Critical properties of Ising model on Sierpinski fractals. A finite size scaling analysis approach

TL;DR

This study investigates the critical behavior of the Ising model on Sierpinski fractals using Monte Carlo simulations and finite size scaling, revealing finite-temperature phase transitions and critical exponents related to the fractals' Hausdorff dimensions.

Contribution

It provides the first detailed numerical analysis of the Ising model's critical properties on Sierpinski carpets with different Hausdorff dimensions, confirming phase transitions and hyperscaling relations.

Findings

Finite-temperature order-disorder transitions observed.

Critical exponents estimated with error analysis.

Hyperscaling relation holds with Hausdorff dimension as the relevant metric.

Abstract

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.