Strong superconducting fluctuations in proximitized fractional (anomalous) Hall edge states

The recent discoveries of

fractional quantum anomalous Hall states in materials such as transition metal dichalcogenides and penta-layer graphene suggest that heterostructures of fractional Hall edge states and superconductors will be experimentally much more realistic. It has been theorized that such heterostructures could host parafermions of interest for topological quantum computing.

Building on these developments, we explore a Z₃ parafermion chain that can be realized using FQH states, subject to fluctuations in the superconducting order parameter. By employing a combination of analytical techniques and numerical methods, including density matrix renormalization group (DMRG), we construct the phase diagram and examine critical behaviour as a function of system parameters. We find various Mott insulating phases and two gapless phases — one with excitations of charge 2e/3 and one with excitations of minimal charge 2e. We compare our results for the transition between these states with the conjecture that the U(1)⋉Z₃ model flows to an emergent SU(2)₃ theory and discuss the appearance of parafermionic domain wall states beyond mean field superconductivity.