Classification of 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 counterpropagating modes may be gapped either by backscattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of Z(m) 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 constituent particles are either fermions or bosons.