Magnon Density Fluctuations in the Heisenberg Ferromagnet

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Abstract
It is shown, by means of a decoupling procedure for the equations of motion of a spin system governed by the Heisenberg Hamiltonian, that the distribution of magnons in phase space satisfies the quantum-mechanical Boltzmann equation at low temperatures. We conclude from this that disturbances in the longitudinal component of the magnetization, which may be thought of as density fluctuations in the gas of magnons, propagate essentially undamped at long wavelengths, in complete analogy with the propagation of sound in a gas of real particles. The interaction of this hydrodynamic mode with phonons is considered, and the damping coefficient of the phonons when their velocity coincides with the magnon sound velocity is obtained.