### 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) |
---|---|

Article number | 494003 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 48 |

Issue number | 49 |

DOIs | |

State | Published - Nov 18 2015 |

### Fingerprint

### 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

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*48*(49), [494003]. https://doi.org/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.** / Gainutdinov, Azat M.; Hao, Wenrui; Nepomechie, Rafael; Sommese, Andrew J.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 48, no. 49, 494003. https://doi.org/10.1088/1751-8113/48/49/494003

}

TY - JOUR

T1 - Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

AU - Gainutdinov, Azat M.

AU - Hao, Wenrui

AU - Nepomechie, Rafael

AU - Sommese, Andrew J.

PY - 2015/11/18

Y1 - 2015/11/18

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.

KW - Bethe ansatz

KW - completeness

KW - quantum group

KW - quantum spin chain

UR - http://www.scopus.com/inward/record.url?scp=84948767654&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84948767654&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/48/49/494003

DO - 10.1088/1751-8113/48/49/494003

M3 - Article

AN - SCOPUS:84948767654

VL - 48

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 49

M1 - 494003

ER -