# 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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#### References

SHOWING 1-10 OF 15 REFERENCES
Correlations in Ising Ferromagnets. I
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Correlations in Ising ferromagnets. III
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Critical Behavior of Several Lattice Models with Long‐Range Interaction
• Mathematics
• 1969
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
Phase Transitions in One‐Dimensional Order—Disorder Systems: Application to Helix—Random‐Coil Transition in Polymers
• Chemistry
• 1962
In this work a discussion is given of the present theory of the helix—random‐coil transition in polypeptides. The seeming conflict between the description of this transition in terms of aExpand
Statistical mechanics of a one-dimensional lattice gas
We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result ofvan Hove we show that, for a large class of interactions, such a system has no phaseExpand
Statistical mechanics of lattice systems
• Mathematics
• 1967
We study the thermodynamic limit for a classical system of particles on a lattice and prove the existence of infinite volume correlation functions for a “large” set of potentials and temperatures.
Analyticity properties of a lattice gas
• Physics
• 1967
Abstract Analyticity properties are given for a system of classical particles on a lattice interacting through many-body potentials.