Structure factors of vector-order-parameter systems containing random topological defects

The spatial correlation function of a vector-order-parameter field and its Fourier transform is derived analytically for a relaxational (N, D) ordering process following a quench (model A) with 1, where N and D are the dimensionality of the order parameter and space, respectively. An assumption that the topological defects are randomly distributed is used. The correlation function C (r) behaves at short distance as 1-ar^, where =1 at N=1 and =2 at N3, and a logarithmic correction exists for N=2 such as C (r) 1- (b-c lnr) r^2. The short-distance behavior is also characterized by a power-law tail of the structure factor, S (k) k^- (N+D). The long-distance behavior is approximately Gaussian. The structure factor agrees with simulations over all length scales.