Slow domain growth in a system with competing interactions

We carry out a numerical study of the growth of domains following a quench in a three-dimensional scalar model with competing ferromagnetic $J_1$ and antiferromagnetic $J_2<J_1$ interactions. A "dynamical phase diagram" separates a region I of algebraic growth from a region II of logarithmic growth across an equilibrium "corner-rounding transition," confirming a previous claim. In region II, up to the late times we study, the correlation functions are anisotropic and violate dynamical scaling. This arises from the presence of two distinct length scales—the distance between interfaces $R$ and the distance between corners $L$, both of which grow logarithmically slowly. In the scaling limit ($t\to \infty$), $\frac{L}{R}\to 0$, restoring scaling and isotropy. Under the assumption of analyticity, the asymptotic scaling function is identical to the pure Ising model. The slow logarithmic growth arises from a renormalization of the kinetic coefficient at the smaller length scale $L$, and can be associated with the dangerously irrelevant operator $J_2$ at the zero-temperature fixed point (ZFP). This implies that at the ZFP, the two models, with and without $J_2$, belong to the same universality class.