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arXiv:hep-th/0504124 · Journal of Statistical Mechanics Theory and Experiment

Bethe ansatz derived from the functional relations of the openXXZchain for new special cases

Rajan Murgan, Rafael I. Nepomechie

Published May 20, 2005• 41 citations
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Abstract

The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe ansatz solution for the transfer matrix eigenvalues for the special cases where all but one of the boundary parameters are zero, and the bulk anisotropy parameter is iπ/3, iπ/5, ....

DOI
10.1088/1742-5468/2005/05/p05007

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The transfer matrix of the general integrable open XXZ quantum spin chain obeys certain functional relations at roots of unity. By exploiting these functional relations, we determine the Bethe ansatz solution for the transfer matrix eigenvalues for the special cases where all but one of the boundary parameters are zero, and the bulk anisotropy parameter is iπ/3, iπ/5, ....
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We propose a Bethe-ansatz-type solution of the open spin-1/2 integrable XXZ quantum spin chain with general integrable boundary terms and bulk anisotropy values iπ/( p +1), where p is a positive integer. All six boundary parameters are arbitrary, and need not satisfy any constraint. The solution is in terms of generalized T – Q equations, having more than one Q function. We find numerical evidence that this solution gives the complete set of 2 N transfer matrix eigenvalues, where N is the number of spins.
We consider the open spin- s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2 s +1) N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.