Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution

TL;DR

This paper provides highly accurate analytical results and conjectures for the ground-state energy and excitation spectrum of the Lieb-Liniger model, a fundamental 1D Bose gas with repulsive interactions, using advanced series expansion techniques.

Contribution

It introduces a novel analytical method based on orthogonal polynomial series expansion to solve Bethe-Ansatz equations and conjectures exact results for the model's properties.

Findings

Conjectured exact zero-temperature ground-state energy.

Derived analytical expressions for excitation spectrum.

Validated results against numerical calculations.

Abstract

We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in \cite{Ristivojevic} and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan's contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.