Signatures of parafermion zero modes in fractional quantum Hall–superconductor heterostructures

Parafermion zero modes can arise in hybrid structures composed of $\ensuremath{\nu}=1/m$ fractional quantum Hall edges' proximity coupled by an $s$-wave superconductor. Here we consider a Josephson junction formed in such hybrid structures in addition to parafermion tunneling, Cooper pair tunneling, and backscattering. We find that the $4\ensuremath{\pi}m$ periodicity due to parafermion-only tunneling reduces, in the presence of backscattering, to $4\ensuremath{\pi}$-periodic at zero temperature and $2\ensuremath{\pi}$-periodic at finite temperature unless the fermion parity is fixed. Nevertheless, a clear signature of parafermion tunneling remains in the shape of the current-phase relation.