Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Abstract

Certain patterns of symmetry fractionalization in (2+1)D topologically ordered phases of matter can be anomalous, which means that they possess an obstruction to being realized in purely (2+1)D. In this paper we demonstrate how to compute the anomaly for symmetry-enriched topological (SET) states of bosons in complete generality. We demonstrate how, given any unitary modular tensor category (UMTC) and symmetry fractionalization class for a global symmetry group $G$, one can define a (3+1)D topologically invariant path integral in terms of a state sum for a $G$ symmetry-protected topological (SPT) state. We present an exactly solvable Hamiltonian for the system and demonstrate explicitly a (2+1)D $G$ symmetric surface termination that hosts deconfined anyon excitations described by the given UMTC and symmetry fractionalization class. We present concrete algorithms that can be used to compute anomaly indicators in general. Our approach applies to general symmetry groups, including anyon-permuting and anti-unitary symmetries. In addition to providing a general way to compute the anomaly, our result also shows, by explicit construction, that every symmetry fractionalization class for any UMTC can be realized at the surface of a (3+1)D SPT state. As a byproduct, this construction also provides a way of explicitly seeing how the algebraic data that defines symmetry fractionalization in general arises in the context of exactly solvable models. In the case of unitary orientation-preserving symmetries, our results can also be viewed as providing a method to compute the ${\mathcal{H}}^4(G,U(1))$ obstruction that arises in the theory of $G$-crossed braided tensor categories, for which no general method has been presented to date.

39+18 pages, 20+13 figures

References in corpus (12)

• Non-Abelian Anyons and Topological Quantum Computation
• Generalized Global Symmetries
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Topological invariants for gauge theories and symmetry-protected topological phases
• Anomalies of discrete symmetries in various dimensions and group cohomology
• Bosonic Topological Insulators and Paramagnets: a view from cobordisms
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Mirror anomaly in fermionic topological orders
• Anomaly indicator of rotation symmetry in (3+1)D topological order

Cited by in corpus (29)

• Global Anomalies on the Hilbert Space
• Symmetric Mass Generation
• Classification of topological phases with finite internal symmetries in all dimensions
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Non-invertible Symmetries and Higher Representation Theory II
• Triangular lattice quantum dimer model with variable dimer density
• Towards a classification of mixed-state topological orders in two dimensions
• Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
• Fermionic symmetry fractionalization in (2+1)D
• Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories
• Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTs
• On the Symmetry TFT of Yang-Mills-Chern-Simons theory
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Anomaly cascade in (2+1)D fermionic topological phases
• Exactly solvable models for U(1) symmetry-enriched topological phases
• Computing classification of interacting fermionic symmetry-protected topological phases using topological invariants
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Classification of symmetry-enriched topological quantum spin liquids
• Towards topological fixed-point models beyond gappable boundaries
• Anomaly constraint on chiral central charge of (2+1)d topological order
• (3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Soft symmetries of topological orders
• One-dimensional symmetric phases protected by frieze symmetries
• Les Houches Lecture Notes on Tensor Networks
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Planar fault-tolerant circuits for non-Clifford gates on the 2D color code