Anomaly Indicators for Time-Reversal Symmetric Topological Orders

Some time-reversal symmetric topological orders are anomalous in that they cannot be realized in strictly two-dimensional systems; instead, they can only be realized on the surface of three-dimensional symmetry-protected topological phases. We propose two quantities, which we call 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, ${\eta }_2$, applies to bosonic systems while the second indicator, ${\eta }_f$, applies to fermionic systems in the DIII class. We conjecture that ${\eta }_2$, together with a previously known indicator ${\eta }_1$, can detect the two known ${\mathbb{Z}}_2$ anomalies in the bosonic case, while ${\eta }_f$ can detect the ${\mathbb{Z}}_{16}$ anomaly in the fermionic case.