On the equivalence of different order parameters for Ising ferromagnets

Abstract

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the “triplet order parameter” 〈σiσjσk〉 are the same as those of the magnetization 〈σi〉. Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureTc, defined as the temperature below which there is a spontaneous magnetization, is for fixed “even” spin interactionsJe independent of the way in which the “odd” interactionsJo approach zero from “above.” This is achieved by using only the “simplest,” Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.