Self consistent Ornstein–Zernike approximation compared with exact results for lattice gases in one and two dimensions

Abstract

We evaluate numerically results for the self consistent Ornstein–Zernike approximation (SCOZA) for the Ising model or the lattice gas in one and two dimensions where exact results are known. The cases we consider thus include the Ising model with nearest-neighbor interaction in two dimensions, and in one dimension the cases with a Kac interaction or exponential potential in the infinite range limit and the one with nearest- and next-nearest neighbor interactions. As earlier found for the three-dimensional Ising model, results with high general accuracy are found, although the phase transition of the two-dimensional Ising model is smeared out a bit, as SCOZA at least in its present form, does not yield a phase transition in two dimensions. In the two-dimensional case more long- range interactions are also considered to see to what extent SCOZA approximates the expected universal critical behavior. By extrapolation we find our numerical results quite consistent with a value near the exact one γ=1.75 for the supercritical exponent of isothermal susceptibility. In the case with the nearest- and next-nearest neighbor interactions a situation that clearly favors ferromagnetic configurations is needed. Otherwise the present version of SCOZA will fail, i.e., the solution becomes less accurate and finally ceases to exist.