Correlations in non-equilibrium Luttinger liquid and singular Fredholm determinants

TL;DR

This paper develops a general asymptotic formula for Fredholm determinants with Fisher-Hartwig singularities, enabling analysis of non-equilibrium correlations in Luttinger liquids with multiple Fermi edges.

Contribution

It introduces a novel asymptotic formula for complex Fredholm determinants and applies it to study non-equilibrium power-law singularities in Luttinger liquids.

Findings

Established non-equilibrium power-law singularities in correlation functions.

Derived analytical and numerical support for the asymptotic formula.

Calculated two-particle distribution functions for fermions leaving the interaction region.

Abstract

We study interaction-induced correlations in Luttinger liquid with multiple Fermi edges. Many-particle correlation functions are expressed in terms of Fredholm determinants ${\rm det}(1+\hat{A}\hat{B})$, where $A(\epsilon)$ and $B(t)$ have multiple discontinuities in energy and time spaces. Such determinants are a generalization of Toeplitz determinants with Fisher-Hartwig singularities. We propose a general asymptotic formula for this class of determinants and provide analytical and numerical support to this conjecture. This allows us to establish non-equilibrium power-law singularities of many-particle correlation functions. As an example, we calculate a two-particle distribution function characterizing correlations between left- and right-moving fermions that have left the interaction region.