Gilbert damping and spin Coulomb drag in a magnetized electron liquid with spin-orbit interaction

We present a microscopic calculation of the Gilbert damping constant for the magnetization of a two-dimensional spin-polarized electron liquid in the presence of intrinsic spin-orbit interaction. First we show that the Gilbert constant can be expressed in terms of the auto-correlation function of the spin-orbit induced torque. Then we specialize to the case of the Rashba spin-orbit interaction and we show that the Gilbert constant in this model is related to the spin-channel conductivity. This allows us to study the Gilbert damping constant in different physical regimes, characterized by different orderings of the relevant energy scales -- spin-orbit coupling, Zeeman coupling, momentum relaxation rate, spin-momentum relaxation rate, spin precession frequency -- and to discuss its behavior in various limits. Particular attention is paid to electron-electron interaction effects,which enter the spin conductivity and hence the Gilbert damping constant via the spin Coulomb drag coefficient.