Upper bounds on the magnetization of arbitrary ferromagnetic spin models are investigated. We discuss two methods by which it was proven that the mean field magnetization was shown to be an upper bound on the true magnetization. These are the Pearce and Slawny proofs. Results are given on analyses of methods attempting to extend the Pearce proof.
Extensions to mean field theory are studied. We present new results which show that two of these extension methods also give upper bounds on the magnetization. We prove that the two-body extension, the Oguchi method, is an upper bound for spin 1/2 Ising models. For those spin 1/2 models where the three-body method predicts a unique magnetization, this too is proven to give an upper bound. The corresponding critical temperatures are proven to fall in the decreasing sequence
Tc (mean field) ≥ Tc (Oguchi) ≥ Tc (3-body) ≥ Tc (true)
where the inequalities are strict if the extension schemes are effectively used. As applications of these methods, we obtain graphical spontaneous magnetization curves for various models and the new upper bound Tc ≤ 2.897 for the one dimensional 1/r2 Ising model, improving the previous mean field upper bound of Tc ≤ 3.290.
