Topological phases of parafermionic chains with symmetries

Abstract

We study the topological classification of parafermionic chains in the presence of a modified time reversal symmetry that satisfies ${\mathcal{T}}^2=1$. 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 ${\mathbb{Z}}_m$ 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 constituents particles are either fermions or bosons.

13 pages, 4 figures

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Cited by in corpus (9)

• Gapless topological phases and symmetry-enriched quantum criticality
• Fermionized parafermions and symmetry-enriched Majorana modes
• Non-topological parafermions in a one-dimensional fermionic model with even multiplet pairing
• Parafermionization, bosonization, and critical parafermionic theories
• Coupled Wire Model of Z4 Orbifold Quantum Hall States
• Majorana zero modes and their bosonization
• Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures
• Classifying parafermionic gapped phases using matrix product states
• Enriched classification of parafermionic gapped phases with time reversal symmetry