Pumping conductance, the intrinsic anomalous Hall effect, and statistics of topological invariants

The pumping conductance of a disordered two-dimensional Chern insulator scales with increasing size and fixed disorder strength to sharp plateau transitions at well-defined energies between ordinary and quantum Hall insulators. When the disorder strength is scaled to zero as system size increases, the "metallic" regime of fluctuating Chern numbers can extend over the whole band. A simple argument leads to a sort of weighted equipartition of Chern number over minibands in a finite system with periodic boundary conditions: even though there must be strong fluctuations between disorder realizations, the mean Chern number at a given energy is determined by the clean Berry curvature distribution, as in the intrinsic anomalous Hall effect formula for metals. This estimate is compared to numerical results using recently developed operator algebra methods, and indeed the dominant variation of average Chern number is explained by the intrinsic anomalous Hall formula. A mathematical appendix provides more precise definitions and a model for the full distribution of Chern numbers.