Cluster analysis and finite-size scaling for Ising spin systems

Area of Science:

• Statistical Mechanics
• Condensed Matter Physics
• Computational Physics

Background:

• The Ising model is a fundamental model in statistical mechanics for studying magnetism.
• Percolation theory describes the formation of connected clusters in random systems.
• Understanding the interplay between these models is crucial for complex systems analysis.

Purpose of the Study:

• To calculate distribution functions for lattice site fractions and magnetization in subgraphs with multiple percolating clusters.
• To investigate the finite-size scaling behavior and universality of these functions.
• To analyze the complex structure of magnetization distribution in systems with varying aspect ratios.

Main Methods:

• Connecting the Ising model with a correlated percolation model.
• Calculating the distribution function f(n)(c) for the fraction of lattice sites in percolating clusters.
• Calculating the distribution function p(n)(m) for magnetization.
• Analyzing finite-size scaling behavior on square, plane triangular, and honeycomb lattices.

Main Results:

• Both f(n)(c) and p(n)(m) exhibit excellent finite-size scaling behavior.
• Universal finite-size scaling functions were identified for specific lattice aspect ratios.
• The complex magnetization distribution in large aspect ratio systems is explained by independent cluster orientations.

Conclusions:

• The connection between Ising and percolation models provides a powerful framework for analyzing complex lattice systems.
• Universality in finite-size scaling functions suggests a common underlying physical mechanism across different lattices.
• The orientation of multiple percolation clusters significantly influences system magnetization.