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

TL;DR

This paper derives formulas for counting solutions and degeneracies in the Bethe ansatz of the open XXZ chain at roots of unity, linking algebraic structures to spectral properties and confirming the completeness of the Bethe ansatz.

Contribution

It introduces explicit formulas for admissible solutions and degeneracies at roots of unity, including corrections for spectrum degeneracies, and verifies completeness through numerical analysis.

Findings

Formulas for admissible solutions at roots of unity

Degeneracy counts linked to tilting module dimensions

Numerical solutions up to N=8 support completeness

Abstract

We consider the sl(2)_q-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 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.