Triangular SOS models and cubic-crystal shapes

Abstract

A solid-on-solid (SOS) model in a field h conjugate to the orientation of the surface is exactly solved with the aid of Pfaffians. The free energy (h) directly gives the equilibrium shape of a finite crystal. The phase diagram exhibits rough and smooth phases, corresponding to rounded and flat portions of the crystal surface. The solid-on-solid model undergoes transitions of the Pokrovsky-Talapov type (1979) characterised by a specific heat exponent alpha =1/2. One special point of the phase diagram corresponds to the appearance of a facet via an alpha =0 transition. Height-height correlations are derived along a special line in the phase diagram. With the aid of the known equivalence of this SOS model with an Ising model, several exponents can be translated from one model to the other. This enables one to derive the topology of the phase diagram of the antiferromagnetic triangular Ising model with first- and second-neighbour couplings in a field.