Topological Structures of Cluster Spins for Ising Models

TL;DR

This paper explores the hierarchical and fractal structures of clusters in Ising models, defining a fractal dimension to identify fixed points and classifying clusters into irreducible and reducible types.

Contribution

It introduces a fractal dimension framework for ordered clusters in Ising models and relates cluster spins to their coordination number and fractal properties.

Findings

Defined a fractal dimension for ordered clusters.

Classified clusters into irreducible and reducible types.

Established relationships between cluster spins, coordination number, and fractal dimension.

Abstract

We discussed hierarchies and rescaling rule of the self similar transformations in Ising models, and define a fractal dimension of an ordered cluster, which minimum corresponds to a fixed point of the transformations. By the fractal structures we divide the clusters into two types: irreducible and reducible. A relationship of cluster spin with its coordination number and fractal dimension is obtained.