An entropic invariant for 2D gapped quantum phases

We introduce an entropic quantity for two-dimensional quantum spin systems to characterize gapped quantum phases modeled by local commuting projector code Hamiltonians. The definition is based on a recently introduced specific operator algebra defined on an annular region, which encodes the superselection sectors of the model. We show that the quantity is invariant under any constant-depth local quantum circuit, and thus an indicator of gapped quantum spin-liquids. We explicitly calculate the quantity for Kitaev's quantum double models, and show that the value is exactly same as the topological entanglement entropy of the models. Our method circumvents some of the problems around extracting the topological entanglement entropy, allowing us to prove invariance under constant-depth quantum circuits.