Abstract
We consider continuous-spin models on the d-dimensional hypercubic lattice with the spins σ x a priori uniformly distributed over the unit sphere in ℝn (with n≥2) 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 ⋅σ y for some exponent s>d and λ≥0. We assume that Φ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For d≥1, s∈(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≥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.
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© 2011 M. Biskup and N. Crawford. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.
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Biskup, M., Crawford, N. Absence of Magnetism in Continuous-Spin Systems with Long-Range Antialigning Forces. J Stat Phys 144, 731–748 (2011). https://doi.org/10.1007/s10955-011-0274-z
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DOI: https://doi.org/10.1007/s10955-011-0274-z