Dynamics near the spin-glass transition

We examine spin dynamics just above a spin-glass transition for two classical soft-spin models with random exchange interactions. The statics of the two models are identical; in mean-field theory we find an Edwards-Anderson transition of the usual sort. The dynamics are described by Langevin equations: in one case we use a purely dissipative equation of motion for a nonconserved scalar spin, while in the other we include a "mode-coupling" term to describe the precession of a vector spin in the field of its neighbors and impose conservation of the total spin. In the first case we find critical slowing down as the transition at $T_g$ is approached from above (and $t^{-\frac{1}{2}}$ tails below $T_g$), in agreement with the result of Ma and Rudnick for a different model. In the second case, we find a divergent spin-diffusion constant: $D\propto {(T-T_g)}^{-\frac{1}{2}}$ ($d>\mathrm{}4\mathrm{}$) and $D\propto {(T-T_g)}^{\frac{2}{d}-1\mathrm{}}$ ($2<d<\mathrm{}4\mathrm{}$), in agreement with our previous calculation for a mode-coupling version of the Ma-Rudnick model.