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Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition

Zhi-Xuan Meng, Shuai-Xia Xu, Yu-Qiu Zhao

TL;DR

This work analyzes the time-dependent correlation function of the one-dimensional impenetrable Bose gas, expressing it as a Fredholm determinant of a time-dependent sine kernel and linked separated NLS equations. By formulating and solving Riemann-Hilbert problems for the determinant and for a Painlevé IV system, the authors derive large time and distance asymptotics in space-like, time-like, and transition regions, uncovering a phase transition described by a special Painlevé IV solution. The space-like and time-like analyses yield explicit leading-term formulas for the determinant derivatives and the associated NLS potentials, with a uniform error control away from cos(2x) zeros, while the transition region is governed by PIV data through a carefully matched RH framework. The results highlight a deep link between integrable kernels, AKNS/NLS hierarchies, and Painlevé transcendents, contributing to a broader understanding of finite-time correlation functions in integrable quantum gases and related models.

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 Painlevé IV equation.

Large time and distance asymptotics of the one-dimensional impenetrable Bose gas and Painlevé IV transition

TL;DR

This work analyzes the time-dependent correlation function of the one-dimensional impenetrable Bose gas, expressing it as a Fredholm determinant of a time-dependent sine kernel and linked separated NLS equations. By formulating and solving Riemann-Hilbert problems for the determinant and for a Painlevé IV system, the authors derive large time and distance asymptotics in space-like, time-like, and transition regions, uncovering a phase transition described by a special Painlevé IV solution. The space-like and time-like analyses yield explicit leading-term formulas for the determinant derivatives and the associated NLS potentials, with a uniform error control away from cos(2x) zeros, while the transition region is governed by PIV data through a carefully matched RH framework. The results highlight a deep link between integrable kernels, AKNS/NLS hierarchies, and Painlevé transcendents, contributing to a broader understanding of finite-time correlation functions in integrable quantum gases and related models.

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 -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 Painlevé IV equation.

Paper Structure

This paper contains 41 sections, 5 theorems, 221 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Statement of results
  3. RH problem for the determinant and the PIV equation
  4. RH problem for the determinant
  5. RH problem for the PIV equation
  6. Asymptotic analysis in the space-like region
  7. Deformation of the jump contour
  8. Global parametrix
  9. Local parametrices near $\lambda=\pm 1$
  10. Local parametrix near $\lambda=-1$
  11. Local parametrix near $\lambda=1$
  12. Local parametrix near the stationary point
  13. RH problem for $M$
  14. Final transformation
  15. Large $x$ asymptotics in the space-like region
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $D(x,t)$ be the Fredholm determinant defined in D, we have the following asymptotic expansions as $x\to+\infty$: where the error terms are uniform for $(x,t)$ in the space-like region and for x bounded away from the zeros of $\cos(2x)$. Moreover, we have the asymptotic expansions of the corresponding solutions of the separated NLS equations $b_{++}$ and $B_{--}$, defined by BC and b++, as $x

Figures (8)

  • Figure 1: The space-like region, time-like region and transition region
  • Figure 3: Deformation of the jump contour
  • Figure 5: Deformation of the jump contour
  • Figure 7: Deformation of the jump contour
  • Figure 10: The jump contour of the RH problem for $T$
  • ...and 3 more figures

Theorems & Definitions (10)

  • Theorem 1.1: Large distance asymptotics in the space-like region
  • Theorem 1.2: Large time asymptotics in the time-like region
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.7: Large $s$ asymptotics of PIV
  • Theorem 1.8: Asymptotics in the transition region
  • Remark 1.9
  • Proposition 2.1