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The eight-vertex and XYZ model

Barry M. McCoy

Published Dec 1, 2009• 0 citations
Publisher

Abstract

Abstract This chapter derives the matrix TQ functional equation for the eight-vertex model. The Bethe's equation for the eigenvalues of Q is derived from the matrix TQ equation. A numerical study is made of the eigenvalues of Q, and the TQ equation is used to compute the free energy of the eight-vertex model. Results on the excitations, order parameters and correlation functions of the six- and eight-vertex models and the related XYZ spin chain are presented.

DOI
10.1093/acprof:oso/9780199556632.003.0014

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Abstract This chapter derives the matrix TQ functional equation for the eight-vertex model. The Bethe's equation for the eigenvalues of Q is derived from the matrix TQ equation. A numerical study is made of the eigenvalues of Q, and the TQ equation is used to compute the free energy of the eight-vertex model. Results on the excitations, order parameters and correlation functions of the six- and eight-vertex models and the related XYZ spin chain are presented.
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.
We consider the interaction-round-a-face version of the isotropic six-vertex model. The associated spin chain is made of two coupled Heisenberg spin chains with different boundary twists. The phase diagram of the model and the long distance correlations were studied in Tavares and Ribeiro (2023) [43] . Here, we compute the short-distance correlation functions of the model in the ground state for finite system sizes via non-linear integral equations and in the thermodynamic limit. This was possible since the model satisfies the face version of the discrete quantum Knizhnik-Zamolodchikov (qKZ) equation. A suitable ansatz for the density matrix is proposed in the form of a direct sum of two Heisenberg density matrices, which allows us to obtain the discrete functional equation for the two-site function ω ( λ 1 , λ 2 ) . Thanks to the known results on the factorization of correlation functions of the Heisenberg chain, we are able to compute the density matrix of the IRF model for up to four sites and its associated spin chain for up to three sites.
The general solutions for the factorization equations of the reflection matrices Kpm(theta)K^{\\pm}(\\theta)Kpm(theta) for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are explicitly derived, finding families dependig on several continuous as well as discrete parameters.
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.