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arXiv · 1505.02104

Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

Azat M. Gainutdinov·Wenrui Hao·Rafael I. Nepomechie·Andrew J. Sommese
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Abstract

We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.

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Azat M. Gainutdinov, Wenrui Hao, Rafael I. Nepomechie, Andrew J. Sommese. 2015-05-08. Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity. https://doi.org/10.1088/1751-8113/48/49/494003

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