On the replica symmetric Ising spin glasses

  • C. De Dominicis
  • P. Mottishaw
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 268)


Confronted with the contradiction between mounting evidence for a phase transition in three dimension with only two states (in zero field) and the necessity for a symmetry breaking à la PARISI when treating an Ising spin glass with a gaussion bond distribution, we look, as an escape, for possible stable replica symmetric solutions, in the tree approximation, associated with non-gaussian distributions.

We first consider a toy model where the bond distribution has one more non vanishing cumulant than the gaussian. We show that there is some theoretical range of stability provided the second cumulant is negative enough. We then turn to a general bond distribution and derive closed expressions for the “masses” that allow to determine a stability region (in dilution/temperature space). We also show that instead of using an infinite number of order parameters qr (generalizing the q2 of EDWARDS ANDERSON one may use a function as MEZARD and PARISI have done for optimization problems. This is illustrated on simpler models due to VIANA and BRAY, and to FU and ANDERSON. Finally we give some hints for the treatment of the general stability problem.


Spin Glass Tree Approximation Closed Expression Replica Symmetry Breaking Bond Distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    See e.g. the review by J.A. Mydosh in Heidelberg Colloqium 1986, Springer Verlag, L. van Hemmen and I. Morgenstern Editors. See also H. Bouchiat, Thesis Orsay (1986) for an extensive discussionGoogle Scholar
  2. 2.
    See e.g. the review by R. Bhatt in Heidelberg Colloqium 1986, and M. Moore ibid. See also K. Binder and A.P. Young, Rev. Mod. Phys. (1986) to appear.Google Scholar
  3. 3.
    S.F. Edwards and P.W. Anderson, J. Phys. F 5 965 (1975)CrossRefGoogle Scholar
  4. 4.
    G. Parisi, Phys. Rev. Lett. 43 1754 (1979)Google Scholar
  5. 5.
    C. De Dominicis and I. Kondor, J. Pysique Lett. 46 L–1037 (1985)Google Scholar
  6. 6.
    C. De Dominicis and I. Kondor, to be publishedGoogle Scholar
  7. 7.
    A. Bovier and J. Frölich, J. Stat. Phys. (1986) to appearGoogle Scholar
  8. 8.
    D.S. Fisher and D.A. Huse, Phys. Rev. Lett. 56 1601 (1986)PubMedGoogle Scholar
  9. 9.
    A.J. Bray and M.A. Moore, J. Phys. C17 L613 (1984)Google Scholar
  10. 10.
    M. Mezard and G. Parisi, J. Physique Lett. 46 L–771 (1985)Google Scholar
  11. 11.
    L. Viana and A.J. Bray, J. Phys. C18, 3037 (1985)Google Scholar
  12. 12.
    Y. Fu and D.W. Anderson, J. Phys. A19, 1605 (1986Google Scholar
  13. 13.
    D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975)Google Scholar
  14. 14.
    E. Pytte and J. Rudnick, Phys. Rev. 19, 3603 (1978)Google Scholar
  15. 15.
    A.J. Bray and M.A. Moore, Phys. Rev. Lett. 41, 1068 (1978)CrossRefGoogle Scholar
  16. 16.
    J.R. de Almeida and D.J. Thouless, J. Phys. A11, 983 (1978)Google Scholar
  17. 17.
    In fact, in order for the series (2.10) to be convergent, it is appropriate to factorize out (expλS)/2 in chλS of eq.(2.9), which only adds-λδr;1, and insures that now 1n(l+exp-4λm) converges for large m. The lnf(O) term in (2.11) is then enough to reconstruct 1nch2λz in (2.12).Google Scholar
  18. 18.
    I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Product, Academic Press, New York 1965Google Scholar
  19. 19.
    C. Itzykson, J.M.P. 10, 1109 (1969)Google Scholar
  20. 20.
    G. Parisi, Phys. Rev. Lett. 50 (1983) 1946Google Scholar
  21. 21.
    I. Kondor and C. De Dominicis, Eur. Lett. (1986) to appear.Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • C. De Dominicis
    • 1
  • P. Mottishaw
    • 1
  1. 1.Service de Physique ThéoriqueCEN SaclayGif-sur-Yvette cedexFrance

Personalised recommendations