Topological phases in two-dimensional arrays of parafermionic zero modes

Abstract

It has recently been realized that zero modes with projective non-Abelian statistics, generalizing the notion of Majorana bound states, may exist at the interface between a superconductor and a ferromagnet along the edge of a fractional topological insulator (FTI). Here we study two-dimensional architectures of these non-Abelian zero modes, whose interactions are generated by the charging and Josephson energies of the superconductors. We derive low-energy Hamiltonians for two different arrays of FTIs on the plane, revealing an interesting interplay between the real-space geometry of the system and its topological properties. On the one hand, in a geometry where the length of the FTI edges is independent on the system size, the array has a topologically ordered phase, giving rise to a qudit toric code Hamiltonian in perturbation theory. On the other hand, in a geometry where the length of the edges scales with system size, we find an exact duality to an Abelian lattice gauge theory and no topological order.

18 pages, 11 figures; new version with additional comments, references, figure and appendix

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Cited by in corpus (39)

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• Majorana nanowires for topological quantum computation
• Stability of zero modes in parafermion chains
• Superconducting analogue of the parafermion fractional quantum Hall states
• Phase diagram of the $Z_3$ Parafermionic Chain with Chiral Interactions
• Quantum Computing with Parafermions
• Fock Parafermions and Self-Dual Representations of the Braid Group
• Assembling Fibonacci Anyons From a ${\mathbb{Z}}_3$ Parafermion Lattice Model
• Anyons in Multichannel Kondo Systems
• Generalized Kitaev Models and Slave Genons
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• Modeling electron fractionalization with unconventional Fock spaces
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• Topological ordering in the Majorana Toric Code
• Disorder-induced topological superconductivity in a spherical quantum-Hall--superconductor hybrid
• Repulsive-Interaction-Driven Topological Superconductivity in a Landau Level Coupled to an $s$-Wave Superconductor