Absence of magnetism in continuous-spin systems with long-range antialigning forces

TL;DR

This paper proves that continuous-spin models with long-range antialigning forces on a lattice do not exhibit spontaneous magnetism, contrasting with ferromagnetic models, by showing all Gibbs states have zero expectation for spins.

Contribution

The paper extends previous results by demonstrating absence of magnetism in continuous-spin systems with long-range antiferromagnetic interactions for a broad range of parameters.

Findings

All Gibbs states have zero spin expectation.

Spontaneous magnetization is absent in these models.

Contrasts with ferromagnetic systems showing magnetic order.

Abstract

We consider continuous-spin models on the $d$-dimensional hypercubic lattice with the spins $\sigma_x$ \emph{a priori} uniformly distributed over the unit sphere in $\R^n$ (with $n\ge2$) and the interaction energy having two parts: a short-range part, represented by a potential $\Phi$, and a long-range antiferromagnetic part $\lambda|x-y|^{-s}\sigma_x\cdot\sigma_y$ for some exponent $s>d$ and $\lambda\ge0$. We assume that $\Phi$ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For $d\ge1$, $s\in(d,d+2]$ and any $\lambda>0$, we then show that the expectation of each $\sigma_x$ vanishes in all translation-invariant Gibbs states. In particular, the spontaneous magnetization is zero and block-spin averages vanish in all (translation invariant or not) Gibbs states. This contrasts the situation of $\lambda=0$ where the ferromagnetic nearest-neighbor systems in $d\ge3$ exhibit strong magnetic order at sufficiently low temperatures. Our theorem extends an earlier result of A. van Enter ruling out magnetized states with uniformly positive two-point correlation functions.