The Mean Field Ising Model trough Interpolating Techniques

Abstract

Aim of this paper is to illustrate how some recent techniques developed within the framework of spin glasses do work on simpler model, focusing on the method and not on the analyzed system. To fulfill our will the candidate model turns out to be the paradigmatic mean field Ising model. The model is introduced and investigated with the interpolation techniques. We show the existence of the thermodynamic limit, bounds for the free energy density, the explicit expression for the free energy with its suitable expansion via the order parameter, the self-consistency relation, the phase transition, the critical behavior and the self-averaging properties. At the end a formulation of a Parisi-like theory is tried and discussed.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Amit, D.J.: Modeling Brain Function: The World of Attractor Neural Network. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  2. 2.

    Agostini, A., Barra, A., De Sanctis, L.: Positive-overlap transition and critical exponents in mean field spin glasses. J. Stat. Mech. P11015 (2006)

  3. 3.

    Aizenman, M., Contucci, P.: On the stability of the quenched state in mean field spin glass models. J. Stat. Phys. 92, 765–783 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  4. 4.

    Aizenman, M., Sims, R., Starr, S.L.: An extended variational principle for the SK spin-glass model. Phys. Rev. B 68, 214403 (2003)

    Article  ADS  Google Scholar 

  5. 5.

    Albert, R., Barabasi, A.-L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  6. 6.

    Barra, A.: Irreducible free energy expansion and overlap locking in mean field spin glasses. J. Stat. Phys. 123, 601–614 (2006)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  7. 7.

    Barra, A., De Sanctis, L.: Overlap fluctuation from Boltzmann random overlap structure. J. Math. Phys. 47, 103305 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  8. 8.

    Barra, A., De Sanctis, L.: Stability properties and probability distributions of multi-overlaps in diluted spin glasses. J. Stat. Mech. P08025 (2007)

  9. 9.

    Barra, A., De Sanctis, L.: Spin-glass transition as the lacking of the volume limit commutativity (2007, to appear)

  10. 10.

    Barra, A., De Sanctis, L., Folli, V.: Critical behavior of random spin systems. J. Phys. A 41(21), 215005 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  11. 11.

    Bovier, A., Kurkova, I.: Rigorous results on some simple spin glass models. Markov Proc. Relat. Fields 9, (2003)

  12. 12.

    Bovier, A., Kurkova, I., Loewe, M.: Fluctuations of the free energy in the REM and the p-spin SK model. Ann. Probab. 30 (2002)

  13. 13.

    Comets, F., Neveu, J.: The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: the high temperature case. Commun. Math. Phys. 166, 549 (1995)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  14. 14.

    Coolen, A.C.C.: The trick which became a theory: a brief history of the replica method. Available at http://www.mth.kcl.ac.uk/~tcoolen/

  15. 15.

    Contucci, P., Ghirlanda, S.: Modeling society with statistical mechanics: an application to cultural contact and immigration. Qual. Quantit. 41, 569–578 (2007)

    Article  Google Scholar 

  16. 16.

    Contucci, P., Giardinà, C.: Spin-glass stochastic stability: a rigorous proof. math-ph/0408002

  17. 17.

    De Sanctis, L.: General structures for spherical and other mean-field spin models. J. Stat. Phys. 126

  18. 18.

    De Sanctis, L., Franz, S.: Self averaging identities for random spin systems. math-ph/0705:2978

  19. 19.

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

    Google Scholar 

  20. 20.

    Ghirlanda, S., Guerra, F.: General properties of overlap distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A 31, 9149–9155 (1998)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  21. 21.

    Fischer, K.H., Hertz, J.A.: Spin Glasses. Cambridge University Press, Cambridge (1991)

    Google Scholar 

  22. 22.

    Gallo, I., Contucci, P.: Bipartite mean field spin system: existence and solution. cond-mat/0710.0800

  23. 23.

    Guerra, F.: Mathematical aspects of mean field spin glass theory. cond-mat/0410435

  24. 24.

