Critical Behavior of an Ising System on the Sierpinski Carpet: A Short-Time Dynamics Study

TL;DR

This study investigates the critical behavior of an Ising model on a Sierpinski carpet fractal using Monte Carlo simulations, revealing power-law dynamics and critical exponents that depend on system size and structure.

Contribution

It provides the first detailed analysis of short-time dynamics and critical exponents for an Ising system on a fractal with noninteger Hausdorff dimension.

Findings

Power-law behavior in physical observables during evolution.

Critical exponents depend on system size and structure.

Effective dimension for phase transition is smaller than Hausdorff dimension.

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 $\theta $ of the critical initial increase in magnetization, as well as the critical temperature, were computed. The exponent $\theta $ 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.