Magnetic critical behavior of the Ising model on fractal structures

Abstract

International audienceThe critical temperature and the set of critical exponents (β,γ,ν) of the Ising model on a fractal structure, namely the Sierpiński carpet, are calculated from a Monte Carlo simulation based on the Wolff algorithm together with the histogram method and finite-size scaling. Both cases of periodic boundary conditions and free edges are investigated. The calculations have been done up to the seventh iteration step of the fractal structure. The results show that, although the structure is not translationally invariant, the scaling behavior of thermodynamical quantities is conserved, which gives a meaning to the finite-size analysis. Although some discrepancies in the values of the critical exponents occur between periodic boundary conditions and free edges, the effective dimension obtained through the Rushbrooke and Josephson's scaling law have the same value in both cases. This value is slightly but significantly different from the fractal dimension

No unitary bootstrap for the fractal Ising model

“…More information about their approach can be found in [57,64]. Other results are available [47,62] but differ from the results of [58] by comparitively minor amounts and do not affect the qualitative conclusions of the comparison with our bounds.…”

“…Initially this seems unlikely, as generic fractal spaces are expected to break translation invariance even in the continuum limit. Nevertheless, such systems do show critical behaviour at finite temperature, and so there have been multiple attempts to compare the Wilson-Fisher critical exponents with those of fractals, with mixed results [47]. Furthermore, it has been suggested that certain fractals [48][49][50] (those with small lacunarity) do recover translational invariance in the continuum.…”

“…periodic vs. free), whether the spins are placed in the center of the squares or at vertices, and the method of calculation chosen (Monte Carlo, real-space renormalization, high-T expansions, etc.). For a thorough review of these intricacies, see [47].…”

“…We are allowed to do this because the bounds become more constraining for higher d and d eff > 1.65 for all theories we considered. We have included only a representative selection JHEP03(2015)167 of fractal Ising models, but we have checked that other results in literature [47,62] fall within this general area and none satisfy our bounds.…”

Critical behavior of an Ising system on the Sierpinski carpet: A short-time dynamics study

“…Based on renormalization methods, it has been shown that a second-order phase transition at nonzero temperature occurs only if the fractal substrate has an infinite ramification order. Moreover, since translational symmetry is a necessary condition to proceed with dimensional perturbation [3], the disagreement between the critical exponents determined by current methods [6,7,8,9,10,11] and those obtained by continuation of ε−expansions to noninteger dimension [12] can be related to the topological features of the fractal structure.…”