets around $Z$, which manifests as a peak at $(1/2,1/2)$ in the bare susceptiblity. I propose that the superconductivity in this compound is mediated by antiferromagnetic spin fluctuations associated with this peak, and the resulting superconductivity has a sign-changing $s_\pm$ symmetry with opposite signs on the nested sheets around $X$ and $Z$. This superconductivity is similar to the one proposed for previously discovered iron-based superconductors.[@mazi08; @kuro08] Furthermore, I find that there are competing magnetic interactions in this compound, and the quantum criticality is due to the fluctuations associated with these magnetic interactions.
The results presented here were obtained within the local density approximation (LDA) using the general full-potential linearized augmented planewave method as implemented in the WIEN2k software package.[@wien2k] Muffin-tin radii of 2.4, 2.2, and 2.1 a.u. for Y, Fe, and Ge, respectively, were used. A $24 \times 24 \times 24$ $k$-point grid was used to perform the Brillouin zone integration in the self-consistent calculations. An equivalently sized or larger grid was used for supercell calcualtions. Some magnetic calculations were also checked with the ELK software package.[@elk] I used the experimental parameters ($a$ = 3.9617 and $c$ = 10.421 Å),[@vent96] but employed the internal coordinate for Ge ${z_{\textrm{Ge}}}$ = 0.3661 obtained via non-spin-polarized energy minimization. The calculated value for ${z_{\textrm{Ge}}}$ is different from the experimentally determined value of ${z_{\textrm{Ge}}}$ = 0.3789. The difference in the Ge height between the calculated and experimental structures is 0.13 Å, which is larger than the typical LDA error in predicting the crystal structure. Such a discrepancy is also found in the iron-based superconductors.[@sing08a] This may suggest that YFe$_2$Ge$_2$ shares some of the underlying physics with the previously discovered iron-based superconductors.
![ Top: LDA non-spin-polarized band structure of YFe$_2$Ge$_2$. Bottom: A blow-up of the band structure around Fermi level. The long $\Gamma$–$Z$ direction is from $(0,0,0)$ to $(1,0,0)$ and the short one is from $(0,0,0)$ to $(0, 0, 1/2)$. The $X$ point is $(1/2,1/2,0)$. The stacking of the Brillouin zone is such that $(1,0,0) = (0,0,1/2)$. See Fig. 1 of Ref. for a particularly illuminating illustration of the reciprocal-space structure. []{data-label="fig:yfg-bnd"}](yfg-bnd-l.ps "fig:"){width="\columnwidth"}\
![ Top: LDA non-spin-polarized band structure of YFe$_2$Ge$_2$. Bottom: A blow-up of the band structure around Fermi level. The long $\Gamma$–$Z$ direction is from $(0,0,0)$ to $(1,0,0)$ and the short one is from $(0,0,0)$ to $(0, 0, 1/2)$. The $X$ point is $(1/2,1/2,0)$. The stacking of the Brillouin zone is such that $(1,0,0) = (0,0,1/2)$. See Fig. 1 of Ref. for a particularly illuminating illustration of the reciprocal-space structure. []{data-label="fig:yfg-bnd"}](yfg-bnd-s.ps "fig:"){width="\columnwidth"}
![ (Color online) Electronic density of states non-spin-polarized of YFe$_2$Ge$_2$ and projections on to the LAPW spheres per formula unit both spin basis. []{data-label="fig:yfg-dos"}](yfg-dos.eps){width="\columnwidth"}
The non-spin-polarized LDA band structure and density of states (DOS) are shown in Figs. \[fig:yfg-bnd\] and \[fig:yfg-dos\], respectively. The lowest band that starts out from $\Gamma$ at $-$5.2 eV relative to the Fermi energy has Ge $4p_z$ character. There is only one band with Ge $4p_z$ character below Fermi level, and there is another band with this character above the Fermi level. This indicates that the Ge ions make covalent bonds along the $c$ axis, which is not surprising given the short Ge–Ge distance in that direction. The four bands between $-$1.2 and $-$4.8 eV that start out from $\Gamma$ at $-$1.5 and $-$2.6 eV have Ge $4p_x$ and $4p_y$ character. Rest of the bands below the Fermi level have mostly Fe $3d$ character. Similar to the other iron-based superconductors,[@sing08a] there is no gap-like structure among the Fe $3d$ bands splitting them into a lower lying $e_g$ and higher lying $t_{2g}$ states. This shows that Fe–Ge covalency is minimal and direct Fe–Fe interactions dominate. Almost all of the Fe $4s$ and Y $4d$ and $5s$ character lie above the Fermi level. This indicates a nominal occupation of Fe $3d^{6.5}$, although the actual occupancy will be different because there is some covalency of Fe $3d$ states with Y $4d$ and Ge $4p$ states.
The electronic states near the Fermi level come from Fe $3d$ derived bands and show a rich structure. The electronic DOS at the Fermi level is $N(E_F) = 4.50$ eV$^{-1}$ on a per formula unit both spin basis corresponding to a calculated Sommerfeld coefficient of 10.63 mJ/mol K$^2$. The Fermi level lies at the bottom of a valley with a large peak due to bands of mostly $d_{xz}$ and $d_{yz}$ characters on the left and a small peak due to a band of mostly $d_{xy}$ character on the right. (The local coordinate system of the Fe site is rotated by 45$^\circ$ in the $xy$ plane with respect to the global cartesian axes such that the Fe $d_{x^2-y^2}$ orbital points away from the Ge $p_x$ and $p_y$ orbitals.) There is a pair of linearly dispersive band with mostly $d_{xz}$ and $d_{yz}$ as well as noticeable Ge $p_z$ characters either side of $Z$. If they are not gapped in the superconducting state, they will provide the system with a massless excitation. In addition to this pair of linearly dispersive bands, there is also a very flat band near the Fermi level along $X$–$\Gamma$. This band has an electron-like nature around $X$ and crosses the Fermi level close to it. Along the $X$–$\Gamma$ direction, it reaches a maximum at 0.08 eV above the Fermi level, turns back down coming within 0.01 eV of touching the Fermi level, and again moves away from the Fermi level. It may be possible to access these band critical points that have vanishing quasiparticle velocities via small perturbations due to impurities, doping, or changes in structural parameters. The role of such band critical points in quantum criticality has been emphasized recently,[@neal11] and similar physics may be relevant in this system.
