Hydrodynamics of Heisenberg Ferromagnets

Abstract

A microscopic derivation is given of hydrodynamic equations and response functions for uniaxial and isotropic exchange Heisenberg ferromagnets. The method is based on the conservation laws and makes no use of a quasiparticle picture. At low temperatures, the hydrodynamic equations involve the $z$ component of the magnetization, the local temperature, and the approximately conserved momentum. In addition to a diffusive mode, there is a propagating mode, the second magnon, which is strongly coupled to the $z$ component of the magnetization and may be observed by neutron scattering, and, in transparent ferromagnets, by Brillouin scattering (e.g., in Eu compounds and Cr${\mathrm{Br}}_3$). The strength of this mode depends on the magnetic field and anisotropy. For zero external field and no anisotropy, the longitudinal susceptibility diverges for wave number $q\to 0$. The velocities and diffusion constants in the hydrodynamic equations become $q$ dependent. Then, also, the transverse spin components perform low-frequency oscillations. For higher temperatures, the momentum is no longer conserved, and the hydrodynamic equations reduce to two coupled diffusion equations for the local temperature and magnetization. For $T>\mathrm{}{T}_c$ and zero external field, the spin density and energy obey uncoupled diffusion equations.