Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlev\'e IV transition

TL;DR

This paper analyzes the large time and distance asymptotics of the correlation function in the one-dimensional impenetrable Bose gas, revealing a phase transition described by Painlevé IV solutions.

Contribution

It derives asymptotic expansions of the correlation function and solutions of NLS equations, identifying a phase transition characterized by Painlevé IV.

Findings

Asymptotic expansions in space-like and time-like regions

Identification of a phase transition between regions

Connection to Painlevé IV equation solutions

Abstract

In the present paper, we study the time-dependent correlation function of the one-dimensional impenetrable Bose gas, which can be expressed in terms of the Fredholm determinant of a time-dependent sine kernel and the solutions of the separated NLS equations. We derive the large time and distance asymptotic expansions of this determinant and the solutions of the separated NLS equations in both the space-like region and time-like region of the $(x,t)$-plane. Furthermore, we observe a phase transition between the asymptotic expansions in these two different regions. The phase transition is then shown to be described by a particular solution of the Painlev\'e IV equation.