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Communications in Mathematical Physics

, Volume 86, Issue 3, pp 337–362 | Cite as

Perturbation about the mean field critical point

  • Jean Bricmont
  • Jean-Raymond Fontaine
  • Eugene Speer
Article

Abstract

We consider two models that are small perturbations of Gaussian or mean field models: the first one is a double well λ/4φ4 — σ/2φ2 perturbation of a massless Gaussian lattice field in the weak coupling limit (λ→0, σ proportional to λ). The other consists of a spin 1/2 Ising model with long-range Kac type interactions; the inverse range of the interaction, γ, is the small parameter. The second model is related to the first one via a sine-Gordon transformation. The lattice ℤ d has dimensiond≧3.

In both cases we derive an asymptotic estimate to first order (in λ or γ2) on the location of the critical point. Moreover, we prove bounds on the remainder of an expansion in λ or γ around the Gaussian or mean field critical points.

The appendix, due to E. Speer, contains an extension of Weinberg's theorem on the divergence of Feynman graphs which is used in the proofs.

Keywords

Neural Network Small Parameter Small Perturbation Quantum Computing Ising Model 
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.
    Fröhlich, J., Simon, B., Spencer, T.: Commun. Math. Phys.50, 79 (1976)Google Scholar
  2. 2.
    Fisher, M.E.: Phys. Rev.162, 480 (1967)Google Scholar
  3. 3.
    Private communicationGoogle Scholar
  4. 4.
    Klein, A., Landau, L.J.: Commun. Math. Phys.43, 143 (1975)Google Scholar
  5. 5.
    Fröhlich, J., Spencer, T.: J. Stat. Phys.24, 617 (1981)Google Scholar
  6. 6.
    Simon, B.: Commun. Math. Phys.77, 111 (1980)Google Scholar
  7. 7.
    Kac, M.: Phys. Fluids2, 8 (1959)Google Scholar
  8. 8.
    Baker, G.A.: Phys. Rev.122, 1477 (1961);126, 2071 (1962)Google Scholar
  9. 9.
    Kac, M., Uhlenbeck, G., Hemmer, P.C.: J. Math. Phys.4, 216 (1963); U.H.K. ibid, p. 229; H.K.U. ibid5, 60 (1964)Google Scholar
  10. 10.
    Lebowitz, J.L., Penrose, O.: J. Math. Phys.7, 8 (1966)Google Scholar
  11. 11.
    Brout, R.: Phys. Rev.115, 824 (1959);118, 1009 (1960);122, 469 (1961)Google Scholar
  12. 12.
    Horwitz, G., Callen, H.B.: Phys. Rev.124, 1757 (1961)Google Scholar
  13. 12a.
    Englert, F.: Phys. Rev.129, 567 (1963)Google Scholar
  14. 12b.
    Coopersmith, M., Brout, R.: Phys. Rev.130, 2539 (1963)Google Scholar
  15. 13.
    Lebowitz, J.L., Stell, G., Baer, S.: J. Math. Phys.6, 1282 (1965)Google Scholar
  16. 13a.
    Stell, G., Lebowitz, J.L., Baer, S., Theumann, W.: J. Math. Phys.7, 1532 (1966)Google Scholar
  17. 14.
    Siegert, A.J.F., Vezetti, D.J.: J. Math. Phys.9, 2173 (1968)Google Scholar
  18. 14a.
    Thompson, C.J., Siegert, A.J.F., Vezetti, D.J.: J. Math. Phys.11, 1018 (1970)Google Scholar
  19. 15.
    Siegert, A.J.F.: In: Statistical mechanics and field theory. Sen, R.N., Weil, C. (eds.). Jerusalem: Israel University Press, 1972Google Scholar
  20. 16.
    Hemmer, P.C., Lebowitz, J.L.: In: Phase transitions and critical phenomena, Vol. 5B, Domb, C., Green, M.S. (eds.). New York: Academic Press 1976Google Scholar
  21. 17.
    Kac, M., Helfand, E.: J. Math. Phys.4, 1078 (1963)Google Scholar
