Logarithmically slow domain growth in nonrandomly frustrated systems: Ising models with competing interactions

We study the growth (‘‘coarsening’’) of domains following a quench from infinite temperature to a temperature T below the ordering transition. The model we consider is an Ising ferromagnet on a square or cubic lattice with weak next-nearest-neighbor antiferromagnetic (AFM) bonds and single-spin-flip dynamics. The AFM bonds introduce free-energy barriers to coarsening and thus greatly slow the dynamics. In two dimensions, the barriers are independent of the characteristic length scale L(t), and therefore the long-time (t→∞) growth of L(t) still obeys the standard ${\mathit{t}}^{1/2}$ law. However, in three dimensions, a simple physical argument suggests that for quenches below the corner-rounding transition temperature, ${\mathit{T}}_{\mathrm{CR}}$, the barriers are proportional to L(t) and thus grow as the system coarsens. Quenches to T<${\mathit{T}}_{\mathrm{CR}}$ should, therefore, lead to L(t)∼ln(t) at long times. Our argument for logarithmic growth rests on the assertion that the mechanism by which the system coarsens involves the creation of a step across a flat interface, which below ${\mathit{T}}_{\mathrm{CR}}$ costs a free energy proportional to its length. We test this assertion numerically in two ways: First, we perform Monte Carlo simulations of the shrinking of a cubic domain of up spins in a larger sea of down spins. These simulations show that, below ${\mathit{T}}_{\mathrm{CR}}$, the time to shrink the domain grows exponentially with the domain size L.