Communications in Mathematical Physics

, Volume 171, Issue 1, pp 203–232 | Cite as

Aggregation and intermediate phases in dilute spin systems

  • L. Chayes
  • R. Kotecky
  • S. B. Shlosman
Article

Abstract

We study a variety of dilute annealed lattice spin systems. For site diluted problems with many internal spin states, we uncover a new phase characterized by the occupation and vacancy of staggered sublattices. In cases where the uniform system has a low temperature phase, the staggered states represent an intermediate phase. Furthermore, in many of these cases, we show that (at least part of) the phase boundary separating the low-temperature and staggered phases is a line of phase coexistence-i.e. the transition is first order. We also study the phenomenon of aggregation (phase separation) in bond diluted models. Such transitions are known, trivially, to occur in the large-q Potts models. However, it turns out that phase separation is typical in bond diluted spin systems with many internal states. (In particular, a bond aggregation transition is not tied to a discontinuous transition in the uniform system.) Along the portions of the phase boundary where any of these phenomena occur, the prospects for a Fisher renormalization effect are deemed to be highly unlikely or are ruled out altogether.

Keywords

Phase Separation Phase Boundary Spin System Lattice Spin Aggregation Transition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [CKS] Chayes, L., Kotecký, R., Shlosman, S.B: Research in Progress.Google Scholar
  2. [D] Dobrushin, R.L.: Problem of Uniqueness of a Gibbs Random Field and Phase Transitions. Funkts. Anal. Prilozh2, 44–57 (1968)Google Scholar
  3. [DS] Dobrushin, R.L., Shlosman, S.B.: Completely Analytical Gibbs Fields. Statistical Physics and Dynamical Systems, Progress in Physics, v.10, Edited by Jaffe, A., Fritz, J., Szasz, D., Boston, Basel, Stuttgart, Birkhäuser, 1985Google Scholar
  4. [EG] Essam, J.W., Garelick, H.: Critical Behavior of a Soluble Model of Dilute Ferromagnetism. Proc. Phys. Soc.92, 136–149 (1967)CrossRefGoogle Scholar
  5. [F] Fisher, M.E.: Renormalization of Critical Exponents by Hidden Variables. Phys. Rev.176, 257–272 (1968)CrossRefGoogle Scholar
  6. [FL] Fröhlich, J., Lieb, E.H.: Phase Transitions in Anisotropic Lattice Spin Systems. Commun. Math. Phys.60, 233–267 (1978)CrossRefGoogle Scholar
  7. [FILS I] Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase Transitions and Reflection Positivity. I. Commun. Math. Phys.62, 1–34 (1978)CrossRefGoogle Scholar
  8. [FILS III] Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase Transitions and Reflection Positivity. III. In preparationGoogle Scholar
  9. [FSS] Fröhlich, J., Simon, B., Spencer, T.: Infra-red Bounds, Phase Transitions and Continuous Symmetry Breaking. Commun. Math. Phys.50, 79–95 (1976)CrossRefGoogle Scholar
  10. [FSS] Hoston, W., Berker, A.N.: Dimensionality Effects on the Multicritical Phase Diagrams of the Blume-Emery-Griffiths Model with Repulsive Biquadratic Couplings: Mean-field and Renormalization Group Studies. J. Appl. Phys.70, 6101–6103 (1991)CrossRefGoogle Scholar
  11. [KS] Kotecký, R., Shlosman, S.B.: First-Order Phase Transitions in Large Entropy Lattice Models. Commun. Math. Phys.83, 493–515 (1982)CrossRefGoogle Scholar
  12. [M] Martirosian, D.H.: Translation Invariant Gibbs States inq-state Potts Model. Commun. Math. Phys.105, 281–290 (1986)Google Scholar
  13. [MS] Minlos, R.A., Sinai, Y.A.: The Phenomenon of “Phase Separation” at Low Temperature in Some Lattice Gas Models I. Mat. Sb. Phys.73, 375–488 (1967)Google Scholar
  14. [NBRS] Nienhuis, B., Berker, A.N., Riedel, E.K., Schick, M.: First- and Second-Order Phase Transitions in Potts Models Renormalization-Group Solution. Phys. Rev. Lett.43, 737–740 (1979)Google Scholar
  15. [RL] Runels, L.K., Lebowitz, J.L.: Phase Transitions of a Multicomponent Widom-Rowlinson Model. J. Math. Phys.15, 1712–1717 (1974)CrossRefGoogle Scholar
  16. [S] Shlosman, S.B.: The Method of Reflection Positivity in the Mathematical Theory of First-Order Phase Transitions. Russ. Math. Surv.41:3, 83–134 (1986)Google Scholar
  17. [S] Stinchcombe, R.B.: Dilute Magnetism. Phase Transitions and Critical Phenomena Vol. 7, Edited by Domb, C., Lebowitz, J.L., London: Academic Press Inc., 1983Google Scholar
  18. [SW] Sarbach, S., Wu, F.Y.: Z Phys. B44, 309 (1981)Google Scholar
  19. [SM] Syozi, I., Miyazima, S.: Prog. Theor. Phys.36, 1803 (1966)Google Scholar
  20. [ST] Southern, B.W., Thorpe, M.F.: J. Phys. C12, 5351 (1979)Google Scholar
  21. [Z] Zahradnik, M.: An Alternate Version of Pirogov-Sinai Theory. Commun. Math. Phys.93, 559–581 (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • L. Chayes
    • 1
  • R. Kotecky
    • 2
  • S. B. Shlosman
    • 3
  1. 1.Department of MathematicsUCLALos AngelesUSA
  2. 2.Center for Theoretical StudyCharles UniversityPraha 3Czech Republic
  3. 3.Department of MathematicsUniversity of CaliforniaIrvineUSA

Personalised recommendations