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arXiv:1905.11144 · Journal of Physics A Mathematical and Theoretical

Towards the solution of an integrable D 2 ( 2 ) spin chain

Rafael I. Nepomechie, Rodrigo A. Pimenta, Ana L. Retore

Published Sep 10, 2019• 12 citations
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Abstract

Abstract Two branches of integrable open quantum-group invariant quantum spin chains are known. For one branch ( ), a complete Bethe ansatz solution has been proposed. However, the other branch ( ) has so far resisted solution. In an effort to address this problem, we consider here the simplest case n = 1. We propose a Bethe ansatz solution, which however is not complete, as it describes only the transfer-matrix eigenvalues with odd degeneracy. We also consider a proposal for the missing eigenvalues.

DOI
10.1088/1751-8121/ab434d

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Abstract Exact solution of the quantum integrable D2(2) {D}_2^{(2)} D2(2)​ D 2 2 spin chain with generic integrable boundary fields is constructed. It is found that the transfer matrix of this model can be factorized as the product of those of two open staggered anisotropic XXZ spin chains. Based on this identity, the eigenvalues and Bethe ansatz equations of the D2(2) {D}_2^{(2)} D2(2)​ D 2 2 model are derived via off-diagonal Bethe ansatz.
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.
We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin- s representation of quantum-deformed s l ( 2 ) . We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.