1D Lieb-Liniger Bose Gas as Non-Relativistic Limit of the Sinh-Gordon Model

TL;DR

This paper establishes a connection between the Lieb-Liniger Bose gas and the Sinh-Gordon model, enabling the calculation of physical quantities and correlators in the Lieb-Liniger model using the integrable structure of the Sinh-Gordon model.

Contribution

It introduces a formalism based on the non-relativistic limit of the Sinh-Gordon model to compute expectation values in the Lieb-Liniger model at various temperatures.

Findings

Derived a correspondence between the models' physical quantities.

Developed a method to compute one-point correlators.

Validated results against existing literature.

Abstract

The repulsive Lieb-Liniger model can be obtained as the non-relativistic limit of the Sinh-Gordon model: all physical quantities of the latter model (S-matrix, Lagrangian and operators) can be put in correspondence with those of the former. We use this mapping, together with the Thermodynamical Bethe Ansatz equations and the exact form factors of the Sinh-Gordon model, to set up a compact and general formalism for computing the expectation values of the Lieb-Liniger model both at zero and finite temperature. The computation of one-point correlators is thoroughly detailed and, when possible, compared with known results in the literature.