Critical Correlations in the Ising Model

The Ising-model correlation function C(R₁₂)=〈μ₁μ₂〉 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)∝R^-1-η for R1κ, and C(R)∝κ^ηR^-1exp(-κR) for R1κ. The correlation length 1κ becomes infinite at the critical point. The calculated value η=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.