The present paper focuses on the order-disorder transition of an Ising model
on a self-similar lattice. We present a detailed numerical study, based on the
Monte Carlo method in conjunction with the finite size scaling method, of the
critical properties of the Ising model on some two dimensional deterministic
fractal lattices with different Hausdorff dimensions. Those with finite
ramification order do not display ordered phases at any finite temperature,
whereas the lattices with infinite connectivity show genuine critical behavior.
In particular we considered two Sierpinski carpets constructed using different
generators and characterized by Hausdorff dimensions d_H=log 8/log 3 = 1.8927..
and d_H=log 12/log 4 = 1.7924.., respectively.
The data show in a clear way the existence of an order-disorder transition at
finite temperature in both Sierpinski carpets.
By performing several Monte Carlo simulations at different temperatures and
on lattices of increasing size in conjunction with a finite size scaling
analysis, we were able to determine numerically the critical exponents in each
case and to provide an estimate of their errors.
Finally we considered the hyperscaling relation and found indications that it
holds, if one assumes that the relevant dimension in this case is the Hausdorff
dimension of the lattice.Comment: 21 pages, 7 figures; a new section has been added with results for a
second fractal; there are other minor change
