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

We study a class of anomalies associated with time-reversal and spatial-reflection symmetry in (2+1)-dimensional bosonic 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 ${\mathbb{Z}}_2^{\mathbf{T}})$ 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 consistent notion of symmetry fractionalization in such systems. We present simple sufficient conditions which determine when ${\mathbb{Z}}_2^{\mathbf{T}}$ symmetry localization anomalies exist in general. We present an infinite series of TQFTs with such anomalies, some examples of which include $\mathrm{USp}{\left(4\right)}_2$ Chern-Simons (CS) theory and $\mathrm{SO}{\left(4\right)}_4$ CS theory. The theories that we find with these ${\mathbb{Z}}_2^{\mathbf{T}}$ anomalies can all be obtained by gauging the unitary ${\mathbb{Z}}_2$ subgroup of a different TQFT with a ${\mathbb{Z}}_4^{\mathbf{T}}$ symmetry. We further show that the anomaly can be resolved in several distinct ways: (1) the true symmetry of the theory is ${\mathbb{Z}}_4^{\mathbf{T}}$, or (2) the theory can be considered to be a theory of fermions, with ${\mathbf{T}}^2={(-1)}^{N_f}$ corresponding to fermion parity. Finally, we demonstrate that theories with the ${\mathbb{Z}}_2^{\mathbf{T}}$ localization anomaly can be compatible with ${\mathbb{Z}}_2^{\mathbf{T}}$ if they are “pseudorealized” at the surface of a (3+1)D symmetry-enriched topological phase. The “pseudorealization” 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 truly confined to the surface.

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 $b\times e$.