Riemann--Hilbert approach to the time-dependent generalized sine kernel

TL;DR

This paper develops a Riemann-Hilbert method to analyze the asymptotic behavior of Fredholm determinants in integrable models, enabling systematic long-time and long-distance correlation function calculations.

Contribution

It introduces a new series representation for Fredholm determinants in integrable models, extending analysis beyond the free fermion point using Riemann-Hilbert techniques.

Findings

Derived leading asymptotics for Fredholm determinants

Constructed a new series representation for correlation functions

Enabled systematic long-time, long-distance asymptotic analysis

Abstract

We derive the leading asymptotic behavior and build a new series representation for the Fredholm determinant of integrable integral operators appearing in the representation of the time and distance dependent correlation functions of integrable models described by a six-vertex R-matrix. This series representation opens a systematic way for the computation of the long-time, long-distance asymptotic expansion for the correlation functions of the aforementioned integrable models away from their free fermion point. Our method builds on a Riemann--Hilbert based analysis.