Effective Gilbert damping in the stochastic Landau-Lifshitz-Gilbert equation

Quasi particle based (e.g. Boltzmann equation) studies of spin wave transport often assume that their scattering rates follow the simple form $\eta=\alpha \omega$, with the Gilbert damping $\alpha$ and frequency $\omega$. In this work, we examine the effective damping $\alpha_{eff,T}=\eta/\omega$ observed in atomistic spin dynamics, when temperature and spin wave interactions are introduced for a 1D spin chain. We extract the dynamical correlation functions from spin trajectories propagated using the stochastic Landau-Lifshitz-Gilbert equation, and fit the dynamical structure factor, yielding the dispersion and scattering rates for a wide range of temperatures. The resulting effective damping can be very different from the initially constant Gilbert value. It exhibits a temperature and crystal momentum scaling which we explain based on interactions with the Gilbert bath and spin wave scattering by changes in local magnetic order.