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Fermi surface versus Fermi sea contributions to intrinsic anomalous and spin Hall effects of multiorbital metals in the presence of Coulomb interaction and spin-Coulomb drag

Anomalous Hall effect (AHE) and spin Hall effect (SHE) are fundamental phenomena, and their potential for application is great. However, we understand the interaction effects unsatisfactorily, and should have clarified issues about the roles of the Fermi sea term and Fermi surface term of the conductivity of the intrinsic AHE or SHE of an interacting multiorbital metal and about the effects of spin-Coulomb drag on the intrinsic SHE. Here we resolve the first issue and provide the first step about the second issue by developing a general formalism in the linear response theory with appropriate approximations and using analytic arguments. The most striking result is that even without impurities the Fermi surface term, a non-Berry-curvature term, plays dominant roles at high or slightly low temperatures. In particular, this Fermi surface term causes the temperature dependence of the dc anomalous Hall or spin Hall conductivity due to the interaction-induced quasiparticle damping and the correction of the dc spin Hall conductivity due to the spin-Coulomb drag. Those results revise our understanding of the intrinsic AHE and SHE. We also find that the differences between the dc anomalous Hall and longitudinal conductivities arise from the difference in the dominant multiband excitations. This not only explains why the Fermi sea term such as the Berry-curvature term becomes important in clean and low-temperature case only for interband transports but also provides the useful principles on treating the electron-electron interaction in an interacting multiorbital metal for general formalism of transport coefficients. Several correspondences between our results and experiments are finally discussed.