Gaussian Random-field Ising Model Research Articles

Nonsteady dynamics at the dynamic depinning transition in the two-dimensional Gaussian random-field Ising model.

With large-scale Monte Carlo simulations, we investigate the nonsteady relaxation at the dynamic depinning transition in the two-dimensional Gaussian random-field Ising model. The dynamic scaling behavior is carefully analyzed, and the transition fields as well as static and dynamic exponents are accurately determined based on the short-time dynamic scaling form. Different from the usual assumption, two distinguished growth processes of spatial correlation lengths for the velocity and height of the domain wall are found. Thus, the universality class of the depinning transition is established, which significantly differs from that of the quenched disorder equationbut agrees with that of the recent experiment as well as other simulations works. Under the influence of the mesoscopic time regime, the crossover from the second-order phase transition to the first-order one is confirmed in the weak-disorder regime, yielding an abnormal disorder-dependent nature of the criticality.

Geometrical control of disorder-induced magnetic domains in planar synthetic antiferromagnets

Magnetic domains play a fundamental role in magnetization processes. However, unlike in ferromagnets (FMs), the formation of domains in antiferromagnets (AFMs) is poorly understood because they are not favored by magnetostatics and are difficult to detect experimentally. In this paper, we create a synthetic planar AFM with tunable lateral coupling between neighboring FM regions to establish the role played by magnetic disorder in the formation of AFM domains. By directly imaging the synthetic AFM in real space, we observe that the AFM lattice spontaneously breaks up into domains following ac demagnetization. These AFM domains nucleate and pin at locally disordered sites that define their size and shape, which is explained with the help of a Gaussian random field Ising model. Furthermore, we can manipulate the AFM domain morphology by varying the interaction strength, which can be tuned with the geometrical parameters of the synthetic AFM.

Absence of replica symmetry breaking in disordered FKG-Ising models under uniform field

We prove that the variance of a spin overlap vanishes in disordered Ising models satisfying the Fortuin–Kasteleyn–Ginibre inequality under a uniform field, such as the generally distributed random field Ising model and site- and bond-diluted Ising models with the Bernoulli distribution. Chatterjee’s proof for the Gaussian random field Ising model is generalized to another independent identically distributed quenched disorder under a uniform field.

Universality in four-dimensional random-field magnets

We investigate the universality aspects of the four-dimensional random-field Ising model (RFIM) using numerical simulations at zero temperature. We consider two different, in terms of the field distribution, versions of the model, namely a Gaussian RFIM and an equal-weight trimodal RFIM. 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 system sizes. Using as finite-size measures the sample-to-sample fluctuations of the order parameter of the system, we propose, for both types of distributions, estimates of the critical field h c and the critical exponent ν of the correlation length, the latter suggesting that the two models in four dimensions share the same universality class.

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.

Universality aspects of the trimodal random-field Ising model

We investigate the critical properties of the d = 3 random-field Ising model with an equal-weight trimodal distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we compute large ensembles of ground states for several values of the disorder strength h and system sizes up to N = 1283. Using a new approach based on the sample-to-sample fluctuations of the order parameter of the system and proper finite-size scaling techniques we estimate the critical disorder strength h c = 2.747(3) and the critical exponents of the correlation length ν = 1.34(6) and order parameter β = 0.016(4). These estimates place the model into the universality class of the corresponding Gaussian random-field Ising model.

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.

Hysteresis in theT=0random-field Ising model: Beyond metastable dynamics

We present a numerical study of the zero-temperature response of the Gaussian random-field Ising model to a slowly varying external field, allowing the system to be trapped in microscopic configurations that are not fully metastable. This modification of the standard single-spin-flip dynamics results in an increase in dissipation (hysteresis) somewhat similar to that observed with a finite driving rate. We then study the distribution of avalanches along the hysteresis loop and perform a finite-size scaling analysis that shows good evidence that the critical exponents associated to the disorder-induced phase transition are not modified.

Hysteresis and avalanches in theT=0random-field Ising model with two-spin-flip dynamics

We study the non-equilibrium behavior of the three-dimensional Gaussian random-field Ising model at T=0 in the presence of a uniform external field using a 2-spin-flip dynamics. The deterministic, history-dependent evolution of the system is compared with the one obtained with the standard 1-spin-flip dynamics used in previous studies of the model. The change in the dynamics yields a significant suppression of coercivity, but the distribution of avalanches (in number and size) stays remarkably similar, except for the largest ones that are responsible for the jump in the saturation magnetization curve at low disorder in the thermodynamic limit. By performing a finite-size scaling study, we find strong evidence that the change in the dynamics does not modify the universality class of the disorder-induced phase transition.

