Anomaly indicators for time-reversal symmetric topological orders

TL;DR

This work introduces two anomaly indicators, η2 for bosonic time-reversal anomalies and ηf for DIII-class fermionic systems, to detect SETs that cannot be realized in strictly 2D with time-reversal symmetry. It provides substantial evidence by evaluating η2 in Kitaev quantum doubles, B × ¯B double-layer systems, and gauged T-Pfaffian states, showing η2 = 1 in non-anomalous cases and η2 distinguishing the ± T-Pfaffian variants. The paper also proves, under broad conditions, that η2 and ηf remain invariant under time-reversal–preserving anyon condensation, strengthening their status as bulk indicators of 3D-SET anomalies. Additionally, it analyzes the structure of ηf in fermionic Abelian orders and derives constraints on its possible values, highlighting a Z16-type classification in the fermionic context and linking to 3D topological superconductors. These results offer a concrete, computable framework for diagnosing anomalies in time-reversal symmetric topological orders and their 3D realizations.

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, $η_2$, applies to bosonic systems while the second indicator, $η_f$, applies to fermionic systems in the DIII class. We conjecture that $η_2$, together with a previously known indicator $η_1$, can detect the two known $\mathbb Z_2$ anomalies in the bosonic case, while $η_f$ can detect the $\mathbb Z_{16}$ anomaly in the fermionic case.