Zero-temperature Random-field Ising Model Research Articles

Finite-dimensional signature of spinodal instability in an athermal hysteretic transition

We study the off-equilibrium critical phenomena across a hysteretic first-order transition in disordered athermal systems. The study focuses on the zero temperature random field Ising model (ZTRFIM) above the critical disorder for spatial dimensions $d=2,3,$ and $4$. We use Monte Carlo simulations to show that disorder suppresses critical slowing down in phase ordering time for finite-dimensional systems. The dynamic hysteresis scaling, the measure of explicit finite-time scaling, is used to subsequently quantify the critical slowing down. The scaling exponents in all dimensions increase with disorder strength and finally reach a stable value where the transformation is no longer critical. The associated critical behavior in the mean-field limit is very different, where the exponent values for various disorders in all dimensions are similar. The non-mean-field exponents asymptotically approach the mean-field value ($\Upsilon \approx 2/3$) with increase in dimensions. The results suggest that the critical features in the hysteretic metastable phase are controlled by inherent mean-field spinodal instability that gets blurred by disorder in low-dimension athermal systems.

Suppression of spinodal instability by disorder in an athermal system

We observed asymmetric critical slowing down and asymmetric dynamical scaling exponent in the superheating and supercooling kinetic processes during the thermally-induced metal-insulator transition of MnNiSn based Heusler alloy. During the transition to the insulator phase, the critical-like features get enhanced compared to the transition back to the metal phase. These experimental findings suggest that the metastable phase in the cooling branch of hysteresis has approached close to the spinodal instability. On the other hand, the extended disorder, generated over and above the intrinsic crystal defects during heating, triggers the excess heterogeneous nucleation before reaching the spinodal point. Zero-temperature random field Ising model (ZTRFIM) simulation, inscribed for the athermal martensitic transitions, supports the argument that the disorder smears the spinodal instabilities as the correlation length is bounded by the average distance between the disorder points.

Critical hysteresis on dilute triangular lattice.

Critical hysteresis in the zero-temperature random-field Ising model on a two-dimensional triangular lattice was studied earlier with site dilution on one sublattice. It was reported that criticality vanishes if less than one-third of the sublattice is occupied. This appears at variance with recently obtained exact solutions of the model on dilute Bethe lattices and prompts us to revisit the problem using an alternate numerical method. Contrary to our speculation that criticality may not be exactly zero below one-third dilution, the present study indicates it is nearly zero if approximately less than two-thirds of the sublattice is occupied. This suggests that hysteresis on dilute periodic lattices is qualitatively different from that on dilute Bethe lattices. Possible reasons are discussed briefly.

Hysteresis in the zero-temperature random-field Ising model on directed random graphs

We use zero-temperature Glauber dynamics to study hysteresis in the random-field Ising model on directed random graphs. The critical behavior of the model depends on the connectivity $z$ of the graph rather differently from that on undirected graphs. Directed graphs and zero-temperature dynamics are relevant to a wide class of social phenomena including opinion dynamics. We discuss the efficacy of increasing external influence in inducing a first-order phase transition in opinion dynamics. The numerical results are supported by an analytic solution of the model.

Threshold-induced correlations in the Random Field Ising Model

We present a numerical study of the correlations in the occurrence times of consecutive crackling noise events in the nonequilibrium zero-temperature Random Field Ising model in three dimensions. The critical behavior of the system is portrayed by the intermittent bursts of activity known as avalanches with scale-invariant properties which are power-law distributed. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations show that the observed correlations emerge upon applying a finite threshold to the pertaining signals when defining events of interest. Such events are called subavalanches and are obtained by separation of original avalanches in the thresholding process. The correlations are evidenced by power law distributed waiting times and are present in the system even when the original avalanche triggerings are described by a random uncorrelated process.

Crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model.

