Topological phases in gapped edges of fractionalized systems

Abstract

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.

11 pages, 7 figures, final version

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• Majorana bound states in semiconducting nanostructures
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• Parafermionic clock models and quantum resonance
• Commensurate and Incommensurate States of Topological Quantum Matter
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• Fragility of Symmetry Protected Topological Order on a Hubbard Ladder
• Non-topological parafermions in a one-dimensional fermionic model with even multiplet pairing
• Globally symmetric topological phase: from anyonic symmetry to twist defect
• Parafermionic wires at the interface of chiral topological states
• Localization-protected order in spin chains with non-Abelian discrete symmetries
• Reflection Positivity for Parafermions
• Parafermionization, bosonization, and critical parafermionic theories
• Free-fermion descriptions of parafermion chains and string-net models
• Matrix product states for topological phases with parafermions
• Families of Gapped Interfaces Between Fractional Quantum Hall States
• Entanglement entropy and massless phase in the antiferromagnetic three-state quantum chiral clock model
• Topological phases of parafermionic chains with symmetries
• Model wavefunctions for interfaces between lattice Laughlin states
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• Accessing topological order in fractionalized liquids with gapped edges
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• Dissipation without resistance: Imaging impurities at quantum Hall edges
• Post quench entropy growth in a chiral clock model
• Classifying parafermionic gapped phases using matrix product states
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• Interrelations among frustration-free models via Witten's conjugation
• Spin chains, defects, and quantum wires for the quantum-double edge
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• Emergent Haldane phase in an alternating bond ${\mathbb{Z}}_3$ parafermion chain
• Phase diagram of an extended parafermion chain