Exceptional Points and Other Properties of Free Parafermions

Abstract

The Baxter-Fendley free parafermion model is a non-Hermitian spin chain with a special exact solution. This solution takes the form of free parafermions and generalises the free fermion solutions found in many spin-half chains. The model is defined using clock states, which are generalisations of spins with discrete rotational Z(N) symmetry. Although not Hermitian, the model is PT-symmetric, which means that unitary time evolution may be defined. The details and history of clock models and PT-symmetry are reviewed in the background theory chapters, as well as some of the numerical methods used throughout this work.

This thesis focuses on new results and variations of the free parafermion model. The most important of these results is the discovery of a series of exceptional points, where two of the quasienergies that define the free spectrum become degenerate. These points appear in the complex plane of the model's magnetic field parameter, but they explain unusual behaviour observed on the real-field line. They also exist in the transverse field Ising model with complex field, which is the limiting case of the free parafermion model. An exact analytic expression describing the location of these exceptional points is derived and their properties are investigated numerically.

The free parafermion model's exact solution only exists for open boundary conditions, and the properties of periodic boundary case remain largely unknown. Preliminary new results on the periodic case are presented, particularly the presence of zero-energy states that exist independently of the magnetic field parameter. Some interesting finite size numerical data is also shown.

By generalising the Pauli y operator to clock states, a new clock model dubbed the XYW model is formulated. The y operator turns out to have multiple distinct generalisations, which when used together produce a commuting structure of terms that leads to a free parafermionic spectrum by satisfying a special exchange algebra. This defines the XYW model, which has a similar but in some cases distinct free spectrum to the Baxter-Fendley model. The XYW model also has an unusual boundary property where periodic boundary conditions may be equivalent to adding a new site to the open boundary case.

Finally, some minor results on the Heisenberg XX model with staggered couplings are presented. The model's phase diagram is determined by examining its Majorana edge zero modes. The central charge of the model in the gapless phase is measured using DMRG and entanglement entropy scaling. A finite size effect where the gapless phase splits into a series of bands with constant order parameter is also observed.