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arXiv:math-ph/0306002

On solutions of Bethe equations for the XXZ model

V. Tarasov

Published Jun 2, 2003• 0 citations
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Abstract

It is shown that solutions of the Bethe ansatz equations for the inhomogeneous arbitrary spin XXX or XXZ model satisfy certain identites which generalize those, recently obtained by K.Fabricius and B.M.McCoy, for solutions of the Bethe ansatz equations for the six-vertex model. Even in the last case, the given proof is simpler than the original one.

DOI
10.48550/arxiv.math-ph/0306002

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It is shown that solutions of the Bethe ansatz equations for the inhomogeneous arbitrary spin XXX or XXZ model satisfy certain identites which generalize those, recently obtained by K.Fabricius and B.M.McCoy, for solutions of the Bethe ansatz equations for the six-vertex model. Even in the last case, the given proof is simpler than the original one.
We give a construction of creation operators responsible for appearance of Bethe vectors with the same eigenvalues of the transfer-matrix for the inhomogeneous arbitrary spin XXZ model at roots of unity with particular quasiperiodic boundary conditions. This construction generalizes the similar one, recently obtained by K.Fabricius and B.M.McCoy, for the homogeneous six-vertex model with periodic boundary conditions. Even in the last case, the given proof for the main formulae are simpler than the original one.
Addendum to the paper J. Phys. A36 (2003) 1391-1401: \nA Bethe Ansatz solution of the open spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al, we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.
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 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 $\psi $ satisfies $H\psi =E\psi $, 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.