# Thermodynamic behavior of a one-dimensional Bose gas at low temperature

## TL;DR

This paper investigates the thermodynamic properties of a one-dimensional Bose gas at low temperatures, revealing a non-monotonic chemical potential behavior linked to superfluidity and phononic excitations, with theoretical and numerical validation.

## Contribution

It provides a detailed analysis of the temperature dependence of the chemical potential in 1D Bose gases, including crossover behavior and finite-size effects, supported by exact solutions and analytic expansions.

## Key findings

- Chemical potential exhibits non-monotonic temperature dependence.
- Coefficient in $T^2$ expansion of chemical potential depends on zero-temperature sound velocity.
- Finite-size effects significantly influence thermodynamic functions at zero temperature.

## Abstract

We show that the chemical potential of a one-dimensional (1D) interacting Bose gas exhibits a non-monotonic temperature dependence which is peculiar of superfluids. The effect is a direct consequence of the phononic nature of the excitation spectrum at large wavelengths exhibited by 1D Bose gases. For low temperatures $T$, we demonstrate that the coefficient in $T^2$ expansion of the chemical potential is entirely defined by the zero-temperature density dependence of the sound velocity. We calculate that coefficient along the crossover between the Bogoliubov weakly-interacting gas and the Tonks-Girardeau gas of impenetrable bosons. Analytic expansions are provided in the asymptotic regimes. The theoretical predictions along the crossover are confirmed by comparison with the exactly solvable Yang-Yang model in which the finite-temperature equation of state is obtained numerically by solving Bethe-{\it ansatz} equations. A 1D ring geometry is equivalent to imposing periodic boundary conditions and arising finite-size effects are studied in details. At $T=0$ we calculated various thermodynamic functions, including the inelastic structure factor, as a function of the number of atoms, pointing out the occurrence of important deviations from the thermodynamic limit.