Probability Surveys

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

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

Full-text: Open access

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
Received: October 2017
First available in Project Euclid: 15 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ps/1521079210

Digital Object Identifier
doi:10.1214/17-PS292

Mathematical Reviews number (MathSciNet)
MR3775121

Zentralblatt MATH identifier
06864439

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B23: Exactly solvable models; Bethe ansatz 82B26: Phase transitions (general)

Keywords
Transfer matrix Bethe ansatz six vertex model XXZ model

Rights
Creative Commons Attribution 4.0 International License.

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


Export citation

References

  • [1] R. J. Baxter. Partition function of the eight vertex lattice model. Annals Phys., 70:193–228, 1972.
  • [2] R. J. Baxter. Exactly solved models in statistical mechanics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989. Reprint of the 1982 original.
  • [3] H. Bethe. Zur Theorie der Metalle I. Eigenwerte und Eigenfunktionen der Hnearen Atomkette. Zeitschrift für Physik, 71(3):205–226, 1931.
  • [4] N. M. Bogoliubov, A. G. Izergin, and V. E. Korepin. Quantum inverse scattering method and correlation functions. Cambridge university press, 1997.
  • [5] H. Duminil-Copin, M. Gagnebin, M. Harel, I. Manolescu, and V. Tassion. Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$. Preprint, https://arxiv.org/abs/1611.09877, 2016.
  • [6] M. Jimbo and T. Miwa. Algebraic Analysis of Solvable Lattice Models. American Mathematical Soc., 1994.
  • [7] M. Karbach, K. Hu, and G. Müller. Introduction to the Bethe ansatz II. Computers in Physics, 12:565–573, 1998.
  • [8] M. Karbach, K. Hu, and G. Müller. Introduction to the Bethe ansatz III. 2000.
  • [9] M. Karbach and G. Müller. Introduction to the Bethe ansatz I. Computers in Physics, 11:36–43, 1997.
  • [10] E. H. Lieb. Residual entropy of square ice. Physical Review, 162(1):162, 1967.
  • [11] N. Reshetikhin. Lectures on the integrability of the 6-vertex model. https://arxiv.org/abs/1010.5031, 2010.
  • [12] C. N. Yang and C. P. Yang. One-dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system. Phys. Rev., 150:321–327, 1966.