The TAP–Plefka Variational Principle for the Spherical SK Model

  • David BeliusEmail author
  • Nicola Kistler


We reinterpret the Thouless–Anderson–Palmer approach to mean field spin glass models as a variational principle in the spirit of the Gibbs variational principle and the Bragg–Williams approximation. We prove this TAP–Plefka variational principle rigorously in the case of the spherical Sherrington–Kirkpatrick model.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The first author thanks Erwin Bolthausen and Giuseppe Genovese for valuable discussions on a draft of this article. The second author wishes to express his gratitude to Markus Petermann for a longstanding discussion on spin glasses, and to Anton Wakolbinger for encouragement.


  1. 1.
    Aizenman M., Lebowitz J.L., Ruelle D.: Some rigorous results on the Sherrington–Kirkpatrick spin glass model. Commun. Math. Phys. 112(1), 3–20 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Auffinger A., Arous G.B., Černý J.: Random matrices and complexity of spin glasses. Commun. Pure Appl. Math. 66(2), 165–201 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Auffinger, A., Jagannath, A.: Thouless–Anderson–Palmer equations for conditional Gibbs measures in the generic p-spin glass model (2016). arXiv preprint arXiv:1612.06359
  4. 4.
    Baik J., Lee J.O.: Fluctuations of the free energy of the spherical Sherrington–Kirkpatrick model. J. Stat. Phys. 165(2), 185–224 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benaych-Georges, F., Knowles, A.: Lectures on the local semicircle law for Wigner matrices (2016). arXiv preprint arXiv:1601.04055
  6. 6.
    Bolthausen, E.: Private communicationGoogle Scholar
  7. 7.
    Bolthausen E.: An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model. Commun. Math. Phys. 325(1), 333–366 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bragg, W.L., Williams, E.J: The effect of thermal agitation on atomic arrangement in alloys. Proc. R. Soc. Lond. Ser. A 145(855), 699–730 (1934). (Containing Papers of a Mathematical and Physical Character) Google Scholar
  9. 9.
    Chatterjee S.: Spin glasses and Stein’s method. Probab. Theory Relat. Fields 148(3), 567–600 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, W.-K., Panchenko, D.: On the TAP free energy in the mixed p-spin models (2017). arXiv preprint arXiv:1709.03468
  11. 11.
    Crisanti A., Sommers H.-J.: Thouless–Anderson–Palmer approach to the spherical p-spin spin glass model. J. Phys. I 5(7), 805–813 (1995)Google Scholar
  12. 12.
    Crisanti A., Sommers H.-J.: The spherical p-spin interaction spin glass model: the statics. Z. Phys. B Condens. Matter 87(3), 341–354 (1992)ADSCrossRefGoogle Scholar
  13. 13.
    Genovese G., Tantari D.: Legendre duality of spherical and Gaussian spin glasses. Math. Phys. Anal. Geom. 18(1), 10 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guerra F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kosterlitz J.M., Thouless D.J., Jones R.C.: Spherical model of a spin-glass. Phys. Rev. Lett. 36(20), 1217 (1976)ADSCrossRefGoogle Scholar
  16. 16.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond, Volume 9 of World Scientific Lecture Notes in Physics. World Scientific Publishing Co., Inc., Teaneck (1987)Google Scholar
  17. 17.
    Panchenko D.: The Parisi ultrametricity conjecture. Ann. Math. (2) 177(1), 383–393 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Panchenko, D.: The Sherrington–Kirkpatrick Model. Springer, Berlin (2013)Google Scholar
  19. 19.
    Plefka T.: A lower bound for the spin glass order parameter of the infinite-ranged Ising spin glass model. J. Phys. A Math. Gen. 15(5), L251 (1982)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Plefka T.: Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model. J. Phys. A Math. Gen. 15(6), 1971 (1982)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Sherrington D., Kirkpatrick S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35(26), 1792 (1975)ADSCrossRefGoogle Scholar
  22. 22.
    Subag E.: The geometry of the Gibbs measure of pure spherical spin glasses. Invent. Math. 210(1), 135–209 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Talagrand, M.: Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models, vol. 46. Springer, Berlin (2003)Google Scholar
  24. 24.
    Talagrand M.: Free energy of the spherical mean field model. Probab. Theory Relat. Fields 134(3), 339–382 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Talagrand M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Tao, T.: Topics in Random Matrix Theory, vol. 132. American Mathematical Soc., Providence (2012)Google Scholar
  27. 27.
    Thouless D.J., Anderson P.W., Palmer R.G.: Solution of ’solvable model of a spin glass’. Philos. Mag. 35(3), 593–601 (1977)ADSCrossRefGoogle Scholar
  28. 28.
    Vilfan, I.: Lecture Notes in Statistical Mechanics. Accessed 21 Jan 2019

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of BaselBaselSwitzerland
  2. 2.University of FrankfurtFrankfurtGermany

Personalised recommendations