PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
arXiv:math-ph/0703085 · doi:10.1088/1751-8113/40/30/016
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.
32 pages with figures, v2: some minor changes and added references, version published in JPA