Deformed Dolan-Grady relations in quantum integrable models

TL;DR

This paper introduces a new hidden symmetry in quantum integrable models via deformed Dolan-Grady relations, leading to novel solutions and algebraic structures in boundary and bulk models, including the sine-Gordon model.

Contribution

It uncovers a dual pair of operators satisfying q-deformed Dolan-Grady relations and constructs a new family of quantum integrable models using inverse scattering.

Findings

Explicit construction of fundamental generators for specific models.

Identification of a dynamical Askey-Wilson symmetry algebra.

Exact solutions for asymptotic boundary states using q-orthogonal polynomials.

Abstract

A new hidden symmetry is exhibited in the reflection equation and related quantum integrable models. It is generated by a dual pair of operators $\{\textsf{A}, \textsf{A}^*\}\in{\cal A}$ subject to $q-$deformed Dolan-Grady relations. Using the inverse scattering method, a new family of quantum integrable models is proposed. In the simplest case, the Hamiltonian is linear in the fundamental generators of ${\cal A}$. For general values of $q$, the corresponding spectral problem is quasi-exactly solvable. Several examples of two-dimensional massive/massless (boundary) integrable models are reconsidered in light of this approach, for which the fundamental generators of ${\cal A}$ are constructed explicitly and exact results are obtained. In particular, we exhibit a dynamical Askey-Wilson symmetry algebra in the (boundary) sine-Gordon model and show that asymptotic (boundary) states can be expressed in terms of $q-$orthogonal polynomials.