Symmetry Fractionalization, Defects, and Gauging of Topological Phases

Abstract

We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological quantum numbers of a topological phase, and we describe its relation with the microscopic symmetry of the underlying physical system. We derive a general framework to characterize and classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter $\mathcal{C}$ with symmetry group $G$. The algebraic theory of the defects, known as a $G$-crossed braided tensor category ${\mathcal{C}}_G^{\times }$, allows one to compute many properties, such as the number of topologically distinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes, braiding exchange transformations, a generalized Verlinde formula for the defects, and modular transformations of the $G$-crossed extensions of topological phases. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic $G$-defects are turned into deconfined quasiparticle excitations, which results in a different topological phase $({\mathcal{C}}_G^{\times }{)}^G$. A number of instructive and/or physically relevant examples are studied in detail.

108 pages, 22 figures; v4: minor corrections and updated references; v3: Additional material on topological charge conjugation symmetry, complete solutions of $G$-crossed extensions for symmetries that do not permute quasiparticle types, and examples; many clarifications, minor corrections, notation changes, and updates throughout

References in corpus (43)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Non-Abelian Anyons and Topological Quantum Computation
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Classification of topological insulators and superconductors in three spatial dimensions
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Condensate induced transitions between topologically ordered phases
• Duality and defects in rational conformal field theory
• Measurement-Only Topological Quantum Computation
• A classification of symmetry enriched topological phases with exactly solvable models
• Genons, twist defects, and projective non-Abelian braiding statistics
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
• Superconducting Proximity Effect on the Edge of Fractional Topological Insulators
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Translational symmetry and microscopic constraints on symmetry-enriched topological phases: a view from the surface
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Fractional topological superconductors with fractionalized Majorana fermions
• Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
• Anomalous Symmetry Fractionalization and Surface Topological Order
• Wire deconstructionism of two-dimensional topological phases
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• A theory of topological edges and domain walls
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Interferometry of non-Abelian Anyons
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Generalizations and limitations of string-net models
• Measurement-Only Topological Quantum Computation via Anyonic Interferometry
• Universal quantum computation with weakly integral anyons
• On Gauging Symmetry of Modular Categories
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions
• Constraints on topological order in Mott Insulators
• Measurement-Only Topological Quantum Computation via Tunable Interactions
• Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases
• Coherent Transmutation of Electrons into Fractionalized Anyons
• Universal Gates via Fusion and Measurement Operations on SU$(2{)}_4$ Anyons
• Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification
• Bridging Fermionic and Bosonic Short Range Entangled States
• Decoherence of Anyonic Charge in Interferometry Measurements
• Accessing topological order in fractionalized liquids with gapped edges

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