Mean field dynamical exponents in finite-dimensional Ising spin glass

Parisi, Giorgio; Ranieri, Paola; Ricci-Tersenghi, Federico; Ruiz‐Lorenzo, J. J.

doi:10.1088/0305-4470/30/20/015

Cited by 28 publications

(41 citation statements)

References 20 publications

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“…While the predicted result, d l ≈ 2.491, is not quite as accurate or stable as above, it is quite consistent. [51] Unlike the consistency between mean-field arguments and numerics for d l , we can observe a large discrepancy for d ≥ d u = 6. An RSB calculations from Ref.…”

Section: Dimension Dcontrasting

confidence: 77%

Stiffness of the Edwards-Anderson Model in all Dimensions

Boettcher

2005

Phys. Rev. Lett.

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A comprehensive description in all dimensions is provided for the scaling exponent y of low-energy excitations in the Ising spin glass introduced by Edwards and Anderson. A combination of extensive numerical as well as theoretical results suggest that its lower critical dimension is exactly d l = 5/2. Such a result would be an essential feature of any complete model of low-temperature spin glass order and imposes a constraint that may help to distinguish between theories.PACS numbers: 05.50.+q , 75.10.Nr , 02.60.Pn Imagining physical systems in non-integer dimensions, such as through an ǫ-expansion near the upper [1] or lower critical dimension [2], has provided many important results for the understanding of the physics in realistic dimensions. Often, peculiarities found in unphysical dimensions may impact real-world physics and enhance our understanding [3]. Here, we will explore the variation in dimension of the low-temperature behavior in the Edwards-Anderson (EA) spin glass model [4].A quantity of fundamental importance for the modeling of amorphous magnetic materials through spin glasses [5] is the "stiffness" exponent y [6,7]. As Hooke's law describes the response in increasing elastic energy imparted to a system for increasing displacement L from its equilibrium position, the stiffness of a spin configuration describes the typical rise in magnetic energy ∆E due to an induced defect-interface of size L. But unlike uniform systems with a convex potential energy function over its configuration space (say, a parabola for the sole variable in Hooke's law), an amorphous many-body system exhibits a function more reminiscent of a highdimensional mountain landscape [8]. Any defect-induced displacement of size L in such a complicated energy landscape may move a system through many ups-and-downs in energy ∆E. Averaging over many incarnations of such a system results in a typical energy scaleThe importance of this exponent for small excitations in disordered spin systems has been discussed in many contexts [5,6,7,9,10,11,12]. Spin systems with y > 0 provide resistance ("stiffness") against the spontaneous formation of defects at sufficiently low temperatures T ; an indication that a phase transition T c > 0 to an ordered state exists. For instance, in an Ising ferromagnet, the energy ∆E is always proportional to the size of the interface, i. e. y = d − 1, consistent with the fact that T c > 0 only when d > 1. When y ≤ 0, a system is unstable (such as the d = 1 ferromagnet) to spontaneous fluctuations which proliferate, preventing any ordered state. Thus, determining the "lower critical dimension" d l , where y d l = 0, is of significant importance [6,13,14,15], and understanding the mechanism leading to d l , however un-natural, provides definite clues to the origin of order [2]. For instance, in homogeneous systems with a continuous symmetry [16], such as the Heisenberg ferromagnet, the possibility of "soft modes" (Goldstone bosons) perpendicular to the direction of magnetization have proved to weaken order furt...

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Section: Dimension Dcontrasting

confidence: 77%

Stiffness of the Edwards-Anderson Model in all Dimensions

Boettcher

2005

Phys. Rev. Lett.

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“…Also the fact that d u = 6 is well supported by numerical results [14,15]. In 6d one determines with good accuracy mean field exponents (γ = 1, β = 1 and z = 4), with logarithmic corrections (that have been detected in the equilibrium simulations).…”

Section: Introductionsupporting

confidence: 63%

3D spin glass and 2D ferromagneticXYmodel: a comparison

Íñiguez¹,

Marinari²,

Parisi³

et al. 1997

J. Phys. A: Math. Gen.

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We compare the probability distributions and Binder cumulants of the overlap in the 3D Ising spin glass with those of the magnetization in the ferromagnetic 2D XY model. We analyze similarities and differences. Evidence for the existence of a phase transition in the spin glass model is obtained thanks to the crossing of the Binder cumulant. We show that the behavior of the XY model is fully compatible with the Kosterlitz-Thouless scenario. Finite size effects have to be dealt with by using great care in order to discern among two very different physical pictures that can look very similar if analyzed without large attention.

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“…We require the off-equilibrium correlation length ξ(t) [8,9,5] to be always much less than the sample size L. This requirement is quite easy to satisfy in a spin glass due to the high value of the dynamical exponent z: working in the frozen phase (T ≃ 3/4 T c ) is In the following we will use a perturbing field of intensity h 0 = 0.1.…”

Section: Site-diluted Ising Modelmentioning

confidence: 99%

“…The presence of a large number of metastable states in all the models does not invalidate the method, which succeed in predicting the right thermodynamical state. Once checked the validity of the method, we can use it to determine whether the frozen phase of the spin glass model in finite dimensions is better described by the mean-field like solution [4,2,8,5,9,10] or by the droplet model [11].…”

Section: Introductionmentioning

confidence: 99%

Generalized off-equilibrium fluctuation-dissipation relations in random Ising systems

1999

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We show that the numerical method based on the off-equilibrium fluctuation-dissipation relation does work and is very useful and powerful in the study of disordered systems which show a very slow dynamics. We have verified that it gives the right information in the known cases (diluted ferromagnets and random field Ising model far from the critical point) and we used it to obtain more convincing results on the frozen phase of finite-dimensional spin glasses. Moreover we used it to study the Griffiths phase of the diluted and the random field Ising models.

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Mean field dynamical exponents in finite-dimensional Ising spin glass

Cited by 28 publications

References 20 publications

Stiffness of the Edwards-Anderson Model in all Dimensions

Stiffness of the Edwards-Anderson Model in all Dimensions

3D spin glass and 2D ferromagneticXYmodel: a comparison

Generalized off-equilibrium fluctuation-dissipation relations in random Ising systems

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