Lectures on the Bethe Ansatz

TL;DR

This paper provides a comprehensive pedagogical overview of Bethe ansatz techniques in integrable quantum field theories and spin chains, illustrating methods through specific models like the SU(2) and SU(3) chiral Gross-Neveu models and quantum mechanics.

Contribution

It introduces and explains Bethe ansatz methods in integrable models, including algebraic and nested algebraic Bethe ansatz, with detailed examples and derivations.

Findings

Derivation of Bethe equations for SU(2) and SU(3) models

Analysis of the Heisenberg XXX spin chain

Application of Bethe ansatz to quantum mechanical oscillators

Abstract

We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S-matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU(2) chiral Gross-Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU(3) chiral Gross-Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU(3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completley different setting, namely for the 1d oscillator in quantum mechanics. This article is part of a collection of introductory reviews originating from lectures given at the YRIS summer school in Durham during July 2015.