Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach

The anomalous Hall effect (AHE) is a consequence of spin-orbit coupling in a ferromagnetic metal and related primarily to density-matrix response to an electric field that is off-diagonal in band index. For this reason disorder contributions to the AHE are difficult to treat systematically using a semiclassical Boltzmann equation approach, even when weak localization corrections are disregarded. In this article we explicitly demonstrate the equivalence of an appropriately modified semiclassical transport theory which includes anomalous velocity and side-jump contributions and microscopic Kubo-Streda perturbation theory, with particular unconventional contributions in the semiclassical theory identified with particular Feynman diagrams when calculations are carried out in a band-eigenstate representation. The equivalence we establish is verified by explicit calculations for the case of the two-dimensional Dirac model Hamiltonian relevant to graphene.