Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product

TL;DR

This paper constructs a new algebraic inner product for the quantum group invariant XXZ spin-chain, conjecturing its unitary equivalence with a PT-symmetric quasi-Hermitian representation, and provides an explicit example.

Contribution

It introduces a novel algebraic approach to define a Hermitian inner product for the XXZ spin-chain using GNS construction and conjectures its equivalence to previous PT-symmetric representations.

Findings

Connected quasi-Hermiticity with GNS construction for Temperley-Lieb algebra

Constructed a self-adjoint representation for specific q values

Presented an explicit example illustrating the conjecture

Abstract

This is a continuation of a previous joint work with Robert Weston on the quantum group invariant XXZ spin-chain (math-ph/0703085). The previous results on quasi-Hermiticity of this integrable model are briefly reviewed and then connected with a new construction of an inner product with respect to which the Hamiltonian and the representation of the Temperley-Lieb algebra become Hermitian. The approach is purely algebraic, one starts with the definition of a positive functional over the Temperley-Lieb algebra whose values can be computed graphically. Employing the Gel'fand-Naimark-Segal (GNS) construction for C*-algebras a self-adjoint representation of the Temperley-Lieb algebra is constructed when the deformation parameter q lies in a special section of the unit circle. The main conjecture of the paper is the unitary equivalence of this GNS representation with the representation obtained in the previous paper employing the ideas of PT-symmetry and quasi-Hermiticity. An explicit example is presented.