DOI: 10.1063/1.1705219. Correlations in Ising Ferromagnets. I

10.1063/1.1705219

Crossref journal-article

AIP Publishing

Journal of Mathematical Physics (317)

Abstract

The following results are proved for a system of Ising spins σi = ±1 in zero magnetic field coupled by a purely ferromagnetic interaction of the form −Σi&lt;j Jijσiσj with Jij ≥ 0, for arbitrary crystal lattice and range of interaction: (1) The binary correlation functions 〈σkσl〉 are always nonnegative (〈〉 denotes a thermal average). (2) For arbitrary i, j, k, and l, 〈σiσjσk σl〉 ≥ 〈σiσj〉〈σkσl〉. Consequences of these results, in particular the second, are: (i) 〈σkσl〉 never decreases if any Jij is increased. (ii) If an Ising model with ferromagnetic interactions exhibits a long-range order, this long-range order increases if additional ferromagnetic interactions are added. This last fact may be used to prove the existence of long-range order in a large class of two- and three-dimensional Ising lattices with purely ferromagnetic interactions of bounded or unbounded range.

Bibliography

Griffiths, R. B. (1967). Correlations in Ising Ferromagnets. I. Journal of Mathematical Physics, 8(3), 478â483.

Authors 1

Robert B. Griffiths (first)

References 11 Referenced 386

10.1103/PhysRev.87.410 / Phys. Rev. (1952)
{'key': '2024020715233457500_r2'}
10.1103/RevModPhys.36.856 / Rev. Mod. Phys. (1964)
10.1063/1.1705220 / J. Math. Phys. (1967)
10.1063/1.1704706 / J. Math. Phys. (1965)
10.1103/PhysRev.65.117 / Phys. Rev. (1944)
10.1073/pnas.55.4.676 / Proc. Natl. Acad. Sci. U.S. (1966)
{'key': '2024020715233457500_r7'}
10.1017/S0305004100019174 / Proc. Cambridge Phil. Soc. (1936)
10.1103/PhysRev.136.A437 / Phys. Rev. (1964)
{'key': '2024020715233457500_r10', 'first-page': '209', 'volume': '10', 'year': '1965', 'journal-title': 'Teoriya Veroyatnostei Primeneniya'} / Teoriya Veroyatnostei Primeneniya (1965)

Dates

Type	When
Created	20 years, 7 months ago (Jan. 6, 2005, 3:06 p.m.)
Deposited	1 year, 6 months ago (Feb. 7, 2024, 4:49 p.m.)
Indexed	2 weeks, 3 days ago (Aug. 6, 2025, 9:01 a.m.)
Issued	58 years, 5 months ago (March 1, 1967)
Published	58 years, 5 months ago (March 1, 1967)
Published Print	58 years, 5 months ago (March 1, 1967)

Funders 0

None

BibTeX

@article{Griffiths_1967, title={Correlations in Ising Ferromagnets. I}, volume={8}, ISSN={1089-7658}, url={http://dx.doi.org/10.1063/1.1705219}, DOI={10.1063/1.1705219}, number={3}, journal={Journal of Mathematical Physics}, publisher={AIP Publishing}, author={Griffiths, Robert B.}, year={1967}, month=mar, pages={478–483} }

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  "abstract": "<jats:p>The following results are proved for a system of Ising spins \u03c3i = \u00b11 in zero magnetic field coupled by a purely ferromagnetic interaction of the form \u2212\u03a3i&amp;lt;j\u2008Jij\u03c3i\u03c3j with Jij \u2265 0, for arbitrary crystal lattice and range of interaction: (1) The binary correlation functions \u3008\u03c3k\u03c3l\u3009 are always nonnegative (\u3008 \u3009 denotes a thermal average). (2) For arbitrary i, j, k, and l, \u3008\u03c3i\u03c3j\u03c3k \u03c3l\u3009 \u2265 \u3008\u03c3i\u03c3j\u3009 \u3008\u03c3k\u03c3l\u3009. Consequences of these results, in particular the second, are: (i) \u3008\u03c3k\u03c3l\u3009 never decreases if any Jij is increased. (ii) If an Ising model with ferromagnetic interactions exhibits a long-range order, this long-range order increases if additional ferromagnetic interactions are added. This last fact may be used to prove the existence of long-range order in a large class of two- and three-dimensional Ising lattices with purely ferromagnetic interactions of bounded or unbounded range.</jats:p>",
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