Dynamics of the Heisenberg Ferromagnet at Low Temperatures

Abstract

Dyson calculated the effect of spin-wave interactions on the static (thermodynamic) properties of the Heisenberg ferromagnet. Within the same approximation, that of including only the contributions of lowest-order (two-magnon) scattering processes and neglecting the kinematic interaction, we have calculated the dynamic properties of this system and find results consistent with Dyson's in the zero-wave-vector limit. In the short-wavelength limit where perturbation theory diverges, we discuss nonperturbatively via the $t$ matrix the influence of the two-spin-wave bound states and the two-spin-wave resonant scattering states on the single-particle spectrum as characterized by the transverse spectral weight function $A_k\left(\omega \right)$. We find that although the total cross section of the bound states is too small for them to be observed directly, the anomalous effect of the bound states and resonant scattering states on the renormalization of the spin-wave energy is observable under favorable conditions. In general, we find the quasiparticle picture to be valid; however, at the highest temperature considered the resonant scattering states cause an extra resonance in the susceptibility. Most of the results for $A_k\left(\omega \right)$ are given numerically and have been checked against the sum rules, although the energy shift and energy width as deduced from ${\Sigma }_k\left({\epsilon }_k\right)$ are given analytically by rather simple expressions. We have obtained for the first time a Green's function that is capable of yielding correctly at low temperatures both the static and dynamic properties for arbitrary spin.