Fractal-to-Euclidean crossover of the thermodynamic properties of the Ising model.

Abstract

In this paper we develop a diagrammatic technique for determining the exact recursive relations for the partition functions of the Ising model in a field, situated on finitely ramified deterministic fractal lattices. Applying this method on the members of the two-dimensional Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gasket type of fractal lattices, which are characterized by generators of side length b, we first recover the known exact-space renormalization-group results for b=2 and 3, in the case of zero field, and for b=2 when H\ensuremath{\ne}0. Then we obtain new results for all b up to b=15, for H=0, and up to b=8 for H\ensuremath{\ne}0. These results enable us to initiate the study of the crossover of thermodynamic properties of the Ising model caused by changing of the underlying fractal lattices towards the Euclidean (triangular) lattice. Accordingly, we calculate the temperature dependence of the specific heat, for b\ensuremath{\le}15, and susceptibility for b\ensuremath{\le}8, and compare these functions with the known results for the triangular lattice. This comparison demonstrates the difference between the standard thermodynamic limit and the fractal-to-Euclidean crossover behavior.