We present extensive numerical studies of the crossover from three-dimensional to two-dimensional systems in the nonequilibrium zero-temperature random-field Ising model with metastable dynamics. Bivariate finite-size scaling hypotheses are presented for systems with sizes L×L×l which explain the size-driven critical crossover from two dimensions (l=const, L→∞) to three dimensions (l∝L→∞). A model of effective critical disorder R_{c}^{eff}(l,L) with a unique fitting parameter and no free parameters in the R_{c}^{eff}(l,L→∞) limit is proposed, together with expressions for the scaling of avalanche distributions bringing important implications for related experimental data analysis, especially in the case of thin three-dimensional systems.

Critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model on a triangular lattice.

We present a numerical study of the critical behavior of the nonequilibrium zero-temperature random field Ising model in two dimensions on a triangular lattice. Our findings, based on the scaling analysis and collapse of data collected in extensive simulations of systems with linear sizes up to L=65536, show that the model is in a different universality class than the same model on a quadratic lattice, which is relevant for a better understanding of model universality and the analysis of experimental data.

Dynamic message-passing equations for models with unidirectional dynamics.

Understanding and quantifying the dynamics of disordered out-of-equilibrium models is an important problem in many branches of science. Using the dynamic cavity method on time trajectories, we construct a general procedure for deriving the dynamic message-passing equations for a large class of models with unidirectional dynamics, which includes the zero-temperature random-field Ising model, the susceptible-infected-recovered model, and rumor spreading models. We show that unidirectionality of the dynamics is the key ingredient that makes the problem solvable. These equations are applicable to single instances of the corresponding problems with arbitrary initial conditions and are asymptotically exact for problems defined on locally treelike graphs. When applied to real-world networks, they generically provide a good analytic approximation of the real dynamics.

Analysis of spanning avalanches in the two-dimensional nonequilibrium zero-temperature random-field Ising model

We present a numerical analysis of spanning avalanches in a two-dimensional (2D) nonequilibrium zero-temperature random field Ising model. Finite-size scaling analysis, performed for distribution of the average number of spanning avalanches per single run, spanning avalanche size distribution, average size of spanning avalanche, and contribution of spanning avalanches to magnetization jump, is augmented by analysis of spanning field (i.e., field triggering spanning avalanche), which enabled us to collapse averaged magnetization curves below critical disorder. Our study, based on extensive simulations of sufficiently large systems, reveals the dominant role of subcritical 2D-spanning avalanches in model behavior below and at the critical disorder. Other types of avalanches influence finite systems, but their contribution for large systems remains small or vanish.

Effect of coordination number on the nonequilibrium critical point

We study the nonequilibrium critical point of the zero-temperature random-field Ising model on a triangular lattice and compare it with known results on honeycomb, square, and simple cubic lattices. We suggest that the coordination number of the lattice rather than its dimension plays the key role in determining the universality class of the nonequilibrium critical behavior. This is discussed in the context of numerical evidence that equilibrium and nonequilibrium critical points of the zero-temperature random-field Ising model belong to the same universality class. The physics of this curious result is not fully understood.

Morphology transition at depinning in a solvable model of interface growth in a random medium

We propose a simple, exactly solvable, model of interface growth in a random medium that is a variant of the zero-temperature random-field Ising model on the Cayley tree. This model is shown to have a phase diagram (critical depinning field vs. disorder strength) qualitatively similar to that obtained numerically on the cubic lattice. We then introduce a specifically tailored random graph that allows an exact asymptotic analysis of the height and width of the interface. We characterize the change of morphology of the interface as a function of the disorder strength, a change that is found to take place at a multicritical point along the depinning-transition line.

Zero-temperature random-field Ising model on a bilayered Bethe lattice

The zero-temperature random-field Ising model is solved analytically for magnetization versus external field for a bilayered Bethe lattice. The mechanisms of infinite avalanches which are observed for small values of disorder are established. The effects of variable interlayer interaction strengths on infinite avalanches are investigated. The spin-field correlation length is calculated and its critical behavior is discussed. Direct Monte Carlo simulations of spin-flip dynamics are shown to support the analytical findings. We find, paradoxically, that a reduction of the interlayer bond strength can cause a phase transition from a regime with continuous magnetization reversal to a regime where magnetization exhibits a discontinuity associated with an infinite avalanche. This effect is understood in terms of the proposed mechanisms for the infinite avalanche.

