Three-dimensional Random-field Ising Model Research Articles

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

Heterogeneous nucleation in the random field Ising model.

We investigate the nucleation dynamics of the three-dimensional random field Ising model under an external field. We use umbrella sampling to compute the free-energy cost of a critical nucleus and use forward flux sampling for the direct estimation of nucleation rates. For moderate to strong disorder, our results indicate that the size of the nucleating cluster is not a good reaction coordinate, contrary to the pure Ising model. We rectify this problem by introducing a coordinate that also accounts for the location of the nucleus. Using the free energy barrier to predict the nucleation rate, we find reasonable agreement, although deviations become stronger as disorder increases. We attribute this effect to cluster shape fluctuations. We also discuss finite-size effects on the nucleation rate.

Spin activity correlations in driven disordered systems

We study the spatio-temporal correlations in the spin flipping activities of the disordered systems realized within the frame of the field-driven three-dimensional zero-temperature nonequilibrium random field Ising model. Our results for the spatial activity correlations reveal that the rate-dependent scaling holds in the full range of rate regimes provided that the system parameters satisfy the finite-size and rate-dependent scaling conditions. Temporal activity correlations show high sensitivity to the driving, allowing the collapsing of waiting time distributions only in the limit of slow driving rates. These findings, based on extensive numerical simulations of large systems, reliably demonstrate the observed behavior and could be relevant for studies on various, particularly experimental, nonequilibrium systems driven at finite rates.

The effect of finite driving rate on avalanche distributions

We study distributions of spin avalanches in the three-dimensional nonequilibrium zero temperature random field Ising model driven by external magnetic field that is increased at constant rate. Our results show that avalanche distributions are affected by the driving regime and although the avalanche properties (e.g. size) remain power-law distributed above critical disorder, the scaling exponents vary with the driving rate. For slow driving, the integral distributions of nonspanning avalanches are almost the same as in the adiabatic and quasi-static driving regime and therefore described by constant values of power-law exponents. Owing to temporal overlapping and merging of avalanches, these distributions become markedly different with the increased driving rate which causes an increase of exponent values in the range of moderate driving rates and their saturation for fast driving. The integral distributions of all avalanches behave similarly but without the saturation of exponents for fast driving. Our results, based on extensive numerical simulations, could be relevant for studies of variety of systems in finite driving regime, particularly those with experimental realizations that are not easily attainable.

Aging in the three-dimensional random-field Ising model.

We studied the nonequilibrium aging behavior of the random-field Ising model in three dimensions for various values of the disorder strength. This allowed us to investigate how the aging behavior changes across the ferromagnetic-paramagnetic phase transition. We investigated a large system size of N=256^{3} spins and up to 10^{8} Monte Carlo sweeps. To reach these necessary long simulation times, we employed an implementation running on Intel Xeon Phi coprocessors, reaching single-spin-flip times as short as 6ps. We measured typical correlation functions in space and time to extract a growing length scale and corresponding exponents.

Revisiting the scaling of the specific heat of the three-dimensional random-field Ising model

We revisit the scaling behavior of the specific heat of the three-dimensional random-field Ising model with a Gaussian distribution of the disorder. Exact ground states of the model are obtained using graph-theoretical algorithms for different strengths 𝒩 = 268 3 spins. By numerically differentiating the bond energy with respect to h, a specific-heat-like quantity is obtained whose maximum is found to converge to a constant in the thermodynamic limit. Compared to a previous study following the same approach, we have studied here much larger system sizes with an increased statistical accuracy. We discuss the relevance of our results under the prism of a modified Rushbrooke inequality for the case of a saturating specific heat. Finally, as a byproduct of our analysis, we provide high-accuracy estimates of the critical field h c = 2.279(7) and the critical exponent of the correlation exponent ν = 1.37(1), in excellent agreement to the most recent computations in the literature.

Multifractal analysis of Barkhausen noise reveals the dynamic nature of criticality at hysteresis loop

The field-driven magnetisation reversal processes in disordered systems exhibit a collective behaviour that is manifested in the scale-invariance of avalanches, closely related to underlying dynamical mechanisms. Using the multifractal time series analysis, we study the structure of fluctuations at different scales in the accompanying Barkhausen noise. The stochastic signal represents the magnetisation discontinuities along the hysteresis loop of a three-dimensional random field Ising model simulated for varied disorder strength and driving rates. The analysis of the spectrum of the generalised Hurst exponents reveals that the dominant segments of the signal with large fluctuations represent two distinct classes of stochastic processes in weak and strong pinning regimes. Furthermore, in the weak pinning regime, the part of the signal originating from the beginning of the hysteresis loop has a different multifractal spectrum than the signal near the coercive field. The enhanced fluctuations (primarily in the central part of the hysteresis loop) for increased driving rate and larger system size, lead to a further broadening of the spectrum. The analysed Barkhausen signals are also shown to exhibit temporal correlations and power-law distributions of the magnetisation discontinuity and avalanche sizes, in agreement with previous studies. The multifractal properties of Barkhausen noise describe the dynamical state of domains and precisely discriminate the weak pinning, permitting the motion of individual walls, from the mechanisms occurring in strongly disordered systems.

