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

Abstract

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...

Marginal Stability in Structural, Spin, and Electron Glasses

“…Instead, for short range interacting spin glasses, no such criticality was found [39,42], despite the fact that the static equilibrium response in small fields is theoretically expected to exhibit power law response, both in spin glasses [17,18] and random ferromagnets [49].…”

“…Even if the equilibrium states display marginality (e.g., in the form of continuous replica symmetry breaking) the corresponding out-of-equilibrium states might not share this property. This is suggested by the absence of crackling in short range spin glasses even in high dimensions d ≥ 8 and on random regular graphs [39] -both of which are believed to exhibit continuous replica symmetry breaking in equilibrium. In Sec.…”

“…of the pseudo-gap, implying that the constraint is just marginally satisfied with θ = 1 [39,43]. In Sec.…”

“…Avalanches in Ising spin glasses have been studied numerically along pseudo-adiabatic hysteresis loops where single spin flip updates in a T = 0 dynamics was considered, both in the SK model [14] and in short range models of finite connectivity in finite dimensions and on random lattices [39]. Spin glasses with continuous XY symmetry have been studied in the fully connected limit in Ref.…”

Chaos properties of the one-dimensional long-range Ising spin-glass

“…A6 and explanations in Appendix A), since the flipping of the spin corresponding to this lowest local field is known to be able to trigger an extensive avalanche [29,[62][63][64].…”