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Forbidden Gap Argument for Phase Transitions Proved by Means of Chessboard Estimates

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Abstract

Chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest. In this note we show that the method not only produces a point in the phase diagram where more than one Gibbs states coexist, but that it can also be used to rule out the existence of shift-ergodic states that differ significantly from those proved to exist. For models depending on a parameter (say, the temperature), this shows that the values of the conjugate thermodynamic quantity (the energy) inside the ``transitional gap'' are forbidden in all shift-ergodic Gibbs states. We point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria.

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References

  1. Alexander, K.S., Chayes, L.: Non-perturbative criteria for Gibbsian uniqueness. Commun. Math. Phys. 189(2), 447–464 (1997)

    Article  MathSciNet  Google Scholar 

  2. Biskup, M., Chayes, L., Crawford, N.: Mean-field driven first-order phase transitions in systems with long-range interactions. J. Statist. Phys. (to appear)

  3. Biskup, M., Chayes, L., Kivelson, S.A.: Order by disorder, without order, in a two-dimensional spin system with O(2)-symmetry. Ann. Henri Poincaré 5(6), 1181–1205 (2004)

    Article  MathSciNet  Google Scholar 

  4. Biskup, M., Chayes, L., Kotecký, R.: Coexistence of partially disordered/ordered phases in an extended Potts model. J. Statist. Phys. 99 (5/6), 1169–1206 (2000)

    Google Scholar 

  5. Biskup, M., Chayes, L., Nussinov, Z.: Orbital ordering in transition-metal compounds: I. The 120-degree model. Commun. Math. Phys. 255, 253–292 (2005)

    Google Scholar 

  6. Biskup, M., Chayes, L., Nussinov, Z.: Orbital ordering in transition-metal compounds: II. The orbital-compass model. In preparation

  7. Borgs, C., Waxler, R.: First order phase transitions in unbounded spin systems. I. Construction of the phase diagram. Commun. Math. Phys. 126, 291–324 (1990)

    ADS  Google Scholar 

  8. Borgs, C., Waxler, R.: First order phase transitions in unbounded spin systems. II. Completeness of the phase diagram. Commun. Math. Phys. 126, 483–506 (1990)

    ADS  MATH  MathSciNet  Google Scholar 

  9. Bricmont, J., Slawny, J.: Phase transitions in systems with a finite number of dominant ground states. J. Statist. Phys. 54(1-2), 89–161 (1989)

    Google Scholar 

  10. Chayes, L., Kotecký, R., Shlosman, S.B.: Aggregation and intermediate phases in dilute spin systems. Commun. Math. Phys. 171, 203–232 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Chayes, L., Kotecký, R., Shlosman, S.B.: Staggered phases in diluted systems with continuous spins. Commun. Math. Phys. 189, 631–640 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Chayes, L., Shlosman, S., Zagrebnov, V.: Discontinuity in magnetization in diluted O(n)-Models. J. Statist. Phys. 98, 537–549 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dinaburg, E.I., Sinai, Ya.G.: An analysis of ANNNI model by Peierls' contour method. Commun. Math. Phys. 98(1), 119–144 (1985)

    Article  MathSciNet  Google Scholar 

  14. Dobrushin, R.L., Shlosman, S.B.: Phases corresponding to minima of the local energy. Selecta Math. Soviet. 1(4), 317–338 (1981)

    MathSciNet  Google Scholar 

  15. Dobrushin, R.L., Zahradník, M.: Phase diagrams for continuous-spin models: an extension of the Pirogov-Sinai theory. In: Dobrushin R.L. (ed.) Mathematical problems of statistical mechanics and dynamics. Math. Appl. (Soviet Ser.), Vol. 6, Dordrecht: Reidel, 1986, pp. 1–123

  16. van Enter, A.C.D., Shlosman, S.B.: First-order transitions for n-vector models in two and more dimensions: Rigorous proof. Phys. Rev. Lett. 89, 285702 (2002)

    Article  Google Scholar 

  17. van Enter, A.C.D., Shlosman, S.B.: Provable first-order transitions for nonlinear vector and gauge models with continuous symmetries. Commun. Math. Phys. 255, 21–32 (2005)

    Article  ADS  MATH  Google Scholar 

  18. Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. I. General theory and long range models. Commun. Math. Phys. 62, 1–34 (1978)

