Exact Results for the Moments of the Rapidity Distribution in Galilean-Invariant Integrable Models

TL;DR

This paper develops a formalism to analyze moments of the rapidity distribution in Galilean-invariant integrable models, providing exact results for the Lieb-Liniger model and ground-state energy at weak interactions.

Contribution

It introduces a difference-differential equation framework for moments of the rapidity distribution in Bethe ansatz models, with explicit solutions for the Lieb-Liniger model.

Findings

Moments satisfy a difference-differential equation.

Analytical moments for the Lieb-Liniger model.

Exact ground-state energy at weak repulsion.

Abstract

We study a class of Galilean-invariant one-dimensional Bethe ansatz solvable models in the thermodynamic limit. Their rapidity distribution obeys an integral equation with a difference kernel over a finite interval, which does not admit a closed-form solution. We develop a general formalism enabling one to study the moments of the rapidity distribution, showing that they satisfy a difference-differential equation. The derived equation is explicitly analyzed in the case of the Lieb-Liniger model and the moments are analytically calculated. In addition, we obtained the exact information about the ground-state energy at weak repulsion. The obtained results directly enter a number of physically relevant quantities.