Disordered and Decohered Emergent Anomalies: The Case for Intrinsic Average Symmetry Protected Topological Phases

Condensed matter systems with emergent anomalies in the infrared(IR) cannot have featureless ground states: they are either spontaneous symmetry breaking, gapless, or gapped but topologically ordered in 2+1 or higher dimensions. In this work, we show that if we loosen the constraint of quantum coherence, there is yet another possibility: a short-range correlated mixed state that consists of a collection of symmetry-broken states so that the whole ensemble is invariant under the IR symmetry group. A natural realization of such an ensemble is a disordered system where the IR symmetry associated with the emergent anomaly is only preserved on average. Despite the short-range correlation, the mixed state of this ensemble can still be topologically non-trivial under the microscopic symmetries due to the emergent anomalies. We explicitly construct lattice models for such non-trivial disordered ensembles starting from two different clean limits with the gapless and topologically ordered ground states respectively. Our study is closely related to the proposed intrinsic average-symmetry-protected topological (iASPT) phases. As an alternative to disorder, we show how symmetric decoherence processes can also evolve the gapless or topologically ordered ground states into the topologically non-trivial mixed states with the average symmetry.