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, η2, applies to bosonic systems while the second indicator, ηf, applies to fermionic systems in the DIII class. We conjecture that η2, together with a previously known indicator η1, can detect the two known Z2 anomalies in the bosonic case, while ηf can detect the Z16 anomaly in the fermionic case.