Critical exponents in Ising spin glasses

Bernardi, L. W.; Campbell, I. A.

doi:10.1103/physrevb.56.5271

Cited by 27 publications

(31 citation statements)

References 22 publications

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“…43 This is in total contrast to the situation for the Bimodal ISG where at d = 5 (so again at ǫ = 1) the FT sum to third order in ǫ gives η(d = 5) = 1.6897, strikingly different from the high temperature series value 12 η(d = 5) = −0.38(7) and the simulation estimate η(d = 5) = −0.39 (2). 1 This implies that a sum including many further terms (oscillating in sign) would be needed to finally obtain stable and accu-rate FT predictions. In practice, establishing such a sum seems entirely ruled out, but the question remains open as to whether the necessary quasi-cancellations among the unknown higher order FT terms could depend on parameters such as the lattice structure or the form of the interactions.…”

Section: Discussioncontrasting

confidence: 39%

“…We have studied numerically critical dynamic behaviour for ISGs having near neighbour interactions in dimensions d = 3 and 4 with Bimodal, Gaussian, or Laplacian distributions of the couplings. 1 We find that the dynamic exponents vary strongly and systematically from one distribution to another. One possible explanation for this could be that for ISGs the universality class depends on the form of the interaction distribution.…”

Section: Introductionmentioning

confidence: 95%

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Dynamic critical behavior in Ising spin glasses

Pleimling

Campbell

2005

Phys. Rev. B

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The critical dynamics of Ising spin glasses with Bimodal, Gaussian, and Laplacian interaction distributions are studied numerically in dimensions 3 and 4. The data demonstrate that in both dimensions the critical dynamic exponent zc, the non-equilibrium autocorrelation decay exponent λc/zc, and the critical fluctuation-dissipation ratio X∞ all vary strongly and systematically with the form of the interaction distribution.

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Section: Discussioncontrasting

confidence: 39%

Section: Introductionmentioning

confidence: 95%

Dynamic critical behavior in Ising spin glasses

Pleimling

Campbell

2005

Phys. Rev. B

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“…The fit has a zero at d l ≈ 2.4986 and yields y 5/2 ≈ 0.0008; strong evidence that d l = 5/2. This value can be corroborated by a similar but independent fit of the existing data for T g [28,29,30]. A speculative calculation based on meanfield arguments [31], recently put on a more rigorous basis [32], has predicted the disappearance of replica symmetry breaking (RSB) [33], taken to be a characteristic of low-T glassy order, for dimensions d < d l = 5/2 exactly.…”

supporting

confidence: 58%

“…(4) fails at d = 1, and its linear form misses the essential features of the data in Fig. 1, but it provides a decent estimate for d = 2 and 3, and predicts d To this end, we have gathered the latest values for T g (d) (using ±J bonds) from the literature, based on simulations [28,29,43] or high-T series expansions [30,44]. We …”

Section: Dimension Dmentioning

confidence: 99%

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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“…Recent extensive MC studies 25,26 have estimated the critical temperature to be 2.0J. The critical exponent ν of the diverging coherence length is also obtained as ν ∼ 0.9 − 1.0 25,26,27 . Another important exponent associated with T = 0 glassy fixed point is the stiffness exponent θ whose value is also obtained around 0.7 by MC simulation 25 and ground state calculation 28 .…”

Section: Model and Simulation Methodsmentioning

confidence: 98%

Extended droplet theory for aging in short-range spin glasses and a numerical examination

2002

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We analyze isothermal aging of a four dimensional Edwards-Anderson model in detail by Monte Carlo simulations. We analyze the data in the view of an extended version of the droplet theory proposed recently (cond-mat/0202110) which is based on the original droplet theory plus conjectures on the anomalously soft droplets in the presence of domain walls. We found that the scaling laws including some fundamental predictions of the original droplet theory explain well our results. The results of our simulation strongly suggest the separation of the breaking of the time translational invariance and the fluctuation dissipation theorem in agreement with our scenario.

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Critical exponents in Ising spin glasses

Cited by 27 publications

References 22 publications

Dynamic critical behavior in Ising spin glasses

Dynamic critical behavior in Ising spin glasses

Stiffness of the Edwards-Anderson Model in all Dimensions

Extended droplet theory for aging in short-range spin glasses and a numerical examination

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