Brownian rotation of classical spins: dynamical equations for non-bilinear spin-environment couplings

Abstract

Dynamical equations for a classical spin interacting with the surrounding medium are derived by means of the formalism of the oscillator-bath environment. The bilinear-coupling treatment of Jayannavar (Z. Phys. B 82, 153 (1991)) is extended to couplings that depend arbitrarily on the spin variables and are linear or linear-plus-quadratic in the environment dynamical variables. The dynamical equations obtained have the structure of generalised Langevin equations, which, in the Markovian approach, formally reduce to known semi-phenomenological equations of motion for classical magnetic moments. Specifically, the generalisation of the stochastic Landau-Lifshitz equation effected by Garanin, Ishchenko, and Panina (Theor. Math. Phys. 82, 169 (1990)) in order to incorporate fluctuations of the magnetic anisotropy of the spin, is obtained for spin-environment interactions including up to quadratic terms in the spin variables. On the other hand, the portion of the coupling quadratic in the environment variables introduces an explicit dependence of the effective damping coefficients on the temperature.