Journal of Statistical Physics

, Volume 36, Issue 5–6, pp 787–793 | Cite as

Hierarchical models and chaotic spin glasses

  • A. Nihat Berker
  • Susan R. McKay


Renormalization-group studies in position space have led to the discovery of hierarchical models which are exactly solvable, exhibiting nonclassical critical behavior at finite temperature. Position-space renormalization-group approximations that had been widely and successfully used are in fact alternatively applicable as exact solutions of hierarchical models, this realizability guaranteeing important physical requirements. For example, a hierarchized version of the Sierpiriski gasket is presented, corresponding to a renormalization-group approximation which has quantitatively yielded the multicritical phase diagrams of submonolayers on graphite. Hierarchical models are now being studied directly as a testing ground for new concepts. For example, with the introduction of frustration, chaotic renormalization-group trajectories were obtained for the first time. Thus, strong and weak correlations are randomly intermingled at successive length scales, and a new microscopic picture and mechanism for a spin glass emerges. An upper critical dimension occurs via a boundary crisis mechanism in cluster-hierarchical variants developed to have well-behaved susceptibilities.

Key words

Hierarchical models renormalization group Sierpiński gasket frustration chaos spin glass boundary crisis 


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  1. 1.
    A. N. Berker and S. Ostlund,J. Phys. C 12;4961 (1979).Google Scholar
  2. 2.
    M. Kaufman and R. B. Griffiths,Phys. Rev. B 24;496 (1981); R. B. Griffiths and M. Kaufman,Phys. Rev. B 26;5022 (1982).Google Scholar
  3. 3.
    S. R. McKay, A. N. Berker, and S. Kirkpatrick,J. Appl. Phys. 53;7974 (1982).Google Scholar
  4. 4.
    K. G. Wilson,Phys. Rev. B 4;3174, 3184 (1971); K. G. Wilson and J. Kogut,Phys. Rep. C 12;75 (1974).Google Scholar
  5. 5.
    M. Nauenberg and B. Nienhuis,Phys. Rev. Lett. 33;1598 (1974).Google Scholar
  6. 6.
    A. A. Migdal,Zh. Eksp. Toar. Fiz. 69;1457 (1975) [Sou.Phys. JETP 42;743 (1976)]; L. P. Kadanoff,Ann. Phys. (N.Y.) 100;359 (1976).Google Scholar
  7. 7.
    M. Kaufman and R. B. Griffiths,Phys. Rev. B 28;3864 (1983).Google Scholar
  8. 8.
    Th. Niemeijer and J. M. J. van Leeuwen,Physica (Utrecht) 71;17 (1974).Google Scholar
  9. 9.
    L. P. Kadanoff,Phys. Rev. Lett. 34;1005 (1974); L. P. Kadanoff, A. Houghton, and M. C. Yalabik,J. Stat. Phys. 14;171 (1976).Google Scholar
  10. 10.
    S. Ostlund and A. N. Berker,Phys. Rev. Lett. 42;843 (1979).Google Scholar
  11. 11.
    B. Mandelbrot,Fractals: Form, Chance, and Dimension (Freeman, San Francisco, 1977).Google Scholar
  12. 12.
    Y. Gefen, B. D. Mandelbrot, and A. Aharony,Phys. Rev. Lett. 45;855 (1980).Google Scholar
  13. 13.
    S. R. McKay, A. N. Berker, and S. Kirkpatrick,Phys. Rev. Lett. 48;767 (1982).Google Scholar
  14. 14.
    M. Kaufman, private communication (1983).Google Scholar
  15. 15.
    S. R. McKay and A. N. Berker,J. Appl. Phys. 55;1646 (1984).Google Scholar
  16. 16.
    M. Kaufman and R. B. Griffiths,Phys. Rev. B 26;5282 (1982);Phys. Rev. B 30:244 (1984).Google Scholar
  17. 17.
    G. Toulouse,Commun. Phys. 2;115 (1977).Google Scholar
  18. 18.
    M. Kaufman and R. B. Griffiths,J. Phys. A 15;L239 (1982).Google Scholar
  19. 19.
    S. R. McKay and A. N. Berker,Phys. Rev. B 29;1315 (1984).Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • A. Nihat Berker
    • 1
  • Susan R. McKay
    • 1
  1. 1.Department of PhysicsMassachusetts Institute of TechnologyCambridge

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