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arXiv:1403.2622 · Journal of Statistical Mechanics Theory and Experiment

The three-dimensional Ising spin glass in an external magnetic field: the role of the silent majority

Marco Baity‐Jesi, R. A. Baños, Andrés De la Cruz, L. A. Fernández, Jose Miguel Gil-Narvion, A. Gordillo-Guerrero, D. Íñiguez, A. Maiorano, Filippo Mantovani, Enzo Marinari, Víctor Martín-Mayor, Jorge Monforte-Garcia, A. Muñoz Sudupe, Denis Navarro, Giorgio Parisi A, Sergio Perez-Gaviro, Marcello Pivanti, Federico Ricci‐Tersenghi, Juan Jesus Ruiz-Lorenzo, Sebastiano Fabio Schifano, Beatriz Seoane, Alfonso Tarancón, R. Tripiccione, David Yllanes

Published May 16, 2014• 51 citations

Abstract

We perform equilibrium parallel-tempering simulations of the 3D Ising Edwards–Anderson spin glass in a field, using the Janus computer. A traditional analysis shows no signs of a phase transition. However, we encounter dramatic fluctuations in the behaviour of the model: averages over all the data only describe the behaviour of a small fraction of the data. Therefore, we develop a new approach to study the equilibrium behaviour of the system, by classifying the measurements as a function of a conditioning variate. We propose a finite-size scaling analysis based on the probability distribution function of the conditioning variate, which may accelerate the convergence to the thermodynamic limit. In this way, we find a non-trivial spectrum of behaviours, where some of the measurements behave as the average, while the majority show signs of scale invariance. As a result, we can estimate the temperature interval where the phase transition in a field ought to lie, if it exists. Although this would-be critical regime is unreachable with present resources, the numerical challenge is finally well posed.

DOI: 10.1088/1742-5468/2014/05/p05014

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We perform equilibrium parallel-tempering simulations of the 3D Ising Edwards–Anderson spin glass in a field, using the Janus computer. A traditional analysis shows no signs of a phase transition. However, we encounter dramatic fluctuations in the behaviour of the model: averages over all the data only describe the behaviour of a small fraction of the data. Therefore, we develop a new approach to study the equilibrium behaviour of the system, by classifying the measurements as a function of a conditioning variate. We propose a finite-size scaling analysis based on the probability distribution function of the conditioning variate, which may accelerate the convergence to the thermodynamic limit. In this way, we find a non-trivial spectrum of behaviours, where some of the measurements behave as the average, while the majority show signs of scale invariance. As a result, we can estimate the temperature interval where the phase transition in a field ought to lie, if it exists. Although this would-be critical regime is unreachable with present resources, the numerical challenge is finally well posed.

We present a massive equilibrium simulation of the three-dimensional Ising spin glass at low temperatures. The Janus special-purpose computer has allowed us to equilibrate, using parallel tempering, L = 32 lattices down to T ≈ 0.64 T c . We demonstrate the relevance of equilibrium finite size simulations to understanding experimental non-equilibrium spin glasses in the thermodynamical limit by establishing a time-length dictionary. We conclude that non-equilibrium experiments performed on a timescale of 1 h can be matched with equilibrium results on L ≈ 110 lattices. A detailed investigation of the probability distribution functions of the spin and link overlap, as well as of their correlation functions, shows that Replica Symmetry Breaking is the appropriate theoretical framework for the physically relevant length scales. Besides, we improve over existing methodologies in ensuring equilibration in parallel tempering simulations.

Using the decay of the out equilibrium spin-spin correlation function we compute the equilibrium Edward-Anderson order parameter in the three dimensional binary Ising spin glass in the spin glass phase. We have checked that the Edward-Anderson order parameter computed from out of equilibrium numerical simulations follows with good precision the critical law as determined in experiments and in numerical studies at equilibrium. We have also studied the dependence of the order parameter with the lattice size. Finally we present a large time study of the scaling of the off-equilibrium fluctuation-dissipation relations.

The three-dimensional ± J Ising spin glass model is investigated by means of Monte Carlo simulation. In uniform fields of various strength, the Edwards-Anderson susceptibility χ EA Q is calculated at T =0.9 J which is lower than the transition temperature in the zero-field case. It is found that χ EA decreases monotonically when the field strength is increased and that the data do not suggest a phase transition in a finite field. The best parameters for the scaling function of the form

\chi _{\rm EA}(H)\simeq l^{\psi}\tilde{\chi}_{\rm EA}((H-H_{\rm c})l^{-\phi})

are estimated to be H c =0.00(10) J , ψ=2.16(22) and φ=-1.13(6). These results support the hypothesis that the phase transition does not exist in a finite magnetic field. \sloppy

We study the off-equilibrium dynamics of the three-dimensional Ising spin glass in the presence of an external magnetic field. We have performed simulations both at fixed temperature and with an annealing protocol. Thanks to the Janus special-purpose computer, based on field-programmable gate array (FPGAs), we have been able to reach times equivalent to 0.01 s in experiments. We have studied the system relaxation both for high and for low temperatures, clearly identifying a dynamical transition point. This dynamical temperature is strictly positive and depends on the external applied magnetic field. We discuss different possibilities for the underlying physics, which include a thermodynamical spin-glass transition, a mode-coupling crossover, or an interpretation reminiscent of the random first-order picture of structural glasses.

We present new numerical simulations of the random field Ising model in three dimensions at zero temperature. The critical exponents are found to agree with previous results. We study the magnetic susceptibility by applying a small magnetic field perturbation. We find the critical amplitude ratio of the magnetic susceptibilities to be very large, equal to 233.1 ± 1.5. We find strong sample to sample fluctuations which obey finite size scaling. The probability distribution of the size of small energy excitations is maximally non-self-averaging, obeying a double peak distribution, and is finite size scaling invariant. We also study the approach to the thermodynamic limit of the ground state magnetization at the phase transition.

We present numerical simulations of the four-dimensional Edwards Anderson Ising spin glass with binary couplings. Our results, in the midst of strong finite-size effects, suggest the existence of a spin-glass phase transition. We present a preliminar determination of critical exponents. We discuss spin-glass susceptibilities, cumulants of the overlap and energy overlap probability distributions, finite-size effects, and the behaviour of the disorder-dependent and integrated probability distributions.