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

2025/05/22 by Meng, Zhi-Xuan, Xu, Shuai-Xia, Zhao, Yu-Qiu
#35Q55 (33E17 34M55 35C20 82C10) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2505.16780

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.

Citations

• Asymptotics of the finite-temperature sine kernel determinant
• On the integrable structure of deformed sine kernel determinants
• On the deformed Pearcey determinant
• Sine-kernel determinant on two large intervals
• Asymptotics of free fermions in a quadratic well at finite temperature and the Moshe-Neuberger-Shapiro random matrix model
• Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump
• The Widom-Dyson constant for the gap probability in random matrix theory
• From Gumbel to Tracy-Widom
• Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle
• Dyson's constant in the asymptotics of the Fredholm determinant of the sine kernel
• Quantum Inverse Scattering Method and Correlation Functions
• Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent
• The Inverse Scattering Transform‐Fourier Analysis for Nonlinear Problems

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• Physics of Low-Dimensional Ultracold Bose Gases
• Large-distance and long-time asymptotic behavior of the reduced density matrix in the non-linear Schrödinger model