Correlation lengths of the repulsive one-dimensional Bose gas

TL;DR

This paper analyzes the asymptotic behavior of correlation functions in the one-dimensional Bose gas at finite temperature, linking it to the XXZ spin chain and expressing correlation lengths via solvable integral equations.

Contribution

It introduces a method to compute correlation lengths of the 1D Bose gas using the eigenvalues of the XXZ spin chain transfer matrix and Yang-Yang equations.

Findings

Correlation lengths are expressed in terms of solutions to non-linear integral equations.

Asymptotic expansions of correlators are obtained via a continuum limit approach.

The method is numerically implementable for both lattice and continuum systems.

Abstract

We investigate the large-distance asymptotic behavior of the static density-density and field-field correlation functions in the one-dimensional Bose gas at finite temperature. The asymptotic expansions of the Bose gas correlators are obtained performing a specific continuum limit in the similar low-temperature expansions of the longitudinal and transversal correlation functions of the XXZ spin chain. In the lattice system the correlation lengths are computed as ratios of the largest and next-largest eigenvalues of the XXZ spin chain quantum transfer matrix. In both cases, lattice and continuum, the correlation lengths are expressed in terms of solutions of Yang-Yang type [C.N. Yang and C.P. Yang, J.Math.Phys. 10, 1151 (1969)] non-linear integral equations which are easily implementable numerically.