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



We consider the sl(2)q-invariant open spin-½ 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.


Attribute NameValues
  • Andrew Sommese

  • Azat M. Gainutdinov

  • Wenrui Hao

  • Rafael I. Nepomechie

  • arXiv

Date Created
  • 2015-08-11

Bibliographic Citation
  • English

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Content License
  • All rights reserved


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