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Communications in Mathematical Physics

, Volume 138, Issue 3, pp 487–505 | Cite as

Low temperature properties of the hierarchical classical vector model

  • Ricardo Schor
  • Michael O'Carroll
Article

Abstract

We obtain low temperature properties of the classical vector model in a hierarchical formulation in three or more dimensions. We consider the lattice model in a zero or non-zero magnetic field, where the single site spin variable ϕ∈Rv has a density proportional to\(e^{ - \lambda (\phi ^2 - 1)^2 } \) for large λ≦∞. Using renormalization group methods we obtain a convergent expansion for the free energy with zero magnetic field. For non-zero fields a shift formula is used to obtain the effective action generated by the renormalization group transformation (RGT). To obtain the pure state zero field free energy and spontaneous magnetization we take the thermodynamic limit together with the zero field limit at a specified rate. The spontaneous magnetization,m, is calculated, is non-zero and the pure state free energy coincides, as expected, with the zero field free energy. Also the sequence of zero field actions does not have a limit but we show that the sequence of actions generated from the original action shifted bym does; the limiting action corresponds to a non-canonical Gaussian fixed point of the RGT.

Keywords

Pure State Spontaneous Magnetization Spin Variable Renormalization Group Method Zero Magnetic Field 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Domb, C., Green, M.S.: Phase transitions and critical phenomena. Vol. 3. New York: Academic Press 1974Google Scholar
  2. 2.
    Parisi, G.: Statistical field theory. New York: Addison-Wesley 1987Google Scholar
  3. 3.
    Fröhlich, J., Spencer, T.: Commun. Math. Phys.81, 527 (1981)Google Scholar
  4. 4.
    Fröhlich, J., Spencer, T.: Commun. Math. Phys.83, 411 (1982)Google Scholar
  5. 5.
    Bricmont, J., Fontaine, J.-R., Lebowitz, J.L., Lieb, E.H., Spencer, T.: Commun. Math. Phys.78, 545 (1981)Google Scholar
  6. 6.
    Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976)Google Scholar
  7. 7.
    Balaban, T.: Renormalization group approach to lattice field theories I. Commun. Math. Phys.109, 249–301 (1987)Google Scholar
  8. 8.
    Balaban, T.: Large field renormalization II. Commun. Math. Phys.122, 355–392 (1989)Google Scholar
  9. 9.
    Gawedzki, K., Kupiainen, A.: Continuum limit of the hierarchicalO(N) non-linear σ-model. Commun. Math. Phys.106, 533–550 (1986)Google Scholar
  10. 10.
    Gawedzki, K., Kupiainen, A.: Asymptotic freedom beyond perturbation theory. In: Les Houches Session XLII, 1984. Phénomènes critiques, Systémes aléatoires, Théories de Jauge Elsevier Science Publishers, B.V. 1986, Osterwalder, K., Stora, R. (eds.)Google Scholar
  11. 11.
    Collet, P., Eckmann, J.P.: A renormalization group analysis of the hierarchical model in statistical mechanics. Lecture Notes in Physics. Vol.74. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  12. 12.
    Bleher, P.M., Major, P.: The large-scale limit of Dyson's hierarchical vector-valued model at low temperatures. Preprint Keldysh Institute of Applied Mathematics. Moscow A-47, 1989Google Scholar
  13. 13.
    Bleher, P.M., Major, P.: The large-scale limit of Dyson's hierarchical vector-valued model at low temperatures. The non-Gaussian case. Ann. Inst. Henri Poincaré, Phys. Théor.49, Vol. 1 (1988)Google Scholar
  14. 14.
    Schor, R., O'Carroll, M.: Correlation functions and the Goldstone picture for the hierarachical classical Vector model at low temperatures in three or more dimensions. June 1990 (to appear in J. Stat. Phys.)Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Ricardo Schor
    • 1
  • Michael O'Carroll
    • 1
  1. 1.Departamento de Física do ICExUniversidade Federal de Minas GeraisBelo HorizonteBrasil

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