# A connection between non-local one-body and local three-body correlations of the Lieb-Liniger model

## TL;DR

This paper establishes a novel link between one-body and three-body correlations in the Lieb-Liniger model, providing analytical expressions valid for all interaction strengths and revealing a sign change in the fourth coefficient at a critical interaction value.

## Contribution

It introduces a new connection between correlation functions in the Lieb-Liniger model and proposes approximate formulas validated by numerical data.

## Key findings

- The fourth coefficient of the correlation function changes sign at .816 interaction strength.
- The first three coefficients maintain their sign across all interaction strengths.
- The derived expressions accurately match numerical simulations.

## Abstract

We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of $\delta$-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasi-momenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength $\gamma_c\simeq 3.816$, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths.