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Low temperature spin glass fluctuations: expanding around a spherical approximation

  • Statistical and Nonlinear Physics
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Abstract

The spin glass behavior near zero temperature is a complicated matter. To get an easier access to the spin glass order parameter Q(x) and, at the same time, keep track of Qab, its matrix aspect, and hence of the Hessian controlling stability, we investigate an expansion of the replicated free energy functional around its “spherical” approximation. This expansion is obtained by introducing a constraint-field and a (double) Legendre Transform expressed in terms of spin correlators and constraint-field correlators. The spherical approximation has the spin fluctuations treated with a global constraint and the expansion of the Legendre Transformed functional brings them closer and closer to the Ising local constraint. In this paper we examine the first contribution of the systematic corrections to the spherical starting point.

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References

  • D. Sherrington, S. Kirkpatrick, Phys. Rev. B 17, 4384 (1978)

    Google Scholar 

  • Notice that Ω[Q] is disorder independent. If we were to study another 2-spin model (ROM, etc.), the Lagrangean would read \({\cal L}[Q] = G[Q] + \Omega[Q]\) where Ω[Q] is as defined in equation (5) and G[Q] determined as in references [3,4]. The Lagrangean can be extended to more complex spin interactions following reference [5]. In this work we keep to the SK model

  • R. Cherrier, D.S. Dean, A. Lefevre, Phys. Rev E 67, 046112 (2003)

    Google Scholar 

  • T. Sarlat, Thesis, 2009

  • A. Crisanti, G. Parisi, L. Leuzzi, J. Phys. A 35, 481 (2002)

    Google Scholar 

  • G. Parisi, Phys. Rev. Lett. 43, 1754 (1979)

    Google Scholar 

  • G. Parisi, J. Phys. A 13, 1101 (1980)

    Google Scholar 

  • B. Duplantier, J. Phys. A 14, 283 (1981)

    Google Scholar 

  • G. Parisi, Phys. Rev. Lett. 50, 1946 (1983)

  • A. Crisanti, T. Rizzo, Phys. Rev. E 65, 46137 (2002)

    Google Scholar 

  • A. Crisanti, T. Rizzo, T. Temesvari, Eur. Phys. J. B 33, 203 (2003)

    Google Scholar 

  • S. Pankov, Phys. Rev. Lett. 96, 197204 (2006)

    Google Scholar 

  • R. Oppermann, D. Sherrington, Phys. Rev. Lett. 95, 197203 (2005)

    Google Scholar 

  • R. Oppermann, M.J. Schmidt, D. Sherrington, Phys. Rev. Lett. 98, 127201 (2007)

    Google Scholar 

  • M.J. Schmidt, R. Oppermann, Phys. Rev. E 77, 061104 (2008)

    Google Scholar 

  • M.J. Schmidt, Dissertation, Würzburg (2008) (unpublished)

  • R. Oppermann, M.J. Schmidt, Phys. Rev. E 78, 061124 (2008)

    Google Scholar 

  • C. De Dominicis, P.C. Martin, J. Math. Phys. 5, 14 (1964)

    Google Scholar 

  • J.M. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. 10, 2428 (1974)

    Google Scholar 

  • R.W. Haymaker, Riv. Nuovo Cimento 14, 1 (1991)

    Google Scholar 

  • If one keeps the 〈σaλb〉 correlators then their equation of motion only admits the trivial null solution

  • We leave aside the possibility of having vectorial replica symmetry breaking

  • For instance in the q≥4 Potts model the symmetry is broken and indeed \(Q_0\not=0\)

  • T Temesvari, C. De Dominicis, I. Kondor, J. Phys. A 27, 7569 (1994)

    Google Scholar 

  • C. De Dominicis, D.M. Carlucci, T. Temesvari, J. Phys. I France 7, 105 (1997)

    Google Scholar 

  • If all 2-PI diagrams coming from \({\cal} K_2\) are omitted the model reduces to the 2-spin spherical model; J.M. Kosterlitz, D.J. Thouless, R.C. Jones, Phys. Rev. Lett. 36, 1217 (1976)

  • T.R. Kirkpatrick, D. Thirumalai, Phys. Rev. B 36, 5388 (1987)

    Google Scholar 

  • A. Crisanti, H.-J. Sommers, Z. Phys. B 87, 341 (1992)

    Google Scholar 

  • A. Crisanti, H. Horner, H.J. Sommers, Z. für Phys. B 92, 257 (1993)

    Google Scholar 

  • R. Monasson, Phys. Rev. Lett. 75, 2847 (1995)

    Google Scholar 

  • C. De Dominicis, T. Temesvari, I. Kondor, J. Phys. IV France 8, 13 (1998), preprint cont-mat/9802166 Equation numbering having been messed up at the editing stage, the reader should rather consult the cond-mat version

  • The RFT was first introduced, directly in the continuum limit (R↦∞) by Mezard and Parisi [33]

  • M. Mezard, G. Parisi, J. Phys. I France 1, 809 (1991)

    Google Scholar 

  • A. Crisanti, L. Leuzzi, Phys. Rev. B 73, 014412 (2006)

    Google Scholar 

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Crisanti, A., De Dominicis, C. & Sarlat, T. Low temperature spin glass fluctuations: expanding around a spherical approximation. Eur. Phys. J. B 74, 139–149 (2010). https://doi.org/10.1140/epjb/e2010-00022-9

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  • DOI: https://doi.org/10.1140/epjb/e2010-00022-9

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