Topological phases in two-dimensional arrays of parafermionic zero modes

TL;DR

This paper explores two-dimensional arrays of parafermionic zero modes at fractional topological insulators' edges, revealing how geometry influences topological order and duality to lattice gauge theories.

Contribution

It derives low-energy Hamiltonians for different array geometries, showing their impact on topological phases and dualities in parafermionic systems.

Findings

Topologically ordered phase with qudit toric code in certain geometries

Duality to Abelian lattice gauge theory in other geometries

Edge length scaling determines topological properties

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.