  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.
