Parafermions in fractional Chern insulator-superconductor heterostructures: the role of spin polarization

Abstract: Most proposals for $Z_3$ parafermions in fractional quantum Hall-superconductor structures used the spin-unpolarized $ν= 2/3$ Halperin $(1,1,2)$ state. The fractional quantum anomalous Hall (FQAH) states of twisted MoTe$_2$ and rhombohedral graphene are believed to be spin- and valley-polarized Jain states, with the same topological order but a different spin structure. We analyze a trench between two polarized $ν= 2/3$ edges. Among the pairing and tunneling bilinears of the edge-mode electrons, exactly two of each have zero conformal spin. Pairing is compatible only with the neutral-mode superconductivity channel and tunneling only with neutral-mode backscattering, whereas in the unpolarized state singlet pairing and tunneling select the same channel. Since the two channels condense the same neutral anyon up to a local operator, the domain walls carry $Z_3$ parafermions, with 3 protected states per pair, whether or not the neutral channel changes between regions; fixing the fermion parity does not enlarge this count. Before the neutral gap forms, pairing is at best marginal for repulsive inter-edge coupling unless an attractive neutral coupling favors its channel; once the gap has formed, both charge terms are relevant for an inter-edge Coulomb coupling below 7/25 of the intra-edge one. The superconductor must supply even-frequency, equal-spin, intravalley pairing, which an s-wave singlet superconductor with only Ising spin-orbit coupling does not provide at leading order, whereas chiral $p+ip$ and $f+if$ states do. The $Z_3$ label is carried by a quasiparticle of the FQAH state, so no process confined to a junction whose tunneling segments are not FQAH can change it, and we expect a $6π$-periodic Josephson current with or without fixed fermion parity.