Topological phases of parafermionic chains with symmetries

We study the topological classification of parafermionic chains in the presence of a modified time reversal symmetry that satisfies T 2 1 Such chains can be realized in one dimensional structures embedded in fractionalized two dimensional states of matter e g at the edges of a fractional quantum spin Hall system where counter propagating modes may be gapped either by back scattering or by coupling to a superconductor In the absence of any additional symmetries a chain of Zm parafermions can belong to one of several distinct phases We find that when the modified time reversal symmetry is imposed the classification becomes richer If m is odd each of the phases splits into two subclasses We identify the symmetry protected phase as a Haldane phase that carries a Kramers doublet at each end When m is even each phase splits into four subclasses The origin of this split is in the emergent Majorana fermions associated with even values of m We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituents particles are either fermions or bosons