Parafermions in polarized $ν=2/3$ Fractional Quantum (Anomalous) Hall Heterostructures

Abstract: Motivated by recent experimental signatures of fractional Chern insulators in 2D quantum materials, we investigate the realization of $\mathbb Z_3$ parafermion zero modes (PZMs) on edges of spin-polarized $ν=2/3$ fractional quantum Hall states which are proximitized by superconductors. Unlike the spin-unpolarized Halperin state, the polarized $ν=2/3$ edge contains counterpropagating electron and fractional quasihole modes, giving rise to a plethora of competing tunneling and pairing processes. We focus on the vicinity of the fixed point where these modes reorganize into decoupled charge and neutral sectors, with the neutral sector describing an emergent dipole degree of freedom. Using bosonization and renormalization group analysis, we lay out the parameter space in which the interedge dipole interactions gap the neutral sector to stabilize one dominant tunneling and one dominant pairing channel resulting in a double sine-Gordon model in the charge sector which supports $\mathbb{Z}_3$ PZMs at domain walls. We discuss both clean and disordered systems. In clean systems this mechanism requires a certain degree of control of the interaction strength and range. In contrast, disorder can provide a more generic route: While short-ranged, edge-uncorrelated disorder % on the two edges suppresses the simultaneous pinning required for stable parafermion modes, when the disorder on the two edges is sufficiently correlated, as expected for smooth disorder potentials, the parameter regime hosting PZMs increases.