Journal of Statistical Physics

, Volume 15, Issue 4, pp 327–341 | Cite as

Inequalities for continuous-spin Ising ferromagnets

  • Garrett S. Sylvester


We investigate correlation inequalities for Ising ferromagnets with continuous spins, giving a simple unified derivation of inequalities of Griffiths, Ginibre, Percus, Lebowitz, and Ellis and Monroe. The single-spin measure and Hamiltonian for which an inequality may be proved become more restricted as the inequality becomes more complex. However, all results hold for a model with ferromagnetic pair interactions, positive (nonuniform) external field, and single-spin measureν eitherv(σ) = [1/(l + 1)] xΣf=0/lδ(−l +2j +σ) (spinl/2) ordv(σ) = exp [−P(σ)], whereP is an even polynomial all of whose coefficients must be positive except the quadratic, which is arbitrary. The Lebowitz correlation inequality is a corollary of the Ellis-Monroe inequality. As an application, we generalize the method of van Beijeren to establish a sharp phase interface at low temperature in nearest neighbor ferromagnets of at least three dimensions with arbitrary (symmetric) single-spin measure.

Key words

Ising model ferromagnetic correlation inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. van Beijeren, Interface Sharpness in the Ising System,Comm. Math. Phys. 40:1 (1975).Google Scholar
  2. 2.
    H. van Beijeren and G. Sylvester, Phase Transitions for Continuous-Spin Ising Ferromagnets, Maryland-Yeshiva preprint.Google Scholar
  3. 3.
    R. S. Ellis, Concavity of Magnetization for a Class of Even Ferromagnets,Bull. AMS 81(5):925 (1975).Google Scholar
  4. 4.
    R. S. Ellis and J. L. Monroe, A Simple Proof of the G.H.S. and Further Inequalities,Comm. Math. Phys. 41:33 (1975).Google Scholar
  5. 5.
    R. S. Ellis, J. L. Monroe, and C. M. Newman, The GHS and Other Correlation Inequalities for a Class of Even Ferromagnets,Comm. Math. Phys., to appear.Google Scholar
  6. 6.
    R. S. Ellis and C. M. Newman, A Tale of Two Inequalities: Concave Force Implies Concave Magnetization for Even Ferromagnets, Northwestern preprint (1975).Google Scholar
  7. 7.
    J. Ginibre, General Formulation of Griffiths' Inequalities,Comm. Math. Phys. 16:310 (1970).Google Scholar
  8. 8.
    J. Glimm and A. Jaffe, A Remark on the Existence of φ4 4,Phys. Rev. Lett. 33:440–442 (1974).Google Scholar
  9. 9.
    R. B. Griffiths, Correlations in Ising Ferromagnets I and II,J. Math. Phys. 8:478, 484 (1967).Google Scholar
  10. 10.
    R. B. Griffiths, Rigorous Results for Ising Ferromagnets of Arbitrary Spin,J. Math. Phys. 10:1559 (1969).Google Scholar
  11. 11.
    R. B. Griffiths, Phase Transitions, inStatistical Mechanics and Quantum Field Theory, De Witt and Stora, eds., Gordon and Breach (1970).Google Scholar
  12. 12.
    R. B. Griffiths, Rigorous Results and Theorems, inPhase Transitions and Critical Phenomena, Vol. I:Exact Results, C. Domb and M. S. Green, eds., Academic, New York (1972).Google Scholar
  13. 13.
    R. B. Griffiths, C. A. Hurst, and S. Sherman, Concavity of Magnetization of an Ising Ferromagnet in a Positive External Field,J. Math. Phys. 11:790 (1970).Google Scholar
  14. 14.
    D. G. Kelly and S. Sherman, General Griffiths Inequalities on Correlations in Ising Ferromagnets,J. Math. Phys. 9:466 (1974).Google Scholar
  15. 15.
    J. L. Lebowitz, G.H.S. and Other Inequalities,Comm. Math. Phys. 35:87 (1974).Google Scholar
  16. 16.
    J. L. Lebowitz, Bounds on Correlations and Analyticity Properties of Ferromagnetic Ising Spin Systems,Comm. Math. Phys. 28:313 (1972).Google Scholar
  17. 17.
    J. L. Lebowitz, Uniqueness, Analyticity and Decay Properties of Correlations in Equilibrium Systems, inInternational Symposium on Mathematical Problems in Theoretical Physics, Springer-Verlag (1975).Google Scholar
  18. 18.
    C. M. Newman, Gaussian Correlation Inequalities for Ferromagnets,Z. Wahrscheinlichkeitstheorie 33:75 (1975).Google Scholar
  19. 19.
    J. K. Percus, Correlation Inequalities for Ising Spin Lattices,Comm. Math. Phys. 40:283 (1975).Google Scholar
  20. 20.
    B. Simon,The P(φ)2 Euclidean (Quantum) Field Theory, Princeton University Press (1974).Google Scholar
  21. 21.
    T. Spencer, The Absence of Even Bound States inφ 2 4,Comm. Math. Phys. 39:77 (1974).Google Scholar
  22. 22.
    G. Sylvester, Continuous-Spin Ising Ferromagnets, MIT Thesis (1976).Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • Garrett S. Sylvester
    • 1
  1. 1.Belfer Graduate School of ScienceYeshiva UniversityNew York

Personalised recommendations