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

Abstract

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\ensuremath{\le}N\ensuremath{\le}D, 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-${\mathit{ar}}^{\mathrm{\ensuremath{\delta}}}$, where \ensuremath{\delta}=1 at N=1 and \ensuremath{\delta}=2 at N\ensuremath{\ge}3, and a logarithmic correction exists for N=2 such as C(r)\ensuremath{\simeq}1-(b-c lnr)${\mathit{r}}^{2}$. The short-distance behavior is also characterized by a power-law tail of the structure factor, S(k)\ensuremath{\sim}${\mathit{k}}^{\mathrm{\ensuremath{-}}(\mathit{N}+\mathit{D})}$. The long-distance behavior is approximately Gaussian. The structure factor agrees with simulations over all length scales.