Abstract
Physicists understand mean-field spin-glass models as possessing a complex free-energy landscape with many equilibrium states. The problem of chaos concerns the evolution of this landscape upon changing the external parameters of the system and is considered relevant for the interpretation of important features of real spin-glass and for understanding the performance of numerical algorithms. The subject is strongly related to that of constrained systems which is considered by mathematicians the natural framework for proving rigorously some of the most peculiar properties of Parisi’s replicasymmetry-breaking solution of mean-field spin-glass models, notably ultrametricity. Many aspects of the problems turned out to possess an unexpected level of difficulty and are still open. We present the results of the physics literature on the subject and discuss the main unsolved problems from a wider perspective.
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References
The problem of chaos has attracted much attention and has been studied by various approaches including: scaling arguments and real space renormalization group analysis A.J. Bray and M.A. Moore, Phys. Rev. Lett. 58, 57 (1987). D.S. Fisher and D.A. Huse, Phys. Rev. B 38, 386 (1988). J.R. Banavar and A.J. Bray, Phys. Rev. B 35, 8888 (1987). M. Nifle and H.J. Hilhorst, Phys. Rev. Lett 68, 2992 (1992); analytical and numerical studies on the mean field models, A. Billoire and B. Coluzzi Phys. Rev. E 67, 036108 (2003); A. Billoire and E. Marinari, cond-mat/020247; T. Rizzo and A. Crisanti, Phys. Rev. Lett. 90, 137201 (2003); numerical studies on finitedimensional Edwards-Andersion models, F. Ritort, Phys. Rev. B 50, 6844 (1994). M. Ney-Nifle, Phys. Rev. B 57, 492 (1997). D.A. Huse and L-F. Ko, Phys. Rev. B 56, 14597 (1997). A. Billoire and E. Marinari, J. Phys. A 33, L265 (2000). T. Aspelmeier, A.J. Bray, and M.A. Moore, Phys. Rev. Lett. 89, 197202 (2002). M. Sasaki, K. Hukushima, H. Yoshino, H. Takayama, cond-mat0411138; analytical and numerical studies on elastic manifolds in random media, D.S. Fisher and D.A. Huse, Phys. Rev. B 43, 10728 (1991). M. Sales and H. Yoshino Phs. Rev. E 65 066131 (2002). A. da Silveira and J.P. Bouchaud, Phys. Rev. Lett. 93, 015901 (2004). Pierre Le Doussal, cond-mat/0505679. It has been proposed that the chaos effect is a possible mechanism of the so-called rejuvenation effect found experimentally. See for example, P.E. Jonsson, R. Mathieu, P. Nordblad, H. Yoshino, H. Aruga Katori, A. Ito, Phys. Rev. B 70, 174402 (2004) and references there in. The problem of the temperature dependence of the Gibbs measure of the Random Energy Model has been also investigated both in the physics literature (M. Sales and J.-P. Bouchaud, Europhys. Lett. 56, 181 (2001)) and in the mathematical physics literature, I. Kurkova, J. Stat. Phys. 111, 35, (2003).
M. Mézard, G. Parisi, and M.A. Virasoro, “Spin glass theory and beyond”, World Scientific (Singapore 1987).
David J. Earl and Michael W. Deem, Phys. Chem. Chem. Phys., 2005, 7, 3910.
T. Rizzo, Europhys. Jour. B 29, 425 (2002).
T. Rizzo and A. Crisanti, Phys. Rev. Lett. 90, 137201 (2003).
H. Yoshino and T. Rizzo, Phys. Rev. B 73, 064416 (2006), Phys. Rev. B 77, 104429 (2008).
T. Rizzo, J. Phys. A: Math. Gen. 34, 5531–5549 (2001).
S. Franz and M. Ney-Nifle, J. Phys. A: Math. Gen. 26, L641 (1993).
I. Kondor, J. Phys. A 22 (1989) L163.
S. Franz, G. Parisi, M.A. Virasoro, J. Physique I 2 (1992) 1969.
T. Nieuwenhuizen, Phys. Rev. Lett. 74 4289 (1995).
T. Aspelmeier, J. Phys. A: Math. Theor. 41 (2008) 205005, J. Stat. Mech. (2008) P04018, Phys. Rev. Lett. 100:117205, 2008.
M. Talagrand, The Parisi formula. Annals of Mathematics 163, no 1, 2006, 221–263.
M. Talagrand, Lecture Notes in Math., 1900, Springer, Berlin, 2007, 63–80.
D. Panchenko and M. Talagrand, Ann. Probab. 35, 2007, no 6, 2321–2355.
S. Chatterjeee, Chaos, concentration, and multiple valleys, arXiv:0810.4221 (2008)
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Rizzo, T. (2009). Chaos in Mean-field Spin-glass Models. In: de Monvel, A.B., Bovier, A. (eds) Spin Glasses: Statics and Dynamics. Progress in Probability, vol 62. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9891-0_6
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