Symmetry fractionalization and anomaly detection in three-dimensional topological phases

Abstract

In a phase with fractional excitations, topological properties are enriched in the presence of global symmetry. In particular, fractional excitations can transform under symmetry in a fractionalized manner, resulting in different Symmetry Enriched Topological (SET) phases. While a good deal is now understood in $2D$ regarding what symmetry fractionalization patterns are possible, the situation in $3D$ is much more open. A new feature in $3D$ is the existence of loop excitations, so to study $3D$ SET phases, first we need to understand how to properly describe the fractionalized action of symmetry on loops. Using a dimensional reduction procedure, we show that these loop excitations exist as the boundary between two $2D$ SET phases, and the symmetry action is characterized by the corresponding difference in SET orders. Moreover, similar to the $2D$ case, we find that some seemingly possible symmetry fractionalization patterns are actually anomalous and cannot be realized strictly in $3D$. We detect such anomalies using the flux fusion method we introduced previously in $2D$. To illustrate these ideas, we use the $3D$ $Z_2$ gauge theory with $Z_2$ global symmetry as an example, and enumerate and describe the corresponding SET phases. In particular, we find four non-anomalous SET phases and one anomalous SET phase, which we show can be realized as the surface of a $4D$ system with symmetry protected topological order.

19 pages, 8 figures

References in corpus (12)

• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• A classification of symmetry enriched topological phases with exactly solvable models
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Anomalies of discrete symmetries in three dimensions and group cohomology
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
• Space group symmetry fractionalization in a chiral kagome Heisenberg antiferromagnet
• Walker-Wang models and axion electrodynamics
• Classification and detection of symmetry fractionalization in chiral spin liquids
• Symmetry Fractionalization in Three-Dimensional ${\mathbb{Z}}_2$ Topological Order and Fermionic Symmetry-Protected Phases

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