Gradient expansion formalism for generic spin torques

We propose a new quantum-mechanical formalism to calculate spin torques based on the gradient expansion, which naturally involves spacetime gradients of the magnetization and electromagnetic fields. We have no assumption in the small-amplitude formalism or no difficulty in the SU($2$) gauge transformation formalism. As a representative, we calculate the spin renormalization, Gilbert damping, spin-transfer torque, and $\beta$-term in a three-dimensional ferromagnetic metal with nonmagnetic and magnetic impurities being taken into account within the self-consistent Born approximation. Our results serve as a first-principles formalism for spin torques.