AIII and BDI topological systems at strong disorder

Using an explicit one-dimensional model, we provide direct evidence that the one-dimensional topological phases from the AIII and BDI symmetry classes follow a Z-classification, even in the strong disorder regime when the Fermi level is embedded in a dense localized spectrum. The main tool for our analysis is the winding number in the noncommutative formulation introduced in I. Mondragon-Shem, J. Song, T. L. Hughes, and E. Prodan, arXiv: 1311. 5233. For both classes, by varying the parameters of the model and/or the disorder strength, a cascade of sharp topological transitions =0=1=2 is generated, in the regime where the insulating gap is completely filled with the localized spectrum. We demonstrate that each topological transition is accompanied by an Anderson localization-delocalization transition. Furthermore, to explicitly rule out a Z₂ classification, a topological transition between =0 and 2 is generated. These two phases are also found to be separated by an Anderson localization-delocalization transition, hence proving their distinct identity.