## Probability Surveys

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

#### 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 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 [5] that amounts to proving that the random-cluster model on $\mathbb{Z}^{2}$ with cluster weight $q>4$ exhibits a first-order phase transition.

#### Article information

Source
Probab. Surveys, Volume 15 (2018), 102-130.

Dates
First available in Project Euclid: 15 March 2018

https://projecteuclid.org/euclid.ps/1521079210

Digital Object Identifier
doi:10.1214/17-PS292

Mathematical Reviews number (MathSciNet)
MR3775121

Zentralblatt MATH identifier
06864439

#### Citation

Duminil-Copin, Hugo; Gagnebin, Maxime; Harel, Matan; Manolescu, Ioan; Tassion, Vincent. The Bethe ansatz for the six-vertex and XXZ models: An exposition. Probab. Surveys 15 (2018), 102--130. doi:10.1214/17-PS292. https://projecteuclid.org/euclid.ps/1521079210

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