Functional Keldysh theory of spin torques
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We present a microscopic treatment of current-induced torques and thermal fluctuations in itinerant ferromagnets based on a functional formulation of the Keldysh formalism. We find that the nonequilibrium magnetization dynamics is governed by a stochastic Landau-Lifschitz-Gilbert equation with spin-transfer torques. We calculate the Gilbert damping parameter alpha and the nonadiabatic spin transfer torque parameter beta for a model ferromagnet. We find that beta not equal alpha, in agreement with the results obtained using imaginary-time methods of Kohno [J. Phys. Soc. Jpn. 75, 113706 (2006)]. We comment on the relationship between s-d and isotropic-Stoner toy models of ferromagnetism and more realistic density-functional-theory models, and on the implications of these relationships for predictions of the beta/alpha ratio which plays a central role in domain-wall motion. Only for a single-parabolic-band isotropic-Stoner model with an exchange splitting that is small compared to the Fermi energy does beta/alpha approach 1. In addition, our microscopic formalism naturally incorporates the fluctuations needed in a nonzero-temperature description of the magnetization. We find that to first order in the applied electric field, the usual form of thermal fluctuations via a phenomenological stochastic magnetic field holds.
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