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Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter

Maissam Barkeshli, Meng Cheng

TL;DR

The work analyzes symmetry-localization anomalies in (2+1)D topological phases with $\mathbb{Z}_2^{\bf T}$ or $\mathbb{Z}_2^{\bf R}$, formalized as an obstruction class $[\textswab{O}]\in \mathcal{H}^3_{[\rho]}(G,\mathcal{A})$ that can obstruct consistent symmetry localization to quasiparticles. It provides practical, modular-data-based diagnostics and shows that many theories (notably USp$(4)_2$ and SO$(4)_4$ CS theories) harbor such anomalies, which can be resolved by enlarging the symmetry to $\mathbb{Z}_4^{\bf T}$, reinterpretation as fermionic spin theories with $T^2=(-1)^{N_f}$, or by realizing the theory as a surface state of a (3+1)D SET via pseudo-realization. The paper also demonstrates an infinite family of root theories and gauging procedures that generate further anomalous descendants, linking the anomalies to higher-dimensional bulk physics and to 2-group symmetry perspectives. These results illuminate when (2+1)D topological phases with parity-reversing symmetries can exist as standalone 2D systems and when they must be realized at the boundary of higher-dimensional topological states.

Abstract

We study a class of anomalies associated with time-reversal and spatial reflection symmetry in (2+1)D topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a $\mathbb{Z}_2$ symmetry which can be associated with either time-reversal (denoted $\mathbb{Z}_2^{\bf T})$ or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a notion of symmetry fractionalization. We present simple sufficient conditions which determine when $\mathbb{Z}_2^{\bf T}$ symmetry localization anomalies exist. We present an infinite series of TQFTs with such anomalies, some examples of which include USp$(4)_2$ and SO$(4)_4$ Chern-Simons (CS) theory. The theories that we find with these $\mathbb{Z}_2^{\bf T}$ anomalies can be obtained by gauging the unitary $\mathbb{Z}_2$ subgroup of a different TQFT with a $\mathbb{Z}_4^{\bf T}$ symmetry. We show that the anomaly can be resolved in several ways: (1) the true symmetry of the theory is $\mathbb{Z}_4^{\bf T}$, or (2) the theory can be considered to be a theory of fermions, with ${\bf T}^2 = (-1)^{N_f}$ corresponding to fermion parity. Finally, we demonstrate that theories with the $\mathbb{Z}_2^{\bf T}$ localization anomaly can be compatible with $\mathbb{Z}_2^{\bf T}$ if they are "pseudo-realized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudo-realization" refers to the fact that the bulk (3+1)D system is described by a dynamical $\mathbb{Z}_2$ gauge theory and thus only a subset of the quasiparticles are confined to the surface.

Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter

TL;DR

The work analyzes symmetry-localization anomalies in (2+1)D topological phases with or , formalized as an obstruction class that can obstruct consistent symmetry localization to quasiparticles. It provides practical, modular-data-based diagnostics and shows that many theories (notably USp and SO CS theories) harbor such anomalies, which can be resolved by enlarging the symmetry to , reinterpretation as fermionic spin theories with , or by realizing the theory as a surface state of a (3+1)D SET via pseudo-realization. The paper also demonstrates an infinite family of root theories and gauging procedures that generate further anomalous descendants, linking the anomalies to higher-dimensional bulk physics and to 2-group symmetry perspectives. These results illuminate when (2+1)D topological phases with parity-reversing symmetries can exist as standalone 2D systems and when they must be realized at the boundary of higher-dimensional topological states.

Abstract

We study a class of anomalies associated with time-reversal and spatial reflection symmetry in (2+1)D topological phases of matter. In these systems, the topological quantum numbers of the quasiparticles, such as the fusion rules and braiding statistics, possess a symmetry which can be associated with either time-reversal (denoted or spatial reflections. Under this symmetry, correlation functions of all Wilson loop operators in the low energy topological quantum field theory (TQFT) are invariant. However, the theories that we study possess a severe anomaly associated with the failure to consistently localize the symmetry action to the quasiparticles, precluding even defining a notion of symmetry fractionalization. We present simple sufficient conditions which determine when symmetry localization anomalies exist. We present an infinite series of TQFTs with such anomalies, some examples of which include USp and SO Chern-Simons (CS) theory. The theories that we find with these anomalies can be obtained by gauging the unitary subgroup of a different TQFT with a symmetry. We show that the anomaly can be resolved in several ways: (1) the true symmetry of the theory is , or (2) the theory can be considered to be a theory of fermions, with corresponding to fermion parity. Finally, we demonstrate that theories with the localization anomaly can be compatible with if they are "pseudo-realized" at the surface of a (3+1)D symmetry-enriched topological phase. The "pseudo-realization" refers to the fact that the bulk (3+1)D system is described by a dynamical gauge theory and thus only a subset of the quasiparticles are confined to the surface.

Paper Structure

This paper contains 30 sections, 104 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Review of the $\mathcal{H}^3_{[\rho]}(G,\mathcal{A})$ anomaly
  3. Review of UMTC notation
  4. Topological symmetry and braided auto-equivalence
  5. Global symmetry
  6. $\mathcal{H}_{[\rho]}^3(G, \mathcal{A})$ obstruction
  7. Symmetry fractionalization
  8. Example: $\mathrm{USp}(4)_2$ Chern-Simons Theory
  9. Sufficient conditions for the presence of $\mathcal{H}^3_{[\rho]}(\mathbb{Z}_2^{\bf T},\mathcal{A})$ anomaly
  10. Overview
  11. Condition (1): (3+1)D Path integral on $\mathbb{RP}^4$
  12. Condition (2): constraints from $\mathbb{RP}^2 \times D^2$
  13. Condition (3): Topological twists and $M_a$
  14. Conflicting constraints on symmetry fractionalization quantum numbers
  15. Non-anomalous cousins
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Illustration of the construction of the anomalous double semion state on the surface of a $\mathbb{Z}_2$ toric code on the mirror plane. Left panel: 3D setup, where a $\mathbb{Z}_2$ gauge theory harbors the mirror plane. Right panel: top view of the surface.
  • Figure 2: Illustration of the layer construction. Each vertical layer denotes a $\mathbb{Z}_2$ toric code, and neighboring layers are coupled such that pairs of $e$ particles are condensed. The surface consists of a double semion state, joined to the vertical layers at each (1+1)D intersection by condensing $b \times e$.