Nonlinear Luttinger liquid: Exact result for the Green function in terms of the fourth Painlev\'e transcendent

TL;DR

This paper derives an exact expression for the Green function in nonlinear Luttinger liquids using Painlevé IV equations, connecting it to integrable systems and confirming asymptotic behaviors.

Contribution

It provides a novel exact formulation of the Green function in terms of Painlevé IV transcendent, extending the Imambekov-Glazman theory.

Findings

Green function expressed via Painlevé IV solution

Asymptotic power law matches mobile impurity results

Full Green function shape obtained through numerical integration

Abstract

We show that exact time dependent single particle Green function in the Imambekov-Glazman theory of nonlinear Luttinger liquids can be written, for any value of the Luttinger parameter, in terms of a particular solution of the Painlev\'e IV equation. Our expression for the Green function has a form analogous to the celebrated Tracy-Widom result connecting the Airy kernel with Painlev\'e II. The asymptotic power law of the exact solution as a function of a single scaling variable $x/\sqrt{t}$ agrees with the mobile impurity results. The full shape of the Green function in the thermodynamic limit is recovered with arbitrary precision via a simple numerical integration of a nonlinear ODE.