Local correlations in the attractive 1D Bose gas: from Bethe ansatz to the Gross-Pitaevskii equation

TL;DR

This paper analyzes the ground-state properties and local correlations of an attractive 1D Bose gas, revealing a quantum phase transition and comparing exact Bethe ansatz results with mean-field predictions from the Gross-Pitaevskii equation.

Contribution

It provides analytic formulas for local correlation functions in the attractive 1D Bose gas and compares finite-size corrections with mean-field results.

Findings

Identifies a quantum phase transition in the attractive 1D Bose gas.

Derives exact formulas for local correlation functions $g_2$ and $g_3$.

Shows differences between finite-size and infinite-size (mean-field) results.

Abstract

We consider the ground-state properties of an extended one-dimensional Bose gas with pointwise attractive interactions. We take the limit where the interaction strength goes to zero as the system size increases at fixed particle density. In this limit the gas exhibits a quantum phase transition. We compute local correlation functions at zero temperature, both at finite and infinite size. We provide analytic formulas for the experimentally relevant one-point functions $g_2$, $g_3$ and analyze their finite-size corrections. Our results are compared to the mean-field approach based on the Gross-Pitaevskii equation which yields the exact results in the infinite system size limit, but not for finite systems.