Electronic contribution to the geometric dynamics of magnetization

To give a general description of the influences of electric fields or currents on magnetization dynamics, we developed a semiclassical theory for the magnetization implicitly coupled to electronic degrees of freedom. In the absence of electric fields, the Bloch electron Hamiltonian changes the Berry curvature, the effective magnetic field, and the damping in the dynamical equation of the magnetization, which we classify into intrinsic and extrinsic effects. Static electric fields modify these as first-order perturbations, with which we were able to give a physically clear interpretation of the current-induced spin-orbit torques. We used a toy model mimicking a ferromagnet-topological-insulator interface to illustrate the various effects, and we predicted an anisotropic gyromagnetic ratio and the dynamical stability for an in-plane magnetization. Our formalism can also be applied to the slow dynamics of other order parameters in crystalline solids.