Communications in Mathematical Physics

, Volume 84, Issue 1, pp 87–101 | Cite as

The phase transition in the one-dimensional Ising Model with 1/r2 interaction energy

  • Jürg Fröhlich
  • Thomas Spencer


We prove the existence of a spontaneous magnetization at low temperature for the one-dimensional Ising Model with 1/r2 interaction energy.


Neural Network Phase Transition Statistical Physic Complex System Nonlinear Dynamics 
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  1. 1.
    Anderson, P. W., Yuval, G., Hamann, D. R.: Phys. Rev.B1, 4464 (1970); Anderson, P. W., Yuval, G.: J. Phys.C4, 607 (1971)Google Scholar
  2. 2.
    Dyson, F. J.: Commun. Math. Phys.12, 212 (1969)Google Scholar
  3. 3.
    Dyson, F. J.: in Statistical mechanics at the turn of the decade, E. G. D. Cohen, (ed.). New York: Marcel Dekker 1971Google Scholar
  4. 4.
    Dyson, F. J.: Commun. Math. Phys.21, 269 (1971)Google Scholar
  5. 5.
    Fröhlich, J., Spencer, T.: Phys. Rev. Lett.46, 1006 (1981); Commun. Math. Phys.81, 527–602 (1981)Google Scholar
  6. 6.
    Landau, L., Lifshitz, E.: Statistical physics. New York: Addison-Wesley 1969Google Scholar
  7. 7.
    Rogers, J., Thompson, C.: PreprintGoogle Scholar
  8. 8.
    Ruelle, D.: Commun. Math. Phys.9, 267 (1968)Google Scholar
  9. 9.
    Simon, B., Sokal, A.: Rigorous energy-entropy arguments. J. Stat. Phys.25, 679 (1981)Google Scholar
  10. 10.
    Thouless, D. J.: Phys. Rev.187, 732 (1969)Google Scholar
  11. 11.
    Griffiths, R. B.: J. Math. Phys.8, 478 (1967);8, 484 (1967); Commun. Math. Phys.6, 121 (1967)Google Scholar
  12. 12.
    Dobrushin, R.: Funct. Anal. Appl.2, 302 (1968)Google Scholar
  13. 13.
    Simon, B.: Commun. Math. Phys.77, 111 (1980)Google Scholar
  14. 14.
    Brydges, D., Fröhlich, J., Spencer, T.: The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys.83, 123–150 (1982)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  • Thomas Spencer
    • 2
  1. 1.Institut des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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