Distribution of metastable states of Ising spin glasses (original) (raw)
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Distribution of metastable states of spin glasses
Journal of Physics: Conference Series
The complex behavior of systems like spin glasses, proteins or neural networks is typically explained in terms of a rugged energy or fitness landscape. Within the highdimensional conformation space of these systems one finds features like barriers, saddle points, and metastable states whose number-at least in the case of spin glasses-grows exponentially with the size of the system. We propose a novel Monte Carlo sampling algorithm that employs an ensemble of short Markovian chains in order to visit all metastable states with equal probability. We apply this algorithm in order to measure the number of metastable states for the twodimensional and the three-dimensional Edwards-Anderson model and compare with theoretical predictions.
Multicanonical Study of the 3D Ising Spin Glass
We simulated the Edwards-Anderson Ising spin glass model in three dimensions via the recently proposed multicanonical ensemble. Physical quantities such as energy density, specific heat and entropy are evaluated at all temperatures. We studied their finite size scaling, as well as the zero temperature limit to explore the ground state properties.
Physical Review B, 2006
We study the three-dimensional (3D) bond-diluted Edwards-Anderson (EA) model with binary interactions at a bond occupation of 45% by Monte Carlo (MC) simulations. Using an efficient cluster MC algorithm we are able to determine the universal finite-size scaling (FSS) functions and the critical exponents with high statistical accuracy. We observe small corrections to scaling for the measured observables. The critical quantities and the FSS functions indicate clearly that the bond-diluted model for dilutions above the critical dilution p * , at which a spin glass (SG) phase appears, lies in the same universality class as the 3D undiluted EA model with binary interactions. A comparison with the FSS functions of the 3D site-diluted EA model with Gaussian interactions at a site occupation of 62.5% gives very strong evidence for the universality of the SG transition in the 3D EA model.
Local phase-space structure and low-temperature dynamics of short-range Ising spin glasses
Physical Review B, 1994
We study the phase-space geometry and the low-temperature relaxation of short range Gaussian Ising spin glasses, by finding all the configurations and their interconnection within a "pocket" of states surrounding a low-lying energy minimum. Thermalization within the pocket is modeled by a master equation, which is solved numerically, yielding the time-dependent propagator. The findings show that the relaxation proceeds by a series of local equilibrations in nested "valleys, " and support the dynamical predictions of hierarchical relaxation models.
Multi-overlap simulations of spin glasses
We present results of recent high-statistics Monte Carlo simulations of the Edwards-Anderson Ising spin-glass model in three and four dimensions. The study is based on a non-Boltzmann sampling technique, the multi-overlap algorithm which is specifically tailored for sampling rare-event states. We thus concentrate on those properties which are difficult to obtain with standard canonical Boltzmann sampling such as the free-energy barriers F q B in the probability density P J (q) of the Parisi overlap parameter q and the behaviour of the tails of the disorder averaged density P (q) = [P J (q)]av.
Lack of Ultrametricity in the Low-Temperature Phase of Three-Dimensional Ising Spin Glasses
Physical Review Letters, 2004
We study the low-temperature spin-glass phases of the Sherrington-Kirkpatrick (SK) model and of the 3-dimensional short-range Ising spin-glass (3DISG). By using clustering to focus on the relevant parts of phase space and reduce finite size effects, we found that for the SK model ultrametricity becomes clearer as the system size increases, while for the short-range case our results indicate the opposite, i.e., lack of ultrametricity. Another method, which does not rely on clustering, indicates that the mean-field solution works for the SK model but does not apply in detail to the 3DISG, for which stochastic stability is also violated.
arXiv Disordered Systems and Neural Networks, 2021
We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phenomenology known as "aging", as a diagnostic tool to find universal features that distinguish between the energy landscapes of Ising spin models and the ferromagnet. According to RD, a non-equilibrium system after a quench relies on fluctuations that randomly generate a sequence of irreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higher barriers of meta-stable states within a complex, hierarchical energy landscape. Once these record events allow the system to overcome such barriers, the system relaxes by tumbling into the following meta-stable state that is marginally more stable. Within this framework, a clear distinction can be drawn between the coarsening dynamics of an Ising ferromagnet and the aging of the spin glass, which are often put in the same category. To that end, we interpolate between the spin glass and ferromagnet by varying the admixture p of ferromagnetic over anti-ferromagnetic bonds from the glassy state (at 50% each) to wherever clear ferromagnetic behavior emerges. The accumulation of record events grows logarithmic with time in the glassy regime, with a sharp transition at a specific admixture into the ferromagnetic regime where such activations saturate quickly. We show this effect both for the Edwards-Anderson model on a cubic lattice as well as the Sherrington-Kirkpatrick (mean-field) spin glass. While this transition coincides with a previously observed zero-temperature equilibrium transition in the former, that transition has not yet been described for the latter.
Computational Complexity and Simulation of Rare Events of Ising Spin Glasses
Computing Research Repository, 2004
We discuss the computational complexity of random 2D Ising spin glasses, which represent an interesting class of constraint satisfaction problems for black box optimization. Two extremal cases are considered: (1) the ± J spin glass, and (2) the Gaussian spin glass. We also study a smooth transition between these two extremal cases. The computational complexity of all studied spin glass systems is found to be dominated by rare events of extremely hard spin glass samples. We show that complexity of all studied spin glass systems is closely related to Fréchet extremal value distribution. In a hybrid algorithm that combines the hierarchical Bayesian optimization algorithm (hBOA) with a deterministic bit-flip hill climber, the number of steps performed by both the global searcher (hBOA) and the local searcher follow Fréchet distributions. Nonetheless, unlike in methods based purely on local search, the parameters of these distributions confirm good scalability of hBOA with local search. We further argue that standard performance measures for optimization algorithms—such as the average number of evaluations until convergence—can be misleading. Finally, our results indicate that for highly multimodal constraint satisfaction problems, such as Ising spin glasses, recombination-based search can provide qualitatively better results than mutation-based search.
Ground-state properties of the three-dimensional Ising spin glass
Physical Review B, 1994
We study zero-temperature properties of the 3d Edwards-Anderson Ising spin glass on finite lattices up to size 12 3. Using multicanonical sampling we generate large numbers of groundstate configurations in thermal equilibrium. Finite size scaling with a zerotemperature scaling exponent y = 0.74 ± 0.12 describes the data well. Alternatively, a descriptions in terms of Parisi mean field behaviour is still possible. The two scenarios give significantly different predictions on lattices of size ≥ 12 3 .