Finite-size corrections in the Sherrington–Kirkpatrick model

Aspelmeier, Timo; Billoire, A.; Marinari, Enzo; Moore, M. A.

doi:10.1088/1751-8113/41/32/324008

Cited by 77 publications

(127 citation statements)

References 40 publications

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“…The argument is however not entirely rigorous. Nevertheless, it has recently been argued by a combination of heuristic arguments and extensive numerical simulations at finite temperature that µ = 1 6 is indeed correct [19]. The bound µ ≤ 1 4 derived here is compatible with this but, unfortunately, does not rule out µ = 1 4 .…”

Section: To Derive This Resultsupporting

confidence: 44%

Free-Energy Fluctuations and Chaos in the Sherrington-Kirkpatrick Model

Aspelmeier

2008

Phys. Rev. Lett.

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The sample-to-sample fluctuations ∆FN of the free energy in the Sherrington-Kirkpatrick model are shown rigorously to be related to bond chaos. Via this connection, the fluctuations become analytically accessible by replica methods. The replica calculation for bond chaos shows that the exponent µ governing the growth of the fluctuations with system size N , ∆FN ∼ N µ , is bounded by µ ≤ 1 4 .The sample-to-sample fluctuations of the free energy in the mean-field Ising spin glass [1] are a long standing unsolved problem in spin glass physics. In addition to their intrinsic interest as a finite size effect in spin glasses, they are of fundamental importance for the physics of finite-dimensional spin glasses. It has been shown [2] that the finite-size scaling of the free energy fluctuations ∆F N in the mean-field spin glass is equal to the scaling of the domain wall energy ∆F DW in finite dimensionswhere N is the total system size and L its linear dimension (in the case of a finite dimensional system). This implies the relationship θ = dµ between the domain wall exponent θ and the fluctuation exponent µ. This highly nontrivial equivalence between a mean-field quantity and a finitedimensional quantity provides a strong test for replica field theory which was used in Ref. 2 to derive this result.Chaos is also a very important aspect of spin glasses. Chaos refers to the property that an infinitesimal change of, for instance, the temperature or the bond strengths results in a complete change of the equilibrium state. Chaos was first suggested in the context of the droplet picture and finite dimensional spin glasses [3] but has also been studied in the mean field model [4,5,6].In this paper we derive a new and exact connection between the free energy fluctuations and the seemingly unrelated phenomenon of chaos. Such a connection has been suggested by Bouchaud et al. [7] as part of a heuristic argument to obtain the free energy fluctuations. Our results partly corroborate the argument but we will see that a crucial ingredient seems to be missing from it. In addition to making the heuristic argument precise, our results provide a new way to access the fluctuations analytically. The fluctuations are a subextensive quantity such that their calculation usually requires higher order terms in the loop expansion. These are, however, inaccessible due to the massless modes present throughout the spin glass phase. Here we will show that it is sufficient to calculate chaos to zero loop order (which is possible) to obtain the fluctuations. We demonstrate this explicitly above and at the critical temperature but believe and present evidence that it also works in the low temperature phase.The method we use to derive the connection between fluctuations and chaos is a variation of the interpolating Hamiltonian method. Our approach is inspired by the work of Billoire [8] where a similar method was introduced to study the finite size corrections to the free energy numerically. In this paper, we will set up a general formalism which will b...

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Section: To Derive This Resultsupporting

confidence: 44%

Free-Energy Fluctuations and Chaos in the Sherrington-Kirkpatrick Model

Aspelmeier

2008

Phys. Rev. Lett.

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“…Qualitatively similar distributions have been observed for the EA and SK models. 18,19,39 It is clear that these inner peaks (that is, peaks away from q ≈ ±1) come from overlaps between states that belong to different basins of attraction. The main aim of this paper is to extract statistical information for these cross-overlap (CO) spikes situated on the interval q ∈ (−Q, Q).…”

Section: A Overlap Distributionsmentioning

confidence: 99%

Low-temperature spin-glass behavior in a diluted dipolar Ising system

Alonso

2015

Phys. Rev. B

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Using Monte Carlo simulations, we study the character of the spin-glass (SG) state of a sitediluted dipolar Ising model. We consider systems of dipoles randomly placed on a fraction x of all L 3 sites of a simple cubic lattice that point up or down along a given crystalline axis. For x 0.65 these systems are known to exhibit an equilibrium spin-glass phase below a temperature Tsg ∝ x. At high dilution and very low temperatures, well deep in the SG phase, we find spiky distributions of the overlap parameter q that are strongly sample-dependent. We focus on spikes associated with large excitations. From cumulative distributions of q and a pair correlation function averaged over several thousands of samples we find that, for the system sizes studied, the average width of spikes, and the fraction of samples with spikes higher than a certain threshold does not vary appreciably with L. This is compared with the behavior found for the Sherrington-Kirkpatrick model.

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“…The "trivial nontrivial" scenario [15][16][17] reconciles these numerical results by postulating that excitations are compact, as in the droplet picture, but their energy cost is independent of system size, as in the RSB picture. In an effort to resolve these discrepancies, here we introduce a statistic obtained from the spin overlap distribution that de-tects sharp peaks in individual samples, inspired by a recent study on the SK model [18]. This statistic clearly differentiates the RSB and droplet pictures: it converges to zero in the large-volume limit if there is a single pair of pure states and to unity if there are countably many.…”

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confidence: 99%

“…In fact, Ref. [18] shows that the number of peaks in P J (q) should scale as N 1/6 for the SK model. On the other hand, for large N and large ∆, the EA contours for q 0 = 0.2 are nearly flat, rising less steeply than for q 0 = 1, suggesting that the number of peaks is either decreasing or staying constant.…”

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confidence: 99%

Evidence of Non-Mean-Field-Like Low-Temperature Behavior in the Edwards-Anderson Spin-Glass Model

2012

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The three-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied via large-scale Monte Carlo simulations at low temperatures, deep within the spin-glass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states.

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Finite-size corrections in the Sherrington–Kirkpatrick model

Cited by 77 publications

References 40 publications

Free-Energy Fluctuations and Chaos in the Sherrington-Kirkpatrick Model

Free-Energy Fluctuations and Chaos in the Sherrington-Kirkpatrick Model

Low-temperature spin-glass behavior in a diluted dipolar Ising system

Evidence of Non-Mean-Field-Like Low-Temperature Behavior in the Edwards-Anderson Spin-Glass Model

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