The topological order of Dirac models in any dimension and symmetry class can be mapped to lattice sites by a universal topological marker. From the topological operator that extracts this marker, one can further define a nonlocal marker that detects topological phase transitions. Several prototype models such as the Su-Schrieffer-Heeger model, Majorana chain, Chern insulator, Bernevig-Hughes-Zhang model, chiral and helical
p-wave superconductors, three-dimensional time-reversal symmetric topological insulators, among many others, are employed to demonstrate the ubiquity of these markers.
We study the critical properties of the topological Anderson phase transition in a strongly disordered Chern insulator, separating the topological phase from a trivial Anderson insulator. We show that the transition is characterized by a non-zero electrical conductance and by the emergence of a critical length scale in the real-space profile of the local Chern marker. From this, we extract the correlation-length and the dynamical critical exponents, which are consistent with those of non-interacting models of the integer quantum Hall effect. We then ramp the disorder strength across the transition and study the ensuing dynamics. In contrast to clean topological systems, we find that the excitation density does not follow the Kibble-Zurek scaling. The non-equilibrium length scale associated with the local Chern marker is decoupled from the generation of excitations. For studied system sizes, we find it to be close to the Kibble-Zurek prediction for the topological-to-trivial quench, while it deviates from it for the reverse direction.
Topological states of matter were first introduced for noninteracting fermions on an infinite uniform lattice. Since then, substantial effort has been made to generalize these concepts to more complex settings. Recently, local markers have been developed that can describe the topological state of systems without translational symmetry and a well-defined gap. However, no local marker for interacting matter has been proposed yet that is capable of directly addressing an interacting system. Here, we suggest such a many-body local marker based on the single-particle Green's function. Using this marker we identify topological transitions in finite lattices of a Chern insulator with Anderson disorder and Hubbard interactions. Importantly, our proposal can be generalized to nonequilibrium systems.