IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v185y1992i1p254-260.html
   My bibliography  Save this article

A class of long range Ising spin models described by Curie-Weiss mean field theory

Author

Listed:
  • Canning, Andrew

Abstract

We show that Ising spin models where the interaction matrix has eigenvalues whose number and values are finite and invariant with respect to N (the system size) in the limit N → ∞ are exactly solvable (subject also to a few other weaker conditions on the eigenvectors). Systems of this type are described by Curie-Weiss mean field type equations (〈Si〉 = tanh β (ΣiJij 〈Sj〉 )), which we shall refer to as the naive mean field type equations (NMFE), to distinguish them from other mean field equations which have an extra “reaction term”. Some models for which NMFE are valid will briefly be reviewed. It is found that, for certain choices of the interaction architecture, ferromagnetic systems (Jij ⩾ 0) exhibit weak spin-glass-like behaviour in the sense that they have many stable states and not just those associated with ferromagnetism. These extra stable states appear discontinuously. The application of these techniques to structured models of spin glasses and neural networks will also be briefly reviewed.

Suggested Citation

  • Canning, Andrew, 1992. "A class of long range Ising spin models described by Curie-Weiss mean field theory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 254-260.
    • Handle: RePEc:eee:phsmap:v:185:y:1992:i:1:p:254-260
    DOI: 10.1016/0378-4371(92)90464-2
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0378437192904642
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/0378-4371(92)90464-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kirkpatrick, Jeane, 1975. "Representation in the American National Conventions: the Case of 1972," British Journal of Political Science, Cambridge University Press, vol. 5(3), pages 265-322, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Akagi, Eiichi & Seino, Mitsuhiro & Katsura, Shigetoshi, 1991. "The free energy at T = 0 of the spin glass in the case of the continuous distribution of the effective field on the Bethe lattice for Z = 4, 5, 6," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 178(3), pages 406-414.
    2. Sibani, Paolo & Andersson, Jan-Olov, 1994. "Excitation morphology of short range Ising spin glasses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 206(1), pages 1-12.
    3. Shirakura, Takayuki & Katsura, Shigetoski, 1983. "Spin glass in the infinitely long-ranged bond-random anisotropic classical planar model-longitudinal and transverse spin glasses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 117(2), pages 281-299.
    4. Morita, T., 1983. "Free energy of the random Ising model in terms of the magnetizations of sites," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 119(1), pages 143-152.
    5. Yokota, Terufumi, 1992. "Numerical study of the TAP equations for infinite-range spin glass models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 336-339.
    6. de Oliveira, Mário J., 1988. "Ising spin glass in a field zero temperature in the Bethe approximation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 150(3), pages 614-626.
    7. Katsura, S., 1977. "Glass-like phase on the close packed lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 88(3), pages 583-590.
    8. Katsura, Shigetoshi & Nagahara, Izuru, 1980. "Effect of the frustration to the ground state energy and entropy of the spin-glass in the random bond Ising model on the square lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 104(3), pages 397-416.
    9. Kanter, I. & Shvartser, M., 1993. "The binary perceptron and general aspects of non-self-averaged quantities," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 200(1), pages 670-678.
    10. Katsura, Shigetoshi, 1980. "Entropy of the spin-glass state in the binary mixture of the ferro- and antiferromagnetic random Ising model at T = 0," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 104(1), pages 333-338.
    11. Anglès d'Auriac, J.C. & Maillard, J.M. & Wu, F.Y., 1991. "Three-state chiral Potts models on the triangular lattice: a Monte Carlo study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 179(3), pages 496-506.
    12. Hassan, R.A. & Cieplak, Marek & Suan Li, Mai, 1993. "Quadrupolar and S = 1 Ising spin glasses in local mean field approximation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 192(1), pages 1-13.
    13. Miyamoto, Nahomi & Katsura, Shigetoshi, 1982. "Entropy and free energy of the spin glass in the random-bond ising model on the square lattice at finite temperatures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 111(1), pages 1-16.
    14. Katsura, Shigetoshi & Inawashiro, Sakari & Fujiki, Sumiyoshi, 1979. "Spin glasses for the infinitely long ranged bond Ising model and for the short ranged binary bond Ising model without use of the replica method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 99(1), pages 193-216.
    15. Chen, Shyh-Tzer, 1996. "Mean-field renormalization group for the q-state clock spin-glass model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 229(2), pages 181-187.
    16. Dotsenko, V.S. & Tirozzi, B., 1992. "Replica-symmetry breaking in neural networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 385-394.
    17. Pirc, R. & Tadić, B. & Blinc, R., 1992. "Proton and deutron glasses," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 322-330.
    18. Morita, T., 1984. "Spin-glass and ferromagnetic phases of the random-bond Ising model on the Bethe lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 125(2), pages 321-343.
    19. Horiguchi, Tsuyoshi, 1981. "On the bethe approximation for the random bond Ising model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 107(2), pages 360-370.
    20. de Oliveira, M.J., 1988. "Ising spin glass in the Bethe approximation at zero temperature," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 148(3), pages 567-574.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:185:y:1992:i:1:p:254-260. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.