Average-exact mixed anomalies and compatible phases

The quantum anomaly of a global symmetry is known to strongly constrain the allowed low-energy physics in a clean and isolated quantum system. However, the effect of quantum anomalies in disordered systems is much less understood, especially when the global symmetry is only preserved on average by the disorder. In this work, we focus on disordered systems with both average and exact symmetries $A\times K$, where the exact symmetry $K$ is respected in every disorder configuration, and the average $A$ is only preserved on average by the disorder ensemble. When there is a mixed quantum anomaly between the average and exact symmetries, we argue that the mixed state representing the ensemble of disordered ground states cannot be featureless. While disordered mixed states smoothly connected to the anomaly-compatible phases in clean limit are certainly allowed, we also found disordered phases that have no clean-limit counterparts, including the glassy states with strong-to-weak symmetry breaking, and average topological orders for certain anomalies. We construct solvable lattice models to demonstrate each of these possibilities. We also provide a field-theoretic argument to provide a criterion for whether a given average-exact mixed anomaly admits a compatible average topological order.