PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain

TL;DR

This paper explores PT-symmetry in the quantum group invariant XXZ spin chain, demonstrating its quasi-Hermiticity and introducing dual C-operators, thus linking integrable lattice models with non-Hermitian quantum mechanics.

Contribution

It provides the first explicit construction of operators ensuring quasi-Hermiticity in the XXZ chain with quantum group symmetry, connecting integrable systems and non-Hermitian Hamiltonians.

Findings

PT-operator commutes with quantum group action

Explicit construction of quasi-Hermiticity operator

Introduction of dual C-operators with algebraic expressions

Abstract

We investigate the PT-symmetry of the quantum group invariant XXZ chain. We show that the PT-operator commutes with the quantum group action and also discuss the transformation properties of the Bethe wavefunction. We exploit the fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order to give an explicit and exact construction of an operator that ensures quasi-Hermiticity of the model. This construction relys on earlier ideas related to quantum group reduction. We then employ this result in connection with the quantum analogue of Schur-Weyl duality to introduce a dual pair of C-operators, both of which have closed algebraic expressions. These are novel, exact results connecting the research areas of integrable lattice systems and non-Hermitian Hamiltonians.