Mathematical Physics Seminar: Free Parafermions and More


In this PhD completion talk I describe the free parafermion model, which is a one-dimensional discrete quantum system built on Zn-symmetric clock states, which generalises the Ising spin chain. This model has a special exact solution generalising free fermions. The model is non-Hermitian, which gives it a complex spectrum and other unusual behaviour, but is also PT-symmetric, which can serve as an analog of Hermiticity and allows a definition of unitary time evolution. In this talk, I report on a newly-discovered series of exceptional points where the Jordan structure of the Hamiltonian becomes non-trivial. These exceptional points accumulate near the critical point and explain some of the model's unusual behaviour. Some other results are also described including a new free parafermionic model constructed with different generalised operators, some results about the free parafermion model with periodic boundaries (which is surprisingly nontrivial), and a determination of the phase diagram of a staggered Heisenberg XX model using Majorana zero modes.


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