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Communications in Mathematical Physics

, Volume 67, Issue 1, pp 17–42 | Cite as

Existence of long-range order in the Migdal recursion equations

  • P. M. Bleher
  • E. Žalys
Article

Abstract

A modification of XY — model is introduced for which Migdal recursion equation are exact. High- and low-temperature fixed points of these equations are investigated. As a result the existence of long-range order at low temperature and its absence at high temperature are proved rigorously for the model under consideration in the case when dimensiond>2.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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References

  1. 1.
    Wilson, K.G., Kogut, I.: Phys. Rep.12C, 75–199 (1974)Google Scholar
  2. 2.
    Wilson, K.G.: Rev. Mod. Phys.47, 773 (1975)Google Scholar
  3. 3.
    Kadanoff, L.P.: Notes on Migdal's recursion formulas. Preprint, IBM Research Lab., Zürich, Switzerland (1976)Google Scholar
  4. 4.
    Patashinskij, A.Z., Pokrovskij, V.L.: Usp. Phys. Nauk.121, 55 (1977) (in Russian)Google Scholar
  5. 5.
    Migdal, A.A.: JETP69, 810 (1975) (in Russian)Google Scholar
  6. 6.
    Migdal, A.A.: JETP69, 1457 (1975) (in Russian)Google Scholar
  7. 7.
    Bleher, P.M., Sinai, Ja.G.: Commun. Math. Phys.33, 23–42 (1973)Google Scholar
  8. 8.
    Bleher, P.M., Sinai, Ja.G.: Commun. Math. Phys.45, 247 (1975)Google Scholar
  9. 9.
    Bleher, P.M.: Tr. Mosc. Math. Ob.33, 155 (1975) (in Russian)Google Scholar
  10. 10.
    Bleher, P.M.: Usp. Math. Nauk.32, 243 (1977) (in Russian)Google Scholar
  11. 11.
    Sinai, Ja.G.: Mathematical problems of the theory of phase transitions. Hungary (1979) (in press)Google Scholar
  12. 12.
    Watson, G.N.: A treatise on the theory of Bessel functions. New York: Repr. Cambridge Univ. Press; Macmillan 1945Google Scholar
  13. 13.
    Bateman, G., Erdelyi, A.: Higher trancendental functions, Vol. 2. New York, Toronto, London: McGraw-Hill 1955Google Scholar
  14. 14.
    Ginibre, J.: Commun. Math. Phys.16, 310 (1970)Google Scholar
  15. 15.
    Bleher, P.M.: The numerical solution of the approximate equations of Migdal's renormgroup. Preprint, Inst. Appl. Math., USSR Academy of Sciences, No. 48 (1976) (in Russian)Google Scholar
  16. 16.
    Asano, T.: Phys. Rev. Letters24, 1409 (1970)Google Scholar
  17. 17.
    Householder, A.S.: Principles of numerical analysis. New York, Toronto, London: McGraw-Hill 1953Google Scholar
  18. 18.
    Sarkar, S.: Phys. Rev. S16 (8), 2666 (1977)Google Scholar
  19. 19.
    Besov, O.V., Nikolskij, S.M., Iljin, V.P.: Integral representations of functions and imbedding theorems. Moscow: Nauka 1975 (in Russian)Google Scholar
  20. 20.
    Bleher, P.M., Žalys, E.: The investigation of low- and high-temperature fixed points in the Migdal's equations. Preprint, Inst. Appl. Math., USSR Academy of Sciences (1978) (in Russian)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • P. M. Bleher
    • 1
  • E. Žalys
    • 2
  1. 1.Institute of Applied MathematicsAcademy of SciencesMoscowUSSR
  2. 2.Institute of Mathematics and CyberneticsAcademy of Sciences of Lithuania SSRVilniusUSSR

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