On the absence of ferromagnetism in typical 2D ferromagnets

TL;DR

This paper investigates the conditions under which 2D ferromagnetic systems with long-range interactions exhibit spontaneous magnetization, proving the absence of ferromagnetism for certain decay rates and its possibility for others.

Contribution

It provides rigorous results on the absence or presence of ferromagnetism in 2D Ising models with long-range interactions based on the decay exponent.

Findings

No spontaneous magnetization when decay exponent is between d and d+1.

Magnetic order can occur when decay exponent exceeds d+1.

Results apply to models with physical relevance such as magnetic dipoles.

Abstract

We consider the Ising systems in $d$ dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions which decay with a power, $s$, of the distance. The physical context of such models is discussed; primarily this is $d=2$ and $s=3$ where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above $d$ and does not exceed $d+1$, then for all temperatures, the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding $d+1$ (with $d\ge2$) magnetic order can occur.