Defect relaxation and coarsening exponents

Coarsening exponents describing the growth of long-range order in systems quenched from a disordered to an ordered phase are discussed in terms of the decay rate $\omega \mathrm{}\left(k\right)\mathrm{}$ for the relaxation of a distortion of wave vector $k$ applied to a topological defect. For systems described by order parameters with $Z\left(2\right)\mathrm{}$ (Ising) and $O\left(2\right)\mathrm{}$ $\mathrm{}\left(\mathrm{XY}\right)\mathrm{}$ symmetry the appropriate defects are domain walls and vortex lines, respectively. From ${\omega }_k\sim k^z,$ we infer $L\left(t\right)\mathrm{}\sim t^{1/z}$ for the coarsening scale, with the assumption that defect relaxation provides the dominant coarsening mechanism. The $O\left(2\right)\mathrm{}$ case requires careful discussion due to infrared divergences associated with the far field of a vortex line. Conserved, nonconserved, and intermediate dynamics are considered, with either short- or long-range interactions. In all cases the results agree with an earlier energy scaling analysis.