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

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.

34 pages; v2: reference added; v3: two more references added and minor corrections

References in corpus (5)

• Associative-algebraic approach to logarithmic conformal field theories
• Modified algebraic Bethe ansatz for XXZ chain on the segment - I - triangular cases
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• The Jordan Structure of Two Dimensional Loop Models

Cited by in corpus (13)

• Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
• On Generalized $Q$-systems
• Scale-free non-Hermitian skin effect in a boundary-dissipated spin chain
• Bethe-State Counting and the Witten Index
• Cylinder partition function of the 6-vertex model from algebraic geometry
• Yang-Baxter Solution of Dimers as a Free-Fermion Six-Vertex Model
• Estimating Bethe roots with VQE
• Extended T-systems, Q matrices and T-Q relations for $s\ell (2)$ models at roots of unity
• Quantum group symmetries and completeness for A_{2n}^(2) open spin chains
• Algebraic Bethe ansatz for the Temperley-Lieb spin-1 chain
• Yang-Baxter Integrable Dimers on a Strip
• Flag Integrable Models and Generalized Graded Algebras
• Exactly Solved Models and Beyond: a special issue in honour of R J Baxter's 75th birthday