On the origin of ultrametricity

Parisi, Giorgio; Ricci-Tersenghi, Federico

doi:10.1088/0305-4470/33/1/307

Cited by 43 publications

(57 citation statements)

References 28 publications

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“…Although it has been shown in [23,24] how to break down these probability distributions, the results only apply for ǫ = 0. We therefore concentrate on the second integral in Eq.…”

Section: To Derive This Resultmentioning

confidence: 99%

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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“…Although it has been shown in [23,24] how to break down these probability distributions, the results only apply for ǫ = 0. We therefore concentrate on the second integral in Eq.…”

Section: To Derive This Resultmentioning

confidence: 99%

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

Aspelmeier

2008

Phys. Rev. Lett.

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“…and from that one deduces that the distribution of the two differences x v ÿ u; y z ÿ v is ~x;y x 1 4 y 3 2 y Z 1 y P a P a ÿ y da ; (3) whose marginals are…”

Section: H Y S I C a L R E V I E W L E T T E R S Week Ending 3 Augustmentioning

confidence: 95%

“…Pierluigi Contucci, 1 Cristian Giardinà, 2 Claudio Giberti, 3 Giorgio Parisi, 4 We test the property of ultrametricity for the spin-glass three-dimensional Edwards-Anderson model in zero magnetic field with numerical simulations up to 20 3 spins. We find an excellent agreement with the prediction of the mean field theory.…”

Section: Ultrametricity In the Edwards-anderson Modelmentioning

confidence: 99%

“…Since ultrametricity is not compatible with a trivial structure of the overlap distribution, our result contradicts the droplet theory. Ultrametricity is a widely accepted property of the mean field spin-glass theory: it is a crucial ingredient in the field theoretical computations of the Sherrington-Kirkpatrick model [1][2][3] as well as a guiding principle for the rigorous proof of its free energy density formula [4,5]. Its relevance in finite dimensional systems is nonetheless still an open matter, subject of intense investigations and debates in the theoretical and mathematical physics communities.…”

Section: Ultrametricity In the Edwards-anderson Modelmentioning

confidence: 99%

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Ultrametricity in the Edwards-Anderson Model

Contucci

Giardinà²,

Giberti

et al. 2007

Phys. Rev. Lett.

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We test the property of ultrametricity for the spin glass three-dimensional Edwards-Anderson model in zero magnetic field with numerical simulations up to $20^3$ spins. We find an excellent agreement with the prediction of the mean field theory. Since ultrametricity is not compatible with a trivial structure of the overlap distribution our result contradicts the droplet theory.Comment: typos correcte

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“…With high accuracy the exponent of the stretched exponential is constant throughout the low temperature phase with value δ = 1.50(1). The dynamical critical exponent dependence on the temperature is very well described by the law z 8.9(2)/T and the replicon exponent is nearly constant with average α 1.06 (6).…”

Section: Resultsmentioning

confidence: 91%

Spatial correlation functions and dynamical exponents in very large samples of four-dimensional spin glasses

2014

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The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite-size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the streaming single instruction, multiple data extensions), we have been able to simulate the four-dimensional Edwards-Anderson model with Gaussian couplings up to sizes L = 70 and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite-size effects are negligible: ξ (t) L/7. Our estimates for the dynamical exponent (z 1/T ) and for the replicon exponent (α 1.0 and T independent), that controls the decay of the spatial correlation in the zero overlap sector, are consistent with the replica symmetry breaking theory, but the latter differs from the theoretically conjectured value.

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On the origin of ultrametricity

Cited by 43 publications

References 28 publications

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

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

Ultrametricity in the Edwards-Anderson Model

Spatial correlation functions and dynamical exponents in very large samples of four-dimensional spin glasses

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