    Google Scholar 

  19. Fröhlich, J., Israel, R., Lieb, E.H., Simon, B.: Phase transitions and reflection positivity. II. Lattice systems with short range and Coulomb interactions. J. Statist. Phys. 22, 297–347 (1980)

    Google Scholar 

  20. Fröhlich, J., Lieb, E.H.: Phase transitions in anisotropic lattice spin systems. Commun. Math. Phys. 60(3), 233–267 (1978)

    Article  Google Scholar 

  21. Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, Vol. 9, Berlin: Walter de Gruyter & Co., 1988

  22. Imbrie, J.Z.: Phase diagrams and cluster expansions for low temperature models. I. The phase diagram. Commun. Math. Phys. 82(2), 261–304 (1981/82)

    Article  MathSciNet  Google Scholar 

  23. Imbrie, J.Z.: Phase diagrams and cluster expansions for low temperature models. II. The Schwinger functions. Commun. Math. Phys. 82(3), 305–343 (1981/82)

    Article  MathSciNet  Google Scholar 

  24. Kotecký, R., Laanait, L., Messager, A., Ruiz, J.: The q-state Potts model in the standard Pirogov-Sinai theory: surface tensions and Wilson loops. J. Statist. Phys. 58(1-2), 199–248 (1990)

    Google Scholar 

  25. Kotecký, R., Shlosman, S.B.: First-order phase transitions in large entropy lattice models. Commun. Math. Phys. 83(4), 493–515 (1982)

    Article  Google Scholar 

  26. Kotecký, R., Shlosman, S.B.: Existence of first-order transitions for Potts models. In: Albeverio, S., Combe, Ph. , Sirigue-Collins M. (eds.), Proc. of the International Workshop — Stochastic Processes in Quantum Theory and Statistical Physics, Lecture Notes in Physics 173, Berlin-Heidelberg-New York: Springer-Verlag, 1982, pp. 248–253

  27. Laanait, L., Messager, A., Miracle-Solé, S., Ruiz, J., Shlosman, S.: Interfaces in the Potts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)

    MathSciNet  Google Scholar 

  28. Martirosian, D.H.: Translation invariant Gibbs states in the q-state Potts model. Commun. Math. Phys. 105(2), 281–290 (1986)

    Article  MathSciNet  Google Scholar 

  29. Messager, A., Nachtergaele, B.: A model with simultaneous first and second order phase transitions. http://arxiv.org/list/cond-mat/0501229, 2005

  30. Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems (Russian). Theor. Math. Phys. 25(3), 358–369 (1975)

    MathSciNet  Google Scholar 

  31. Pirogov, S.A., Sinai, Ya.G.: Phase diagrams of classical lattice systems. Continuation (Russian). Theor. Math. Phys. 26(1), 61–76 (1976)

    MathSciNet  Google Scholar 

  32. Shlosman, S.B.: The method of reflective positivity in the mathematical theory of phase transitions of the first kind (Russian). Uspekhi Mat. Nauk 41(3(249)), 69–111, 240 (1986)

    MathSciNet  Google Scholar 

  33. Shlosman, S., Zagrebnov, V.: Magnetostriction transition. J. Statist. Phys. 114, 563–574 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Zahradník, M.: An alternate version of Pirogov-Sinai theory. Commun. Math. Phys. 93, 559–581 (1984)

    Article  ADS  Google Scholar 

  35. Zahradník, M.: Contour methods and Pirogov-Sinai theory for continuous spin lattice models. In: R.A. Minlos, S. Shlosman Yu.M. Suhov (eds.), On Dobrushin's way. From probability theory to statistical physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 198, Providence, RI: Amer. Math. Soc., 2000, pp. 197–220

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, California, USA

    Marek Biskup

  2. Center for Theoretical Study, Charles University, Prague, Czech Republic

    Roman Kotecký

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  1. Marek Biskup
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  2. Roman Kotecký
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Communicated by M. Aizenman

© 2006 by M. Biskup and R. Kotecký. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.

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Biskup, M., Kotecký, R. Forbidden Gap Argument for Phase Transitions Proved by Means of Chessboard Estimates. Commun. Math. Phys. 264, 631–656 (2006). https://doi.org/10.1007/s00220-006-1523-x

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  • DOI: https://doi.org/10.1007/s00220-006-1523-x

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