Superconducting proximity effect on the edge of fractional topological insulators

We study the superconducting proximity effect on the helical edge states of time-reversal-symmetric fractional topological insulators (FTI). The Cooper pairing of physical electrons results in many-particle condensation of the fractionalized excitations on the edge. We find localized zero-energy modes emerge at interfaces between superconducting regions and magnetically insulating regions, which are responsible for the topological degeneracy of the ground states. By mapping the low-energy effective Hamiltonian to the quantum chiral Potts model, we determine the operator algebra of the zero modes and show that they exhibit nontrivial braiding properties. We then demonstrate that the Josephson current in the junction between superconductors mediated by the edge states of the FTI exhibit fractional Josephson effect with period as multiples of 4 pi.