Hall and Faraday effects in interacting multi-band systems with arbitrary band topology and spin-orbit coupling

A formula for the Hall response of interacting multi-band systems with arbitrary band topology and spin-orbit coupling is derived. The formula is valid at finite frequency, which is relevant for Faraday rotation, and it takes into account all particle-hole vertex corrections. The formula includes both three-leg (triangular) and four-leg (rectangular) diagrams. The latter diagrams are absent in the single band case. We show that the rectangular diagrams are necessary to recover the semiclassical formula for the Hall effect from the Kubo formula in the DC limit. They also give the linear response of the anomalous Hall effect to an external magnetic field, an effect which goes beyond the semiclassical theory. Three- and four-particle scatterings are neglected.