Large Deviations for Quantum Spin Systems


We consider high temperature KMS states for quantum spin systems on a lattice. We prove a large deviation principle for the distribution of empirical averages \(\bar X_\Lambda : = \frac{1}{{|\Lambda |}}\Sigma _{i \in \Lambda } X_i \), where the X i 's are copies of a self-adjoint element X (level one large deviations). From the analyticity of the generating function, we obtain the central limit theorem. We generalize to a level two large deviation principle for the distribution of \(\frac{1}{{|\Lambda |}}\Sigma _{i \in \Lambda } \delta _{X_i } \)

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


  1. 1.

    W. Bryc, A remark on the connection between the large deviation principle and the central limit theorem, Stat. Prob. Lett. 18:253–256 (1993).

    Google Scholar 

  2. 2.

    O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2 (Springer, Berlin, 1996).

    Google Scholar 

  3. 3.

    A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, (Springer-Verlag, New York, 1998).

    Google Scholar 

  4. 4.

    Enter A. C. D. van, R. Fern´andez, and A. D. Sokal, Regularity properties and pathologies of position-space renormalization group transformations: Scope and limitations of Gibbsian theory, J.Stat.Phys. 72:879–1167 (1993).

    Google Scholar 

  5. 5.

    H.O. Georgii, Gibbs Measures and Phase Transitions(Walter de Gruyter & Co., Berlin, 1988).

    Google Scholar 

  6. 6.

    R. S. Ellis, Large Deviations and Statistical Mechanics(Springer-Verlag, New York, 1985).

    Google Scholar 

  7. 7.

    R. B. Israel, Convexity in the Theory of Lattice Gases(Princeton University Press, Princeton, 1979).

    Google Scholar 

  8. 8.

    R. V. Kadison, J. R. Ringrose, Fundamentals of the Theory of Operator Algebras(Academic Press, New York, London, 1983).

    Google Scholar 

  9. 9.

    R. Kotecky and D. Preiss, Cluster expansion for abstract polymer models, Commun.Math.Phys. 103:491–498 (1986).

    Google Scholar 

  10. 10.

    J. L. Lebowitz, M. Lenci and H. Spohn, Large deviations for ideal quantum systems, Math-Phys Archive 9906014.

  11. 11.

    G. Gallavotti, J. L. Lebowitz, and V. Mastropietro, Large deviations in rarefied quantum gases, J.Statist.Phys. 108:(5–6): 831–861 (2002).

    Google Scholar 

  12. 12.

    S. Miracle-Sol´e, On the convergence of cluster expansions, Physica A 279:244–249 (2000).

    Google Scholar 

  13. 13.

    S. Olla, Large deviations for Gibbs random fields, Prob.Theory.Related.Fields 77:343–357 (1988).

    Google Scholar 

  14. 14.

    Y. M. Park, The cluster expansion for classical and quantum lattice systems, J.Stat.Phys. 27:553–576 (1982).

    Google Scholar 

  15. 15.

    B. Simon, The Statistical Mechanics of Lattice Gases(Princeton University Press, Princeton, 1993).

    Google Scholar 

  16. 16.

    D. Goderis and P. Vets P, Central limit theorem for mixing quantum systems and the CCR-algebra of fluctuations, Common.Math.Phys. 122:249–265 (1989).

    Google Scholar 

  17. 17.

    D. Goderis, A. Verbeure, P. Vets, Noncommutative central limits, Prob.Theory Related Fields 82:527–544 (1989).

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Netočný, K., Redig, F. Large Deviations for Quantum Spin Systems. Journal of Statistical Physics 117, 521–547 (2004).

Download citation

  • Large deviation principle
  • central limit theorem
  • boundary terms
  • cluster expansion
  • Goldon–Thompson inequality