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arXiv:cond-mat/9808140 · Physical Review Letters

Evidence for the Droplet Picture of Spin Glasses

M A Moore, Hemant Bokil, Barbara Drossel

Published Nov 9, 1998• 113 citations
arXivHTMLPDF

Abstract

We have studied the Parisi overlap distribution for the three-dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures $T\ensuremath{\simeq}{0.7T}_{c}$ and system sizes up to $L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}32$, we found a $P(q)$ as expected for full Parisi replica-symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behavior in Monte Carlo simulations is due to the fact that all existing simulations are done at temperatures too close to the transition temperature so that system sizes larger than the correlation length have not been achieved.

DOI
10.1103/physrevlett.81.4252

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We have studied the Parisi overlap distribution for the three-dimensional Ising spin glass in the Migdal-Kadanoff approximation. For temperatures T\ensuremath≃0.7TcT\ensuremath{\simeq}{0.7T}_{c}T\ensuremath≃0.7Tc​ and system sizes up to L=32L\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}32L=32, we found a P(q)P(q)P(q) as expected for full Parisi replica-symmetry breaking, just as was also observed in recent Monte Carlo simulations on a cubic lattice. However, for lower temperatures our data agree with predictions from the droplet or scaling picture. The failure to see droplet model behavior in Monte Carlo simulations is due to the fact that all existing simulations are done at temperatures too close to the transition temperature so that system sizes larger than the correlation length have not been achieved.
We have argued in recent papers that the Monte Carlo results for the equilibrium properties of the Edwards-Anderson spin glass in three dimensions, which had been interpreted earlier as providing evidence for replica symmetry breaking, can be explained quite simply within the droplet model once finite-size effects and proximity to the critical point are taken into account. In this paper we show that similar considerations are sufficient to explain the Monte Carlo data in four dimensions. In particular, we study the Parisi overlap and the link overlap for the four-dimensional Ising spin glass in the Migdal-Kadanoff approximation. Similar to what is seen in three dimensions, we find that temperatures well below those studied in the Monte Carlo simulations have to be reached before the droplet model predictions become apparent. We also show that the double-peak structure of the link overlap distribution function is related to the difference between domain-wall excitations that cross the entire system and droplet excitations that are confined to a smaller region.
Using Monte Carlo simulations, we study in detail the overlap distribution for individual samples for several spin-glass models including the infinite-range Sherrington-Kirkpatrick model, short-range Edwards-Anderson models in three and four space dimensions, and one-dimensional long-range models with diluted power-law interactions. We study three long-range models with different powers as follows: The first is approximately equivalent to a short-range model in three dimensions, the second to a short-range model in four dimensions, and the third to a short-range model in the mean-field regime. We study an observable proposed earlier by some of us which aims to distinguish the ``replica symmetry breaking'' picture of the spin-glass phase from the ``droplet picture,'' finding that larger system sizes would be needed to unambiguously determine which of these pictures describes the low-temperature state of spin glasses best, except for the Sherrington-Kirkpatrick model, which is unambiguously described by replica symmetry breaking. Finally, we also study the median integrated overlap probability distribution and a typical overlap distribution, finding that these observables are not particularly helpful in distinguishing the replica symmetry breaking and the droplet pictures.
Simulational studies of spin glasses since the early 2010s have focused on the so-called replicon exponent α as a means of determining whether the low-temperature phase of spin glasses is described by the replica symmetry breaking picture of Parisi or by the droplet-scaling picture. On the latter picture, it should be zero, but we shall argue that it will only be zero for systems of linear dimension L>L^{*}. The crossover length L^{*} may be of the order of hundreds of lattice spacings in three dimensions and approach infinity in six dimensions. We use the droplet-scaling picture to show that the apparent nonzero value of α when L<L^{*} should be 2θ, where θ is the domain wall energy scaling exponent. This formula is in reasonable agreement with the reported values of α.
We have studied the diluted Heisenberg spin glass model in a three-component random field for the commonly used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r^{2σ}, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξ_{SG} varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξ_{SG} becomes comparable to the system size L, there are departures which we attribute to the features deriving from the replica symmetry breaking picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L^{*}, where L^{*} is large (≈4000-8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L^{*}. We sketch a renormalization group scenario to explain how these features could arise. In this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.
We study the low-energy excitations of the three dimensional spin glass through a large-scale Monte Carlo simulation on lattices up to L=18L=18L=18. We find smooth extrapolations down to zero temperature, which, in the case of the energy and of the link overlap, can be directly -- and favourably -- compared with previous investigations featuring ground states (i.e., at zero temperature). The best fit for the fractal dimension of the excitations is provided by Replica-Symmetry Breaking theory, but we also consider the alternative TNT description. The P(q)P(q)P(q) is found to verify the Parisi-Toulouse temperature scaling. Our data provides a spectacular confirmation of the overlap-equivalence hypothesis.
We have studied the diluted Heisenberg spin glass model in a 3-component random field for the commonly-used one-dimensional long-range model where the probability that two spins separated by a distance rrr interact with one another falls as 1/r2σ1/r^{2 σ}1/r2σ, for two values of σσσ, 0.750.750.75 and 0.850.850.85. No de Almeida-Thouless line is expected at these σσσ values. The spin glass correlation length ξSGξ_{\text{SG}}ξSG​ varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξSGξ_{\text{SG}}ξSG​ becomes comparable to the system size LLL, there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case σ=0.85σ=0.85σ=0.85 these features go away for system sizes with L>L∗L >L^*L>L∗, where L∗L^*L∗ is large (≈4000−8000\approx 4000-8000≈4000−8000 lattice spacings). In the case of σ=0.75σ= 0.75σ=0.75 we have been unable to study large enough systems to determine its value of L∗L^*L∗. We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.