![(Color online) Top: LDA Fermi surface of YFe$_2$Ge$_2$. Bottom: The Fermi surface without the large sheet. The shading is by velocity.[]{data-label="fig:yfg-fs"}](yfg-fs1v2 "fig:"){width="0.8\columnwidth"} ![(Color online) Top: LDA Fermi surface of YFe$_2$Ge$_2$. Bottom: The Fermi surface without the large sheet. The shading is by velocity.[]{data-label="fig:yfg-fs"}](yfg-fs2v2 "fig:"){width="0.8\columnwidth"}
The Fermi surface of this compound is shown in Fig. \[fig:yfg-fs\]. There is an open very two dimensional tetragonal electron cylinder around $X$. This has mostly $d_{xz}$ and $d_{yz}$ character. There are four closed sheets around $Z$. One of them is a large three dimensional sheet with the shape like the shell of a clam with $d_{xz}$, $d_{yz}$, $d_{xy}$, and $d_{z^2}$ characters. There are two almost spherical hole sheets. These have mostly $d_{xz}$ and $d_{yz}$ characters, with the smaller one also containing noticeable Ge $p_z$ character. These two spherical sheets are enclosed by a closed cylindrical hole sheet that has mostly $d_{xy}$ character.
The cylindrical and larger spherical sheets centered around $Z$ touch at isolated points. Otherwise, the Fermi surface is comprised of disconnected sheets. If one considers the $\Gamma$–$Z$–$\Gamma$ path along the $k_z$ direction, there is a series of box-shaped cylindrical hole sheet that encloses the two spherical sheets. Although there are no sections around $\Gamma$, these sheets around $Z$ enclose almost two-third of the $\Gamma$–$Z$–$\Gamma$ path. Therefore, there is likely to be substantial nesting between the sheets around $Z$ and $X$ that will lead to a peak in the susceptibility at the wave vector $(1/2,1/2)$.
I have calculated the Lindhard susceptibility $$\chi_0(q,\omega) = \sum_{k,m,n} |M_{k,k+q}^{m,n}|^2
\frac{f(\epsilon_k^m) - f(\epsilon_{k+q}^n)}{\epsilon_k^m -
\epsilon_{k+q}^n - \omega - \imath \delta}$$ at $\omega \to 0$ and $\delta \to 0$, where $\epsilon_k^m$ is the energy of a band $m$ at wave vector $k$ and $f$ is the Fermi distribution function. $M$ is the matrix element, which is set to unity. The real part of the susceptibility is shown in Fig. \[fig:yfg-suscep\], and it shows peaks at $\Gamma$, $Z$, and $X$ with the peak at $X$ having the highest magnitude. Note, however, that the cylinders around $Z$ and $X$ have different characters, which should reduce the peak $X$ and make it broader as well. The peak at $\Gamma$ is equal to the DOS $N(E_F)$. The peak at $Z$ reflect the nesting along the flat sections of the sheets along $(0,0,1/2)$ direction, while the peak at $X$ is due to the nesting between the hole cylinder and spheres centered around $Z$ and the electron cylinder centered around $X$.
The bare Lindhard susceptibility is further enhanced due to the RPA interaction, and its real part is related to magnetism and superconductivity. It is found experimentally that pure YFe$_2$Ge$_2$ does not order magnetically down to a temperature of 2 K although it shows non-Fermi liquid behavior in the transport and heat capacity measurements that is likely due to proximity to a magnetic quantum critical point.[@zou13] As the temperature is lowered further, superconductivity manifests in the resistivity measurements at $T_c^\rho$ = 1.8 K and DC magnetization at $T_c^{\textrm{mag}}$ = 1.5 K. This superconductivity can be due spin fluctuations associated with the peak in the susceptibility.[@berk66; @fay80] The pairing interaction has the form $$V(q=k-k') = - \frac{I^2(q) \chi_0(q)}{1 - I^2(q) \chi_0^2(q)}$$ in the singlet channel and is repulsive. (In the triplet channel, the interaction is attractive and also includes an angular factor.) Here $I(q)$ is the Stoner parameter which microscopically derives from Coulomb repulsion between electrons.
![The real part of bare susceptibility calculated with the matrix element set to unity.[]{data-label="fig:yfg-suscep"}](yfg-suscep-color){width="0.6\columnwidth"}
In the present case, the structure of the calculated susceptibility leads to the off-diagonal component of the interaction matrix to have a large negative value $-\lambda$ for the pairing between the hole sheets at $Z$ and electron cylinder at $X$ in the singlet channel. The diagonal component of the interaction matrix $\lambda_d$ pairing interactions on the hole and electron sheets are small and ferromagnetic. (For simplicity, I have assumed that the density of states are same for the hole and electron sections.) The eigenvector corresponding to the largest eigenvalue of this interaction matrix has opposite signs between the hole sheets around $Z$ and electron cylinder around $X$, and this is consistent with a singlet $s_\pm$ superconductivity with a wave vector $(1/2,1/2)$. This superconductivity is similar to the previously discovered iron-based superconductors.[@mazi08; @kuro08]
The proposed superconducvity in YFe$_2$Ge$_2$ and the previously discovered iron-based superconductor is similar, but the $T_c$ = 1.8 K for YFe$_2$Ge$_2$ is much smaller than those reported for other iron-based superconductors. One reason for this may be the smaller nesting in this compound leading to a smaller peak in susceptibility. The hole cylinder around $Z$ has mostly $d_{xy}$ character whereas the hole spheres around $Z$ and the electron cylinder around $X$ have mostly $d_{xz}$ and $d_{yz}$ character. These factors should lead to a slightly smaller and broader peak at $X$. I note, however, that nesting in the other iron-based superconductors is also not perfect[@mazi08] and the band characters between the nested sheets also vary.[@kuro08]
Another reason for the smaller $T_c$ in YFe$_2$Ge$_2$ may be due to the existence of competing magnetic fluctuations associated with the proximity to quantum criticality. The DOS from non-spin-polarized calculation is $N(E_F)$ = 1.125 eV$^{-1}$ per spin per Fe, which puts this material on the verge of a ferromagnetic instability according to the Stoner criterion. Ferromagnetism is pair-breaking for the singlet pairing and will suppress the $T_c$ in this compound. Furthermore, there is a peak in the susceptibility at $Z$ as well. The presence of additional antiferromagnetic interactions might reduce the phase space available for the spin fluctuation associated with the pairing channel and may be pair-breaking as well.
