Exactly Solvable Quasi-hermitian Transverse Ising Model

TL;DR

This paper introduces an exactly solvable non-Hermitian deformation of the transverse Ising model that maintains real spectra and physical observables through a constructed metric, revealing differences in transverse correlations and phase transition behavior.

Contribution

It presents a quasi-Hermitian transverse Ising model with explicit metric construction, showing how to preserve real eigenvalues and correlation functions despite non-Hermiticity.

Findings

Eigen-spectra of the non-Hermitian and Hermitian models are identical.

Transverse correlation functions are complex and differ from the Hermitian case.

A set of Hermitian operators in the non-Hermitian model's Hilbert space makes all correlation functions real.

Abstract

A non-hermitian deformation of the one-dimensional transverse Ising model is shown to have the property of quasi-hermiticity. The transverse Ising chain is obtained from the starting non-hermitian Hamiltonian through a similarity transformation. Consequently, both the models have identical eigen-spectra, although the eigen-functions are different. The metric in the Hilbert space, which makes the non-hermitian model unitary and ensures the completeness of states, has been constructed explicitly. Although the longitudinal correlation functions are identical for both the non-hermitian and the hermitian Ising models, the difference shows up in the transverse correlation functions, which have been calculated explicitly and are not always real. A proper set of hermitian spin operators in the Hilbert space of the non-hermitian Hamiltonian has been identified, in terms of which all the correlation functions of the non-hermitian Hamiltonian become real and identical to that of the standard transverse Ising model. Comments on the quantum phase transitions in the non-hermitian model have been made.