Parafermion zero modes and fractional Josephson tunneling in fractional quantum Hall-superconductor hybrid structures

We consider a hybrid structure of two narrow fractional topological superconductors, consisting of a Laughlin ν=1/m fractional quantum Hall state with two narrow s-wave superconductors. The edge states of the Laughlin state become superconducting by the proximity effect. The effective field theory for such system consists of two sine-Gordon theories coupled by parafermion tunneling. We identify the parafermion zero modes at the ends of fractional topological superconductors as "half" of a quasiparticle pair. We compute the parafermion tunneling current and show that it exhibits an anomalous periodicity of 4mπ in the Josephson phase which is a tell-tale signature of the existence of parafermion zero modes. We also discuss other signatures such as the dissipative part of the admittance which is accessible to experiment.