Spanning avalanches in the three-dimensional Gaussian random-field Ising model with metastable dynamics: Field dependence and geometrical properties

Spanning avalanches in the 3D Gaussian Random Field Ising Model (3D-GRFIM) with metastable dynamics at $T=0$ have been studied. Statistical analysis of the field values for which avalanches occur has enabled a Finite-Size Scaling (FSS) study of the avalanche density to be performed. Furthermore, a direct measurement of the geometrical properties of the avalanches has confirmed an earlier hypothesis that several types of spanning avalanches with two different fractal dimensions coexist at the critical point. We finally compare the phase diagram of the 3D-GRFIM with metastable dynamics with the same model in equilibrium at $T=0$.

Statistical properties of pinning fields in the 3d-Gaussian RFIM

We have defined pinning fields as those random fields that keep some of the magnetic moments unreversed in the region of negative external applied field during the demagnetizing process. An analysis of the statistical properties of such pinning fields is presented within the context of the Gaussian Random Field Ising Model. We show that the average of the pinning fields exhibits a drastic increase close to the coercive field and that such an increase is discontinuous for low degrees of disorder. This behaviour can be described with standard finite-size scaling assumptions. Furthermore, we also show that the pinning fields corresponding to states close to coercivity exhibit strong statistical correlations.

Finite-size scaling analysis of the avalanches in the three-dimensional Gaussian random-field Ising model with metastable dynamics

A numerical study is presented of the 3d Gaussian Random Field Ising Model at T=0 driven by an external field. Standard synchronous relaxation dynamics is employed to obtain the magnetization versus field hysteresis loops. The focus is on the analysis of the number and size distribution of the magnetization avalanches. They are classified as being non-spanning, 1d-spanning, 2d-spanning or 3d-spanning depending on whether or not they span the whole lattice in the different space directions. Moreover, finite-size scaling analysis enables identification of two different types of non-spanning avalanches (critical and supercritical) and two different types of 3d-spanning avalanches (critical and subcritical), whose numbers increase with L as a power-law with different exponents. We conclude by giving a scenario for the avalanches behaviour in the thermodynamic limit.

“Avalanches” in the ground state of the 3D Gaussian random field Ising model driven by an external field

Abstract We present a numerical study of the exact ground states of the 3D Gaussian random field Ising model (G-RFIM) with an applied external field B . We combine a max-flow min-cut algorithm with an optimal procedure for determining all the ground states when B is swept from −∞ to ∞. The magnetization of finite lattices ( L 3 ) is studied as a function of the degree of disorder in the system σ (standard deviation of the Gaussian random fields). The magnetization evolves as a sequence of jumps or “avalanches” with a certain size s . The statistical distribution p ( s ) becomes a power law p ( s )∼ s − τ for a certain degree of disorder σ c ( L ). The extrapolation of the results to L →∞ renders σ c ≃2.4±0.1 and τ ≃1.70±0.07.

Phase diagram of the three-dimensional Gaussian random field Ising model: A Monte Carlo renormalization-group study

With the help of the replica exchange Monte Carlo method and the improved Monte Carlo renormalization-group scheme, we investigate over a wide area in the phase diagram of the Gaussian random field Ising model on the simple cubic lattice. We found that the phase transition at a weak random field belongs to the same universality class as the zero-temperature phase transition. We also present a possible scenario for the replica symmetry breaking transition.

Efficient Algorithm for Finding Ground-States in the Random Field Ising Model with an External Field

We present an efficient algorithm that, combined with a max-flow, min-cut minimization algorithm, makes it possible to find the ground states of the Gaussian Random Field Ising model when the external applied field B is continuously varied from −∞ to +∞. The algorithm exactly finds all the possible ground states and their limiting range (Bmin, Bmax). Examples of the dependence of the magnetization and energy with B are shown for the 2d-RFIM.

Gaussian random field p-spin-interaction ising model in a transverse field

The Gaussian random field Ising model with p-spin interactions in the presence of a transverse field is studied by combining the Suzuki-Trotter approach and the thermodynamic perturbation theory. The first-order phase transitions are found in the limit p → ∞, in contrast to the cases with p=2.