Mechanisms of evolution of avalanches in regular graphs

A mapping of avalanches occurring in the zero-temperature random-field Ising model to life periods of a population experiencing immigration is established. Such a mapping allows the microscopic criteria for the occurrence of an infinite avalanche in a q-regular graph to be determined. A key factor for an avalanche of spin flips to become infinite is that it interacts in an optimal way with previously flipped spins. Based on these criteria, we explain why an infinite avalanche can occur in q-regular graphs only for q>3 and suggest that this criterion might be relevant for other systems. The generating function techniques developed for branching processes are applied to obtain analytical expressions for the durations, pulse shapes, and power spectra of the avalanches. The results show that only very long avalanches exhibit a significant degree of universality.

Avalanche distributions in the two-dimensional nonequilibrium zero-temperature random field Ising model

We present in detail the scaling analysis and data collapse of avalanche distributions and joint distributions that characterize the recently evidenced critical behavior of the two-dimensional nonequilibrium zero-temperature random field Ising model. The distributions are collected in extensive simulations of systems with linear sizes up to L=131072.

Zero-temperature random field Ising model on a Bethe lattice: Correlation functions along the hysteresis loop

We consider the Gaussian random field Ising model (RFIM) on the Bethe lattice at zero temperature in the presence of a uniform external field and derive the exact expressions of the two-point spin-spin and spin-random field correlation functions along the saturation hysteresis loop. To complete the analytical description and suggest possible approximations for the RFIM on Euclidian lattices we also compute the corresponding direct correlation functions (or proper vertices) and show that they decay rapidly with the distance in the weak-coupling/large-disorder regime; their range, however, is not limited to the nearest-neighbor distance.

Numerical Evidence for Critical Behavior of the Two-Dimensional Nonequilibrium Zero-Temperature Random Field Ising Model

We give numerical evidence that the two-dimensional nonequilibrium zero-temperature random field Ising model exhibits critical behavior. Our findings are based on the results of scaling analysis and collapsing of data, obtained in extensive simulations of systems with sizes sufficiently large to clearly display the critical behavior.

Numerical approach to metastable states in the zero-temperature random-field Ising model

We study numerically the number of single-spin-flip stable states in the $T=0$ random field Ising model on random regular graphs of connectivity $z=2$ and $z=4$ and on the cubic lattice. The annealed and quenched complexities (i.e., the entropy densities) of the metastable states with given magnetization are calculated as a function of the external magnetic field. The results show that the appearance of a (disorder-induced) out-of-equilibrium phase transition in the magnetization hysteresis loop at low disorder can be ascribed to a change in the distribution of the metastable states in the field-magnetization plane.

Seed size strongly affects cascades on random networks

The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions-these are typically continuous at low z and discontinuous at high z; here it is demonstrated that the low- z transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.

Hysteresis behavior of the random-field Ising model with 2-spin-flip dynamics: exact results on a Bethe lattice

We present an exact treatment of the hysteresis behavior of the zero-temperature random-field Ising model on a Bethe lattice when it is driven by an external field and evolved according to a 2-spin-flip dynamics. We focus on lattice connectivities z=2 (the one-dimensional chain) and z=3. For the latter case, we demonstrate the existence of an out-of-equilibrium phase transition, in contrast with the situation found with the standard 1-spin-flip dynamics. We discuss the influence of the degree of cooperativity of the (local) spin dynamics of the nonequilibrium response on the system.

History-induced critical behavior in disordered systems

Barkhausen noise as found in magnets is studied both with and without the presence of long-range (LR) demagnetizing fields using the non-equilibrium, zero-temperature random-field Ising model. Two distinct subloop behaviors arise and are shown to be in qualitative agreement with experiments on thin film magnets and soft ferromagnets. With LR fields present subloops resemble a self-organized critical system, while their absence results in subloops that reflect the critical point seen in the saturation loop as the system disorder is changed. In the former case, power law distributions of noise are found in subloops, while in the latter case history-induced critical scaling is studied in avalanche size distributions, spin-flip correlation functions, and finite-size scaling of the second moments of the size distributions. Results are presented for simulations of over 10^8 spins.