# Exact perturbative results for the Lieb-Liniger and Gaudin-Yang models

## TL;DR

This paper develops a systematic method to derive and analyze the divergent perturbative series for the ground state energies of the Lieb-Liniger and Gaudin-Yang models, revealing their factorial divergence and connection to non-perturbative effects.

## Contribution

It introduces a procedure to explicitly compute perturbative series coefficients from Bethe ansatz solutions for these models, and studies their large order behavior and non-perturbative implications.

## Key findings

- Both series diverge factorially and are not Borel summable.
- The first Borel singularity in the Gaudin-Yang model relates to the non-perturbative energy gap.
- Provides insights into the Cooper instability through perturbative series analysis.

## Abstract

We present a systematic procedure to extract the perturbative series for the ground state energy density in the Lieb-Liniger and Gaudin-Yang models, starting from the Bethe ansatz solution. This makes it possible to calculate explicitly the coefficients of these series and to study their large order behavior. We find that both series diverge factorially and are not Borel summable. In the case of the Gaudin-Yang model, the first Borel singularity is determined by the non-perturbative energy gap. This provides a new perspective on the Cooper instability.