Topological phases with parafermions: theory and blueprints

We concisely review the recent evolution in the study of parafermions -- exotic emergent excitations that generalize Majorana fermions and similarly underpin a host of novel phenomena. First we illustrate the intimate connection between Z_3-symmetric "spin" chains and one-dimensional parafermion lattice models, highlighting how the latter host a topological phase featuring protected edge zero modes. We then tour several blueprints for the laboratory realization of parafermion zero modes -- focusing on quantum Hall/superconductor hybrids, quantum Hall bilayers, and two-dimensional topological insulators -- and describe striking experimental fingerprints that they provide. Finally, we discuss how coupled parafermion arrays in quantum Hall architectures yield topological phases that potentially furnish hardware for a universal, intrinsically decoherence-free quantum computer.

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