Dynamics of magnetization on the topological surface

We investigate theoretically the dynamics of magnetization coupled to the surface Dirac fermions of a three dimensional topological insulator, by deriving the Landau-Lifshitz-Gilbert (LLG) equation in the presence of charge current. Both the inverse spin-Galvanic effect and the Gilbert damping coefficient $\alpha$ are related to the two-dimensional diagonal conductivity $\sigma_{xx}$ of the Dirac fermion, while the Berry phase of the ferromagnetic moment to the Hall conductivity $\sigma_{xy}$. The spin transfer torque and the so-called $\beta$-terms are shown to be negligibly small. Anomalous behaviors in various phenomena including the ferromagnetic resonance are predicted in terms of this LLG equation.