Phase transitions and scale invariance in topological Anderson insulators

We investigate disorder-driven transitions between trivial and topological insulator (TI) phases in two-dimensional (2D) systems. Our study primarily focuses on the Bernevig-Hughes-Zhang (BHZ) model with Anderson disorder, while other standard 2D TI models exhibit equivalent features. The analysis is based on the local Chern marker (LCM), a local quantity that allows for the characterization of topological transitions in finite and disordered systems. Our simulations indicate that disorder-driven trivial to topological insulator transitions are nicely characterized by C-0, the disorder-averaged LCM near the central cell of the system. We show that C-0 is characterized by a single-parameter scaling, namely, C-0(M, W, L) equivalent to C-0(z), with z = [W-mu - W-c(mu)(M)]L, where M is the Dirac mass, W is the disorder strength, and L is the system size, while W-c(M) proportional to root M and mu approximate to 2 stand for the critical disorder strength and the critical exponent, respectively. Our numerical results are in agreement with a theoretical prediction based on a first-order Born approximation analysis. These observations lead us to speculate that the universal scaling function we have found is rather general for amorphous and disorder-driven topological phase transitions.