Crossover from a Fractal Lattice to an Euclidean Lattice for theThermodynamic Properties of a Four-Spin Interaction Ising Model

We study how thermodynamic properties of the four-spin interaction Ising model on a family of Sierpinski carpets fractals cross over to that of the Ising model on square lattice with four-spin interaction in half of the square faces. The exact free energy f_s for the square model and f_f for the fractals are obtained in a closed form and found to be analytic in temperature. The free energy varies smoothly with the geometrical parameter b (which labels each member of the fractal family) and for large b the difference between f_s and f_f is asymptotically proportional to l/b².