Wiener-Hopf算子的迹公式及其对自由费米子平衡态熵的应用
Trace formulas for Wiener--Hopf operators with applications to entropies of free fermionic equilibrium states
摘要：
考虑Hilbert空间$L^2(\mathbb R^d)$上的（截断）Wiener-Hopf型算子的非光滑函数.我们的主要结果是迹范数的一致估计（$d\ge 1$）和所得算子迹的拟经典渐近公式（$d=1$）.在这里，我们遵循Harold Widom的开创性思想，他在几十年前证明了光滑函数的公式。扩展到非光滑函数和各种（物理）参数估计的一致性取决于作者之一（AVS）的最新进展。利用我们的结果得到了位置空间$\mathbb R^d$（$d\ge 1$）中非相互作用费米子在正温度下热平衡态的局域熵和空间二分纠缠熵的大尺度行为，$T>0$ .特别是，我们的热EE的定义导致同时尖锐的小$T$和大尺度参数$α>0$的产品$Tα$提供的估计仍然有界从下面。这里$α$是倒数准经典参数。对于$d=1$，我们得到的热EE的渐近公式，这是一致的大尺度行为的基态EE（在$T=0$），以前建立的作者为$d\ge 1$。
英文摘要：
We consider non-smooth functions of (truncated) Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. Our main results are uniform estimates for trace norms ($d\ge 1$) and quasiclassical asymptotic formulas for traces of the resulting operators ($d=1$). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space $\mathbb R^d$ ($d\ge 1$) at positive temperature, $T>0$. In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small $T$ and large scaling parameter $α>0$ provided that the product $Tα$ remains bounded from below. Here $α$ is the reciprocal quasiclassical parameter. For $d=1$ we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at $T=0$), previously established by the authors for $d\ge 1$.
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