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 generalize the intimate connection between the Z(2)-symmetric Ising quantum spin chain and Majorana fermions to Z(3)-symmetric chains and parafermions. In particular, we highlight how parafermion chains 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.