Random-field Ising model: Insight from zero-temperature simulations

We enlighten some critical aspects of the three-dimensional ($d=3$) random-field Ising model from simulations performed at zero temperature. We consider two different, in terms of the field distribution, versions of model, namely a Gaussian random-field Ising model and an equal-weight trimodal random-field Ising model. By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes $\mathcal{V}=L\times L\times L$, where $L$ denotes linear lattice size and $L_{\rm max}=156$. Using as finite-size measures the sample-to-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field $h_{\rm c}$ and the critical exponent $\nu$ of the correlation length, the latter clearly suggesting that both models share the same universality class. Additional simulations of the Gaussian random-field Ising model at the best-known value of the critical field provide the magnetic exponent ratio $\beta/\nu$ with high accuracy and clear out the controversial issue of the critical exponent $\alpha$ of the specific heat. Finally, we discuss the infinite-limit size extrapolation of energy- and order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well as the critical slowing down aspects of the algorithm.

Critical aspects of the random-field Ising model

We investigate the critical behavior of the three-dimensional random-field Ising model (RFIM) with a Gaussian field distribution at zero temperature. By implementing a computational approach that maps the ground-state of the RFIM to the maximum-flow optimization problem of a network, we simulate large ensembles of disorder realizations of the model for a broad range of values of the disorder strength h and system sizes 𝒱 = L 3, with L ≤ 156. Our averaging procedure outcomes previous studies of the model, increasing the sampling of ground states by a factor of 103. Using well-established finite-size scaling schemes, the fourth-order’s Binder cumulant, and the sample-to-sample fluctuations of various thermodynamic quantities, we provide high-accuracy estimates for the critical field h c, as well as the critical exponents ν, β/ν, and γ/ν of the correlation length, order parameter, and disconnected susceptibility, respectively. Moreover, using properly defined noise to signal ratios, we depict the variation of the self-averaging property of the model, by crossing the phase boundary into the ordered phase. Finally, we discuss the controversial issue of the specific heat based on a scaling analysis of the bond energy, providing evidence that its critical exponent α ≈ 0−.

Nonequilibrium phase transitions and stationary-state solutions of a three-dimensional random-field Ising model under a time-dependent periodic external field

Nonequilibrium behavior and dynamic phase-transition properties of a kinetic Ising model under the influence of periodically oscillating random fields have been analyzed within the framework of effective-field theory based on a decoupling approximation. A dynamic equation of motion has been solved for a simple-cubic lattice (q=6) by utilizing a Glauber-type stochastic process. Amplitude of the sinusoidally oscillating magnetic field is randomly distributed on the lattice sites according to bimodal and trimodal distribution functions. For a bimodal type of amplitude distribution, it is found in the high-frequency regime that the dynamic phase diagrams of the system in the temperature versus field amplitude plane resemble the corresponding phase diagrams of the pure kinetic Ising model. Our numerical results indicate that for a bimodal distribution, both in the low- and high-frequency regimes, the dynamic phase diagrams always exhibit a coexistence region in which the stationary state (ferro or para) of the system is completely dependent on the initial conditions, whereas for a trimodal distribution, the coexistence region disappears depending on the values of the system parameters.

Dynamical Monte Carlo studies of the three-dimensional bimodal random-field Isingmodel

The nature of the three-dimensional random-field Ising model with a bimodal probabilitydistribution is investigated using finite-time scaling combined with the Monte Carlorenormalization group method, in the presence of a linearly varying temperature. Our resultssupport the existence of a first-order phase transition for this model, obtained throughreducing the influence of finite-size effects. The critical exponents are estimated to beν = 0.74(3), β = 0.27(1), α = − 0.023(5), andγ = 1.48(3) with corrections to the scaling. The Rushbrooke scaling law is satisfied with theseexponents, and in turn identifies the asymptotic critical behavior.

Effective-field-theory analysis of the three-dimensional random-field Ising model on isometric lattices

An Ising model with quenched random magnetic fields is examined for single-Gaussian, bimodal, and double-Gaussian random-field distributions by introducing an effective-field approximation that takes into account the correlations between different spins that emerge when expanding the identities. Random-field distribution shape dependencies of the phase diagrams and magnetization curves are investigated for simple cubic, body-centered-cubic, and face-centered-cubic lattices. The conditions for the occurrence of reentrant behavior and tricritical points on the system are also discussed in detail.

Scaling and self-averaging in the three-dimensional random-field Ising model

We investigate, by means of extensive Monte Carlo simulations, the magnetic critical behavior of the three-dimensional bimodal random-field Ising model at the strong disorder regime. We present results in favor of the two-exponent scaling scenario, $\bar{\eta}$ = 2η, where η and $\bar{\eta}$ are the critical exponents describing the power-law decay of the connected and disconnected correlation functions and we illustrate, using various finite-size measures and properly defined noise to signal ratios, the strong violation of self-averaging of the model in the ordered phase.

