Exceptional Points in the Baxter-Fendley Free Parafermion Model

TL;DR

This paper investigates the emergence of exceptional points in a non-Hermitian generalization of the Baxter-Fendley free parafermion model, revealing how complex parameters induce degeneracies in the spectrum and impact related models.

Contribution

It analytically locates exceptional points in the free parafermion spectrum by extending the magnetic field parameter into the complex plane, providing new insights into spectral degeneracies.

Findings

Exceptional points occur where quasienergies degenerate.

Locations of exceptional points are derived analytically.

Proximity of exceptional points affects the PT-symmetric real line.

Abstract

Certain spin chains, such as the quantum Ising chain, have free fermion spectra which can be expressed as the sum of decoupled two-level fermionic systems. Free parafermions are a simple generalisation of this idea to $Z(N)$-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but $PT$-symmetric model directly generalising the Ising chain, which was much later recognised by Fendley to be a free parafermion spectrum. By extending the model's magnetic field parameter to the complex plane, it is shown that a series of exceptional points emerges, where the quasienergies defining the free spectrum become degenerate. An analytic expression for the locations of these points is derived, and various numerical investigations are performed. These exceptional points also exist in the Ising chain with a complex transverse field. Although the model is not in general $PT$-symmetric at these exceptional points, their proximity can have a profound impact on the model on the $PT$-symmetric real line. Furthermore, in certain cases of the model an exceptional point may appear on the real line (with negative field).