On finite-temperature Fredholm determinants

TL;DR

This paper develops a method to analyze finite-temperature Fredholm determinants related to free fermion models, providing explicit asymptotics and addressing cases with non-zero winding numbers.

Contribution

It introduces an elementary approach to derive large-distance asymptotics for finite-temperature Fredholm determinants with complex phase shifts, using an effective kernel and Riemann-Hilbert problem.

Findings

Derived explicit large-distance asymptotics for finite-temperature determinants

Extended classical asymptotic formulas to non-zero winding number cases

Provided explicit resolvent and subleading correction calculations

Abstract

We consider finite-temperature deformation of the sine kernel Fredholm determinants acting on the closed contours. These types of expressions usually appear as static two-point correlation functions in the models of free fermions and can be equivalently presented in terms of Toeplitz determinants. The corresponding symbol, or the phase shift, is related to the temperature weight. We present an elementary way to obtain large-distance asymptotic behavior even when the phase shift has a non-zero winding number. It is done by deforming the original kernel to the so-called effective form factors kernel that has a completely solvable matrix Riemann-Hilbert problem. This allows us to find explicitly the resolvent and address the subleading corrections. We recover Szego, Hartwig and Fisher, and Borodin-Okounkov asymptotic formulas.