The Bethe ansatz for the six-vertex and XXZ models: an exposition

TL;DR

This paper provides a detailed, pedagogical exposition of the Bethe ansatz for the six-vertex and XXZ models, explaining the construction of eigenvectors and eigenvalues crucial for exactly solvable models in statistical mechanics.

Contribution

It offers a clear, detailed presentation of the Bethe ansatz construction for the six-vertex and XXZ models aimed at a mathematical audience, facilitating understanding of these fundamental techniques.

Findings

Explicit construction of Bethe ansatz vectors and energies

Demonstration that these vectors satisfy key eigenvalue equations

Framework setting for future analysis of phase transitions in related models

Abstract

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi$ and energy $\Lambda$, which satisfy $V \psi = \Lambda \psi$, where $V$ is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a= b=1$ and $c > 0$. We also show that the same vector $\psi$ satisfies $H \psi = E \psi$, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on $\mathbb{Z}^2$ with cluster weight $q >4$ exhibits a first-order phase transition.