Pair approximation equations for interfaces and free surfaces in the Ising model

Abstract

We present a simple derivation of the pair approximation equations for spin distribution functions in the Ising model. In a uniform system, the pair equations are equivalent to those found by the Bethe or the quasichemical methods. Our derivation requires self-consistency between approximations for the singlet and pair distribution functions and is easily generalized to study inhomogeneous systems. A simple and efficient numerical method for solution of the equations is presented. We study in particular properties of the interface or free surface in an anisotropic Ising model with coupling between spins in adjacent planes given by J⊥, which may differ from the coupling between spins in the same plane. In the limit J⊥→∞, the model is equivalent to the ’’solid-on-solid’’ model used in theories of crystal growth. In this limit the mean field theory for the interface fails completely, while the pair approximation gives results at low temperatures in good agreement with previous calculations. Other possible applications for the pair equations are briefly discussed.