Thermal Behavior of Spin Clusters and Interfaces in two-dimensional Ising Model on Square Lattice

TL;DR

This study uses Monte Carlo simulations to analyze the thermal behavior of spin clusters and interfaces in the 2D Ising model, revealing temperature-dependent fractal dimensions and confirming finite-size scaling near criticality.

Contribution

It introduces a tie-breaking rule for defining interfaces and demonstrates their conformal invariance at critical temperature, providing new insights into fractal dimensions across temperatures.

Findings

Fractal dimension of spin clusters varies linearly with temperature.

Interface fractal dimension sharply crosses over at critical temperature.

Finite-size scaling matches theoretical predictions.

Abstract

Extensive Monte Carlo study of two-dimensional Ising model is done to investigate the statistical behavior of spin clusters and interfaces as a function of temperature, $T$. We use a \emph{tie-breaking} rule to define interfaces of spin clusters on square lattice with strip geometry and show that such definition is consistent with conformal invariant properties of interfaces at critical temperature, $T_c$. The \emph{effective} fractal dimensions of spin clusters and interfaces ($d_c$ and $d_I$, respectively) are obtained as a function of temperature. We find that the effective fractal dimension of the spin clusters behaves almost linearly with temperature in three different regimes. It is also found that the effective fractal dimension of the interfaces undergoes a sharp crossover around $T_c$, between values 1 and 1.75 at low and high temperatures, respectively. We also check the finite-size scaling hypothesis for the percolation probability and the average mass of the largest spin-cluster in a good agreement with the theoretical predictions.