A Unified Framework of Topological Phases with Symmetry

Abstract

In topological phases in $2+1$ dimensions, anyons fall into representations of quantum group symmetries. As proposed in our work (arXiv:1308.4673), physics of a symmetry enriched phase can be extracted by the Mathematics of (hidden) quantum group symmetry breaking of a "parent phase". This offers a unified framework and classification of the symmetry enriched (topological) phases, including symmetry protected trivial phases as well. In this paper, we extend our investigation to the case where the "parent" phases are non-Abelian topological phases. We show explicitly how one can obtain the topological data and symmetry transformations of the symmetry enriched phases from that of the "parent" non-Abelian phase. Two examples are computed: (1) the $\text{Ising}\times \overline{\text{Ising}}$ phase breaks into the ${\mathbb{Z}}_2$ toric code with ${\mathbb{Z}}_2$ global symmetry; (2) the $SU(2{)}_8$ phase breaks into the chiral Fibonacci $\times$ Fibonacci phase with a ${\mathbb{Z}}_2$ symmetry, a first non-Abelian example of symmetry enriched topological phase beyond the gauge theory construction.

18 pages. Minor changes made, and references added

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Cited by in corpus (20)

• Anyon condensation and its applications
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• Boson Condensation in Topologically Ordered Quantum Liquids
• Cage-Net Fracton Models
• Ground State Degeneracy of Topological Phases on Open Surfaces
• Generalized ADE Classification of Gapped Domain Walls
• Electric-magnetic duality in the quantum double models of topological orders with gapped boundaries
• Entanglement Entropy, Quantum Fluctuations, and Thermal Entropy in Topological Phases
• Symmetry Enrichment in Three-Dimensional Topological Phases
• Fermion Condensation and Gapped Domain Walls in Topological Orders
• Symmetry fractionalization and anomaly detection in three-dimensional topological phases
• Anyon Condensation: Coherent states, Symmetry Enriched Topological Phases, Goldstone Theorem, and Dynamical Rearrangement of Symmetry
• Symmetry reduction induced by anyon condensation: a tensor network approach
• Symmetry Fractionalized (Irrationalized) Fusion Rules and Two Domain-Wall Verlinde Formulae
• Characteristic Properties of a Composite System of Topological Phases Separated by Gapped Domain Walls via an Exactly Solvable Hamiltonian Model
• Topological symmetry breaking: Domain walls and partial instability of chiral edges
• Time reversal invariant gapped boundaries of the double semion state
• Nonabelian Anyon Condensation in 2+1d topological orders: A String-Net Model Realization
• Nonlinear Symmetry-Fragmentation of Nonabelian Anyons In Symmetry-Enriched Topological Phases: A String-Net Model Realization
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries