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


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.


Phase Separation Phase Boundary Spin System Lattice Spin Aggregation Transition 
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  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

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