FRACTAL-TO-EUCLIDEAN CROSSOVER OF THE THERMODYNAMIC PROPERTIES OF THEISING-MODEL

In this paper we develop a diagrammatic technique for determining the exact recursive relations for the partition functions of the Ising mod el in a field, situated on finitely ramified deterministic fractal lat tices. Applying this method on the members of the two-dimensional Sier pinski gasket type of fractal lattices, which are characterized by gen erators of side length b, we first recover the known exact-space renor malization-group results for b = 2 and 3, in the case of zero field, a nd for b = 2 when H not-equal 0. Then we obtain new results for all b up to b = 15, for H = 0, and up to b = 8 for H not-equal 0. These resu lts enable us to initiate the study of the crossover of thermodynamic properties of the Ising model caused by changing of the underlying fra ctal lattices towards the Euclidean (triangular) lattice. Accordingly, we calculate the temperature dependence of the specific heat, for b l ess-than-or-equal-to 15, and susceptibility for b less-than-or-equal-t o 8, and compare these functions with the known results for the triang ular lattice. This comparison demonstrates the difference between the standard thermodynamic limit and the fractal-to-Euclidean crossover be havior.