  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.
