On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary

TL;DR

This paper constructs new crossing tensor space representations of Hecke algebras, leading to novel solutions of the Yang-Baxter and Boundary Yang-Baxter equations, and analyzes the spectrum and Bethe ansatz for related quantum spin chains.

Contribution

It introduces new representations of Hecke and blob algebras that generate solutions to integrable models with boundary conditions, expanding the understanding of quantum group symmetries.

Findings

All constructed Hamiltonians share the same spectrum up to multiplicity.

The models have the same spectrum as the open XXZ chain with nondiagonal boundary.

Derived Bethe ansatz equations describe the spectrum of the models.

Abstract

Motivated by a study of the crossing symmetry of the `gemini' representation of the affine Hecke algebra we give a construction for crossing tensor space representations of ordinary Hecke algebras. These representations build solutions to the Yang--Baxter equation satisfying the crossing condition (that is, integrable quantum spin chains). We show that every crossing representation of the Temperley--Lieb algebra appears in this construction, and in particular that this construction builds new representations. We extend these to new representations of the blob algebra, which build new solutions to the Boundary Yang--Baxter equation (i.e. open spin chains with integrable boundary conditions). We prove that the open spin chain Hamiltonian derived from Sklyanin's commuting transfer matrix using such a solution can always be expressed as the representation of an element of the blob algebra, and determine this element. We determine the representation theory (irreducible content) of the new representations and hence show that all such Hamiltonians have the same spectrum up to multiplicity, for any given value of the algebraic boundary parameter. (A corollary is that our models have the same spectrum as the open XXZ chain with nondiagonal boundary -- despite differing from this model in having reference states.) Using this multiplicity data, and other ideas, we investigate the underlying quantum group symmetry of the new Hamiltonians. We derive the form of the spectrum and the Bethe ansatz equations.