A consistent theory to describe the correlated dynamics of quantum mechanicalitinerant spins and semiclassical local magnetization is given. We consider theitinerant spins as quantum mechanical operators, whereas local moments areconsidered within classical Lagrangian formalism. By appropriately treatingfluctuation space spanned by basis functions, including a zero-mode wavefunction, we construct coupled equations of motion for the collectivecoordinate of the center-of-mass motion and the localized zero-mode coordinateperpendicular to the domain wall plane. By solving them, we demonstrate thatthe correlated dynamics is understood through a hierarchy of two time scales:Boltzmann relaxation time when a non-adiabatic part of the spin-transfer torqueappears, and Gilbert damping time when adiabatic part comes up.
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