Advertisement

Communications in Mathematical Physics

, Volume 66, Issue 2, pp 181–196 | Cite as

Critical point inequalities and scaling limits

  • Charles M. Newman
Article

Abstract

A refined and extended version of the Buckingham-Gunton inequality relating various pairs of critical exponents is shown to be valid for a large class of statistical mechanical models. If this inequality is an equality (in the refined sense) and one of the critical exponents has a non-Gaussian value, then any scaling limit must be non-Gaussian. This result clarifies the relationship between the nontriviality or triviality of the scaling limit for ordinary critical points in four dimensions (or tricritical points in three dimensions) and the existence of logarithmic factors in the asymptotics which define the two critical exponents.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Large Class 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    deAngelis, G.F., deFalco, D., Guerra, F.: Note on the abelian Higgs-Kibble model on a lattice: absence of spontaneous magnetization. Phys. Rev. D17, 1624–1628 (1978)Google Scholar
  2. 2.
    Baker, G.A., Jr.: Selfinteracting boson quantum field theory and the thermodynamic limit ind dimensions. J. Math. Phys.16, 1324–1346 (1975)Google Scholar
  3. 3.
    Baker, G.A., Jr.: Renormalization group structure for translationally invariant ferromagnets. J. Math. Phys.18, 590–607 (1977)Google Scholar
  4. 4.
    Blume, M., Emery, V.J., Griffiths, R.B.: Ising model for the λ transition and phase separation in He3-He4 mixtures. Phys. Rev. A4, 1071–1077 (1971)Google Scholar
  5. 5.
    Brydges, D., Fröhlich, J., Seiler, E.: On the construction of quantized gauge fields, I. general results. Preprint I.H.E.S. (1978)Google Scholar
  6. 6.
    Buckingham, M.J., Gunton, J.D.: Correlations at the critical point of the Ising model. Phys. Rev.178, 848–853 (1969)Google Scholar
  7. 7.
    Benguigui, L., Schulman, L.S.: Topological classification of phase transitions. Phys. Lett.45A, 315–316 (1973)Google Scholar
  8. 8.
    Ellis, R.S., Newman, C.M.: Fluctuationes in Curie-Weiss exemplis. In: Mathematical problems in theoretical physics. Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds.). Berlin, Heidelberg, New York: Springer 1978Google Scholar
  9. 9.
    Fisher, M.E.: Rigorous inequalities for critical-point correlation exponents. Phys. Rev.180, 594–600 (1969)Google Scholar
  10. 10.
    Fisher, M.E.: General scaling theory for critical points. In: Collective properties of physical systems. Lundqvist, B., Lundqvist, S. (eds.). New York and London: Academic Press 1973Google Scholar
  11. 11.
    Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. I. General theory and long range lattice models. Commun. Math. Phys.62, 1–34 (1978)Google Scholar
  12. 12.
    Fortuin, C., Kastelyn, P., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys.22, 89–103 (1971)Google Scholar
  13. 13.
    Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys.8, 484–489 (1967)Google Scholar
  14. 14.
    Gallavotti, G.: Probabilistic aspects of critical fluctuations. In: Critical phenomena. Brey, J., Jones, R.B. (eds.). Berlin, Heidelberg, New York: Springer 1976Google Scholar
  15. 15.
    Glimm, J., Jaffe, A.: Critical problems in quantum fields. In: Mathematical problems in theoretical physics. Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds.). Berlin, Heidelberg, New York: Springer 1978Google Scholar
  16. 16.
    Glimm, J., Jaffe, A.: Critical exponents and elementary particles. Commun. Math. Phys.52, 203–209 (1977)Google Scholar
  17. 17.
    Glimm, J., Jaffe, A.: Quark trapping for latticeU(1) gauge fields. Phys. Lett.66B, 67–69 (1977)Google Scholar
  18. 18.
    Glimm, J., Jaffe, A., Spencer, T.: Phase transitions for φ24 quantum fields. Commun. Math. Phys.45, 203–216 (1975)Google Scholar
  19. 19.
    Hecht, R.: Correlation functions for the two-dimensional Ising model. Phys. Rev.158, 557–561 (1967)Google Scholar
  20. 20.
    Hegerfeldt, G.C., Nappi, C.: Mixing properties in lattice systems. Commun. Math. Phys.53, 1–7 (1977)Google Scholar
  21. 21.
    Jona-Lasinio, G.: Probabalistic approach to critical behavior. In: New developments in quantum field theory and statistical mechanics. Lévy, M., Mitter, P. (eds.). New York: Plenum 1977Google Scholar
  22. 22.
    Kadanoff, L.P.: Scaling laws for Ising models nearT c. Physics 2, 263–272 (1966)Google Scholar
  23. 23.
    Kelly, D.G., Sherman, S.: General Griffith's inequalities on correlations in Ising ferromagnets. J. Math. Phys.9, 466–484 (1968)Google Scholar
  24. 24.
    Larkin, A.I., Khmel'nitskii, D.E.: Phase transition in uniaxial ferroelectrics. Sov. Phys. JETP29, 1123–1128 (1969)Google Scholar
  25. 25.
    Newman, C.M.: Short distance scaling and the maximal degree of a field theory. Phys. Lett. B (to appear)Google Scholar
  26. 26.
    Nelson, D.: Private communicationGoogle Scholar
  27. 27.
    Osterwalder, K., Schrader, R.: Axioms for Euclidean Green's functions. Commun. math. Phys.31, 83–112 (1973), Axioms for Euclidean Green's functions. II. Commun. Math. Phys.42, 281–305 (1975)Google Scholar
  28. 28.
    Osterwalder, K., Seiler, E.: Gauge field theories on a lattice. Ann. Phys.110, 440–471 (1978)Google Scholar
  29. 29.
    Pfeuty, P., Toulouse, G.: Introduction to the renormalization group and to critical phenomena. New York: Wiley 1977 (Sect. 5.4)Google Scholar
  30. 30.
    Riedel, E.K.: Private communicationGoogle Scholar
  31. 31.
    Schrader, R.:49, 131–153 (1976); A possible constructive approach to φ44. Commun. math. Phys. III. Commun. Math. Phys.50, 97–102 (1976)Google Scholar
  32. 32.
    Simon, B.: TheP(φ)2 Euclidean (quantum) field theory. Princeton: Princeton University Press 1974Google Scholar
  33. 33.
    Stell, G.: Extension of the Ornstein-Zernike theory of the critical region. Phys. Rev. Lett.20, 533–536 (1968); Extension of the Ornstein-Zernike theory of the critical region. II. Phys. Rev. B1, 2265–2270 (1970)Google Scholar
  34. 34.
    Thompson, C.: Mathematical statistical mechanics. New York: Macmillan 1972 (Sect. 5-5)Google Scholar
  35. 35.
    Wilson, K.G., Fisher, M.E.: Critical exponents in 3.99 dimensions. Phys. Rev. Lett.28, 240–243 (1972)Google Scholar
  36. 36.
    Wilson, K.G., Kogut, J.: The renormalization group and the ε expansion. Phys. Rep.12, 75–200 (1975)Google Scholar
  37. 37.
    Wegner, F.J., Riedel, E.K.: Logarithmic corrections to the molecular-field behavior of critical and tricritical systems. Phys. Rev. B7, 248–256 (1973)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Charles M. Newman
    • 1
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA

Personalised recommendations