Three-dimensional random-field Ising model phase transition in virginSr0.4Ba0.6Nb2O6: Overcoming aging

Isothermal aging above the ferroelectric phase-transition temperature, ${T}_{\text{c}}\ensuremath{\approx}410\text{ }\text{K}$, causes the charge-disordered uniaxial crystal ${\text{Sr}}_{0.40}{\text{Ba}}_{0.60}{\text{Nb}}_{2}{\text{O}}_{6}$ to become trapped in metastable nanopolar states at any finite wait time. However, backward extrapolation of the isothermal relaxation of its axial linear susceptibility ${\ensuremath{\chi}}^{\ensuremath{'}}$ under various frequency, time, and temperature protocols allows disclosing its unaged virgin state. In this limit it is supposed to undergo a sharp weakly first-order transition within the framework of the three-dimensional random-field Ising model.

Low-energy excitations in the three-dimensional random-field Ising model

The random-field Ising model (RFIM), one of the basic models for quenched disorder, can be studied numerically with the help of efficient ground-state algorithms. In this study, we extend these algorithm by various methods in order to analyze low-energy excitations for the three-dimensional RFIM with Gaussian distributed disorder that appear in the form of clusters of connected spins. We analyze several properties of these clusters. Our results support the validity of the droplet-model description for the RFIM.

Work distributions in theT=0random field Ising model

We perform a numerical study of the three-dimensional random-field Ising model at T=0. We compare work distributions along metastable trajectories obtained with the single-spin flip dynamics with the distribution of the internal energy change along equilibrium trajectories. The goal is to investigate the possibility of extending the Crooks fluctuation theorem [G. E. Crooks, Phys. Rev. E. 60, 2721 (1999)] to zero temperature when, instead of the standard ensemble statistics, one considers the ensemble generated by the quenched disorder. We show that a simple extension of Crooks fails close to the disordered induced equilibrium phase transition due to the fact that work and internal energy distributions are very asymmetric.

Wang–Landau study of the critical behavior of the bimodal 3D random field Ising model

We apply the Wang–Landau method to the study of the critical behavior of the three-dimensional random field Ising model with a bimodal probability distribution. For high values of the random field intensity we find that the energy probability distribution at the transition temperature is double peaked, suggesting that the phase transition is of first order. On the other hand, the transition looks continuous for low values of the field intensity. In spite of the large sample to sample fluctuations observed, the double peak in the probability distribution is always present for high fields.

On the nature of the phase transition in the three-dimensional random field Isingmodel

A brief survey of the theoretical, numerical and experimental studies of the random fieldIsing model (RFIM) during the last three decades is given. The nature of the phasetransition in the three-dimensional RFIM with Gaussian random fields is discussed. Usingsimple scaling arguments it is shown that if the strength of the random fields is nottoo small (bigger than a certain threshold value), the finite temperature phasetransition in this system is equivalent to the low temperature order–disordertransition which takes place with variations of the strength of the random fields.A detailed study of the zero-temperature phase transition in terms of simpleprobabilistic arguments and a modified mean field approach (which take intoaccount nearest neighbor spin–spin correlations) is given. It is shown that if allthermally activated processes are suppressed, the ferromagnetic order parameterm(h) as a functionof the strength h of the random fields becomes history dependent. In particular, the behavior of the magnetization curvesm(h) for increasingand decreasing h reveals a hysteresis loop.

Evolution of nanodomains in the uniaxial relaxor Sr/sub 0.61/Ba/sub 0.39/Nb/sub 2/O/sub 6/:Ce

The evolution of the nanodomain pattern of the uniaxial relaxor ferroelectric strontium barium niobate doped with cerium was studied by piezoresponse force microscopy (PFM). The fractal-like nanodomains observed at room temperature decay on heating. At temperatures up to about 15 K above the Curie temperature, Tc = 320 K, areas of correlated polarization are still visible. On cooling from the paraelectric state to below Tc, a slow isothermal growth of nanodomain was found. The mean domain size increases according to a logarithmic law as predicted for the three-dimensional random field Ising model.

Quasistationary criticality of the order parameter of the three-dimensional random-field Ising antiferromagnetFe0.85Zn0.15F2: A synchrotron x-ray scattering study

The critical exponent beta =0.17(1) for the three-dimensional random-field Ising model (RFIM) order parameter upon zero-field cooling (ZFC) has been determined using extinction-free magnetic x-ray scattering techniques for Fe(0.85)Zn(0.15)F2. This result is consistent with other exponents determined for the RFIM in that Rushbrooke scaling is satisfied. Nevertheless, there is poor agreement with equilibrium computer simulations, and the ZFC results do not agree with field-cooling (FC) results. We present details of hysteresis in Bragg scattering amplitudes and line shapes that help elucidate the effects of thermal cycling in the RFIM, as realized in dilute antiferromagnets in an applied field. We show that the ZFC critical-like behavior is consistent with a second-order phase transitions, albeit quasi-stationary rather than truly equilibrium in nature, as evident from the large thermal hysteresis observed near the transition.