Approximating the Ising model on fractal lattices of dimension less than two

We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of a zero external magnetic field, based on the combinatorial method of Feynman and Vdovichenko. We show that the procedure is applicable to any fractal obtained by the removal of sites from a periodic two-dimensional lattice. As a first application, we compute estimates for the critical temperatures of many different Sierpinski carpets and we compare them to known Monte Carlo estimates. The results show that our method is capable of determining the critical temperature with, possibly, arbitrary accuracy and paves the way for determination T-c of any fractal of dimension less than two. Critical exponents are more difficult to determine since the free energy of any periodic approximation still has a logarithmic singularity at the critical point implying alpha = 0. We also compute the correlation length as a function of the temperature and extract the relative critical exponent. We find nu = 1 for all periodic approximations, as expected from universality.