Folding approach to topological orders enriched by mirror symmetry

Abstract

We develop a folding approach to study two-dimensional symmetry-enriched topological (SET) phases with the mirror reflection symmetry. Our folding approach significantly transforms the mirror SETs, such that their properties can be conveniently studied through previously known tools: (i) it maps the nonlocal mirror symmetry to an onsite ${\mathbb{Z}}_2$ layer-exchange symmetry after folding the SET along the mirror axis, so that we can gauge the symmetry; (ii) it maps all mirror SET information into the boundary properties of the folded system, so that they can be studied by the anyon condensation theory---a general theory for studying gapped boundaries of topological orders; and (iii) it makes the mirror anomalies explicitly exposed in the boundary properties, i.e., strictly 2D SETs and those that can only live on the surface of a 3D system can be easily distinguished through the folding approach. With the folding approach, we derive a set of physical constraints on data that describes mirror SET, namely mirror permutation and mirror symmetry fractionalization on the anyon excitations in the topological order. We conjecture that these constraints may be complete, in the sense that all solutions are realizable in physical systems. Several examples are discussed to justify this. Previously known general results on the classification and anomalies are also reproduced through our approach.

73 pages, 5 figures; v2: updated to the published version

References in corpus (22)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• 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
• Condensate induced transitions between topologically ordered phases
• 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
• Topological phases protected by point group symmetry
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Symmetry enriched string-nets: Exactly solvable models for SET phases
• Exactly Solvable Models for Symmetry-Enriched Topological Phases
• Topological invariants for gauge theories and symmetry-protected topological phases
• Constraints on topological order in Mott Insulators
• Anomaly indicators for time-reversal symmetric topological orders
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• More on time-reversal anomaly of 2+1d topological phases
• Detecting crystal symmetry fractionalization from the ground state: Application to ${\mathbb{Z}}_2$ spin liquids on the kagome lattice
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• Fermion Condensation and Gapped Domain Walls in Topological Orders
• Space group symmetry fractionalization in a family of exactly solvable models with Z2 topological order
• Anomalies and symmetry fractionalization in reflection-symmetric topological order

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• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• Pyrochlore U(1) spin liquid of mixed symmetry enrichments in magnetic fields
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Average-exact mixed anomalies and compatible phases
• Classification of symmetry-enriched topological quantum spin liquids
• Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase
• Mirror anomaly in fermionic topological orders
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Anomalies in mirror symmetry enriched topological orders
• Loops in 4+1d Topological Phases