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

A formula for the Hall response of interacting multiband 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.

Figure 1

Diagrammatic expansion of the change in the current due to the presence of a magnetic field, evaluated in the zero-field limit. Lines show the fully dressed Green's function, cirles and squares with their adjacent shaded area denote the dressed current vertex and its derivative, respectively. We also define ${\mathbf{k}}_{\pm }^{\prime }\equiv \mathbf{k}\pm {\mathbf{q}}^{\prime }/2$ and ${\omega }_m^{\pm }={\omega }_m\pm {\nu }_n$. Only diagrams at the second line contribute to the Hall response. The vertex ${\mathrm{\Lambda }}_{\gamma \beta }$, represented by an attached green square box, must be treated carefully, as discussed in Appendix pp2. The final results include triangular (three-leg) and rectangular (four-leg) diagrams shown in the next two figures.

Figure 2

The triangular diagrams appearing at the Hall response of multiband systems. The arrows on the vertices show the incoming or outcoming bosonic frequency associated with the electric field frequency. The circle and square vertices demonstrate current vertex and current vertex derivative (effective mass), respectively.

Figure 3

The rectangular diagrams appearing at the Hall response of multiband systems. These diagrams are absent in the single-band system. The arrows on the vertices show the incoming or outcoming bosonic frequency associated with the electric field frequency. All the vertices are current vertices.