Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result

TL;DR

This paper provides a rigorous proof of spontaneous magnetization at finite temperature for the Ising model on the Sierpinski carpet fractal, extending the understanding of phase transitions in low-dimensional fractal structures.

Contribution

It offers the first rigorous proof of spontaneous magnetization in the Ising model on a fractal with dimension less than two, using a Peierls argument adaptation.

Findings

Proves spontaneous magnetization exists on the Sierpinski carpet.

Extends phase transition theory to fractal geometries.

Demonstrates low-dimensional fractals can exhibit magnetic order.

Abstract

We give a rigorous proof of the existence of spontaneous magnetization at finite temperature for the Ising spin model defined on the Sierpinski carpet fractal. The theorem is inspired by the classical Peierls argument for the two dimensional lattice. Therefore, this exact result proves the existence of spontaneous magnetization for the Ising model in low dimensional structures, i.e. structures with dimension smaller than 2.