Energy (meV/Fe) Moment ($\mu_B$/Fe)
----------------- ----------------- ---------------------
NSP 0 0
FM $-$6.29 0.59
AFM (0,0,1/2) $-$11.63 0.64
SDW (1/2,1/2,0) $-$6.52 0.72
: \[tab:mag\] The relative energies of various magnetic orderings and the moments within the Fe spheres. These are almost degenerate, indicating the proximity to quantum criticality is due to competing magnetic interactions.
I performed magnetic calculations with various orderings on $(1 \times
1 \times 2)$ and $(\sqrt{2} \times \sqrt{2} \times 2)$ supercells to check the strength of competing magnetic interactions. The relative energies and the Fe moments are summarized in Table \[tab:mag\]. I was able to stabilize various magnetic configurations, and their energies are close to that of the non-spin-polarized configuration. However, I was not able to stabilize the checkerboard antiferromagnetic order in the $ab$ plane. When the magnetic order is stabilized, the magnitude of the Fe moment is less than 1 $\mu_{B}$, and the magnitudes vary between different orderings. This indicates that the magnetism is of itinerant nature. It is worthwhile to note that LDA calculations overestimate the magnetism in this compound as it does not exhibit any magnetic order experimentally. This disagreement between LDA and experiment is different from that for the Mott insulating compounds where LDA in general underestimates the magnetism.
Although this compound does not magnetically order experimentally, it nonetheless shows proximity to magnetism. It is found that partial substitution of Y by isovalent Lu causes the system to order antiferromagnetically, with 81% Lu substitution being the critical composition.[@ran11] At substitution values below the critical composition, the system shows non-Fermi liquid behavior in the heat capacity and transport measurements.[@zou13] The unusually high Sommerfeld coefficient of $\sim$90 mJ/mol K$^2$ at 2 K further increases as the temperature is lowered and the resistivity varies as $\rho \propto T^{3/2}$ up to around 10 K. This non-Fermi liquid behavior and the large renormalization of the magnetic moments may happen because there is a large phase for competing magnetic tendencies in this compound. This is due to the fluctuation-dissipation theorem, which relates the fluctuation of the moment to the energy and momentum integrated imaginary part of the susceptibility.[@mori85; @solo95; @ishi98; @agua04; @lars04] If the quantum criticality is due to competing magnetic interactions, the inelastic neutron scattering experiments, which measures the imaginary part of the susceptibility, would exhibit the structure related to the competing interactions. Therefore, even though this compound does not show magnetic ordering, it would be useful to perform such experiments and compare with the results presented here.
In any case, I indeed find that various magnetic orderings and the non-spin-polarized configuration are close in energy (see Table \[tab:mag\]). The energy of the lowest magnetic configuration is only 11.6 meV/Fe lower than the non-spin-polarized one, and the energies of the different magnetic orderings are within 6 meV/Fe of each other. As a comparison, the difference in energy between the non-magnetic configuration and the most stable magnetic ordering in BaFe$_2$As$_2$ is 92 meV/Fe, and the energy of the magnetic ordering closest to the most stable one is higher by 51 meV.[@sing08b] Signatures of quantum criticality has been reported for BaFe$_2$As$_2$ and related compounds.[@ning09; @jian09; @kasa10] YFe$_2$Ge$_2$ should show pronounced effects of proximity to quantum criticality as the competition between magnetic interactions is even stronger.
In summary, I have discussed the superconductivity and quantum criticality in YFe$_2$Ge$_2$ in terms of its electronic structure and competing magnetic interactions. The electronic states near the Fermi level are derived from Fe $3d$ bands and show a rich structure with the presence of both linearly dispersive and heavy bands. The Fermi surface consists of five sheets. There is an open rectangular electron cylinder around $X$. A big sheet shaped like the shell of a clam encloses a hole cylinder and two hole spheres around $Z$. There is a peak in the bare susceptibility at $(1/2,1/2)$ due to nesting between the hole sheets around $Z$ and the electron cylinder around $X$. I propose that the superconductivity in YFe$_2$Ge$_2$ is due to antiferromagnetic spin fluctuations associated with this peak. The resulting superconducting state has a $s_\pm$ state similar to that of previously discovered iron-based superconductors. I also find that different magnetic configurations are close in energy, which suggests the presence of competing magnetic interactions that are responsible for the proximity to quantum criticality observed in this compound.
I am grateful to Antoine Georges for helpful comments and suggestions. This work was partially supported by a grant from Agence Nationale de la Recherche (PNICTIDES).
