PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields

Abstract

The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to be quasi-Hermitian for special values of the boundary parameters. This is proved by explicit construction of a new inner product employing a "quasi-fermion" algebra in momentum space where creation and annihilation operators are not related via Hermitian conjugation. For a special example, when the boundary fields lie on the imaginary axis, we show the spectral equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion model, where long range hopping of the particles occurs as the non-Hermitian boundary fields increase in strength. The corresponding Hamiltonian interpolates between the open XX and the quantum group invariant XXZ model at the free fermion point. For an even number of sites the former is known to be related to a CFT with central charge c=1, while the latter has been connected to a logarithmic CFT with central charge c=-2. We discuss the underlying algebraic structures and show that for an odd number of sites the superalgebra symmetry U(gl(1|1)) can be extended from the unit circle along the imaginary axis. We relate the vanishing of one of its central elements to the appearance of Jordan blocks in the Hamiltonian.

37 pages, 5 figures

References in corpus (5)

• Making Sense of Non-Hermitian Hamiltonians
• Associative-algebraic approach to logarithmic conformal field theories
• Time evolution of non-Hermitian Hamiltonian systems
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product

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