Magnetic critical behavior of fractals in dimensions between 2 and 3

Abstract

International audienceWe report the critical exponent values of the Ising model, n21, g /n , and b/n , and the critical temperatures of three Sierpı'nski fractals with Hausdorff dimensions df equal to 2.966, 2.904, and 2.631. The results are calculated from finite-size scaling analysis by Monte Carlo simulations. They are precise enough to show that the hyperscaling relation df52b/n1g /n is satisfied. Furthermore, the discrepancy between the values provided by e expansions and by Monte Carlo simulations shows that the critical behavior of fractals cannot be fully understood in the framework of strong universality

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

“…The relative difference between d eff and the Hausdorff fractal dimension d f ϭ1.9746 is smaller than 0.2%. No straight evidence for a discrepancy between these dimensions can be brought out from these results: As already showed in the case of fractal dimensions between two and three, 16 the hyperscaling relation is satisfied with d eff ϭd f .…”

“…͑In other words, an infinitely narrow transition occurs right at Tϭ0.͒ As a matter of fact, a full understanding of second order phase transitions in fractals needs the investigation of dimensions higher than 2. Very recently, Hsiao et al 16 studied three fractal dimensions between 2 and 3. They showed that scaling corrections vanish much more quickly than in the present case, and were able to give evidence that the hyperscaling relation is satisfied when the space dimension is replaced by the Hausdorff one.…”

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

“…At the present stage, we are only able to provide an upper bound for the Sierpiński carpet of infinite size: t Ͻ1.781 (10) or y t Ͻ0.5254(51). The same situation is also observed at T sim ϭ1.4813.…”

“…Re-cently, due to the progress in simulation methods and the growth of computer power, the critical behavior of the Ising model on fractals of Hausdorff dimension d f between 1 and 3 has been studied numerically in a much more precise way. [8][9][10][11] The results showed that the finite-size scaling ͑FSS͒ analysis works in the case of fractals, although the convergence towards the thermodynamical limit can be very slow when d f Ͻ2, and that the hyperscaling law d f ϭ2␤/ϩ␥/ is verified. Moreover, discrepancies with the predictions of the ⑀ expansions 13 were observed.…”

Effects of random and deterministic discrete scale invariance on the critical behavior of the Potts model

“…As a main result, it turns out that besides the space dimension, topological details of the fractal structure (lacunarity, connectivity, ramification order) have an influence on the values of the critical exponents. It is now well established that the critical behavior on deterministic fractal structures can be understood in the framework of a weak universality [3][4][5][6][7]: a universality class does not only depend on the order parameter dimension, the space dimension, and the interaction range, but also on topological details of the fractal structure. Since no analytical general theory is hitherto available, most results have been obtained with the help of numerical simulations; in the case of deterministic fractals, these methods come up against peculiar difficulties which have been discussed in Ref.…”