Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Auffinger A., Arous G.B., Černý J.: Random matrices and complexity of spin glasses. Commun. Pure Appl. Math. 66(2), 165–201 (2013)
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
Baik J., Lee J.O.: Fluctuations of the free energy of the spherical Sherrington–Kirkpatrick model. J. Stat. Phys. 165(2), 185–224 (2016)
Benaych-Georges, F., Knowles, A.: Lectures on the local semicircle law for Wigner matrices (2016). arXiv preprint arXiv:1601.04055
Bolthausen, E.: Private communication
Bolthausen E.: An iterative construction of solutions of the TAP equations for the Sherrington–Kirkpatrick model. Commun. Math. Phys. 325(1), 333–366 (2014)
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)
Chatterjee S.: Spin glasses and Stein’s method. Probab. Theory Relat. Fields 148(3), 567–600 (2010)
Chen, W.-K., Panchenko, D.: On the TAP free energy in the mixed p-spin models (2017). arXiv preprint arXiv:1709.03468
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)
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)
Genovese G., Tantari D.: Legendre duality of spherical and Gaussian spin glasses. Math. Phys. Anal. Geom. 18(1), 10 (2015)
Guerra F.: Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233(1), 1–12 (2003)
Kosterlitz J.M., Thouless D.J., Jones R.C.: Spherical model of a spin-glass. Phys. Rev. Lett. 36(20), 1217 (1976)
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)
Panchenko D.: The Parisi ultrametricity conjecture. Ann. Math. (2) 177(1), 383–393 (2013)
Panchenko, D.: The Sherrington–Kirkpatrick Model. Springer, Berlin (2013)
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)
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)
Sherrington D., Kirkpatrick S.: Solvable model of a spin-glass. Phys. Rev. Lett. 35(26), 1792 (1975)
Subag E.: The geometry of the Gibbs measure of pure spherical spin glasses. Invent. Math. 210(1), 135–209 (2017)
Talagrand, M.: Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models, vol. 46. Springer, Berlin (2003)
Talagrand M.: Free energy of the spherical mean field model. Probab. Theory Relat. Fields 134(3), 339–382 (2006)
Talagrand M.: The Parisi formula. Ann. Math. (2) 163(1), 221–263 (2006)
Tao, T.: Topics in Random Matrix Theory, vol. 132. American Mathematical Soc., Providence (2012)
Thouless D.J., Anderson P.W., Palmer R.G.: Solution of ’solvable model of a spin glass’. Philos. Mag. 35(3), 593–601 (1977)
Vilfan, I.: Lecture Notes in Statistical Mechanics. http://www-f1.ijs.si/~vilfan/SM/. Accessed 21 Jan 2019
Acknowledgement
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Belius, D., Kistler, N. The TAP–Plefka Variational Principle for the Spherical SK Model. Commun. Math. Phys. 367, 991–1017 (2019). https://doi.org/10.1007/s00220-019-03304-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03304-y