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

Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Ioan Manolescu, Vincent Tassion

Research output: Contribution to journalArticlepeer-review


In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector ψ and energy Λ, which satisfy Vψ = Λψ, 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 ψ satisfies Hψ = Eψ, 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 pedagogicallyminded 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 ℤ2 with cluster weight q > 4 exhibits a first-order phase transition.

Original languageEnglish
Pages (from-to)102-130
Number of pages29
JournalProbability Surveys
StatePublished - 2018
Externally publishedYes


  • Bethe ansatz
  • Six vertex model
  • Transfer matrix
  • XXZ model


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