  22. 18.
    Kac, M., Thompson, C.J.: J. Math. Phys.10, 1373 (1969)Google Scholar
  23. 19.
    Stell, G., Theumann, W.K.: Phys. Rev.186, 581 (1969)Google Scholar
  24. 20.
    Griffiths, R.B.: In: Les Houches lectures, 1970; de Witt, C., Stora, R. (eds.). N.Y.: Gordon and Breach, 1971; and In: Phase transitions and critical phenomena, Vol. 1; Domb, C., Green, M.S. (eds.). New York: Academic Press 1972Google Scholar
  25. 21.
    Simon, B.: Ann. Math.101, 260 (1975)Google Scholar
  26. 21a.
    Guerra, F., Rosen, L., Simon, B.: Commun. Math. Phys.41, 19 (1975)Google Scholar
  27. 22.
    Nelson, E.: In: Constructive quantum field theory. In: Lecture Notes in Physics, Vol. 25. Velo, G., Wightman, A.S. (eds.). Berlin, Heidelberg, New York: Springer 1973Google Scholar
  28. 23.
    Ruelle, D.: Commun. Math. Phys.50, 189 (1976). For a similar result in field theory, see, Glimm, J., Jaffe, A.: J. Math. Phys.13, 1568 (1972)Google Scholar
  29. 24.
    Glimm, J., Jaffe, A., Spencer, T.: Commun. Math. Phys.45, 203 (1975)Google Scholar
  30. 25.
    Ellis, R., Monroe, J.L., Newman, C.: Commun. Math. Phys.46, 167 (1976)Google Scholar
  31. 26.
    Bricmont, J.: J. Stat. Phys.17, 289 (1977)Google Scholar
  32. 27.
    Sokal, A.D.: Princeton University Thesis. Ann. Inst. H. Poincaré (to appear)Google Scholar
  33. 28.
    Bricmont, J., Fontaine, J.-R., Lebowitz, J.L., Spencer, T.: Commun. Math. Phys.78, 363 (1981)Google Scholar
  34. 29.
    Griffiths, R.B., Hurst, C., Sherman, S.: Math. Phys.11, 790 (1970)Google Scholar
  35. 30.
    Newman, C.: Commun. Math. Phys.41, 1 (1975) and J. Math. Phys.16, 1956 (1975)Google Scholar
  36. 31.
    Private communicationGoogle Scholar
  37. 32.
    Dreissler, W., Landau, L., Perez, J.: J. Stat. Phys.20, 123 (1979)Google Scholar
  38. 33.
    Griffiths, R.B.: Commun. Math. Phys.6, 121 (1967)Google Scholar
  39. 34.
    Fröhlich, J., Israel, R., Lieb, E., Simon, B.: Commun. Math. Phys.62, 1 (1978)Google Scholar
  40. 35.
    Fröhlich, J., Israel, R., Lieb, E., Simon, B.: J. Stat. Phys.22, 297 (1980)Google Scholar
  41. 36.
    Hegerfeldt, G.C., Nappi, C.: Commun. Math. Phys.53, 1 (1977)Google Scholar
  42. 37.
    Newman, C.N.: Commun. Math. Phys.41, 1 (1975); J. Math. Phys.16, 1956 (1975)Google Scholar
  43. 38.
    Feldman, J., Osterwalder, K.: Ann. Phys.97, 80 (1976)Google Scholar
  44. 39.
    Magnen, J., Sénéor, R.: Ann. Inst. Henri Poincaré24, 95 (1976)Google Scholar
  45. 40.
    Brydges, D., Fröhlich, J., Spencer, T.: Commun. Math. Phys.83, 123 (1982)Google Scholar
  46. 41.
    Ma, S.K.: Modern theory of critical phenomena. Reading, M.A.: Benjamin 1976Google Scholar
  47. 42.
    Schor, R.: Commun. Math. Phys.53, 213 (1978)Google Scholar
  48. 43.
    Nakanishi, N.: Graph theory and Feynman integrals. New York: Gordon and Breach 1971Google Scholar
  49. 44.
    Bricmont, J., Fontaine, J.-R.: Infrared bounds and the Peierls argument in two dimensions. PreprintGoogle Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Jean Bricmont
    • 1
  • Jean-Raymond Fontaine
    • 2
  • Eugene Speer
    • 2
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

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