Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1611.09909 · Probability Surveys

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

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

Published Jan 1, 2018• 16 citations
arXivHTMLPDF

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.

DOI
10.1214/17-ps292

Related papers

From algebraic to coordinate Bethe ansatz for square ice

Silvère Gangloff

In this text, we provide a detailed exposition of the Algebraic Bethe ansatz for square ice (or six vertex model), which allows the construction of candidate eigenvectors for the transfer matrices of this model. We also prove some formula of V.E. Korepin for these vectors, which leads to an identification, up to a non-zero complex factor, with the vector obtained by coordinate Bethe ansatz.

arXiv:1904.09282•Apr 16, 2019•0 citations

An introduction to the algebraic Bethe ansatz

James Plunkett, Omar E. Foda

The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)

•Jan 1, 2006•2 citations

Introduction to solvable lattice models in statistical and mathematical physics

Tetsuo Deguchi

This chapter introduces the six-vertex model defined on a two-dimensional square lattice. It describes the model in detail, since it gives an important prototype of many solvable lattice models defined on two-dimensional lattices. The eight-vertex model, which generalizes the six-vertex model directly, may be considered as the most important exactly solvable model in statistical mechanics. The chapter explores some features of the six-vertex model defined on a square lattice. It reviews a method for diagonalizing the transfer matrix, which is called the coordinate Bethe ansatz, and provides the expressions of the free energy per site in the ferroelectric, the antiferroelectric and the disordered phases, respectively. The chapter discusses some mathematical theories associated with integrable models such as the braid group and the quantum groups. It outlines the Ising model, the Potts models and the chiral Potts model and then the eight-vertex model and the interaction-round-a-face models.

CRC Press eBooks•Apr 23, 2019•1 citation

States with $v_1 = \lambda, v_2 = -\lambda$ and reciprocal equations in the six-vertex model(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

M. J. Rodríguez-Plaza

The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of $n$ constants $v_{1},$ $\ldots,$ $v_{n}$ , the zeros of the function $Q(v)$ .A peculiar class of eigenvalues are those in which two of the constants $v_{1},$ $v_{2}$ are equal to $\lambda,$ $-\lambda$ , with $\Delta=-\cosh$ A and $\Delta$ related to the Boltzmann weights of the six-vertex model by the usual combination $\Delta=(a^{2}+b^{2}-c^{2})/2ab$ .The eigenvectors associated to these eigenvalues are Bethe states (although they seem not).We count the number of such states (eigenvectors) for $n=2,3,4,5$ when $N$ , the columns in a row of a square lattice, is arbitrary.The number obtained is independent of the value of $\Delta$ , but depends on $N$ .We give the explicit expres- sion of the eigenvalues in terms of $a,$ $b,$ $c$ (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary $N$ in the blocks $n$ considered.For real $a,$ $b,c$ such eigenvalues are real.

数理解析研究所講究録•Apr 1, 2006•0 citations

Corner transfer matrix eigenstates for the six-vertex model

Holger Frahm, H. B. Thacker

Eigenstates of the corner transfer matrix (CTM) for the six-vertex model are constructed and their relation to the Bethe ansatz eigenstates of the XXZ Hamiltonian is discussed. In the ferromagnetic regime ( Delta >1), eigenstates with any finite number of overturned arrows are constructed. They are related to the Bethe ansatz states by Fourier transformation over spectral (rapidity) variables. This exhibits the role of the CTM as a lattice boost operator. For the antiferromagnetic regime, the relation between CTM eigenstates and those of the XXZ Hamiltonian is complicated by the filling of the sea and associated Bethe ansatz integral equations. For this case, the authors construct CTM eigenstates by first showing that the Hamiltonian ground state is also an eigenstate of the CTM. Other CTM eigenstates are constructed from excited eigenstates of the Hamiltonian by Fourier transforming over the rapidity of each dressed excitation. They discuss the relationship between CTM eigenstates and a Heisenberg algebra of bosonic oscillators.

