Exceptional points in the Baxter-Fendley free parafermion model

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 generalisation of this idea to ZN-symmetric clock models. In 1989 Baxter discovered a non-Hermitian but PT-symmetric model directly generalising the Ising chain, which was later described by Fendley as 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 an exceptional point may appear on the real line (with negative field).