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Journal of Physics A Mathematical and General

Upper bounds on Tcfor many-body, ferromagnetic, Ising systems

James L. Monroe

Published Oct 21, 1986• 1 citation
Publisher

Abstract

A method used previously to obtain upper bounds on the critical temperature T c for ferromagnetic Ising systems with pair interactions is generalised to treat systems with odd or even many-body interactions. The results are compared to recent work of Horiguchi and Morita (1985) who, using a different method, found bounds on T c for systems with even many-body interactions. An interesting feature of the method presented here involves the new use of some correlation inequalities.

DOI
10.1088/0305-4470/19/15/032

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A method used previously to obtain upper bounds on the critical temperature T c for ferromagnetic Ising systems with pair interactions is generalised to treat systems with odd or even many-body interactions. The results are compared to recent work of Horiguchi and Morita (1985) who, using a different method, found bounds on T c for systems with even many-body interactions. An interesting feature of the method presented here involves the new use of some correlation inequalities.
Recent results on the upper-bound property of the magnetization in certain approximations to ferromagnetic spin systems are here used to derive a rigorous upper bound for the critical temperature of the one-dimensional ferromagnetic 1n2\frac{1}{{n}^{2}}n21​ Ising spin system. We find that ${\ensuremath{\beta}}_{c}J\ensuremath{\ge}0.441({\ensuremath{\beta}}_{c}=\frac{1}{k{T}_{c}})$. The best previous bound was 0.360. A further improvement by the same technique is possible.
For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.
We use an effective field model which explicitly incorporates correlations to determine the critical temperatures for Ising ferromagnets on the common lattices. We specifically retain only the first nontrivial odd correlation function, i.e., the triplet correlation function, while neglecting higher order correlations. Within this approximation, assuming the equivalence of triplet correlations involving both central and noncentral sites in the nearest neighbor cluster yields values for the critical temperature below the Bethe approximations for z⩾4. Likewise, assuming the inequivalence of triplet correlations involving the central and noncentral sites yields values of the critical temperature only slightly higher than the Bethe approximation for z⩾4. For lattices with four- and six-fold coordination this equivalence and inequivalence of triplet correlation functions is sufficient to cause the critical temperatures to approach their two- and three-dimensional values. For instance, for quadratic lattices we obtain values for the critical temperature of 2.5284 and 2.9796 as compared to the exact and series results of 2.2692 and 2.7044 for the simple quadratic and diamond lattices, respectively, and the Bethe approximation of 2.8854.
For pt.I see ibid., vol.14, p.57 (1981). The recently developed theory of correlation reduction method near transition temperature is generalised to the Ising model with arbitrary spin. The generalisation from the case of spin 1 / 2 is natural and straightforward, although some non-trivial technicalities are involved for the detailed applications. The transition temperatures for the spin 1 and spin 3 / 2 Ising ferromagnets are obtained explicitly, and are shown to be consistent with the general trend exhibited by other calculations.