Using the techniques introduced by Honmura and Kaneyoshi, we show that two sets of identities for the two-state Ising model with
z nearest neighbors can be used as a basis for various approximation schemes which can systematically include correlation effects. In one dimension the results in zero field are exact for the magnetization, nearest-neighbor correlation function, and critical temperature. By neglecting correlations of more than three sites we introduce a simple approximation in two and three dimensions for the critical temperature. This simple approximation yields results which improve on those of the Bethe approximation. We discuss how the approximation may be improved and extended to other Ising problems.
The Ising-model correlation function $C({R}_{12})=〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}〉$ is studied in terms of a novel
N-fold integral representation. This formula stems from a procedure proposed by Montroll and Berlin. The integral is estimated by maximizing the integrand, an approximation related to the spherical-model assumptions. The correlation function is not of the Ornstein-Zernike type, just above the critical point, but rather $C(R)\ensuremath{\propto}{R}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\eta}}$ for $R\ensuremath{\ll}\frac{1}{\ensuremath{\kappa}}$, and $C(R)\ensuremath{\propto}{\ensuremath{\kappa}}^{\ensuremath{\eta}}{R}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\kappa}R)$ for $R\ensuremath{\gg}\frac{1}{\ensuremath{\kappa}}$. The correlation length $\frac{1}{\ensuremath{\kappa}}$ becomes infinite at the critical point. The calculated value $\ensuremath{\eta}=0.646$ is too large, reflecting the omission of important terms in the evaluation of the integral. The unusual mechanism inducing the nonclassical behavior is carefully examined.