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We construct a variational wave function to study whether a fully polarized Fermi sea of ultracold atoms is energetically stable against a single spin flip. Our variational wavefunction contains short-range correlations at least to the same level as Gutzwiller’s projected wavefunction. For the Hubbard lattice model and the continuum model with pure repulsive interaction, we show a fully polarized Fermi sea is generally unstable even for infinite repulsive strength. By contrast for a resonance model, the ferromagnetic state is possible if the $s$-wave scattering length is positive and sufficiently large and the system is prepared to be orthogonal to molecular bound state. However, we can not rule out the possibility that more exotic correlation can destabilize the ferromagnetic state.'
author:
- 'Xiaoling Cui$^{1}$ and Hui Zhai$^{2}$'
title: Stability of a Fully Magnetized Ferromagnetic state in Repulsively Interacting Ultracold Fermi Gases
---
Whether a fermion system with repulsive interaction will become ferromagnetic is a long-standing problem in condensed matter physics. Early in 1930s’, Stoner used a simple mean-field theory to predict that ferromagnetism will always take place with sufficient large repulsive interaction [@Stoner]. However, this conclusion is later challenged by Gutzwiller who took the short-range correlation into account [@Gutzwiller]. So far, except a few specific cases [@Nagaoka; @specific], there is no conclusive result on itinerant ferromagnetism. Recently, MIT group reports an experiment on itinerant fermions in an ultracold Fermi gas with large positive scattering length close to a Feshbach resonance [@Ketterle], and they attribute their observations to Stoner ferromagnetism by comparing to theories [@theory; @MacDonald]. However, these theories are basically mean-field theory or a second-order perturbation, which neither include the Gutzwiller type short-range correlation nor consider the unitary limited interaction nearby Feshbach resonance. Moreover many of the experimental signatures can be reproduced qualitatively by a non-magnetic correlated state [@Zhai]. Thus, it calls for a serious study including the effects of both correlation and unitarity in this problem.
In this Rapid Communication we address the question whether a fully magnetized state is stable against a single spin flip. We compare the energy of $N+1$ spin-up particles with that of one spin-down particle and $N$ spin-up particles. A fully magnetized ferromagnetic state is definitely unstable if we can find a variational state of the latter whose energy is lower. Similar idea has been used previously in studying the stability of Nagaoka ferromagnetism in the Hubbard model [@Anderson; @Basile; @Roth; @Nagaoka], and attractively interacting Fermi gases with large population imbalance [@Chevy; @imbalance]. In this work, we will explore different realizations of “repulsive interactions" in ultracold Fermi gases:
([**I**]{}) Single-band Hubbard model in a two-dimensional (2D) square or three-dimensional (3D) cubic lattice. The Hamiltonian $\hat{H}=\hat{H}_{\text{t}}+\hat{H}_{\text{int}}$, $\hat{H}_{\text{t}}=-t\sum_{\langle
ij\rangle,\sigma}c^\dag_{i\sigma} c_{j\sigma}+\text{h.c}.$, where $\langle
ij\rangle$ are nearest-neighbor sites, and $H_{\text{int}}=U\sum_{i}n_{i\uparrow}n_{i\downarrow}$ ($U>0$).
([**II**]{}) Continuum model with finite-range interaction potential in three dimension. $\hat{H}=\sum_{{\bf k}\sigma}\epsilon_{{\bf
k}}c^\dag_{{\bf k}\sigma}c_{{\bf k}\sigma}+(1/\Omega)\sum_{{\bf
q,k,k^\prime}}V({\bf k-k^\prime})c^\dag_{{\bf
q+k}\uparrow}c^\dag_{{\bf q-k}\downarrow}c_{{\bf
q-k^\prime}\downarrow}c_{{\bf q+k^\prime}\uparrow}$. Here $\epsilon_{{\bf k}}= {\bf k}^2/(2m)$, $\Omega$ is system volume, and $V({\bf k})$ is the Fourier transformation of real space interaction potential $V({\bf r})$. Let $r_0$ be the interaction range, and thus $V({\bf k})\rightarrow 0$ for $|{\bf k}|\gg k_0=1/r_0$. For the convenience of later calculations, we adopt $s$-wave separable potential $V({\bf k-k^\prime})=Uw({\bf k})w({\bf k^\prime})$, and approximate $w({\bf k})=1/\sqrt{1+e^{\alpha(|{\bf k}|-k_0)/k_0}}$ with $\alpha\gg 1$. The $s$-wave scattering length $a_{\text{s}}$ is related to $U$ as $m/(4\pi a_{\text{s}})=1/U+(1/\Omega)\sum_{|{\bf
k}|=0}^{k_c}1/(2\epsilon_{{\bf k}})$, where $k_c=k_0 \ln
(1+e^\alpha)/\alpha$. For a repulsive interaction $U>0$, $a_s$ is positive but upper bounded by $\pi/(2k_c)$ at $U\rightarrow +\infty$, and no bound state exists; for an attractive $U<0$, $a_{\text{s}}$ diverges at $U_c=-2\pi^2/(mk_c)$, and only a sufficient attraction $U<U_c$ with $a_{\text{s}}>0$ can support a two-body bound state.
([**III**]{}) Continuum model with zero-range interaction potential in three dimension. $\hat{H}=\sum_{{\bf k}}\epsilon_{{\bf k}}c^\dag_{{\bf
k}\sigma}c_{{\bf k}\sigma}+g\sum_{{\bf q,k,k^\prime}}c^\dag_{{\bf
q+k}\uparrow}c^\dag_{{\bf q-k}\downarrow}c_{{\bf
q-k^\prime}\downarrow}c_{{\bf q+k^\prime}\uparrow}$, and $g$ is related to $a_{\text{s}}$ by $m/(4\pi
a_{\text{s}})=1/g+(1/\Omega)\sum_{|{\bf
k}|=0}^{\infty}1/(2\epsilon_{{\bf k}})$. There is always a two-body bound state when $a_{\text{s}}>0$.
The single-band Hubbard model, as a simplified model for cold atoms in optical lattices (valid when interaction smaller than band gap) and many correlated materials, has been extensively studied before[@Anderson; @Basile; @Roth; @Nagaoka]. The comparison to previous known results justifies the validity of our approach and calibrates the correlation incorporated in our variational wavefunction (w.f.). Then we apply our method to continuum models which are commonly-used for quantum gases and are of our primary interests.
