Anomaly indicators for topological orders with $U(1)$ and time-reversal symmetry

Abstract

We study anomalies in time-reversal (${\mathbb{Z}}_2^T$) and $U(1)$ symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-D system but can be realized on the surface of a $(3+1)$-D symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators -- functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e. $U(1)⋊{\mathbb{Z}}_2^T$ and $U(1)\times {\mathbb{Z}}_2^T$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)$-D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)$-D SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U(1)$ symmetry.

arXiv commentv2: 21 pages, 2 figures, minor changes from the first version. Version published in PRB. v3: corrected a minor error in the proof that ${\eta }_{\text{ETI}}=1$ for Abelian topological orders

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• Relative Anomalies in (2+1)D Symmetry Enriched Topological States
• Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions
• Symmetry-preserving boundary of (2+1)D fractional quantum Hall states
• Minimal Fractional Topological Insulator in half-filled conjugate moiré Chern bands
• Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries
• Fractionalization as an alternate to charge ordering in electronic insulators
• Mirror anomaly in fermionic topological orders
• (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
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Ordering the topological order in the fractional quantum Hall effect
• Higher Hall conductivity from a single wave function: Obstructions to symmetry-preserving gapped edge of (2+1)D topological order