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

Abstract

We consider non-smooth functions of (truncated) Wiener–Hopf type operators on the Hilbert space L2(Rd). Our main results are uniform estimates for trace norms (d≥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 Rd (d≥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≥1.