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

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 ${\mathbb{R}}^n$ (with $n\ge 2$) 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 $\lambda \ge 0$. We assume that $\Phi$ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For $d\ge 1$, $s\in (d,d+2]$ 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 $d\ge 3$ 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.

17 pages, fixed typos and improved presentation; version to appear in J. Statist. Phys

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