Skip to main content
SpringerLink
Log in

The Phase Diagram of the Quantum Curie-Weiss Model

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aizenman, M.: Geometric analysis of φ 4 fields and Ising models. I, II. Commun. Math. Phys. 86, 1–48 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Statist. Phys. 44, 393–454 (1986)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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. Aizenman, M., Nachtergaele, B.: Geometric aspects of quantum spin states. Commun. Math. Phys. 164, 17–63 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Baldi, P.: Large deviations and stochastic homogenization. Ann. Mat. Pura Appl. 151(4), 161–177 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Crawford, N., Ioffe, D.: On a switching lemma for quantum Ising models in transverse field, preprint (2008)

  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. Fannes, M., Spohn, H., Verbeure, A.: Equilibrium states for mean field models. J. Math. Phys. 21, 355–360 (1980)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  12. Grimmett, G.: Space-time percolation. Preprint, arXiv:0705.0506v1 [math.PR]

  13. Guerra, F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. Ioffe, D.: Stochastic geometry of classical and quantum Ising models. In: Proceedings of the 5th Prague Summer School, LNM. Springer, Berlin (2008)

  15. Ioffe, D., Levit, A.: Long range order and giant components of quantum random graphs. Mark. Proc. Rel. Fields 13(3), 469–492 (2007)

    MATH  MathSciNet  Google Scholar 

  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)

  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. 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. Parisi, G.: Field theory, disorder and simulations. In: World Scientific Lecture Notes in Physics, vol. 49, World Scientific, River Edge (1992)

  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)

  21. Talagrand, M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at Los Angeles, Los Angeles, USA

    Lincoln Chayes

  2. Department of Statistics, University of California at Berkeley, Berkeley, USA

    Nicholas Crawford

  3. Department of Industrial Engineering, The Technion, Haifa, Israel

    Dmitry Ioffe & Anna Levit

Authors
  1. Lincoln Chayes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas Crawford
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dmitry Ioffe
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anna Levit
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicholas Crawford.

Additional information

© 2008 by L. Chayes, N. Crawford, D. Ioffe, A. Levit. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.

The research of Nicholas Crawford, Dmitry Ioffe and Anna Levit was partly supported by the German-Israeli Foundation under the grant I-870-58.6/2005.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chayes, L., Crawford, N., Ioffe, D. et al. The Phase Diagram of the Quantum Curie-Weiss Model. J Stat Phys 133, 131–149 (2008). https://doi.org/10.1007/s10955-008-9608-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9608-x

Keywords

Search

Navigation