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

TL;DR

This paper explores the PT symmetry of a non-Hermitian XX spin-chain with complex boundary fields, revealing non-local interactions and spectral equivalences to fermionic models, with implications for conformal field theories.

Contribution

It introduces a new inner product for quasi-Hermitian XX spin-chains with complex boundaries and establishes their spectral equivalence to non-local fermion models, linking to conformal field theories.

Findings

Quasi-Hermitian XX spin-chain constructed with a new inner product.

Spectral equivalence to a non-local fermion model with long-range hopping.

Connection to conformal field theories with different central charges.

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.