EXACT EQUILIBRIUM CRYSTAL SHAPES IN 2 DIMENSIONS FOR FREE-FERMION MODELS

Using a conceptually novel approach that maps a two-dimensional interface exactly onto a Feynman-Vdovichenko lattice walker, we derive an exact and general solution for the equilibrium crystal shapes (ECSs) of free-fermion models, i.e., models that are solvable via the Feynman-Vdovichenko or (equivalently) Pfaffian methods. 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, simply be read off from the analytical expression for the bulk free energy. From these shapes one can then obtain numerically (but to arbitrary accuracy) the high-temperature direction-dependent correlation length of the dual system. We give several examples of previously unknown Ising ECSs, 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 2×1 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. © 1990 The American Physical Society.