Anomaly manifestations in topological insulator surfaces with strong correlations and disorder

Three dimensional topological insulators exhibit a bulk quantum anomaly that protects certain properties of their surface. For example, theoretical studies of weakly interacting strong and weak topological insulators show that their surface stays conducting, even with arbitrarily strong disorder, as long as the relevant symmetries are preserved on average. However, the combination of strong interactions and disorder offers theoretical questions which have not yet been addressed. Here we address this interplay and show that it can lead to a type of localized surface state, even while the disorder ensemble preserves all symmetries. We discuss how the quantum anomaly is manifested within the localized surface. These minimal manifestations of the quantum anomaly can constrain possible Lieb-Schultz-Mattis-type theorems for disordered systems.