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

doi:10.1088/1751-8113/48/49/494003

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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17 Scopus citations

Abstract

We consider the U_qsl (2)-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 U_qsl (2)-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q = e^iπ/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.

Original language	English (US)
Article number	494003
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	48
Issue number	49
DOIs	https://doi.org/10.1088/1751-8113/48/49/494003
State	Published - Nov 18 2015

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Statistics and Probability
Modeling and Simulation
Mathematical Physics
General Physics and Astronomy

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10.1088/1751-8113/48/49/494003

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title = "Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity",

abstract = "We consider the Uqsl (2)-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 Uqsl (2)-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q = eiπ/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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AU - Sommese, Andrew J.

PY - 2015/11/18

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N2 - We consider the Uqsl (2)-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 Uqsl (2)-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q = eiπ/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.

AB - We consider the Uqsl (2)-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 Uqsl (2)-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q = eiπ/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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Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

Abstract

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