The variational w.f. we used for one down-spin system is similar to that used in the discussion of imbalanced Fermi gases [@Chevy], which is $$|\Psi\rangle=\left(\phi_0 c^\dag_{{\bf
q_0}\downarrow}+\sum\limits_{{\bf k>k_{\text{F}}},{\bf
q<k_{\text{F}}}}\phi_{{\bf k}{\bf q}}c^\dag_{{\bf q_0+
q-k}\uparrow}c^\dag_{{\bf k}\uparrow}c_{{\bf
q}\uparrow}+\dots\right)|N\rangle\label{wf}$$ where $|N\rangle$ represents a Fermi sea of $N$-spin up particles with Fermi momentum $k_{\text{F}}$. “$\dots$" in (\[wf\]) represents terms contains more than one particle-hole pairs of spin-up particles. We compute the energy $\mathcal{E}$ (measured from energy of $|N\rangle$) of $|\Psi\rangle$, and compare it with $E_{\text{F}}$ ($=E_{|N+1\rangle}-E_{|N\rangle}$). Our main results are summarized as follows.
First, for the single-band Hubbard model, we show that under certain conditions, $H_{\text{int}}|\Psi\rangle=0$ and $\mathcal{E}=\langle
\Psi|H_{\text{t}}|\Psi\rangle<E_{\text{F}}$ for most range of particle filling, except for nearby half-filling where the ground state is rigorously proved to be ferromagnetic by Nakaoka [@Nagaoka]. (See Fig. 1). This result agrees with previous studies by various other methods [@Anderson; @Basile; @Roth]; We show that the real space representation of $|\Psi\rangle$ corresponds to Gutzwiller’s w.f. with optimized “backflow" type corrections, which implies $|\Psi\rangle$ includes short-range correlations at least as the Gutzwiller projection. Similarly for the finite-range continuum model, we find $\mathcal{E}<E_{\text{F}}$ for all range of $U>0$. (See Fig. 2). This shows that generally in a purely repulsive interaction model, a fully magnetized ferromagnetic state can not be ground state even for infinite $U$, which is in sharp contrast to Stoner’s mean-field conclusion.
Second, for the zero-range continuum model, apart from the polaron state with negative energy discussed before in Ref. [@Chevy; @imbalance; @MC; @polaron], we find the w.f. orthogonal to the polaron state has a positive $\mathcal{E}$ for $a_{\text{s}}>0$. At small $k_{\text{F}}a_{\text{s}}$, $\mathcal{E}$ as a function of $k_{\text{F}}a_{\text{s}}$ follows the prediction of perturbation expansion very well. At large $k_{\text{F}}a_{\text{s}}$, $\mathcal{E}$ saturates (due to unitarity) to $1.82E_{\text{F}}$. Below $k_{\text{F}}a^{\text{c}}_{\text{s}}=2.35$, we can find variational state with which $\mathcal{E}<E_{\text{F}}$, but fail to find such cases otherwise. (See Fig. 3). This result contradicts to the previous prediction $k_{\text{F}}a^{\text{c}}_{\text{s}}=1.40$ based on perturbation theory [@MacDonald; @notation]. Thus the necessary condition for ferromagnetism is a large enough $a_{\text{s}}\gtrsim 2.35/k_{\text{F}}$, which requires a sufficiently attractive interaction potential to support a bound state and cause resonant scattering, and more importantly the system has to be prepared in the metastable scattering state that is orthogonal to molecule state. Nevertheless, our method can not prove this is a sufficient condition. Similar result is found for the finite-range model with $U<0$ and nearby scattering resonance. The details of our calculation are explained below.
[*Single-band Hubbard model:*]{} Here we can first prove
Theorem 1: Under two conditions that (i) $\phi_{{\bf k
q}}\equiv\phi_{\bf k}$ independent of ${\bf q}$, and (ii) $\phi_0=-\sum_{{\bf k}>{\bf k_F}} \phi_{\bf k}$, the w.f. $$|\psi\rangle_{1}=\left(\phi_0 c^\dag_{{\bf
q_0}\downarrow}+\sum_{{\bf k}>{\bf k_F}, {\bf q}<{\bf
k_F}}\phi_{{\bf k q}}c^\dag_{{\bf q_0}+{\bf q}-{\bf
k}\downarrow}c^\dag_{{\bf k}\uparrow}u_{{\bf q}\uparrow} \right)
|N\rangle,\label{general_psi}$$ is an exact eigenstate of $\hat{H}_{\text{int}}$, and $\hat{H}_{\text{int}}|\psi\rangle_{1}=0$.
This theorem can be verified straightforwardly. The subscript $_1$ of $|\psi\rangle$ means it contains one particle-hole pairs of up-spins, and the best variational energy for this state is denoted by $\mathcal{E}^{(1)}$. Condition (i) ensures that $\hat{H}_{\text{int}}$ acting onto $|\psi\rangle_{1}$ will not generate two particle-hole term, and condition (ii) ensures zero interaction energy. For an intuitive understanding of this zero interaction energy, we can Fourier transfer it to real space. Using condition (ii), one can show the w.f. (\[general\_psi\]) is equivalent to $$\begin{aligned}
\left(\frac{\phi_0}{\sqrt{N_s}}\sum\limits_{{\bf m}}c^\dag_{{\bf
m}\downarrow}\mathcal{P}_{{\bf m}} +\sum\limits_{{\bf n}\neq {\bf
m}}\phi_{{\bf mn}}c^\dag_{{\bf m}\downarrow}c^\dag_{{\bf n}\uparrow}
c_{{\bf m}\uparrow}\right)e^{i{\bf q_0}{\bf m}}|N\rangle\nonumber.\end{aligned}$$ Here $\mathcal{P}_{{\bf m}}=1-c^\dag_{{\bf m}\uparrow}c_{{\bf
m}\uparrow}$ presents standard Gutzwiller projection operator. $N_{\text{s}}$ is the number of lattice sites. $\phi_{{\bf mn}}=\sum_{{\bf k}>{\bf k_F}}\phi_{{\bf k}}e^{i{\bf
k}{\bf (m-n)}}$, and the second term presents “backflow" type corrections. It becomes obvious that there will be no double occupancy and no interaction energy. Hence, we have established a momentum space w.f. representation of short-range correlation, which can be generalized to free space straightforwardly.