Journal of Physics A Mathematical and General•Dec 7, 1991•13 citations

Six-Vertex Model

Hans-Peter Eckle

Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.

•Jul 25, 2019•12 citations
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\psi ψ and energy Λ\Lambda Λ, which satisfy Vψ=ΛψV\psi =\Lambda \psi Vψ=Λψ, where VVV is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1a=b=1a=b=1 and c>0c>0c>0. We also show that the same vector ψ\psi ψ satisfies Hψ=EψH\psi =E\psi Hψ=Eψ, where HHH is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value EEE 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\mathbb{Z}^{2}Z2 with cluster weight q>4q>4q>4 exhibits a first-order phase transition.
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
This chapter introduces the six-vertex model defined on a two-dimensional square lattice. It describes the model in detail, since it gives an important prototype of many solvable lattice models defined on two-dimensional lattices. The eight-vertex model, which generalizes the six-vertex model directly, may be considered as the most important exactly solvable model in statistical mechanics. The chapter explores some features of the six-vertex model defined on a square lattice. It reviews a method for diagonalizing the transfer matrix, which is called the coordinate Bethe ansatz, and provides the expressions of the free energy per site in the ferroelectric, the antiferroelectric and the disordered phases, respectively. The chapter discusses some mathematical theories associated with integrable models such as the braid group and the quantum groups. It outlines the Ising model, the Potts models and the chiral Potts model and then the eight-vertex model and the interaction-round-a-face models.
States with v1=λ,v2=−λv_1 = \lambda, v_2 = -\lambdav1​=λ,v2​=−λ and reciprocal equations in the six-vertex model(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )
The eigenvalues of the transfer matrix in a six-vertex model (with periodic boundary conditions) can be written in terms of nnn constants v1,v_{1},v1​, …,\ldots,…, vnv_{n}vn​ , the zeros of the function Q(v)Q(v)Q(v) .A peculiar class of eigenvalues are those in which two of the constants v1,v_{1},v1​, v2v_{2}v2​ are equal to λ,\lambda,λ, −λ-\lambda−λ , with Δ=−cosh⁡\Delta=-\coshΔ=−cosh A and Δ\DeltaΔ related to the Boltzmann weights of the six-vertex model by the usual combination Δ=(a2+b2−c2)/2ab\Delta=(a^{2}+b^{2}-c^{2})/2abΔ=(a2+b2−c2)/2ab .The eigenvectors associated to these eigenvalues are Bethe states (although they seem not).We count the number of such states (eigenvectors) for n=2,3,4,5n=2,3,4,5n=2,3,4,5 when NNN , the columns in a row of a square lattice, is arbitrary.The number obtained is independent of the value of Δ\DeltaΔ , but depends on NNN .We give the explicit expres- sion of the eigenvalues in terms of a,a,a, b,b,b, ccc (when possible) or in terms of the roots of a certain reciprocal polynomial, being very simple to reproduce numerically these special eigenvalues for arbitrary NNN in the blocks nnn considered.For real a,a,a, b,cb,cb,c such eigenvalues are real.
Eigenstates of the corner transfer matrix (CTM) for the six-vertex model are constructed and their relation to the Bethe ansatz eigenstates of the XXZ Hamiltonian is discussed. In the ferromagnetic regime ( Delta >1), eigenstates with any finite number of overturned arrows are constructed. They are related to the Bethe ansatz states by Fourier transformation over spectral (rapidity) variables. This exhibits the role of the CTM as a lattice boost operator. For the antiferromagnetic regime, the relation between CTM eigenstates and those of the XXZ Hamiltonian is complicated by the filling of the sea and associated Bethe ansatz integral equations. For this case, the authors construct CTM eigenstates by first showing that the Hamiltonian ground state is also an eigenstate of the CTM. Other CTM eigenstates are constructed from excited eigenstates of the Hamiltonian by Fourier transforming over the rapidity of each dressed excitation. They discuss the relationship between CTM eigenstates and a Heisenberg algebra of bosonic oscillators.
Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.