Efficient algorithms for computing ground states of the 2D random-field Ising model

We discuss some aspects of the problem of the equivalence of dilute antiferromagnets and random field Ising models. We first investigate for dilute antiferromagnets the validity of the arguments of Imry and Ma. It turns out that they are applicable, but some care is required concerning the role played by the so-called internal Peierls contours. Next we consider a hierarchical version of a dilute antiferromagnetic Ising model in the presence of a uniform magnetic field and show that a renormalization group transformation maps it exactly into a hierarchical version of the random field Ising model, thus proving their equivalence as far as the critical behavior is concerned. In particular this implies that phase transition with spontaneous magnetization occurs only for dimensiond>2. Finally we show that in the absence of internal Peierls contours both models, in their hierarchical versions, exhibit phase transition already in dimensiond=2.

We consider geometrical or Ising clusters (i.e., domains of parallel spins) in the square lattice random-field Ising model by varying the strength of the Gaussian random field Delta . In agreement with the conclusion of a previous investigation [Phys. Rev. E 63, 066109 (2001)], the geometrical correlation length, i.e., the average size of the clusters xi is finite for Delta>Delta_{c} approximately 1.65 and divergent for DeltaDelta_{c} . The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand, the divergence of the correlation length, xi(Delta) approximately (Delta-Delta_{c});{-nu} , with nu approximately 2 , is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.

Random fields play a central role in the analysis of spatially correlated data and, as a result, have a significant impact on a broad array of scientific applications. This paper studies the cepstral random field model, providing recursive formulas that connect the spatial cepstral coefficients to an equivalent moving-average random field, which facilitates easy computation of the autocovariance matrix. We also provide a comprehensive treatment of the asymptotic theory for two-dimensional random field models: we establish asymptotic results for Bayesian, maximum likelihood and quasi-maximum likelihood estimation of random field parameters and regression parameters. The theoretical results are presented generally and are of independent interest, pertaining to a wide class of random field models. The results for the cepstral model facilitate model-building: because the cepstral coefficients are unconstrained in practice, numerical optimization is greatly simplified, and we are always guaranteed a positive definite covariance matrix. We show that inference for individual coefficients is possible, and one can refine models in a disciplined manner. Our results are illustrated through simulation and the analysis of straw yield data in an agricultural field experiment.

I will give in this lecture an introductory overview of our present understanding of the Random Field Ising Model (RFIM). As this is a subject of a large literature and a long, turbulent history, I will restrict myself to a small number of topics, whose choice is very subjective. In particular, I will neither discuss experimental nor numerical work, except when it cannot be avoided. It is, however, important to realize why such a disproportionately large amount of attention has been devoted to this model. The first reason is that the RFIM is the simplest model which incorporates the dramatic effect of quenched-in disorder coupled linearly to an order-parameter. This effect has been seen in a steadily growing list of experiments (1). It was extensively studied for the first time in diluted antiferromagnets in a magnetic field (2) but the RFIM is now used to explain how impurities affect displacive transitions, how phase separation of binary fluids proceeds in porous media or gels, and how hydrogen dissolves in metallic alloys. Random fields also play an important role in charge-density wave systems.

Monte Carlo simulation of very large dilute random-field Ising models using multi-spin coding

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.

The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results).

We study a zero-temperature phase transition in the random field Ising model on scale-free networks with the degree exponent gamma. Using an analytic mean-field theory, we find that the spins are always in the ordered phase for gamma<3. On the other hand, the spins undergo a phase transition from an ordered phase to a disordered phase as the dispersion of the random fields increases for gamma>3. The phase transition may be either continuous or discontinuous depending on the shape of the random field distribution. We derive the condition for the nature of the phase transition. Numerical simulations are performed to confirm the results.

The random field Ising model with four spin interactions is studied within the finite cluster approximation (F.C.A), on square and simple cubic lattices. We have investigated the influence of the four spin interactions on the phase diagrams of the system. We have shown that for the simple cubic lattice, the tricritical behaviour due to the random field does not exist for every strength of the four spin interactions, and that he tricritical point due to four spin intenactions disappears when the value of the random field is large.

In this note we study the mean field equations for the $3d$ Random Field Ising Model. We discuss the phase diagram of the model, and we address the problem of finding if such equations admit more than one solution. We find two different critical values of $\beta$: one where the magnetization takes a non-zero expectation value, and one where we start to have more than one solution to the mean field equation. We find that, inside a given solution, there are no divergent correlation lengths.

Recent work on random field Ising model is described briefly emphasizing exact solutions of the model in simple cases and their relevance in understanding equilibrium and non-equilibrium properties of systems with quenched disorder.

We extend the self-consistent Ornstein–Zernike approximation (SCOZA), first formulated in the context of liquid-state theory, to the study of the random field Ising model. Within the replica formalism, we treat the quenched random field just as another spin variable, thereby avoiding the usual average over the random field distribution. This allows us to study the influence of the distribution on the phase diagram in finite dimensions. The thermodynamics and the correlation functions are obtained as solutions of a set a coupled partial differential equations with magnetization, temperature, and disorder strength as independent variables. A preliminary analysis based on high-temperature and 1/d series expansions shows that the theory can predict accurately the dependence of the critical temperature on disorder strength (no sharp transition, however, occurs for d≤4). For the bimodal distribution, we find a tricritical point which moves to weaker fields as the dimension is reduced. For the Gaussian distribution, a tricritical point may appear for d around 4.

Two-dimensional hidden periodic model is an important model in random fields. The model is used in the field of two-dimensional signal processing, prediction and spectral analysis. A method of estimating the parameters for the model is designed. The strong consistency of the estimators is proved.

We have performed measurements of the critical neutron scattering on CsCo0.83Mg0.17Br3, a dilute stacked triangular lattice (STL) Ising antiferromagnet (AF). A two component line shape associated with the critical fluctuations appears at a temperature coincident with T(N1) observed in pure CsCoBr3. Such scattering is indicative of fluctuations in prototypical random field Ising model (RFIM) systems. The random field domain state arises in this case due to geometrical frustration within the STL Ising AF, which gives rise to a three sublattice Néel state, in which one sublattice is disordered. Magnetic vacancies nucleate AF domains in which the vacancies reside on the disordered sublattice thereby generating a RFIM state in the absence of an applied magnetic field.

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