![$\delta E=\mathcal{E}^{(1)}-E_{\text{F}}$(in units of $t$) as a function of particle density $\rho=N/N_{\text{s}}$ in a 3D cubic (solid) and a 2D square (dashed line) lattice. \[Hubbard\]](Hubbard.eps){height="3.5cm" width="5.5cm"}
Next we try to find the minimum total(=kinetic) energy by introducing a Lagrange multiplier that can be proved as just the energy $\mathcal{E}^{(1)}$ (counted from the band bottom), $\mathcal{F}=\langle\psi|\hat{H}_t|\psi\rangle-\mathcal{E}^{(1)}
\langle\psi|\psi\rangle$. Minimization of $\mathcal{F}$ gives ${\bf
q_0}=0$ and the self-consistent equation $$\sum\limits_{{\bf
k>k_{\text{F}}}}\frac{\mathcal{E}^{(1)}}{\sum_{{\bf q<k_{\text{F}}}}
(\epsilon_{{\bf k q}}-\mathcal{E}^{(1)})}=1,$$ in which $\epsilon_{{\bf k q}}=\epsilon_{{\bf q-k}}+\epsilon_{{\bf
k}}-\epsilon_{{\bf q}}$. Its solution gives $\mathcal{E}^{(1)}$.
$\delta E=\mathcal{E}^{(1)}-E_{\text{F}}$ as a function of particle density $\rho$ in a cubic and a square lattice is plotted in Fig. \[Hubbard\]. We find for $\rho<\rho_c$, $\delta E<0$ means the ferromagnetic state is always unstable even for infinite $U$. In fact, for a cubic lattice, when $\rho\rightarrow 0$, $\mathcal{E}^{(1)}$ goes like $\rho$ while $E_{\text{F}}$ goes like $\rho^{2/3}$, hence $\delta E$ is negative and shows a much rapid decrease compared with the square lattice. While for $\rho>\rho_c$, $\delta E>0$, this is consistent with Nagaoka theorem, which forces $\delta E$ to be positive when $\rho\rightarrow 1$. $\rho_c$ obtained as $0.59\ (0.76)$ for the square (cubic) lattice agree with previous studies by a finite-size real space evaluation[@Basile] or by using Green functions[@Roth]. By contrast, our variational w.f. (Eq. \[general\_psi\]) greatly simplifies the calculation, and also enable calculations in the thermodynamic limit simply by employing density of state in a one-dimensional integral equation. Moreover, this w.f. can be systematically improved by including multiple particle-hole contributions.
Theorem 2: Consider the wave function $|\psi\rangle_{n}$ that contains up to $n$ particle-hole pairs $$\begin{aligned}
\sum\limits_{m=0}^{n}\frac{1}{(m!)^2}\sum_{\{ {\bf k_i} \}_{1}^{m};
\{ {\bf q_j}\}_{1}^{m}}\phi^{(m)}_{\{ {\bf k_i}\}_{1}^{m};\{ {\bf
q_j}\}_{1}^{m}}c^\dag_{{\bf
p_m}\downarrow}\prod_{i=1}^{m}c^\dag_{{\bf
k_i}\uparrow}\prod\limits_{j=1}^{m}c_{{\bf q_j}\uparrow}
|N\rangle\nonumber\end{aligned}$$ with ${\bf p_m}={\bf q_0}+\sum_{i=1}^m({\bf q_i}-{\bf k_i})$, where $\{ {\bf k_i} \}_{1}^{m}$ denotes a set $\{ {\bf k_1},\dots,{\bf
k_m}\}$ and $\{ {\bf q_j} \}_{1}^{m}$ denotes $\{ {\bf
q_1},\dots,{\bf q_m}\}$. It satisfies $H_{\text{int}}|\psi\rangle_{n}=0$ under the condition (i) $\phi^{(m)}_{\{ {\bf k_i}\}_{1}^{m};\{ {\bf q_j} \}_{1}^{m}}$ can be expressed as $B^{(m)}_{\{ {\bf k_i} \}_{1}^{m};{\bf
q_{m-1}},\dots,{\bf q_1}}-B^{(m)}_{\{ {\bf k_i} \}_{1}^{m};{\bf
q_{m}},{\bf q_{m-2}},\dots,{\bf q_1}}+\dots+(-1)^{m-1}B^{(m)}_{\{
{\bf k_i} \}_{1}^{m};{\bf q_{m}},{\bf q_{m-1}},\dots,{\bf q_2}}
+C^{(m)}_{\{ {\bf k_i} \}_{1}^{m};\{ {\bf q_j}\}_{1}^{m}}, $ with $B^{(0)}=0$, $C^{(0)}=\phi^{(0)}$; $C^{(n)}_{\{ {\bf
k_i}\}_{1}^{m};\{ {\bf q_j}\}_{1}^{m}}=0$; and (ii) $C^{(m-1)}_{\{
{\bf k_i} \}_{1}^{m-1};\{ {\bf q_j} \}_1^{m-1}}+\sum_{{\bf
k_{m}}}B^{(m)}_{\{{\bf k_i} \}_{1}^{m-1},{\bf k_m}; \{ {\bf
q_j}\}_{1}^{m-1}}=0$ for any $\{ {\bf k_i} \}_{1}^{m-1}$ and $\{
{\bf q_j}\}_{1}^{m-1}$.
