Dynamics of classicalXYspins in one and two dimensions

Abstract

We suggest that low-temperature spin waves in classical spin systems can be understood in terms of a "fixed-length" hydrodynamic theory. A theory is constructed along these lines which is exactly soluble in one and two dimensions for models with an easy-plane anisotropy. The results should apply at low temperatures to one-dimensional ferromagnets such as CsNi${\mathrm{F}}_3$, and agree with a microscopic truncated-spin-wave theory proposed by Villain. In two dimensions, we expect the calculations to be valid in a band of temperatures for $\mathrm{XY}$ magnets with an underlying hexagonal symmetry. The calculations should describe in addition the long-wavelength, low-frequency dynamics of third-sound propagation in films of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^3{}_{}{}^{}\mathrm{He}$-${}_{}{}^4{}_{}{}^{}\mathrm{He}$ mixtures. We also show that the critical exponent $\nu$ is $\nu \approx \frac{1}{2\sqrt{\epsilon }}$ for $\mathrm{XY}$ models in $2+\epsilon$ dimensions. Some results for dynamics in three dimensions are presented as well.