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Journal of Physics A Mathematical and General

Corner transfer matrix eigenstates for the six-vertex model

Holger Frahm, H. B. Thacker

Published Dec 7, 1991• 13 citations

Abstract

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.

DOI: 10.1088/0305-4470/24/23/023

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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.

We consider corner transfer matrices of the six-vertex model and of its spin-l/2 generalizations, truncated to a finite lattice, in the anti-ferromagnetic regime. We show that, in the sense of finite size approximation, the low-lying eigenvectors approximate weight vectors of a level-l highest weight U q (sl 2 ) module, whilst the truncated corner transfer matrices approximate the role of a derivation operator. Our method is to first use the theory of the crystal base to prove the result exactly at q=0, and then extend to q not=0. We also consider the relationship between truncated corner transfer matrices, the vertex operator construction of Frenkel and Reshetikhin (1992), and Baxter's original argument for the properties of CTMs (1976).

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.

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.

The Hamiltonian limit of the corner transfer matrix (CTM) of a generalised free Fermion vertex system of finite size leads to a quantum spin Hamiltonian of the particular form:

{\cal H}_N=-\sum_{n=1}^{N-1}\left\{ n\left( σ_n^xσ_{n+1}^x +λσ_n^yσ_{n+1}^y +h(σ_n^z+σ_{n+1}^z) \right)\right\}

Diagonalisation may be achieved for all pairs of parameters

(λ,h)

with the use of some new elliptic polynomials which extend the class of special polynomials known so far in the context of CTM.

We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1/2 Heisenberg chains using the algebraic Bethe ansatz. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on

{\bar {\mathscr H}}

, the tensor product of auxiliary spaces. By changing the basis in

{\bar {\mathscr H}}

, we derive explicit finite-dimensional representations for the matrices. These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo [Alcaraz F C and Lazo M J 2006 {\it J. Phys. A: Math. Gen.} \textbf{39} 11335.] apart from normalization factors. We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions [Korepin V E 1982 {\it Commun. Math. Phys.}, \textbf{86} 391.] and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.