On the Ising Model with Long-Range Interaction. II. Critical-Region Analysis

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Abstract
The high‐temperature expansion for the free energy in powers of γ (the reciprocal of the range of interaction) developed in a previous paper is studied in the critical region. In terms of the graphology introduced in the previous paper, it is proved that the ring diagrams give the dominant contribution in the critical region, if and only if the integral R(νγ)= ( 1 2π ) D ∫ 2π … ∫ 0 g(ω)d D ω 1−νγg(ω) diverges no worse than logarithmically at [the Curie‐Weiss (CW) point] νCW = J/kT CW = [γg(0)]−1, where D denotes dimensionality and γg(ω) the Fourier transform of the interaction potential. The results are in agreement with various model results, and it is conjectured that the above condition is also a necessary and sufficient condition for the existence of a phase transition.