### Abstract

We consider the U_{q}sl (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_{q}sl (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) |
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Article number | 494003 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 49 |

DOIs | |

State | Published - Nov 18 2015 |

### Keywords

- Bethe ansatz
- completeness
- quantum group
- quantum spin chain

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Modeling and Simulation
- Statistics and Probability

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## Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*48*(49), [494003]. https://doi.org/10.1088/1751-8113/48/49/494003