    Guerra, F.: About the cavity fields in mean field spin glass models. cond-mat/0307673

  25. 25.

    Guerra, F.: Fluctuations and thermodynamic variables in mean field spin glass models. In: Albeverio, S., et al. (eds.) Stochastic Provesses, Physics and Geometry, II. Singapore (1995)

  26. 26.

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

    MATH  Article  MathSciNet  ADS  Google Scholar 

  27. 27.

    Guerra, F.: About the overlap distribution in mean field spin glass models. Int. J. Mod. Phys. B 10, 1675–1684 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  28. 28.

    Guerra, F., Albeverio, S. et al.: The cavity method in the mean field spin glass model. Functional representations of thermodynamic variables. In: Albeverio, S., et al. (eds.) Advances in Dynamical Systems and Quantum Physics. Singapore (1995)

  29. 29.

    Guerra, F.: Sum rules for the free energy in the mean field spin glass model. In: Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects. Fields Institute Communications, vol. 30. American Mathematical Society, Providence (2001)

    Google Scholar 

  30. 30.

    Guerra, F.: Private communications

  31. 31.

    Guerra, F.: An introduction to mean field spin glass theory: methods and results. In: Lecture at Les Houches Winter School (2005)

  32. 32.

    Guerra, F., Toninelli, F.L.: The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230(1), 71–79 (2002)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  33. 33.

    Guerra, F., Toninelli, F.L.: The high temperature region of the Viana-Bray diluted spin glass model. J. Stat. Phys. 115 (2004)

  34. 34.

    Guerra, F., Toninelli, F.L.: Central limit theorem for fluctuations in the high temperature region of the Sherrington-Kirkpatrick spin glass model. J. Math. Phys. 43, 6224–6237 (2002)

    MATH  Article  MathSciNet  ADS  Google Scholar 

  35. 35.

    Guerra, F., Toninelli, F.L.: The infinite volume limit in generalized mean field disordered models. Markov Process. Relat. Fields 9(2), 195–207 (2003)

    MATH  MathSciNet  Google Scholar 

  36. 36.

    Kuttner, J.: Some theorems on the Cesaro limit of a function. Lond. Math. Soc. s1-33, 107–118 (1958)

    Article  MathSciNet  Google Scholar 

  37. 37.

    Mertens, S., Mezard, M., Zecchina, R.: Threshold values of random K-SAT from the cavity method. Random Struct. Algorithms 28, 340–373 (2006)

    MATH  Article  MathSciNet  Google Scholar 

  38. 38.

    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. World Scientific, Singapore (1987)

    Google Scholar 

  39. 39.

    Mézard, M., Parisi, G., Sourlas, N., Toulouse, G., Virasoro, M.A.: Replica symmetry breaking and ultrametricity. J. Phys. 45, 843 (1984)

    Google Scholar 

  40. 40.

    Pagnani, A., Parisi, G., Ricci-Tersenghi, F.: Glassy transition in a disordered model for the RNA secondary structure. Phys. Rev. Lett. 84, 2026 (2000)

    Article  ADS  Google Scholar 

  41. 41.

    Parisi, G.: Stochastic stability. In: Proceedings of the Conference Disordered and Complex Systems, London (2000)

  42. 42.

    Parisi, G.: Statistical Field Theory. Addison-Wesley, New York (1988)

    Google Scholar 

  43. 43.

    Talagrand, M.: Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin (2003)

    Google Scholar 

  44. 44.

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

    MATH  MathSciNet  Article  Google Scholar 

  45. 45.

    Viana, L., Bray, A.J.: Phase diagrams for dilute spin-glasses. J. Phys. C 18, 3037 (1985)

    Article  ADS  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Adriano Barra.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Barra, A. The Mean Field Ising Model trough Interpolating Techniques. J Stat Phys 132, 787 (2008). https://doi.org/10.1007/s10955-008-9567-2

Download citation

Keywords

  • Cavity field
  • Spin glasses
  • Interpolating techniques