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Inequalities and many phase transitions in ferromagnetic systems

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Abstract

A general construction of ferromagnetic systems with many phase transitions is given. It is based on two new results: an extension of one of the GKS inequalities to not necessarily ferromagnetic interactions, and a uniqueness of the Gibbs state theorem for perturbations of some simple systemsat all temperatures.

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References

  1. Alcaraz, F.C., Koberle, R.: Duality and the phases ofZ(N) spin systems. J. Phys. A13, L153 (1980)

  2. Alcaraz, F.C., Koberle, R.: The phases of two-dimensional spin and four-dimensional gauge systems withZ(N) symmetry. J. Phys. A14, 1169 (1981)

    Google Scholar 

  3. Bricmont, J., Kuroda, K., Lebowitz, J.L.: First order phase transitions in lattice and continuous systems: extension of the Pirogov-Sinai theory. Commun. Math. Phys.101, 501 (1985)

    Google Scholar 

  4. Bricmont, J., Slawny, J.: First order phase transitions and perturbation theory. In: Statistical mechanics and field theory: mathematical aspects. Hugenholtz, N.M., Winnink, M. (eds.). Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  5. Briemont, J., Slawny, J.: Phase transitions with a finite number of dominant ground states. To appear in J. Stat. Phys.

  6. Dobrushin, R.L.: The existence of a phase transition in the two- and three-dimensional Ising models. Teorija Verojatn. Prim.10, 209 (1965)

    Google Scholar 

  7. Dobrushin, R.L., Pecherski, E.A.: Uniqueness conditions for finitely dependent random fields. In: Colloquia mathematica societatis Janos bolyai, Vol. 27. Random Fields, Hungary 1979

  8. Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of Gibbs fields. Statistical physics and dynamical systems, pp. 347–370. Fritz, J., Jaffe, A., Szasz, D. (eds.). Boston, Basel, Stuttgart: Birkhäuser 1985

    Google Scholar 

  9. Dobrushin, R.L., Zahradnik, M.: Phase diagrams for the continuous spin models. Extension of Pirogov-Sinai theory. In: Mathematical problems of statistical physics and dynamical problems. Dobrushin, R.L. (ed.). Dordrecht: Reidel

  10. Dudley, R.M.: Probabilities and metrics. Aarhus University, Lecture Notes, Vol. 45, 1976

  11. Elitzur, E., Pearson, R.B., Shigemitsu, J.: Phase structure of discrete abelian spin and gauge systems. Phys. Rev. D19, 3698 (1979)

    Google Scholar 

  12. Fisher, M.E.: Critical temperatures of anisotropic Ising lattices. II. General upper bounds. Phys. Rev.162, 480–485 (1967)

    Google Scholar 

  13. Fernández, R.: Study of ferromagnetic systems with many phase transitions. Thesis, Virginia Polytechnic Institute and State University (1984)

  14. Fröhlich, J., Simon, B., Spencer, T.: Infrared bounds, phase transitions and continuous symmetry breaking. Commun. Math. Phys.50, 79 (1976)

    Google Scholar 

  15. 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 (1978)

    Google Scholar 

  16. Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: J. Stat. Phys.22, 297 (1980)

    Google Scholar 

  17. Ginibre, J.: Correlations in Ising ferromagnets. Cargèse Lectures in Physics, Vol. 4, Kastler, D. (ed.). New York, London, Paris: Gordon and Breach

  18. Ginibre, J.: General formulation of Griffith's inequalities. Commun. Math. Phys.16, 310 (1970)

    Google Scholar 

  19. Griffiths, R.B.: Peierls proof of spontaneous magnetization in a two-dimensional Ising ferromagnet. Phys. Rev.136A, 437 (1964)

    Google Scholar 

  20. Griffiths, R.B.: Correlations in Ising ferromagnets. I. J. Math. Phys.8, 478 (1967)

    Google Scholar 

  21. Gruber, C., Hinterman, A., Merlini, D.: Group analysis of classical lattice systems. Lecture Notes in Physics, Vol. 60. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  22. Holsztynski, W.: Communique at Yeshiva University Meeting on Statistical Mechanics, December 1976

  23. Holsztynski, W., Slawny, J.: Phase transitions in ferromagnetic spin systems at low temperatures. Commun. Math. Phys.66, 147 (1979)

    Google Scholar 

  24. Kelly, D.G., Sherman, S.: General Griffith's inequalities on correlations in Ising ferromagnets. J. Math. Phys.9, 466 (1968)

    Google Scholar 

  25. Kotecky, R., Preiss, D.: An inductive approach to Pirogov-Sinai theory. Proc. Winter School on Abstract Analysis 1983. [Suppl.] Ai. Rend del Circ. Mat. di Palermo (1983)

  26. Miekisz, J.: The decomposition property and equilibrium states of ferromagnetic lattice systems. Commun. Math. Phys.109, 353 (1987)

    Google Scholar 

  27. Peierls, R.: On Ising's model ferromagnetism. Proc. Camb. Phil. Soc.32, 477 (1936)

    Google Scholar 

  28. Pfister, C.-E.: Translation invariant equilibrium states of ferromagnetic abelian lattice systems. Commun. Math. Phys.86, 375 (1982)

    Google Scholar 

  29. Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. I, II. Theor. Math. Phys.25, 1185 (1976);26, 39

    Google Scholar 

  30. Slawny, J.: A family of equilibrium states. Commun. Math. Phys.35, 297 (1974)

    Google Scholar 

  31. Slawny, J.: Ferromagnetic systems with local symmetries (unpublished) (1979)

  32. Slawny, J.: Low temperature properties of classical lattice systems: Phase transitions and phase diagrams. In: Phase transitions and critical phenomena, Vol. 11. Domb, C., Lebowitz, J.L. (eds.). London: Academic Press 1987

    Google Scholar 

  33. Wegner, F.: Duality in generalized Ising models and phase transitions without local order parameters. J. Math. Phys.12, 2259 (1971)

    Google Scholar 

  34. Zahradnik, M.: Low temperature continuous spin Gibbs states on a lattice and the interfaces between them — A Pirogov-Sinai type approach. To appear in Proc. Intern. Conf. “Statistical Mechanics and Field Theory; Mathematical Aspects,” Hugenholtz, N.M., Winnik, M. (eds.). Groningen, Aug. 1985

  35. Zahradnik, M.: An alternate version of Pirogov-Sinai theory. Commun. Math. Phys.93, 559 (1984)

    Google Scholar 

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Communicated by M. Aizenman

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Fernández, R., Slawny, J. Inequalities and many phase transitions in ferromagnetic systems. Commun.Math. Phys. 121, 91–120 (1989). https://doi.org/10.1007/BF01218626

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  • DOI: https://doi.org/10.1007/BF01218626

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