Mean-field glassy phase of the random-field Ising model

Pastor, A. A.; Dobrosavljević, Vladimir; Horbach, M. L.

doi:10.1103/physrevb.66.014413

Cited by 26 publications

(28 citation statements)

References 58 publications

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“…Existing work has already established that the single-particle density of states, which represents the direct analogue of P (h) in this Letter, opens a power-law "Efros-Shklovskii" gap within the Coulomb glass phase [42][43][44]. Given our result that the vanishing of P (0) is a direct manifestation of SOC, our findings strongly suggest that in the presence of frustrating fully connected longrange Coulomb interactions, SOC may survive [45,46], even in physically-relevant space dimensions.…”

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confidence: 70%

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

et al. 2013

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We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established only for the fully-connected infiniterange Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034]. Here we study both avalanche and magnetization jump distributions triggered by an external magnetic field, as well as internal field distributions in the short-range Edwards-Anderson Ising spin glass for various space dimensions between 2 and 8, as well as the fixed-connectivity mean-field Viana-Bray model. Our numerical results, obtained on systems of unprecedented size, demonstrate that self-organized criticality is recovered only in the strict limit of a diverging number of neighbors, and is not a generic property of spin-glass models in finite space dimensions. PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, Self-organized criticality (SOC) refers to the tendency of large dissipative systems to drive themselves into a scaleinvariant critical state without any special parameter tuning [1,2]. These phenomena are of crucial importance because fractal objects displaying SOC are found everywhere [3], e.g., in earthquakes, in the structure of dried-out rivers, in the meandering of sea coasts, or in the structure of galactic clusters. Understanding its origin, however, represents a major unresolved puzzle because in most equilibrium systems critical behavior featuring scale-free (fractal) patterns is found only at isolated critical points and is not a generic feature across phase diagrams.Pioneering work in the 1980s provided insights into the possible origin of SOC by identifying a few theoretical examples that display it. The "sandpile" [4] and forest-fire models [5] are hallmark examples of dynamical systems that exhibit SOC. However, these models feature ad hoc dynamical rules, without showing how these can be obtained from an underlying Hamiltonian. Major questions thus remain: Can one obtain SOC from a Hamiltonian system, beyond invasion percolation [6,7]? Is this behavior a feature of high-dimensional models, models with a diverging number of neighbors and/or long-range interactions, or is it a generic property of a broad class of systems?Work in the 1990s offered a glint of hope. The first Hamiltonian model displaying SOC without any parameter tuning was studied in detail by Pazmandi et al. [8]: the infinite-range fully connected Sherrington-Kirkpatrick (SK) model [9]. Outof-equilibrium avalanches at zero temperature (T = 0) triggered by varying the magnetic field were numerically studied along the hysteresis loop. A distinct power-law behavior in the distribution of spin avalanches, as well as of the magnetization jumps, was established, i.e., SOC.The possible existence of SOC was also tested in several finite-dimensional models, but in all these cases, at least one parameter has to be tuned. The best-studied such model is the random-field Ising model where ferromagnetic Ising spins are coupled to a random field of ave...

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confidence: 70%

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

et al. 2013

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“…On the other hand, the random-field Ising model (RFIM) has received a great amount of interest [10][11][12][13][14] in the last few years because it helps to simulate many interesting but complicated problems. A dilute uniaxial two-sublattice antiferromagnet in a uniform magnetic field fits this model in that random local fields couple linearly to the antiferromagnetic order parameter [15].…”

Section: Introductionmentioning

confidence: 99%

Effect of the transverse field on the site-diluted Ising model in a longitudinal random field

Liang

Wei

Zhang

2008

Journal of Magnetism and Magnetic Materials

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“…The question we address in the following is whether a similar connection can be established for a more general class of disordered systems and for more general, non mean-field, conditions. Such a connection would be especially valuable in the case of the random field O(N ) model for which the existence of a spontaneous RSB phenomenon has been suggested 30,31,32,33 whereas recent FRG studies have shown that a cusp singularity in the field dependence of the cumulants of the renormalized random field appear below a critical dimension 25,26,27,28,34 . It would be interesting too for spin glasses.…”

Section: Introductionmentioning

confidence: 99%

“…involves a 2-point, 2-replica function, as in the Parisi solution of the SK model 2 . As developed in the theory of spin glasses and discussed more recently for the random field O(N ) model 30,31,32,33 , spontaneous RSB is signaled by an instability of the replica-symmetric solution which is characterized by the appearance of a zero eigenvalue in an appropriate stability operator. The natural framework to investigate such phenomena involving 2-point, 2-replica functions is the so-called 2-particle irreducible (2PI) formalism 35,36,37 : by introducing sources that couple not only to the fundamental fields but also to composite bilinear fields and then performing a double Legendre transform, one obtains a generating functional that has for argument both fields (magnetizations) and 2-point correlation functions.…”

Section: Introductionmentioning

confidence: 99%

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

Mouhanna

Tarjus²

2010

Phys. Rev. E

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We investigate the relation between spontaneous and explicit replica symmetry breaking in the theory of disordered systems. On general ground, we prove the equivalence between the replicon operator associated with the stability of the replica symmetric solution in the standard replica scheme and the operator signaling a breakdown of the solution with analytic field dependence in a scheme in which replica symmetry is explicitly broken by applied sources. This opens the possibility to study, via the recently developed functional renormalization group, unresolved questions related to spontaneous replica symmetry breaking and spin-glass behavior in finite-dimensional disordered systems.

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Mean-field glassy phase of the random-field Ising model

Cited by 26 publications

References 58 publications

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

Effect of the transverse field on the site-diluted Ising model in a longitudinal random field

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

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