Anomaly indicators for time-reversal symmetric topological orders

Abstract

Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensions without breaking time reversal symmetry; instead, they can only be realized on the surface of certain three-dimensional systems. We propose two quantities, which we call {\it anomaly indicators}, that can detect if a time-reversal symmetric topological order is anomalous in this sense. Both anomaly indicators are expressed in terms of the quantum dimensions, topological spins, and time-reversal properties of the anyons in the given topological order. The first indicator, $\hat{I}{\cdot }_2$, applies to bosonic systems while the second indicator, $\hat{I}{\cdot }_f$, applies to fermionic systems in the DIII class. We conjecture that $\hat{I}{\cdot }_2$, together with a previously known indicator $\hat{I}{\cdot }_1$, can detect the two known ${\mathbb{Z}}_2$ anomalies in the bosonic case, while $\hat{I}{\cdot }_f$ can detect the ${\mathbb{Z}}_{16}$ anomaly in the fermionic case.

11 pages