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The RG-transformation for the hierarchical model

Part I. Heuristics
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Part of the Lecture Notes in Physics book series (LNP, volume 74)

Keywords

Critical Temperature Renormalization Group Central Limit Theorem Hierarchical Model Critical Index 
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References

The Hierarchical Model has been invented by Dyson to show that one-dimensional systems may exhibit phase transitions if they have long-range forces.

  1. [9]
    F.J. DYSON: Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math.Phys. 12, 91 (1969).CrossRefGoogle Scholar
  2. [10]
    F.J.DYSON: An Ising ferromagnet with discontinuous longrange order. Commun. Math. Phys. 21, 269 (1971).CrossRefGoogle Scholar

Baker reinvented the model and pointed out that the RG acted on the single spin distribution. He also calculated critical indices.

  1. [11]
    G.A. BAKER, Jr: Ising model with a scaling interaction. Phys. Rev. B5, 2622, (1972)CrossRefGoogle Scholar
  2. [12]
    G.A. BAKER, Jr; G.R. GOLNER: Spin-spin correlations in an Ising model for which scaling is exact. Phys. Rev. Lett. 31, 22 (1973).CrossRefGoogle Scholar
  3. [13]
    G.A. BAKER, Jr, S. KRINSKY: Renormalization group structure for translationally invariant ferromagnets. Journ. math. Phys. 18, 590 (1977).CrossRefGoogle Scholar

The first rigorous mathematical work was done in the paper by Bleher and Sinai [5], on the case of Gaussian fixedpoint (with 21/2 < c < 2). The situation at that point was then clarified and reviewed in the following papers.

  1. [14]
    G. GALLAVOTTI, H. KNOPS: The Hierarchical Model and the renormalization group. Nuovo Cimento 5, 341–368 (1975).Google Scholar
  2. [15]
    H. van BEYEREN, G. GALLAVOTTI, H. KNOPS: Conservation laws in the Hierarchical Model. Physica 78, 541 (1974).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1978

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