Defect coarsening and spin waves in the nonlinear σ model

We propose a phenomenological equation of motion for the nonlinear σ model undergoing defect coarsening at low temperatures, in which the thermal fluctuation degrees of freedom (spin waves) are explicitly separated from the defect degrees of freedom. The Gaussian closure approximation is used to find the zero temperature order parameter correlation function. The main result of this paper is the calculation of the spin wave correlation function, which is the O(T) correction to the order parameter correlation function. The spin wave correlation function shows the scaling behavior predicted by Bray [Phys. Rev. Lett. 62, 2841 (1989)] for N>2, and is exact in the t→∞ and N→∞ limits. Our results for the XY model in two dimensions clarify the nature of coarsening observed at the lower critical dimension.