Inspired by the models of Rebei and Parker [Phys. Rev. B 67, 104434 (2003)] and [Rebei, Hitchon and Parker Phys. Rev. B 72, 064408 (2005)], we study a physical model which describes the behavior of magnetic moments in a ferromagnet. The magnetic moments are associated to $3d$ electrons which interact with conduction-band electrons and with phonons. We study each interaction separately and then collect the results, assuming that the electron-phonon interaction can be neglected. For the case of the spin-phonon interaction, we study the derivation of the equations of motion for the classical spin vector and find that the correct behavior, as given by the Brown equation for the spin vector and the Bloch equation, using the results obtained by Garanin [Phys. Rev. B 55, 3050 (1997)] for the average over fluctuations of the spin vector, can be obtained in the high-temperature limit. At finite temperatures, we show that the Markovian approximation for the fluctuations is not correct for time scales below some thermal correlation time ${\tau }_{\text{th}}$. For the case of electrons we work a perturbative expansion of the Feynman-Vernon influence functional. We find the expression for the random field correlation function which exhibits the properties of the electron bath; namely, we observe Friedel oscillations at small temperatures. The composite model (as well as the individual models) is shown to satisfy a fluctuation-dissipation theorem for all temperature regimes if the behavior of the coupling constants of the phonon-spin interaction remain unchanged with the temperature. The equations of motion are derived.

• Received 16 July 2012

DOI:https://doi.org/10.1103/PhysRevB.88.184419