Communications in Mathematical Physics

, Volume 82, Issue 2, pp 261–304 | Cite as

Phase diagrams and cluster expansions for low temperature Open image in new window models

I. The phase diagram
  • John Z. Imbrie


Low temperature phase diagrams of two-dimensional
quantum field models are constructed. Let
lie in an (r−1)-dimensional space of perturbations of a polynomial withr degenerate minima. Perform a scaling
and assume λ«1. We constructk distinct states on\(\left( {\begin{array}{*{20}c} r \\ k \\ \end{array} } \right)\) hypersurfaces of codimensionk−1 in the space of perturbations. An expansion is used to exhibit exponential clustering of the Schwinger functions of each of these states. At the core of the construction is a general technique for finding the thermodynamically stable phases from a collection of competing minima. We draw on ideas of Pirogov and Sinai [24] for this problem.


Neural Network Statistical Physic Phase Diagram Complex System Nonlinear Dynamics 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bałaban, T., Gawedzki, K.: A low temperature expansion for the pseudoscalar Yukawa model of quantum fields in two space-time dimensions. Ann. Inst. Henri Poincaré (to appear)Google Scholar
  2. 2.
    Bricmont, J., Fontaine, J., Landau, L.: Commun. Math. Phys.64, 49–72 (1978)CrossRefGoogle Scholar
  3. 3.
    Brydges, D.: Commun. Math. Phys.58, 313–350 (1978)CrossRefGoogle Scholar
  4. 4.
    Brydges, D., Federbush, P.: Commun. Math. Phys.73, 197–246 (1980)CrossRefGoogle Scholar
  5. 5.
    Constantinescu, F., Ruck, H.: Ann. Phys.115, 474–495 (1978)CrossRefGoogle Scholar
  6. 6.
    Dimock, J.: Commun. Math. Phys.35, 347–356 (1974)CrossRefGoogle Scholar
  7. 7.
    Dimock, J., Glimm, J.: Adv. Math.12, 58–83 (1974)CrossRefGoogle Scholar
  8. 8.
    Eckmann, J.-P., Magnen, J., Sénéor, R.: Commun. Math. Phys.39, 251–271 (1975)CrossRefGoogle Scholar
  9. 9.
    Fröhlich, J.: Acta Phys. Austr. Suppl.15, 133–269 (1976)Google Scholar
  10. 10.
    Fröhlich, J., Simon, B.: Ann. Math.105, 493–526 (1977)Google Scholar
  11. 11.
    Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79–95 (1976)CrossRefGoogle Scholar
  12. 12.
    Fröhlich, J., Spencer, T.: Phase transitions in statistical mechanics and quantum field theory. In: New developments in quantum field theory and statistical mechanics — Cargèse 1976. Levy, M., Mitter, P. (eds.): New York: Plenum 1977Google Scholar
  13. 13.
    Gawedzki, K.: Commun. Math. Phys.59, 117–142 (1978)CrossRefGoogle Scholar
  14. 14.
    Glimm, J., Jaffe, A.: Acta Math.125, 203–261 (1970)Google Scholar
  15. 15.
    Glimm, J., Jaffe, A.: Quantum Physics. Berlin, Heidelberg, New York: Springer 1981Google Scholar
  16. 16.
    Glimm, J., Jaffe, A., Spencer, T.: Ann. Math.100, 585–632 (1974)Google Scholar
  17. 17.
    Glimm, J., Jaffe, A., Spencer, T.: The particle structure of the weakly coupledP(φ)2 model and other applications of high temperature expansions. In: Constructive quantum field theory. Lecture Notes in Physics, Vol. 25. Velo, G., Wightman, A. (eds.). Berlin, Heidelberg, New York: Springer 1973Google Scholar
  18. 18.
    Glimm, J., Jaffe, A., Spencer, T.: Commun. Math. Phys.45, 203–216 (1975)CrossRefGoogle Scholar
  19. 19.
    Glimm, J., Jaffe, A., Spencer, T.: Ann. Phys.101, 610–669 (1976)CrossRefGoogle Scholar
  20. 20.
    Imbrie, J.: Commun. Math. Phys.78, 169–200 (1980)CrossRefGoogle Scholar
  21. 21.
    Imbrie, J.: Cluster expansions and mass spectra forP(φ)2 models possessing many phases. Harvard University thesis, 1980Google Scholar
  22. 22.
    Osterwalder, K., Schrader, R.: Commun. Math. Phys.31, 83–112 (1973) and42, 281–305 (1975)CrossRefGoogle Scholar
  23. 23.
    Pirogov, S., Sinai, Ya.: Funct. Anal. Appl.8, 21–25 (1974)CrossRefGoogle Scholar
  24. 24.
    Pirogov, S., Sinai, Ya.: Theor. Math. Phys.25, 1185–1192 (1975) and26, 39–49 (1976)CrossRefGoogle Scholar
  25. 25.
    Pirogov, S., Sinai, Ya.: Ann. Phys.109, 393–400 (1977)CrossRefGoogle Scholar
  26. 26.
    Ruelle, D.: Statistical Mechanics. New York: Benjamin 1969Google Scholar
  27. 27.
    Spencer, T.: Commun. Math. Phys.39, 63–76 (1974)CrossRefGoogle Scholar
  28. 28.
    Summers, S.: The phase diagram for a two dimensional bose quantum field model. Harvard University thesis, 1979Google Scholar
  29. 29.
    Summers, S.: Ann. Inst. Henri Poincaré34, 173–229 (1981)Google Scholar
  30. 30.
    Summers, S.: Helv. Phys. Acta53, 1–30 (1980)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • John Z. Imbrie
    • 1
  1. 1.Department of PhysicsHarvard UniversityCambridgeUSA

Personalised recommendations