A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field

TL;DR

This paper explores a non-Hermitian quantum spin chain model with PT-symmetry, demonstrating how to maintain real spectra and constructing its Hermitian equivalent, with applications to magnetization analysis.

Contribution

It introduces a novel lattice implementation of PT-symmetry for non-Hermitian spin chains and constructs the Hermitian counterpart using Dyson maps and perturbation theory.

Findings

PT-symmetry guarantees real spectra in certain parameter regions.

Hermitian counterparts are constructed for small system sizes.

Magnetization in z and x directions is computed.

Abstract

We investigate a lattice version of the Yang-Lee model which is characterized by a non-Hermitian quantum spin chain Hamiltonian. We propose a new way to implement PT-symmetry on the lattice, which serves to guarantee the reality of the spectrum in certain regions of values of the coupling constants. In that region of unbroken PT-symmetry we construct a Dyson map, a metric operator and find the Hermitian counterpart of the Hamiltonian for small values of the number of sites, both exactly and perturbatively. Besides the standard perturbation theory about the Hermitian part of the Hamiltonian, we also carry out an expansion in the second coupling constant of the model. Our constructions turns out to be unique with the sole assumption that the Dyson map is Hermitian. Finally we compute the magnetization of the chain in the z and x direction.