Critical behavior of the ferromagnetic Ising model on a Sierpiński carpet: Monte Carlo renormalization group study

Hsiao, Pai‐Yi; Monceau, Pascal

doi:10.1103/physrevb.67.064411

Cited by 12 publications

(1 citation statement)

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“…Less clear is the situation in dimensions less than two and greater than one. Continuous methods [13,14] give a fairly good description, and in some dimensions real space renormalization group studies are available [15,16], but an explicit solution of the Ising model in some fractal cases will provide a strong indication regarding the reliability or not of continuous methods in dimensions less than two. In fact it is not completely clear if there is a dierence between the values of critical exponents one can obtain with continuous methods, which usually make a continuation of the integer number of dimensions to fractional values, and the actual values obtained by studying the analogous systems defined directly on lattices of non-integer fractal dimension.…”

Section: Universalitymentioning

confidence: 99%

Approximating the Ising model on fractal lattices of dimension less than two

Codello¹,

Drach²,

Hietanen³

2015

J. Stat. Mech.

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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 α = 0. We also compute the correlation length as a function of the temperature and extract the relative critical exponent. We find ν = 1 for all periodic approximation, as expected from universality.

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