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

Azat M. Gainutdinov, Wenrui Hao, Rafael Nepomechie, Andrew J. Sommese

Research output: Contribution to journalArticle

9 Citations (Scopus)

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.

Original languageEnglish (US)
Article number494003
JournalJournal of Physics A: Mathematical and Theoretical
Volume48
Issue number49
DOIs
StatePublished - Nov 18 2015

Fingerprint

Roots of Unity
Quantum Groups
unity
Counting
counting
Bethe Ansatz
Invariant
Algebra
Quantum Spin Chain
Tilting Modules
modules
Tilting
Transfer Matrix
Irreducible Representation
Explicit Formula
Numerical Solution
Eigenvalue
Module
algebra
eigenvalues

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

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.

In: Journal of Physics A: Mathematical and Theoretical, Vol. 48, No. 49, 494003, 18.11.2015.

Research output: Contribution to journalArticle

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