How disordered integer topological insulators become trivial: Prominent case studies in static and Floquet crystals

The landscape of electrical and thermal topological insulator (TIs) can be fragmented into two categories, respectively featuring integer $Z$ and $Z_2$ topological invariant. But, irrespective of these details, while all TIs are stable against sufficiently weak disorder, a normal insulator (NI) sets in for strong enough disorder. Such a seemingly featureless quantum phase diagram, however, unfolds an unprecedented rich structure when we consider disordered integer TIs with $Z>1$. As such with increasing strength of disorder, we show that such systems undergo a cascade of quantum phase transitions, fostering disorder-stabilized plateaus of TIs with intermediate integer invariant $Z-1, Z-2, \cdots, 1$ before ultimately giving way to a NI. In contrast to our anticipation, a similar outcome is also observed in crystalline topological insulators where the invariant $Z>1$ is protected by discrete rotational symmetry. We anchor these (possibly) generic outcomes by considering a variety of square lattice-based quantum anomalous Hall insulator models and computing the disorder-averaged Bott index therein. Finally, we extend the jurisdiction of such disorder-driven intermediate topological phases to dynamic or Floquet quantum crystals.