Similar to Theorem 1, this theorem can be verified straightforwardly, and condition (i) ensures $H_{\text{int}}$ acting on $|\psi\rangle_n$ will not generate $n+1$ particle-hole terms, and condition (ii) ensures zero-energy. Obviously, $|\psi\rangle_{n-1}$ is a special case of $|\psi\rangle_n$, and hence $\mathcal{E}^{(n)}\leq \mathcal{E}^{(n-1)}$, where $\mathcal{E}^{(n)}$ is the best variational energy for $|\psi\rangle_n$. By including more particle-hole terms, one can always further lower $\mathcal{E}$, and make $\rho_c$ more close to unity. On the other hand, one can also prove $\lim_{\rho\rightarrow
0}(\mathcal{E}^{(n)}-\mathcal{E}^{(n-1)})/\mathcal{E}^{(1)}
\rightarrow 0$, namely, the multiple particle-hole contribution vanishes and $\mathcal{E}^{(1)}$ becomes exact at low density limit.
[*Finite-range continuum model:*]{} We consider the same variational w.f. as Eq. \[general\_psi\] with $\phi_{{\bf
k}{\bf q}}\equiv\phi_{\bf k}$. The saddle point equation from energy minimization is $$\begin{aligned}
\frac{U}{\Omega}\sum_{\bf q}\left( \phi_0 +\sum_{{\bf k}}w_{\bf k}\phi_{{\bf k}}\right)&=&\mathcal{E}\phi_0,\nonumber\\
\frac{U}{\Omega}w_{{\bf k}}\sum_{{\bf
q}}\left(\phi_0+\sum\limits_{{\bf k^\prime}}w_{{\bf
k^\prime}}\phi_{{\bf k^\prime}}\right)&=&\sum_{{\bf
q}}(\mathcal{E}-\epsilon_{{\bf k q}}) \phi_{{\bf k}},\label{sadd}\end{aligned}$$ where we have adopted the approximation $w_{{\bf q}}\approx 1$ for $|{\bf q}|<|{\bf k_{\text{F}}}|<k_0$. From Eqs. \[sadd\] one can obtain a self-consistent equation $$\frac{2E_{\text{F}}}{3\mathcal{E}}-\frac{1}{U_0}=
\int_{1}^{+\infty}\frac{\tilde{k}^2 w^2(\tilde{k})
d\tilde{k}}{\tilde{k}^2-\mathcal{E}/(2E_{\text{F}})},\label{finite}$$ where $\tilde{k}=k/k_{\text{F}}$ and $U_0=U mk_{\text{F}}/(2\pi^2)$ are dimensionless. The solution to Eq. \[finite\] is plotted in Fig. \[finiterange\](a). Fig. \[finiterange\](b) shows the results are insensitive to the choice of the free parameter $\alpha$ in the separable potential. As one can see, for $U_0\rightarrow
+\infty$, $\mathcal{E}$ saturates to a finite value, which is smaller than $E_{\text{F}}$. This saturation precisely reflects the physics that the repulsive interaction can be strongly renormalized by short-range correlations, whose upper bound is the kinetic energy cost for the screening, and is always finite even when the bare interaction diverges. In fact, for $k_0\gg k_{\text{F}}$, the integral at r.h.s. of Eq. \[finite\] is approximately $k_0/k_{\text{F}}$, hence at $U\rightarrow +\infty$, $\mathcal{E}/E_{\text{F}}= 2k_{\text{F}}/(3k_0)<1$. For a general case, we show in Fig. \[finiterange\](c) $\mathcal{E}/E_{\text{F}}$ at $U_0=+\infty$ as a function of $k_0/k_{\text{F}}$ at different $\alpha$, and this ratio is always below unity.
![(color online). $\mathcal{E}/E_{\text{F}}$ for the continuum model with finite-range interactions. (a) $\mathcal{E}$ as a function of $U_0$ for $k_{\text{0}}/k_{\text{F}}=2$ or $5$, $\alpha=10$. Arrows denote the corresponding critical $U_c$ that can afford a two-body bound state. (b) the same plot as (a) for $k_{\text{0}}/k_{\text{F}}=5$ and $\alpha=2$ or $10$. (c) $\mathcal{E}/E_{\text{F}}$ at $U_0\rightarrow +\infty$ as a functions $k_0$ for $\alpha=2,3,10$. \[finiterange\]](finiterange.eps){height="4.5cm" width="8.2cm"}
[*Zero-range continuum model:*]{} This model is in fact the $k_0\rightarrow
+\infty$ limit of the finite-range model with $U<0$. For the w.f. of Eq. \[general\_psi\], $H|\psi\rangle=\mathcal{E}|\psi\rangle$ gives the same self-consistent equation as in Ref. [@Chevy], $$\frac{\mathcal{E}}{E_{\text{F}}}=2\int_0^1d\tilde{q}\tilde{q}^2\left[\frac{\pi}{2}\frac{1}{k_{\text{F}}a_{\text{s}}}-1+g\left(\frac{\mathcal{E}}{E_{\text{F}}},\tilde{q}\right)\right]^{-1}$$ where $g(\mathcal{E}/E_{\text{F}},\tilde{q})$ is given by the integral $$\begin{aligned}
\int_1^{+\infty}d\tilde{k}\left(\int_0^{\pi}d\theta\frac{\tilde{k}^2
\sin\theta}{2\tilde{k}^2-2\tilde{q}\tilde{k}\cos\theta-\mathcal{E}/E_{\text{F}}}-1\right)\nonumber\end{aligned}$$ where $\tilde{q}=q/k_{\text{F}}$ and $\tilde{k}=k/k_{\text{F}}$. For this model there is always a bound state in a two-body problem for $a_{\text{s}}>0$, and correspondingly, there is always a polaron solution with negative $\mathcal{E}$ when $a_{\text{s}}>0$, which has been extensively discussed in Ref. [@Chevy; @imbalance; @MC; @polaron]. In addition, there is always a positive energy solution orthogonal to the polaron solution. For small $k_Fa_s$ applying the second-order perturbation theory[@LHY] to this case, one will obtain $\mathcal{E}/E_{\text{F}}=4k_{\text{F}}a_{\text{s}}/(3\pi)+2(k_{\text{F}}a_{\text{s}})^2/\pi^2$. It is found our variational results fit quite well to this expansion in the regime $|k_{\text{F}}a_{\text{s}}|\lesssim 0.6$ (see inset of Fig. \[zero-range\]), and start to deviate substantially when $k_{\text{F}}a_{\text{s}}\gtrsim0.6$. At $k_{\text{F}}a_{\text{s}}\rightarrow+\infty$, we find the energy under this trial w.f. saturates at $1.82E_{\text{F}}$ instead of diverging as perturbation theory predicts. This saturation is due to unitary limit of resonance interaction. We find $\mathcal{E}<E_{\text{F}}$ for $k_{\text{F}}a_{\text{s}}<2.35$, which means a fully polarized ferromagnetism is definitely unstable below a critical $k_{\text{F}}a^{\text{c}}_{\text{s}}=2.35$. It contradicts to previous results based on a second-order perturbation [@MacDonald] which predicts the system becomes fully magnetized at $k_{\text{F}}a_{\text{s}}=1.40$ [@notation]. The reason for this discrepancy is because the second order perturbation overestimates the interaction effects in the regime $k_{\text{F}}a_{\text{s}}\gtrsim 1$. Similar behavior is found for a resonance scattering with finite interaction range, as shown in the $U_0<0$ part of Fig. \[finiterange\](a).
