Dynamical properties of critical and paramagnetic spin fluctuations in anisotropic Heisenberg magnets

Decay rates for long-wavelength spin fluctuations in the standard x-y model, and a second model described by the sum of the full Heisenberg exchange interaction and a single-site anisotropy, are calculated on the basis of a coupled mode theory in which the isothermal susceptibilities are consistent with corresponding spherical models (in consequence, quantum fluctuations are absent). Results for the decay rates are provided for two- and three-dimensional lattices and ferromagnetic and antiferromagnetic exchange interactions. Their dependence on the wave vector, q, and the inverse correlation lengths are derived from integral equations that admit homogeneous functions as solutions. The decay rates may thus be expressed as qd2/R1(q/ kappa 0) and q(d+2/2)R2(q/ kappa ,q/ kappa 0) where R1 and R2 are, respectively, scaling functions for the x-y and ferromagnetic single-site anisotropy models, d is the spatial dimension, and kappa (KO) is the out of plane (in plane) inverse correlation length. Analytical and numerical values of the scaling functions are presented for the limits of main physical interest.