The Bethe Ansatz for the six-vertex and XXZ models: An exposition
- Duminil-Copin, Hugo Institut des Hautes Etudes Scientifiques and University of Geneva, Switzerland
- Gagnebin, Maxime Institut des Hautes Etudes Scientifiques and University of Geneva, Switzerland
- Harel, Matan Institut des Hautes Etudes Scientifiques / Tel Aviv University, Israel
- Manolescu, Ioan University of Fribourg, Switzerland
- Tassion, Vincent ETH Zurich, Switzerland
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2018
Published in:
- Probability Surveys. - 2018, vol. 15, p. 102–130
English
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 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 Z2 with cluster weight q>4 exhibits a first-order phase transition.
- Faculty
- Faculté des sciences et de médecine
- Department
- Département de Mathématiques
- Language
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- English
- Classification
- Mathematics
- License
- License undefined
- Identifiers
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- RERO DOC 309506
- DOI 10.1214/17-PS292
- Persistent URL
- https://folia.unifr.ch/unifr/documents/306708
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