# 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} }

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

- Physics
- 1967

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 cannot… Expand

Correlations in Ising ferromagnets. III

- Physics
- 1967

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 cannot… Expand

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 of… Expand

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 a… Expand

Statistical mechanics of a one-dimensional lattice gas

- Mathematics
- 1968

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 phase… Expand

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.

The Institute for Advanced Study

- Nature
- 1933

THE Institute for Advanced Study in the United States was founded in 1930 by a grant of 5,000,000 dollars from Louis Bamberger and Mrs. Felix Fuld. Dr. Abraham Flexner, director of the Institute, has… Expand

"J."

- 1890

however (for it was the literal soul of the life of the Redeemer, John xv. io), is the peculiar token of fellowship with the Redeemer. That love to God (what is meant here is not God’s love to men)… Expand

Commun

- Math. Phys. 9, 267
- 1968