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The coupled kinetic equation for density and spin Wigner functions arederived including spin-orbit coupling, electric and magnetic field as well asselfconsistent Hartree meanfields suited for SU(2) transport. The interactionsare assumed to be with scalar and magnetic impurities as well as scalar andspin-flip potentials among the particles. The spin-orbit interaction is used ina form suitable to solid state physics with Rashba or Dresselhaus coupling,graphene, extrinsic spin-orbit coupling as well as effective nuclear mattercoupling. The deficiencies of the two-fluid model are worked out consisting inthe appearance of an effective in-medium spin-precession. The stationarysolution of all these systems shows a band splitting controlled by an effectivemedium-dependent Zeeman field. The selfconsistent precession direction isdiscussed and a cancellation of linear spin-orbit coupling at zero temperatureis reported. The precession of spin around this effective direction caused byspin-orbit coupling leads to anomalous charge and spin currents in an electricfield. Anomalous Hall conductivity is shown to consists of the known resultsobtained from Kubo formula or Berry phases and a new symmetric part interpretedas inverse Hall effect. Analogously the spin-Hall and inverse spin-Hall effectof spin currents are discussed which are present even without magnetic fieldsshowing a spin accumulation triggered by currents. The analytical dynamicalexpressions for zero temperature are derived and discussed in dependence on themagnetic field and effective magnetizations. The anomalous Hall and spin-Halleffect changes sign at higher than a critical frequency dependent on therelaxation time.

physical review. b, condensed matter and materials physics

Kinetic theory of spin-polarized systems in electric and magnetic fields  with spin-orbit coupling: I. Kinetic equation and anomalous Hall and  spin-Hall effects