Topological defects, correlation functions, and power-law tails in phase-ordering kinetics

The correlation functions ${\mathit{C}}_{\mathrm{\phi }}$ and ${\mathit{C}}_{\mathrm{\phi }}^2$, associated with the order-parameter field φ→(r,t) and its square, respectively, are discussed using heuristic arguments and an approximate analytical approach. Topological defects (walls, strings, monopoles) in the field, seeded by a quench from the high- to the low-temperature phase, lead to singular short-distance behavior in the scaling functions, and power-law tails in the corresponding structure factors. For superfluid helium, the structure factor ${\mathit{S}}_{\mathrm{\phi }}^2$(k,t) is measurable in principle using small-angle scattering (whereas ${\mathit{S}}_{\mathrm{\phi }}$ is inaccessible). It is predicted to exhibit a power-law tail, ∼[${\mathit{a}}^4$/L(t${)}^2$](lnka${)}^2$/k, where L(t) is the characteristic scale at time t after the quench and a is the core size of a vortex line. Correlation functions for the defect density are also discussed.