Topological Phases in gapped edges of fractionalized systems

We present an extension of the classification scheme for topological phases in interacting one-dimensional fermionic systems to parafermionic chains. We find that the parafermions support both topological as well as symmetry broken phases in which the parafermions condense. In a series of recent works an experimental way of creating parafermions had been proposed: they can arise on the edge of a two-dimensional fractional topological insulator when coupled to superconducting and ferromagnetic domains. The low-energy edge degrees of freedom are described by a chain of coupled parafermions. As a concrete example of our classification we consider the $\nu=1/3$ fractional topological insulator for which we calculate the phase diagram and study the entanglement spectra. We furthermore discuss a concrete physical realization which allows us to tune between the different topological phases.