Average-exact mixed anomalies and compatible phases

Quantum anomalies strongly constrain possible low-energy physics of condensed matter systems. In particular, it prohibits the existence of a trivially gapped phase. Here, the authors extend the notion of quantum anomalies to disordered systems, where symmetries can be preserved on average by disorder. Through physical arguments and solvable lattice models, the authors discover a series of phases intrinsic to disordered systems that are compatible with quantum anomalies involving average symmetries. These results serve as compelling evidence of the nontriviality of quantum anomalies, even with loosened requirements on symmetry and quantum coherence.