Hydrodynamic theory of vorticity-induced spin transport

Electron spin transport in a disordered ferromagnetic metal is theoretically studied from the hydrodynamic viewpoint, focusing on role of electron vorticity. The spin-resolved momentum flux density electrons calculated microscopically, taking account spin-orbit interaction and uniform magnetization, expression for motive force obtained as linear response to driving electric field. It shown that are characterized by troidal moments, vector products applied external field polarization or which act effective fields when anomalous Hall effects taken into account. spin-vorticity magnetization-vorticity couplings arise naturally conservative forces driven toroidal nonconservative also found exist. Spin accumulation induced flow vorticity-induced torque relaxation discussed. torque, effect interaction, argued be larger than conventional torques acting magnetization structures such nonadiabatic ($\ensuremath{\beta}$) current-induced torque. direct inverse discussed context coupling.