Topologically localized insulators

We show that fully localized, three-dimensional, time-reversal-symmetry-broken Anderson insulators support topologically distinct phases that can be labelled by integers. Any two such topologically localized phases are separated by a metallic phase. We find that these novel topological phases are fundamentally distinct from insulators without disorder: they are guaranteed to host delocalized states along their insulating boundaries, giving rise to the quantized boundary Hall conductance whose value is determined by the integer invariant assigned to the bulk.