Approximating the Ising model on fractal lattices of dimension below two

TL;DR

This paper develops a method to approximate the free energies of Ising models on fractal lattices of dimension less than two, enabling accurate estimation of critical temperatures and critical exponents.

Contribution

It introduces a generalized combinatorial approach to compute free energies on fractal lattices and estimates critical temperatures with high accuracy, extending analysis to any fractal of dimension below two.

Findings

Accurately estimates critical temperatures for Sierpinski carpets.

Demonstrates the method's potential for arbitrary accuracy in $T_c$ estimation.

Finds a universal critical exponent $ u=1$ for all approximations.

Abstract

We construct periodic approximations to the free energies of Ising models on fractal lattices of dimension smaller than two, in the case of zero external magnetic field, using a generalization of the combinatorial method of Feynman and Vodvickenko. Our procedure is applicable to any fractal obtained by the removal of sites of 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 to determine $T_c$ for any fractal of dimension below 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 approximation, as expected from universality.