Magnetization dynamics: path-integral formalism for the stochastic Landau-Lifshitz-Gilbert equation

摘要

We construct a path-integral representation of the generating functional forthe dissipative dynamics of a classical magnetic moment as described by thestochastic generalization of the Landau-Lifshitz-Gilbert equation proposed byBrown, with the possible addition of spin-torque terms. In the process ofconstructing this functional in the Cartesian coordinate system, we criticallyrevisit this stochastic equation. We present it in a form that accommodates forany discretization scheme thanks to the inclusion of a drift term. Thegeneralized equation ensures the conservation of the magnetization modulus andthe approach to the Gibbs-Boltzmann equilibrium in the absence of non-potentialand time-dependent forces. The drift term vanishes only if the mid-pointStratonovich prescription is used. We next reset the problem in the morenatural spherical coordinate system. We show that the noise transformsnon-trivially to spherical coordinates acquiring a non-vanishing mean value inthis coordinate system, a fact that has been often overlooked in theliterature. We next construct the generating functional formalism in thissystem of coordinates for any discretization prescription. The functionalformalism in Cartesian or spherical coordinates should serve as a startingpoint to study different aspects of the out-of-equilibrium dynamics of magnets.Extensions to colored noise, micro-magnetism and disordered problems arestraightforward.