Abstract
We give a short review about recent results in the study of the mean field Sherrington-Kirkpatrick model for a spin glass. Our methods are essentially based on interpolation and comparison arguments for families of Gaussian random variables. In particular we show how to control the infinite volume limit for the free energy density, and how to relate the model to its replica symmetric approximation. We discuss also the mechanism of replica symmetry breaking, by using suitable interpolation methods. Our results are in agreement with those obtained in the frame of the replica trick through the Parisi Ansatz. Finally, we point out some possible further developments of the theory.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
M. Aizenman, R. Sims, and S. Starr, Extended variational principle for the Sherrington-Kirkpatrick spin-glass model. Phys. Rev. B 68, 214403 (2003)
A. Barra, The mean field Ising model trough interpolating techniques. J. Stat. Phys. 132, 787–809 (2008)
L. De Sanctis, Structural approaches to spin glasses and optimization problems. Ph.D. Thesis, Department of Mathematics, Princeton University (2005)
L. De Sanctis and F. Guerra, Mean field dilute ferromagnet: High temperature and zero temperature behavior. J. Stat. Phys. 132, 759–785 (2008)
S.F. Edwards and P.W. Anderson, Theory of spin glasses. J. Phys. F, Met. Phys. 5, 965–974 (1975)
S. Franz and M. Leone, Replica bounds for optimization problems and diluted spin systems. J. Stat. Phys. 111, 535–564 (2003)
S. Franz and F.L. Toninelli, The Kac limit for finite-range spin glasses. Phys. Rev. Lett. 92, 030602 (2004)
S. Franz and F.L. Toninelli, Finite-range spin glasses in the Kac limit: Free energy and local observables. J. Phys. A, Math. Gen. 37, 7433 (2004)
F. Guerra, Fluctuations and thermodynamic variables in mean field spin glass models. In: Albeverio, S., Cattaneo, U., Merlini, D. (eds.) Stochastic Processes, Physics and Geometry, II. World Scientific, Singapore (1995)
F. Guerra, Sum rules for the free energy in the mean field spin glass model. Fields Inst. Commun. 30, 161 (2001)
F. Guerra, About the cavity fields in mean field spin glass models, invited lecture at the International Congress of Mathematical Physics, Lisboa (2003). Available on http://arxiv.org/abs/cond-mat/0307673
F. Guerra, Broken replica symmetry bounds in the mean field spin glass model. Commun. Math. Phys. 233, 1–12 (2003)
F. Guerra, An introduction to mean field spin glass theory: Methods and results. In: Bovier, A., et al. (eds.) Mathematical Statistical Physics, pp. 243–271. Elsevier, Oxford (2006)
F. Guerra and S. Ghirlanda, General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity. J. Phys. A, Math. Gen. 31, 9149–9155 (1998)
F. Guerra and F.L. Toninelli, The thermodynamic limit in mean field spin glass models. Commun. Math. Phys. 230, 71–79 (2002)
F. Guerra and F.L. Toninelli, Some comments on the connection between disordered long range spin glass models and their mean field version. J. Phys. A, Math. Gen. 36, 10987–10995 (2003)
F. Guerra and F.L. Toninelli, The high temperature region of the Viana-Bray diluted spin glass model. J. Stat. Phys. 115, 531–555 (2004)
S. Kirkpatrick and D. Sherrington, Infinite-ranged models of spin-glasses. Phys. Rev. B 17, 4384–4403 (1978)
E. Marinari, G. Parisi, and J.J. Ruiz-Lorenzo, Numerical simulations of spin glass systems, pp. 59–98, in [7]
M. Mézard, G. Parisi, and M.A. Virasoro, Spin Glass Theory and Beyond. World Scientific, Singapore (1987)
E. Marinari, G. Parisi, F. Ricci-Tersenghi, J.J. Ruiz-Lorenzo, and F. Zuliani, Replica symmetry breaking in short range spin glasses: A review of the theoretical foundations and of the numerical evidence. J. Stat. Phys. 98, 973–1074 (2000)
M. Mézard, G. Parisi, and R. Zecchina, Analytic and algorithmic solution of random satisfiability problems. Science 297, 812 (2002)
C.M. Newman and D.L. Stein, Simplicity of state and overlap structure in finite-volume realistic spin glasses. Phys. Rev. E 57, 1356–1366 (1998)
D. Panchenko and M. Talagrand, Bounds for diluted mean-field spin glass models. Probab. Theory Relat. Fields 130, 319–336 (2004)
G. Parisi, A sequence of approximate solutions to the S-K model for spin glasses. J. Phys. A 13, L-115 (1980)
D. Ruelle, Statistical Mechanics. Rigorous Results. Benjamin, New York (1969)
D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass. Phys. Rev. Lett. 35, 1792–1796 (1975)
H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena. Oxford University Press, New York (1971)
D.L. Stein, Disordered systems: mostly spin glasses. In: Stein, D.L. (ed.) Lectures in the Sciences of Complexity. Addison–Wesley, New York (1989)
M. Talagrand, The generalized Parisi formula. C. R. Acad. Sci., Paris 337, 111–114 (2003)
M. Talagrand, Spin Glasses: A Challenge for Mathematicians. Mean Field Models and Cavity Method. Springer, Berlin (2003)
M. Talagrand, The Parisi formula. Ann. Math. 163, 221–263 (2006)
P. Young (ed.), Spin Glasses and Random Fields. World Scientific, Singapore (1987)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer Science+Business Media B.V.
About this paper
Cite this paper
Guerra, F. (2009). Spontaneous Replica Symmetry Breaking in the Mean Field Spin Glass Model. In: Sidoravičius, V. (eds) New Trends in Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-2810-5_21
Download citation
DOI: https://doi.org/10.1007/978-90-481-2810-5_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-2809-9
Online ISBN: 978-90-481-2810-5
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)