Fluctuations and correlations of local topological order parameters in disordered two-dimensional topological insulators

Two-dimensional topological insulators are characterized by an insulating bulk and conductive edge states protected by the nontrivial topology of the bulk electronic structure. They remain robust against moderate disorder until Anderson localization occurs and destroys the topological phase. Interestingly, disorder can also induce a topological phase—known as a topological Anderson insulator—starting from an otherwise pristine trivial phase. While topological invariants are generally regarded as global quantities, we argue that space-resolved topological markers can act as local order parameters, revealing the role of fluctuations and correlations in the local topology under Anderson disorder and vacancies. With this perspective, we perform numerical simulations of disorder-driven topological phase transitions in the Haldane and Kane-Mele models, using supercells with both open and periodic boundary conditions. We find that short-scale fluctuations of topological markers vanish upon coarse-graining, except at the topological phase transition, where their correlation length peaks and large-scale fluctuations remain. Notably, such a topological correlation function is characterized by critical exponents that appear universal across disorder types, yet they can resolve different topological phase transitions.