Critical behavior of the geometrical spin clusters and interfaces in the two-dimensional thermalized bond Ising model

TL;DR

This study numerically investigates the fractal and percolation properties of geometrical spin clusters in a two-dimensional thermalized bond Ising model, confirming universality with the standard Ising model through critical exponents and interface analysis.

Contribution

It provides the first detailed numerical analysis of geometrical cluster properties in the thermalized bond Ising model, demonstrating universality with the regular Ising model via critical exponents and fractal dimensions.

Findings

Fractal dimensions of clusters match those of the standard Ising model.

Winding angle variance indicates a diffusivity of 3, placing it in the Ising universality class.

Percolation exponents are consistent with the regular Ising model, supporting universality.

Abstract

The fractal dimensions and the percolation exponents of the geometrical spin clusters of like sign at criticality, are obtained numerically for an Ising model with temperature-dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify non-intersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters $D_{c}$, and their interfaces $D_{I}$, are in perfect agreement with those reported for the standard two-dimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles, results in a diffusivity $\kappa=3$ for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters, results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained, are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.