Quasi-classical asymptotics for functions of Wiener-Hopf operators: smooth vs non-smooth symbols

TL;DR

This paper studies the asymptotic behavior of Wiener-Hopf operators with symbols transitioning from smooth to discontinuous, revealing how trace asymptotics change and applying results to fermionic entanglement entropy at low temperatures.

Contribution

It introduces a two-parameter asymptotic analysis capturing the transition from smooth to discontinuous symbols in Wiener-Hopf operators, with applications to quantum entanglement entropy.

Findings

Asymptotic formulas describe the transition regime between smooth and discontinuous symbols.

The results elucidate the impact of symbol smoothness on trace asymptotics.

Application to low-temperature entanglement entropy scaling in fermionic systems.

Abstract

We consider functions of Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. It has been known for a long time that the quasi-classical asymptotics for traces of resulting operators strongly depend on the smoothness of the symbol: for smooth symbols the expansion is power-like, whereas discontinuous symbols (e.g. indicator functions) produce an extra logarithmic factor. We investigate the transition regime by studying symbols depending on an extra parameter $T\ge 0$ in such a way that the symbol tends to a discontinuous one as $T\to 0$. The main result is two-parameter asymptotics (in the quasi-classical parameter and in $T$), describing a transition from the smooth case to the discontinuous one. The obtained asymptotic formulas are used to analyse the low-temperature scaling limit of the spatially bipartite entanglement entropy of thermal equilibrium states of non-interacting fermions.