Introduction to the thermodynamic Bethe ansatz

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.

arXiv commentintroductory review article, v3, 46 pages, 9 figures, minor typographical changes to match published version

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• An introduction to integrable techniques for one-dimensional quantum systems
• Superuniversality of superdiffusion
• Lectures on Yangian Symmetry
• Lectures on the Bethe Ansatz
• Rule 54: Exactly solvable model of nonequilibrium statistical mechanics
• Structure Constants in $\mathcal{N}=4$ SYM at Finite Coupling as Worldsheet $g$-Function
• Generalized hydrodynamics of the KdV soliton gas
• An Integrability Primer for the Gauge-Gravity Correspondence: an Introduction
• Flow equations for generalised $T\bar{T}$ deformations
• The low-energy limit of AdS${}_3$/CFT${}_2$ and its TBA
• Mirror Thermodynamic Bethe Ansatz for AdS3/CFT2
• Lectures on S-matrices and Integrability
• Introduction to classical and quantum integrability
• Lectures on Integrable Structures in Quantum Field Theory and Massive ODE/IM Correspondence
• Integrable Kondo problems
• TBA Equations and Quantization Conditions
• Thermodynamic Bethe Ansatz past turning points: the (elliptic) sinh-Gordon model
• Resurgence in the Bi-Yang-Baxter Model
• Exact solutions and critical chaos in dilaton gravity with a boundary
• Bethe Ansatz, Quantum Circuits, and the F-basis
• Lectures on Classical Integrability
• Reducing Circuit Depth in Quantum State Preparation for Quantum Simulation Using Measurements and Feedforward
• The Theory of Generalised Hydrodynamics for the One-dimensional Bose Gas
• Free energy asymptotics of the quantum Heisenberg spin chain
• Parameters of Tomonaga-Luttinger liquid in a quasi-1D material with Coulomb interactions
• Particle Scattering and Fusion for the Ablowitz-Ladik Chain
• Growth of Rényi Entropies in Interacting Integrable Models and the Breakdown of the Quasiparticle Picture
• Free energy from forward scattering in 1+1d
• Finite volume corrections of non-diagonal form factors