Low-temperature expansions for the step free energy and facet shape of the simple-cubic Ising model

We report here the first systematic low-temperature expansion of the step free energy for the simple-cubic nearest-neighbor Ising model and we obtain the facet shapes for the associated equilibrium crystal-shape problem. An expansion scheme is developed which makes explicit use of the conjugacy between the step free energy and the facet shape. The calculation is structured as a perturbation expansion about the exact solution for the interfacial free energy of the two-dimensional square Ising model. We find that the facet shape is approximated to better than 1% by the equilibrium crystal shape of the two-dimensional Ising model for temperatures less than about 72% of the roughening temperature. In that temperature range overhangs and bubbles contribute less than 0.1% to the step free energy. At higher temperatures the facet shape is nearly circular with anisotropies of less than 0.4% and a ratio of facet diameter to crystal diameter of less than 0.4. Extrapolations into the isotropic region give critical roughening amplitudes consistent with recent Monte Carlo data.