Journal of Statistical Physics

, Volume 77, Issue 1–2, pp 77–87 | Cite as

A goldstone mode in the Kawasaki-Ising model

  • Claudio Albanese


The hydrodynamic regime of superfluids is dominated by a Goldstone mode corresponding to a spontaneously brokenU(1) symmetry. In this article we map the Kawasaki-Ising model for a classical lattice gas into a quantum model for a superfluid and establish a connection between the normal density fluctuations of the first and the Goldstone mode of the second. The fact that the quantum model we obtain describes a superfluid derives from an inequality by Penrose and Onsager which gives a lower bound to the Bose-Einstein condensate density. Mathematically, the Goldstone mode can be described by means of a “quantum” extension of the local algebra of the Ising model. The classification of its irreducible representations requires an additionalU(1) phase factor and the correspondingU(1) gauge symmetry is spontaneously broken for all finite values of the temperature and of the density.

Key Words

Ising model Monte Carlo dynamics spontaneous gauge symmetry breaking 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. Spohn,Large Scale Dynamics of Interacting Particles (Springer-Verlag, Berlin, 1991).Google Scholar
  2. 2.
    A. De Masi and E. Presutti,Mathematical Methods for Hydrodynamic Limits (Springer-Verlag, Berlin, 1991).Google Scholar
  3. 3.
    S. L. Lu and H. T. Yau, Spectral gap and logarithmic Sobolev inequality of Kawasaki and Glauber dynamics, preprint (1993).Google Scholar
  4. 4.
    O. Penrose and L. Onsager, On the quantum mechanics of helium II,Phys. Rev. 104:576 (1956).CrossRefGoogle Scholar
  5. 5.
    L. Reatto, Bose-Einstein condensation for a class of wavefunctions,Phys. Rev. 183:334 (1969).CrossRefGoogle Scholar
  6. 6.
    D. Ruelle, Classical statistical mechanics of a system of particles,Helv. Phys. Acta 36:183 (1963).Google Scholar
  7. 7.
    K. Symanzik, Euclidean proof of the Goldstone Theorem,Commun. Math. Phys. 6:228 (1967).CrossRefGoogle Scholar
  8. 8.
    C. Albanese and M. Isopi, Long time asymptotics of infinite particle systems, preprint (1994).Google Scholar
  9. 9.
    R. P. Feynman,Phys. Rev. 91:1291 (1953).CrossRefGoogle Scholar
  10. 10.
    K. Binder,Monte Carlo Methods in Statistical Physics, (Berlin, Springer, 1979).Google Scholar
  11. 11.
    P. W. Anderson,Basic Notions of Condensed Matter Physics (Benjamin-Cummings, London, 1984).Google Scholar
  12. 12.
    D. J. Amit,Field Theory, the Renormalization Group and Critical Phenomena (McGraw-Hill, 1978).Google Scholar
  13. 13.
    O. A. McBrian and T. Spencer,Commun. Math. Phys. 53:299 (1977).CrossRefGoogle Scholar
  14. 14.
    D. Forster,Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (Benjamin, New York, 1975).Google Scholar
  15. 15.
    P. Nozières and D. Pines,The Theory of Quantum Liquids, Vol. 2 (Benjamin, New York, 1966–1990).Google Scholar
  16. 16.
    J. Feldman, J. Magnen, V. Rivasseau, and E. Trubowitz,Helv. Phys. Acta 66 (1993).Google Scholar
  17. 17.
    J. Goldstone,Nuovo Cimento 19:154 (1961).Google Scholar
  18. 18.
    G. Parisi and N. Sourlas,Nucl. Phys. B 206:321 (1982).CrossRefGoogle Scholar
  19. 19.
    J. R. Schrieffer,Theory of Superconductivity (Benjamin, New York, 1964).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Claudio Albanese
    • 1
  1. 1.Institute for Advanced StudySchool of MathematicsPrinceton

Personalised recommendations