Critical behavior of the Ising model on fractal structures in dimensions between one and two: Finite-size scaling effects

Abstract

The magnetic critical behavior of Ising spins located at the sites of deterministic Sierpinski carpets is studied within the framework of a ferromagnetic Ising model. A finite-size scaling analysis is performed from Monte Carlo simulations. We investigate four different fractal dimensions between 1.9746 and 1.7227, up to the sixth and eighth iteration step of the fractal structure in one case. It turns out that the finite-size scaling behavior of most thermodynamical quantities is affected by scaling corrections increasing as the fractal dimension decreases, tending towards the lower critical dimension of the Ising model. These corrections are related to the topology of the fractal structure and to the scale invariance. Nevertheless the maxima of the susceptibility follow power laws in a very reliable way, which allows us to calculate the ratio of the exponents $\ensuremath{\gamma}/\ensuremath{\nu}.$ Moreover, the fixed point of the fourth order cumulant at ${T}_{c}$ exhibited by Binder on translation invariant lattices is replaced by a decreasing sequence of intersection points converging towards the critical temperature. The convergence towards the thermodynamical limit as the size of the networks increases is slowed down as the fractal dimension decreases. At last, the evolution of the discrepancies between Monte Carlo simulations and $\ensuremath{\epsilon}$ expansions with the fractal dimension is set out.