The Phase Diagram of the Quantum Curie-Weiss Model

  • Lincoln Chayes
  • Nicholas Crawford
  • Dmitry Ioffe
  • Anna Levit
Article
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Abstract

This paper studies a generalization of the Curie-Weiss model (the Ising model on a complete graph) to quantum mechanics. Using a natural probabilistic representation of this model, we give a complete picture of the phase diagram of the model in the parameters of inverse temperature and transverse field strength. Further analysis computes the critical exponent for the vanishing of the order parameter in the approach to the critical curve and gives useful stability properties for a variational problem associated with the representation.

Keywords

Quantum spin systems Phase diagrams Mean field theory Large deviations Random current representation Ising model Feynman-Kac transformation 
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References

  1. 1.
    Aizenman, M.: Geometric analysis of φ 4 fields and Ising models. I, II. Commun. Math. Phys. 86, 1–48 (1982) MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Aizenman, M., Chayes, J.T., Chayes, L., Newman, C.M.: Discontinuity of the magnetization in one-dimensional 1/| x−y | 2 Ising and Potts models. J. Statist. Phys. 50(1–2), 1–40 (1988) MATHCrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Statist. Phys. 44, 393–454 (1986) MATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Aizenman, M., Klein, A., Newman, C.: Percolation methods for disordered quantum Ising models. In: Kotecky, R. (ed.) Phase Transitions: Mathematics, Physics, Biology, …, pp. 1–26. World Scientific, Singapore (1993) Google Scholar
  5. 5.
    Aizenman, M., Nachtergaele, B.: Geometric aspects of quantum spin states. Commun. Math. Phys. 164, 17–63 (1994) MATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Baldi, P.: Large deviations and stochastic homogenization. Ann. Mat. Pura Appl. 151(4), 161–177 (1988) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Campanino, M., Klein, A., Perez, J.F.: Localization in the ground state of the Ising model with a random transverse field. Commun. Math. Phys. 135, 499–515 (1991) MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Crawford, N., Ioffe, D.: On a switching lemma for quantum Ising models in transverse field, preprint (2008) Google Scholar
  9. 9.
    Dorlas, T.C.: Probabilistic derivation of a noncommutative version of Varadhan’s theorem, unpublished, June 2002. http://www.stp.dias.ie/~dorlas/tony_index2.html
  10. 10.
    Fannes, M., Spohn, H., Verbeure, A.: Equilibrium states for mean field models. J. Math. Phys. 21, 355–360 (1980) MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    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), 1–34 (1978) CrossRefADSGoogle Scholar
  12. 12.
    Grimmett, G.: Space-time percolation. Preprint, arXiv:0705.0506v1 [math.PR]
  13. 13.
    Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003) MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Ioffe, D.: Stochastic geometry of classical and quantum Ising models. In: Proceedings of the 5th Prague Summer School, LNM. Springer, Berlin (2008) Google Scholar
  15. 15.
    Ioffe, D., Levit, A.: Long range order and giant components of quantum random graphs. Mark. Proc. Rel. Fields 13(3), 469–492 (2007) MATHMathSciNetGoogle Scholar
  16. 16.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin glass theory and beyond. In: World Scientific Lecture Notes in Physics, vol. 9. World Scientific Publishing, Teaneck (1987) (preprint) Google Scholar
  17. 17.
    Nachtergaele, B.: Quasi-state decompositions for quantum spin systems in Probability Theory and Mathematical Statistics. In: Grigelionis, B., et al. (eds.) Proceedings of the 6th Vilnius Conference, pp. 565–590. VSP/TEV, Utrecht, Tokyo, Vilnius (1994) Google Scholar
  18. 18.
    Nachtergaele, B.: A stochastic geometric approach to quantum spin systems. In: Probability and Phase transition, Cambridge, 1993. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 420, pp. 237–246. Kluwer Acad. Publ., Dordrecht (1994) Google Scholar
  19. 19.
    Parisi, G.: Field theory, disorder and simulations. In: World Scientific Lecture Notes in Physics, vol. 49, World Scientific, River Edge (1992) Google Scholar
  20. 20.
    Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. A Series of Modern Surveys in Mathematics, vol. 46. Springer, Berlin (2003) Google Scholar
  21. 21.
    Talagrand, M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006) MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Toland, J.F.: A duality principle for non-convex optimization in the calculus of variations, F.M.R.I. (University of Essex), Arch. Rational Mech. Analysis (1979) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Lincoln Chayes
    • 1
  • Nicholas Crawford
    • 2
  • Dmitry Ioffe
    • 3
  • Anna Levit
    • 3
  1. 1.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA
  2. 2.Department of StatisticsUniversity of California at BerkeleyBerkeleyUSA
  3. 3.Department of Industrial EngineeringThe TechnionHaifaIsrael

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