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

The landscape of electrical and thermal topological insulators (TIs) can be partitioned into two categories, respectively featuring strong topological indices valued in $\mathbb{Z}$ and $\mathbb{Z}_2$. Irrespective of these details, while all TIs are stable against sufficiently weak disorder, a normal insulator (NI) emerges with sufficiently strong disorder. Such a seemingly featureless quantum phase diagram, however, unfolds an unprecedented rich structure when we consider disordered integer TIs with $I_{\rm clean}>1$. 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 $I=I_{\rm clean}-1, I_{\rm clean}-2, \cdots, 1$ before ultimately giving way to an NI. In contrast to our anticipation, a similar outcome is also observed in crystalline topological insulators where the invariant satisfies $I_{\rm clean}>1$ and is protected by discrete rotational symmetry. We anchor these (possibly) generic outcomes by considering a variety of square-lattice quantum anomalous Hall models and computing the disorder-averaged Bott index $B$ therein. Finally, we extend the jurisdiction of such disorder-driven intermediate topological phases to dynamic (Floquet) quantum crystals.