Existence of a phase-transition in a one-dimensional Ising ferromagnet

@article{Dyson1969ExistenceOA,
  title={Existence of a phase-transition in a one-dimensional Ising ferromagnet},
  author={Freeman J. Dyson},
  journal={Communications in Mathematical Physics},
  year={1969},
  volume={12},
  pages={91-107}
}
  • F. Dyson
  • Published 1 June 1969
  • Mathematics
  • Communications in Mathematical Physics
AbstractExistence of a phase-transition is proved for an infinite linear chain of spins μj=±1, with an interaction energy $$H = - \sum J(i - j)\mu _i \mu _j ,$$ whereJ(n) is positive and monotone decreasing, and the sums ΣJ(n) and Σ (log logn) [n3J(n)]−1 both converge. In particular, as conjectured byKac andThompson, a transition exists forJ(n)=n−α when 1 < α < 2. A possible extension of these results to Heisenberg ferromagnets is discussed. 
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An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannotExpand
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An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannotExpand
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We consider a one‐dimensional model with infinite‐range interaction, a two‐dimensional model, and a three‐dimensional model, whose free energies can be expressed in terms of the largest eigenvalue ofExpand
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Commun
  • Math. Phys. 9, 267
  • 1968
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