Fermion parity flips and Majorana bound states at twist defects in superconducting fractional topological phases

In this paper, we consider a layered heterostructure of an Abelian topologically ordered state (TO), such as a fractional Chern insulator (FCI)/quantum Hall state (FQH) with an s-wave superconductor in order to explore the existence of non-Abelian defects. To uncover such defects, we note that the ground state corresponds to a charge 2e Cooper pair, the electron can no longer be treated as a local particle, and hence we must consider a larger TO due to the presence of h/2e flux vortices, which strictly speaking are not deconfined. Quantum dimension and species of the defects follow directly from the fusion algebra. For FCI/Laughlin states, we show the presence of three kinds of defects, two of which had been previously ignored. They owe their origin to a general anyon permutation symmetry (AS) that exists in any fermionic Abelian TO state in contact with an s-wave superconductor. Physically, this permutation corresponds to adding a fermion to odd flux vortices (in units of h/2e) as they travel around the associated topological (twist) defect. As such, we call it a fermion parity flip AS. We show that calculations can be handled more simply, by considering an equivalent fermion parity gauged theory, where the original TO is suitably augmented by a Z2 gauge sector coming from the s-wave SC, but with identical fusion structure. This trick makes our approach useful for analyzing a wide variety of FQH/FCI heterostructures. We give examples of the fermion parity gauging procedure for a large number of hierarchy and spin singlet states. We consider twist defects which mutate anyons according to the fermion parity flip symmetry and show that they can be realized at domain walls between distinct gapped edges or interfaces of the TO superconducting state. We analyze the properties of such defects and show that fermion parity flip twist defects are always associated with Majorana zero modes. When defects corresponding to AS which is a combination of fermion parity flip and charge conjugation are considered, they lead to Z2n+1 parafermions in Laughlin 1/(2n+1) states. Our formalism also reproduces known results such as Majorana/parafermionic bound states at superconducting domain walls of topological/fractional Chern insulators when twist defects are constructed based on charge conjugation symmetry. Finally, we briefly describe more exotic twist liquid phases obtained by gauging the AS where the twist defects become deconfined anyonic excitations.