Emergent fractal at the brink of disorder-driven quantum phase transition in topological insulators

Gapped topological phases are robust against weak disorder. However, when subjected to strong disorder — relative to the unperturbed band gap — they undergo a disorder-driven quantum phase transition into a trivial Anderson insulator. At the critical point of this transition, the local topological properties in the bulk develop fractal clusters. Within these cluster distributions, the topological markers across all sites are approximately evenly distributed between trivial and non-trivial values, mirroring the disorder-free global invariant. We demonstrate this phenomenon in two-dimensional disordered Chern insulators, observing the emergence of fractal formations that feature universal critical exponents, such as the anomalous and fractal dimensions.