Trace formulas for Wiener--Hopf operators with applications to entropies of free fermionic equilibrium states

TL;DR

This paper extends trace formulas for Wiener--Hopf operators to non-smooth functions, providing uniform estimates and asymptotic formulas that are applied to analyze the large-scale behavior of entropies in free fermionic equilibrium states across various dimensions.

Contribution

It introduces uniform trace norm estimates and quasiclassical asymptotics for non-smooth functions of Wiener--Hopf operators, and applies these to study entanglement and local entropies in fermionic systems.

Findings

Derived uniform trace norm estimates for non-smooth functions of Wiener--Hopf operators.

Established quasiclassical asymptotic formulas for traces in one dimension.

Analyzed the large-scale behavior of entanglement entropy at positive temperature.

Abstract

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 $\alpha>0$ provided that the product $T\alpha$ remains bounded from below. Here $\alpha$ 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 $d\ge 1$.