Exact Equilibrium Crystal Shapes in Two Dimensions and Perturbation Expansions for the Facet Shape and Step Free Energy of a Three-Dimensional Equilibrium Crystal

The fundamentals of the theory of equilibrium crystal shapes (ECS's) are reviewed. The concepts developed are then applied to two model calculations:. Using a conceptually novel approach which maps a two-dimensional (2D) interface exactly onto a Feynman -Vdovichenko lattice walker, we derive an exact and general solution for the ECS of free-fermion models. The ECS for these models is given by the locus of purely imaginary poles of the determinant of the "momentum-space" lattice -path propagator. The ECS may, therefore, be read off simply from the analytical expression for the bulk free energy. From these shapes one can then obtain numerically (but to arbitrary accuracy) the anisotropic interfacial free energy per unit length and, therefore, the high-temperature direction-dependent correlation length of the dual system. We give several examples of previously unknown Ising ECS's, and we examine in detail the free-fermion case of the eight-vertex model. The free-fermion eight-vertex model includes the modified potassium dihydrogen phosphate (KDP) model, which is not in the Ising universality class. The ECS of the modified KDP model is shown to be the limiting case of the ECS of an antiferromagnetic 2 x 1 phase on a triangular lattice in the limit of infinite interactions. The ECS of the modified KDP model is lenticular at finite temperature and has sharp corners. We explain the physics of this lens shape from an elementary calculation. To obtain the facet shapes and anisotropic step free energies for the 3D simple-cubic nearest-neighbour Ising model, we develop systematic low-temperature perturbation expansions about the exact solution for the ECS's and interfacial free energies of the 2D square Ising model. An expansion scheme is developed which makes explicit use of the conjugacy between the step free energy and the facet shape. We find that the facet shape is approximated to better than 1% by the equilibrium crystal shape of the corresponding 2D 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.