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Journal of Physics A Mathematical Nuclear and General

Baxter's method for XXZ model

Robert B. Jones

Published Mar 1, 1974• 2 citations
Publisher

Abstract

It is shown that Baxter's method when applied to the XXZ hamiltonian gives a set of eigenvectors and wavenumbers different from those of the familiar Bethe solution. In this limiting case of Baxter's results for the eight-vertex model the author finds that one of Baxter's parameters corresponds to the freedom of making arbitrary rotations about the z axis. It is suggested that for the general XXZ hamiltonian there may be a generalization of the ordinary z component of spin which would be conserved in addition to Baxter's quantum number.

DOI
10.1088/0305-4470/7/4/013

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It is shown that Baxter's method when applied to the XXZ hamiltonian gives a set of eigenvectors and wavenumbers different from those of the familiar Bethe solution. In this limiting case of Baxter's results for the eight-vertex model the author finds that one of Baxter's parameters corresponds to the freedom of making arbitrary rotations about the z axis. It is suggested that for the general XXZ hamiltonian there may be a generalization of the ordinary z component of spin which would be conserved in addition to Baxter's quantum number.
We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q -operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q -operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.
The method of separation of variables (SoV) is employed for the spectral problem of the XXZ model. The Baxter difference equation is resolved by means of a special isotropic asymptotic expansion. States are identified by multiplicities of limiting values of the Bethe parameters. As an application, the statistical properties of integral spectra are investigated. It is shown that the power function gives the more correct description of nearest-neighbour spacing distribution density at intermediate spacings as compared with the exponential.
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.