Introduction to the thermodynamic Bethe ansatz

TL;DR

This paper provides a pedagogical introduction to the thermodynamic Bethe ansatz, illustrating its application to various integrable models and emphasizing the role of particle and hole densities in their thermodynamics.

Contribution

It offers a comprehensive, step-by-step explanation of the thermodynamic Bethe ansatz applied to multiple integrable models, including derivations and simplifications of TBA equations.

Findings

Derivation of Fermi-Dirac distribution for free electrons

Application of TBA to 1D Bose gas, XXX spin chain, and SU(2) chiral Gross-Neveu model

Discussion of TBA to Y systems and integral equation simplifications

Abstract

We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi-Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing on the one dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the SU(2) chiral Gross-Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross-Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples.