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 $d_H=\ln 8/\ln 3=1.8927\dots$ and $d_H=\ln 12/\ln 4=1.7924\dots ,$ 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.

• Received 2 February 1998

DOI:https://doi.org/10.1103/PhysRevB.58.14387