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

Abstract

The short-time dynamic evolution of an Ising model embedded in an infinitely ramified fractal structure with noninteger Hausdorff dimension was studied using Monte Carlo simulations. Completely ordered and disordered spin configurations were used as initial states for the dynamic simulations. In both cases, the evolution of the physical observables follows a power-law behavior. Based on this fact, the complete set of critical exponents characteristic of a second-order phase transition was evaluated. Also, the dynamic exponent θ of the critical initial increase in magnetization, as well as the critical temperature, were computed. The exponent θ exhibits a weak dependence on the initial (small) magnetization. On the other hand, the dynamic exponent z shows a systematic decrease when the segmentation step is increased, i.e., when the system size becomes larger. Our results suggest that the effective noninteger dimension for the second-order phase transition is noticeably smaller than the Hausdorff dimension. Even when the behavior of the magnetization (in the case of the ordered initial state) and the autocorrelation (in the case of the disordered initial state) with time are very well fitted by power laws, the precision of our simulations allows us to detect the presence of a soft oscillation of the same type in both magnitudes that we attribute to the topological details of the generating cell at any scale.

No unitary bootstrap for the fractal Ising model

“…They studied a range of Sierpinski carpets using Monte Carlo simulations with the Metropolis algorithm, periodic boundary conditions, and spins placed at lattice vertices. 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.…”

“…This was precisely the approach first attempted by Mandelbrot and collaborators [50], who considered the Ising model on various kinds of fractal lattices. An incomplete list of later work includes [57][58][59][60][61][62]. These works show that for large classes of fractals it is possible to find critical points and associated critical exponents.…”

Log-periodic modulation in one-dimensional random walks

“…It is in general believed that they appear because of an inherent self-similarity [12], responsible for a discrete scale invariance (DSI) [13]. Examples of these oscillations have been detected in earthquakes [14,15], escape probabilities in chaotic maps close to crisis [16], biased diffusion [17,18], kinetic and dynamic processes on random quenched and fractal media [19][20][21][22], and stock markets near a financial crash [23][24][25][26].In this work we analyse a minimal model of RW, which results in log-periodic modulations of some observables. The main objective is to investigate the underlying physics of the oscillatory behaviour mentioned above.…”

Tsallis Distribution Decorated with Log-Periodic Oscillation

“…It is is usually taken in the form [56]: Before proceeding any further, let us remember that such log-periodic oscillations are widely know in all situations in which one encounters power distributions. In fact, such behavior has been found in earthquakes [57,58], escape probabilities in chaotic maps close to a crisis [59], biased diffusion of tracers on random systems [60][61][62], kinetic and dynamic processes on random quenched and fractal media [63][64][65][66], when considering the specific heat associated with self-similar [67] or fractal spectra [68], diffusion-limited-aggregate clusters [69], growth models [70] or stock markets near financial crashes [71][72][73][74], to name only a few examples. However, in all of these cases, the basic distributions were a scale-free power laws, without any scale parameter (here T ) and without a constant term governing their X < nT behavior.…”