Topological phases in gapped edges of fractionalized systems

Recently, it has been proposed that exotic one-dimensional phases can be realized by gapping out the edge states of a fractional topological insulator. The low-energy edge degrees of freedom are described by a chain of coupled parafermions. We introduce a classification scheme for the phases that can occur in parafermionic chains. We find that the parafermions support both topological symmetry fractionalized phases as well as phases in which the parafermions condense. In the presence of additional symmetries, the phases form a non-Abelian group. As a concrete example of the classification, we consider the effective edge model for a $\nu= 1/3$ fractional topological insulator for which we calculate the entanglement spectra numerically and show that all possible predicted phases can be realized.