Anatomy of topological Anderson transitions

We study mesoscopic signatures of the topological Anderson transitions in topological disordered chains. To this end we introduce an integer-valued sample-specific definition of the topological index in finite size systems. Its phase diagram exhibits a fascinating structure of intermittent topological phases, dubbed topological islands. Their existence is rooted in the real zeros of the underlying random polynomial. Their statistics exhibits finite-size scaling, pointing to the location of the bulk topological Anderson transition. While the average theories in AIII and BDI symmetry classes are rather similar, the corresponding patterns of topological islands and their statistics are qualitatively different. We also discuss observable signatures of sharp topological transitions in mesoscopic systems, such as persistent currents and entanglement spectra.