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

The pumping conductance of a disordered two-dimensional Chern insulator\nscales with increasing size and fixed disorder strength to sharp plateau\ntransitions at well-defined energies between ordinary and quantum Hall\ninsulators. When the disorder strength is scaled to zero as system size\nincreases, the "metallic" regime of fluctuating Chern numbers can extend over\nthe whole band. A simple argument leads to a sort of weighted equipartition of\nChern number over minibands in a finite system with periodic boundary\nconditions: even though there must be strong fluctuations between disorder\nrealizations, the mean Chern number at a given energy is determined by the {\\it\nclean} Berry curvature distribution expected from the intrinsic anomalous Hall\neffect formula for metals. This estimate is compared to numerical results using\nrecently developed operator algebra methods, and indeed the dominant variation\nof average Chern number is explained by the intrinsic anomalous Hall effect. A\nmathematical appendix provides more precise definitions and a model for the\nfull distribution of Chern numbers.\n