Hall and Faraday effects in interacting multiband systems

The Hall conductivity is widely used as a probe of Fermi surface evolution. However, in the case of interacting multi-band systems, calculations of the Hall conductivity are challenging because the application of the semi-classical single-band formula to many bands is ambiguous. Here, we derive a formula for the Hall response of interacting multi-band systems with arbitrary band topology and spin-orbit coupling. The formula is valid beyond the semiclassical approximation and at finite frequency, which is relevant for Faraday rotation, and it takes into account vertex corrections. In addition to the triangular diagrams, the formula includes rectangular diagrams that are absent in the single-band case. We show that these diagrams are necessary to recover a valid semiclassical formula for the Hall effect from the Kubo formula in the DC limit.