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Finite-size critical scaling in Ising spin glasses in the mean-field regime

T. Aspelmeier, Helmut G. Katzgraber, Derek Larson, M. A. Moore, Matthew Wittmann, and Joonhyun Yeo
Phys. Rev. E 93, 032123 – Published 14 March 2016
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  • INTRODUCTION
  • MODEL, OBSERVABLES, AND SCALING…
  • UNIVERSALITY OF THE FINITE-SIZE SCALING…
  • CALCULATION OF THE SCALING FUNCTION f(x)
  • FINITE-SIZE SCALING AT THE…
  • NUMERICAL SEARCHES FOR THE…
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida–Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida–Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida–Thouless transition even though it is absent.

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    • Received 17 September 2015

    DOI:https://doi.org/10.1103/PhysRevE.93.032123

    ©2016 American Physical Society

    Physics Subject Headings (PhySH)

    1. Techniques
    Ising model
    Statistical Physics & Thermodynamics

    Authors & Affiliations

    T. Aspelmeier1,2,3, Helmut G. Katzgraber4,5,6, Derek Larson7, M. A. Moore8, Matthew Wittmann7, and Joonhyun Yeo9

    • 1Felix Bernstein Inst. Math. Stat. Biosci., Göttingen, Germany
    • 2Univ. Göttingen, Inst. Math. Stochast., D-37073 Göttingen, Germany
    • 3Max Planck Inst. Biophys. Chem., Stat. Inverse Problems Biophys. Grp., D-37077 Göttingen, Germany
    • 4Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA
    • 5Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
    • 6Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, England
    • 7Department of Physics, University of California, Santa Cruz, California 95064, USA
    • 8School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
    • 9Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul 143-701, Korea

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    Issue

    Vol. 93, Iss. 3 — March 2016

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    Images

    • Figure 1
      Figure 1

      (a) Critical scaling form ξL/L vs temperature T for the fully connected (complete) system with σ=0.55 in zero random field H. (b) Mean-field scaling form ξL/L vs temperature T for the fully connected (complete) system with σ=0.55 in zero random field H.

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    • Figure 2
      Figure 2

      (a) Critical scaling form ξL/L vs temperature T for the diluted model with σ=0.55 in zero random field H. (b) Mean-field scaling form ξL/L vs temperature T for the diluted model with σ=0.55 in zero random field H.

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    • Figure 3
      Figure 3

      (a) Critical scaling form of the susceptibility χ/L2σ−1 vs temperature T for the fully connected (complete) system with σ=0.55 when the random field H=0. (b) Mean-field scaling form χ/L1/3 of the susceptibility χ/L2σ−1 vs temperature T for the fully connected (complete) system with σ=0.55 when the random field H=0. Note that χ≡χSG.

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    • Figure 4
      Figure 4

      Reduced spin-glass susceptibility χSG/N1/3 vs x=N1/3(T/Tc−1) (recall N≡L), i.e., the finite-size scaling function f(x) in zero field for all system sizes N studied. For the SK model (σ=0) we simulated N=1024,...,4096. For the diluted model and σ=0 we studied N=2048,...,16384. For σ=0.25 we studied N=512,...,4096 for the complete (fully-connected) model and N=2048,...,16384 for the diluted model. Data taken from Ref. [34]. For σ=0.55 we studied N=32,...,512 for the complete (fully connected) model and N=128,...,2048 for the diluted model. Data taken from Refs. [26, 27]. For σ=0.55 for the complete (fully connected) case we have taken Tc≈0.94, while for the diluted case we use Tc≈1.98. Note the vertical logarithmic scale.

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    • Figure 5
      Figure 5

      Reduced spin-glass susceptibility χSG/N1/3 vs x=N1/3(T/Tc−1), i.e., the finite-size scaling function f(x) in zero field for the SK model (σ=0). The data are taken from Ref. [34]. For this model Tc=1 [6]. The data for x>1 are strongly affected by finite-size effects. The solid curve shows our approximation based on Eq. (29) for the scaling function f(x) based on solving the TAPP equations. It gives, at large positive x,f(x)→1/(2x), while at large negative x,f(x)→x2. The blue dashed curve is the asymptotic limit f(x)→x2 for negative x values.

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    • Figure 6
      Figure 6

      Reduced spin-glass susceptibility χSG/N1/3 vs x=N1/3[T/Tc(H)−1], that is the finite-size scaling function fH(x) in a random field of standard deviation 0.10 when σ=0.55. For this model Tc(H=0.1)≈0.815.

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    • Figure 7
      Figure 7

      Correlation length ξL/L over a large range of field values for L=1024,T=0.48, and σ=0.75. The horizontal dashed line is a guide to the eye marking the point where ξL=L. A change in behavior for ξ<L is apparent. The solid (blue) line marks the regime where the Imry-Ma argument [55] is valid.

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