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- Title
Absence of Magnetism in Continuous-Spin Systems with Long-Range Antialigning Forces.
- Authors
Biskup, Marek; Crawford, Nicholas
- Abstract
We consider continuous-spin models on the d-dimensional hypercubic lattice with the spins σ a priori uniformly distributed over the unit sphere in ℝ (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| σ⋅ σ 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 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.
- Subjects
MAGNETISM; FERROMAGNETIC resonance; ROTATIONAL motion; LONG range order (Solid state physics); GIBBS phenomenon; GIBBS' equation; LINEAR free energy relationship
- Publication
Journal of Statistical Physics, 2011, Vol 144, Issue 4, p731
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-011-0274-z