Spin glasses on the hypercube

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We present a mean field model for spin glasses with a natural notion of distance built in, namely, the Edwards-Anderson model on the diluted D-dimensional unit hypercube in the limit of large D. We show that finite D effects are strongly dependent on the connectivity, being much smaller for a fixed coordination number. We solve the nontrivial problem of generating these lattices. Afterward, we numerically study the nonequilibrium dynamics of the mean field spin glass. Our three main findings are the following: i the dynamics is ruled by an infinite number of time sectors, ii the aging dynamics consists of the growth of coherent domains with a nonvanishing surface-volume ratio, and iii the propagator in Fourier space follows the p4 law. We study as well the finite D effects in the nonequilibrium dynamics, finding that a naive finite size scaling ansatz works surprisingly well.
© 2010 The American Physical Society. We thank Federico Ricci-Tersenghi for discussions. Computations have been carried out in PC clusters at BIFI and DFTI-UCM. We have been partly supported by MICINN, Spain through Research Contracts No. FIS2006-08533 and No. FIS2009-12648-C03 and by UCM-Banco de Santander through Grant No. GR58/08. B.S. was supported by the FPU program Spain.
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1) J. A. Mydosh, Spin Glasses: An Experimental Introduction (Taylor & Francis, London), 1993. 2) K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud, P. Nordblad, Phys. Rev. Lett., 81, 3243, 1998. 3) G. F. Rodríguez, G. G. Kenning, R. Orbach, Phys. Rev. Lett., 91, 037203, 2003. 4) G. G. Kenning, G. F. Rodríguez, R. Orbach, Phys. Rev. Lett., 97, 057201, 2006. 5) K. Gunnarsson, P. Svedlindh, P. Nordblad, L. Lundgren, H. Aruga, A. Ito, Phys. Rev. B, 43, 8199, 1991. 6) S. Franz, M. Mézard, G. Parisi, L. Peliti, Phys. Rev. Lett., 81, 1758, 1998 -- ibid., J. Stat. Phys., 97, 459, 1999. 7) F. Belletti, M. Cotallo, A. Cruz, L. A. Fernández, A. Gordillo Guerrero, M. Guidetti, A. Maiorano, F. Mantovani, E. Marinari, V. Martín-Mayor, A. M. Sudupe, D. Navarro, G. Parisi, S. Pérez-Gaviro, J. J. Ruiz-Lorenzo, S. F. Schifano, D. Sciretti, A. Tarancón, R. Tripiccione, J. L. Velasco, D. Yllanes, Phys. Rev. Lett., 101, 157201, 2008. 8) R. Álvarez Baños, et al., JANUS collaboration, arXiv:1003.2569. 9) D. S. Fisher, D. A. Huse, Phys. Rev. Lett., 56, 1601, 1986 -- ibid., Phys. Rev. B, 38, 386, 1988. 10) E. Marinari, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz Lorenzo, F. Zuliani, J. Stat. Phys., 98, 973, 2000. 11) D. S. Fisher, D. A. Huse, Phys. Rev. B, 38, 373, 1988. 12) F. Krzakala, O. C. Martin, Phys. Rev. Lett,. 85, 3013, 2000 -- M. Palassini, A. P. Young, ibid., 85, 3017, 2000. 13) D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett., 35, 1792, 1975. 14) M. Mézard, G. Parisi, Eur. Phys. J. B, 20, 217, 2001. 15) A nondiluted spin glass model on a D-dimensional hypercube, in the limit of large D, was considered by G. Parisi, F. Ritort, and J. M. Rubi, J. Phys. A: Math. Gen., 24, 5307, 1991. While a fully frustrated model was studied by E. Marinari, G. Parisi, and F. Ritort, ibid., 28, 327, 1995. 16) C. De Dominicis, I. Kondor, T. Temesvari, Int. J. Mod. Phys. B, 7, 986, 1993 -- Spin Glasses and Random Fields, edited by P. Young (World Scientific, Singapore), 1997 -- C. De Dominicis, I. Giardina, Random Fields and Spin Glasses: A Field Theory Approach (Cambridge University Press, Cambridge), 2006. 17) G. Parisi, Proceedings of the 2006 Les Houches Summer School (unpublished). 18) D. J. Thouless, Phys. Rev. Lett., 56, 1082, 1986. 19) F. Ricci-Tersenghi, private communication. 20) This movement keeps 21) See, e.g., D. J. Amit, V. Martín-Mayor, Field Theory: The Renormalization Group and Critical Phenomena, 3rd ed (World Scientific, Singapore), 2005. 22) M. Hasenbusch, A. Pelissetto, E. Vicari, Phys. Rev. B, 78, 214205, 2008. 23) JANUS Collaboration, J. Stat. Phys., 135, 1121, 2009. 24) G. Parisi, F. Zamponi, J. Stat. Phys., 123, 1145, 2006. 25) L. F. Cugliandolo, J. Kurchan, J. Phys. A, 27, 5749, 1994. 26) J. P. Bouchaud, L. Cugliandolo, J. Kurchan, M. Mèzard, in Spin Glasses and Random Fields, edited by P. Young (World Scientific, Singapore), 1997. 27) E. Bertin, J. P. Boucheaud, J. Phys. A, 35, 3039, 2002. 28) M. Mèzard, G. Parisi, M. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore), 1987. 29) L. Berthier, J. L. Barrat, J. Kurchan, Phys. Rev. E, 63, 016105, 2000. 30) P. Granberg, P. Svedlindh, P. Nordblad, L. Lundgren, H. S. Chen, Phys. Rev. B, 35, 2075, 1987 -- J. J. Préjean, J. Souletie, ibid., 577, 1988, and references therein. 31) G. Parisi, P. Ranieri, F. Ricci-Tersenghi, J. J. Ruiz Lorenzo, J. Phys. A, 30, 7115, 1997. 32) L. Leuzzi, G. Parisi, F. Ricci-Tersenghi, J. J. Ruiz Lorenzo, Phys. Rev. Lett., 101, 107203, 2008. 33) G. Parisi, Statistical Field Theory (Addison-Wesley, Reading, MA), 1988.