On the Ising Model with Long Range Interaction III: A Rigorous Lower Bound of the Free Energy.
NORTHWESTERN UNIV EVANSTON ILL DEPT OF PHYSICS
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In the Weiss theory of ferromagnetism the interaction between the elementary magnets is replaced by a properly chosen spacially homogeneous field. It is known that the results of the Weiss theory applied to Ising models become exact in the limit of infinitely weak ferromagnetic interaction of infinite range, provided certain conditions hold. Even before the result had been rigorously proved, Brout started a program of finding correction terms to the Weiss theory for small values of a parameter gamma, defined so that the interaction energy of a pair of spins is proportional to gamma, and the range of interaction is Gamma to the power -1D, where D is the dimensionality of the lattice. The complete expansion of the free energy in a series, in which the dominant order in gamma of each term is easily obtained, was given in part I of the present paper for any fixed temperature T Tw, where Tw is the Curie temperature of the Weiss theory. In part 2 it was shown that these results can be obtained by a resummation of the most divergent terms of the original expansion, if a certain integral diverges no worse than logarithmically at Tw. Convergence properties of these expansions are known only for the one dimensional model with exponential interaction, which has been solved exactly. In the present paper the author obtains a rigorous lower bound for the free energy, to supplement the known upper bound. Author
- Solid State Physics