We would like to point out several intrinsic relations between (a) single-band Hubbard model at $U\gg t$, (b) finite-range continuum model at $U_0=+\infty$ and (c) zero-range continuum model at $a_{\text{s}}=+\infty$. First is the relation between the coefficient of different terms in the w. f. (Eq. [\[general\_psi\]]{}): for (b) we have $\phi_0+\sum_{{\bf
k>k_{\text{F}}}}w_{{\bf k}}\phi_{{\bf k}}=0$; this is equivalent to (a) with all $w_{{\bf k}}=1$ and a momentum cut-off imposed by $\pi/a_{\text{L}}$ instead of $k_0$ ($a_{\text{L}}$ is the lattice constant); for (c) we can define $\chi_{{\bf q}}=\phi_0+\sum_{{\bf
k>k_{\text{F}}}}\phi_{{\bf k}{\bf q}}$ then we have $\phi_{{\bf
k}{\bf q}}=\chi^0_{{\bf q}}/(\mathcal{E}-\epsilon_{{\bf k}{\bf q}})$ with $\chi^0_{{\bf q}}=g\chi_{{\bf q}}/\Omega$[@tan], so at unitary limit one can get similar relation as $\phi_0+\sum_{{\bf
k>k_{\text{F}}}}(\phi_{{\bf k}{\bf q}}+\chi^0_{{\bf
q}}/(2\epsilon_{{\bf k}}))+\sum_{{\bf
q<k_{\text{F}}}}1/(2\epsilon_{{\bf q}})=0$. Second, for (a) at low density limit and (b) with $k_F\ll k_0$ we can show $\mathcal{E}/E_F\sim k_Fr_{\text{eff}}$, here the effective range $r_{\text{eff}}=a_{\text{L}}$ for (a), and $1/k_0$ for (b), which directly leads to the instability of ferromagnetism in these limits; while (c) does not fall into the class of pure repulsive interactions.
![$\mathcal{E}/E_{\text{F}}$ as a function of $k_{\text{F}}a_{\text{s}}$ for the continuum model with zero range interactions. Dashed horizontal lines denote saturated values of $\mathcal{E}/E_{\text{F}}$ at unitarity. Inset is fit to the perturbation results (see text) for small $k_{\text{F}}a_{\text{s}}$. \[zero-range\]](zerorange.eps){height="4.2cm" width="6.3cm"}
The calculation above uses the w.f. that contains only one particle-hole pairs of up-spins. Including multiple particle-hole pairs can systematically improve the results, lead to further lower $\mathcal{E}$ and increased $k_{\text{F}}a^{\text{c}}_{\text{s}}$. However, from the experience in studies of polaron branch, it is found the single particle-hole w.f. can already produce a result very close to Monte Carlo simulation [@MC] and later experiments [@polaron], and the reason for this perfect agreement is understood as a nearly perfect destructive interference of higher-order particle-hole contributions [@imbalance]. Hence it is unlikely $\mathcal{E}/E_{\text{F}}$ at resonance can be reduced from $1.82$ to below unity. Even though, we only prove ferromagnetic is stable against single spin flip above a critical $k_{\text{F}}a^{\text{c}}_{\text{s}}$, and we can not rule out the possibility that a state with more down-spin can be energetically more favorable due to more exotic correlations.
Our results bring forward two intriguing issues. (i) for $0.6\lesssim
k_{\text{F}}a_{\text{s}}<2.35$, the system can neither be fully ferromagnetic nor be well described by perturbation theory. It is in a very interesting strongly interacting quantum phase with large ferromagnetic and/or short-range fluctuations; (ii) for $k_{\text{F}}a_{\text{s}}>2.35$, our approach predicts the system will become ferromagnetic. If it is true, what is the smoking gun experimental evidence? If future experiments find it is not true, then it means the system contains much stronger correlations than discussed here.
[*Acknowledgment*]{}: We thank Z. Y. Weng, Y. P. Wang, S. Chen, J. L. Song for valuable discussions. H.Z. is supported by the Basic Research Young Scholars Program of Tsinghua University, NSFC Grant No. 10944002 and 10847002. X.L.C. is supported by NSFC, CAS and 973-project of MOST.
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$k_{\text{F}}$ in our paper is the Fermi momentum for a fully polarized state, thus differs by a factor of $2^{1/3}$ from that previously defined for each component in a non-magnetic state [@Ketterle; @theory; @MacDonald]. Hence the prediction for fully polarized state $k_{\text{F}}a_{\text{s}}=1.112$[@MacDonald] is translated to $1.40$ to compare with our result (2.35). Our result should be translated to 1.87 to compare with MIT experiment [@Ketterle].
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