Cluster Analysis and Finite-Size Scaling for Ising Spin Systems

TL;DR

This paper investigates the distribution functions of percolating clusters and magnetization in Ising spin systems, revealing universal finite-size scaling behaviors across different lattice types and explaining complex magnetization structures.

Contribution

It introduces a method to calculate distribution functions for percolating clusters and magnetization, demonstrating their universal finite-size scaling properties across various lattice geometries.

Findings

Distribution functions exhibit good finite-size scaling behavior.

Universal scaling functions are consistent across different lattice types.

Complex magnetization structures can be explained by independent cluster orientations.

Abstract

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction ($c$) of lattice sites in percolating clusters in subgraphs with $n$ percolating clusters, $f_n(c)$, and the distribution function for magnetization ($m$) in subgraphs with $n$ percolating clusters, $p_n(m)$. We find that $f_n(c)$ and $p_n(m)$ have very good finite-size scaling behavior and they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:$\sqrt 3$/2:$\sqrt 3$. The complex structure of the magnetization distribution function $p(m)$ for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such system.