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

We consider continuous-spin models on the $d$-dimensional hypercubic lattice with the spins $σ_x$ \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 $Φ$, and a long-range antiferromagnetic part $λ|x-y|^{-s}σ_x\cdotσ_y$ for some exponent $s>d$ and $λ\ge0$. We assume that $Φ$ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For $d\ge1$, $s\in(d,d+2]$ and any $λ>0$, we then show that the expectation of each $σ_x$ 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 $λ=0$ 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.