The random field Ising model

We discuss the current status of random field systems, particularly those with Ising symmetry. Both theory and experiment agree that, in the equilibrium state, there is a transition to an ordered state in three dimensions and no such transition in two dimensions. The critical behavior in three dimensions is, however, not very well understood. More work remains to be done to understand the dynamics, both in the critical region and the low temperature phase.     

The hierarchical random field Ising model

We introduce and solve explicitly a hierarchical approximation to the random field Ising model. This approximation is defined in terms of Peierls' contours. It exhibits a spontaneous magnetization ind>2 and illustrates some of the ideas used in the proof of that result for the real RFIM. Ind=2, there is no spontaneous magnetization, but a very slow decay of correlations. However, we argue that this latter property is an artifact of the approximation. For the real RFIM, we expect exponential decay of correlations for any value of the disorder.     

Dynamics of the random field Ising model

Dynamics of the kinetic Ising model in the presence of static random fields is investigated using a self-consistent method. It is shown that if the interface fluctuations of the low temperature phase are small the system at low temperatures stays in a state without long range order. For this state the spin correlation function 〈S_q(t)S_?q(O)> averaged over all configurations of random fields decays exponentially in time with a single wavevector dependent relaxation time which is finite at the transition temperature T₀ and remains very long below T₀. In the mean field approximation the correlation time at the magnetic Bragg peak and at T₀ scales with the magnitude of the random field as τ∞h^?z_h with z_h = 1 for d = 2 and $\text{z}{}_{\mathrm{h}}\text{=}\text{4}\text{3}$ for d = 3, respectively.     

The Ising model in a random magnetic field

The existence of a spontaneous magnetization in the three-dimensional Ising model in a weak random magnetic field (RFIM) is investgated. Following Imry and Ma, we consider the energy change, E, from the fully aligned ferromagnetic state caused by flipping all the spins inside a connected surface, . It is proved rigorously that with high probability, E is positive forall enclosing the origin. Under the unproven assumption that the expectation value of the spin at one site is weakly correlated with the random fields at far away sites (which is true if surfaces within surfaces can be ignored) it follows that the three-dimensional RFIM has a spontaneous magnetization at low temperatures. The proof works for all dimensions greater than two, providing support for the conjecture that two is the lower critical dimension.Work supported in part by NSF grant No. DMR 8100417.     

Ising model in a transverse random field

The transverse random-field Ising model with a trimodal distribution is studied within mean-field and mean-field renormalization-group approaches. The phase diagram is obtained and all the transition lines are second order. An ordered phase persists for large random fields provided that the probability of the zero transverse field is greater than the site-percolation threshold.     

Dynamical phase diagram of the random field Ising model

The stationary states of the random-field Ising model are determined through the master equation approach, where the contact with the heat bath is simulated by the Glauber stochastic dynamics. The phase diagram of the model is constructed from the stationary values of the magnetization as a function of temperature and field amplitude. The continuous phase transitions coincide with the equilibrium ones, while the first-order transitions occur at fields larger than the corresponding values at equilibrium. The difference between the fields at the limit of stability of the ordered phase and that of the equilibrium is maximum at zero temperature and vanishes at the tricritical point. We also find the mean field time auto-correlation function at the stationary states of the model. Received: 4 June 1997 / Revised: 5 August 1997 / Accepted: 10 November 1997     

Random infinite-volume Gibbs states for the Curie-Weiss random field Ising model

An approach to the definition of infinite-volume Gibbs states for the (quenched) random-field Ising model is considered in the case of a Curie-Weiss ferromagnet. It turns out that these states are random quasi-free measures. They are random convex linear combinations of the free product-measures shifted by the corresponding effective mean fields. The conditional self-averaging property of the magnetization related to this randomness is also discussed.This paper is dedicated to Robert A. Minlos on the occasion of his 60th birthday.     

The normal random field for the Ising model. The evaluation of the critical indexes

The thermodynamic functions and magnetization are calculated for the normal random field. The critical index η is obtained from the minimum of free energy. Other critical indexes have also been evaluated.     

The random field Ising model with an asymmetric trimodal probability distribution

The Ising model in the presence of a random field is investigated within the mean field approximation based on Landau expansion. The random field is drawn from the trimodal probability distribution P(h_i)=pδ(h_i−h₀)+qδ(h_i+h₀)+rδ(h_i), where the probabilities p,q,r take on values within the interval [0,1] consistent with the constraint p+q+r=1 (asymmetric distribution), h_i is the random field variable and h₀ the respective strength. This probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites. The current random field Ising system displays second order phase transitions, which, for some values of p,q and h₀, are followed by first order phase transitions, thus confirming the existence of a tricritical point and in some cases two tricritical points. Also, reentrance can be seen for appropriate ranges of the aforementioned variables. Using the variational principle, we determine the equilibrium equation for magnetization, solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h₀.     

Phase transition in the 3d random field Ising model

We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,d _l for the theory isd _l 2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.     

Interactions of several replicas in the random field Ising model

The replicated field theory of the random field Ising model involves the couplings of replicas of different indices. The resulting correlation functions involve a superposition of different types of long distance behaviours. However the n = 0 limit allows one to discuss the renormalization group properties in spite of this phenomenon. The attraction of pairs of replicas is enhanced under renormalization flow and no stable fixed point is found. Consequently, an instability occurs in the paramagnetic region, before one reaches the Curie line, signalling the onset of replica symmetry breaking. Received 28 July 2000     

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.     

One-dimensional random field Ising model and discrete stochastic mappings

Previous results relating the one-dimensional random field Ising model to a discrete stochastic mapping are generalized to a two-valued correlated random (Markovian) field and to the case of zero temperature. The fractal dimension of the support of the invariant measure is calculated in a simple approximation and its dependence on the physical parameters is discussed.Contribution to the symposium Statistical Mechanics of Phase Transitions—Mathematical and Physical Aspects, Trebo, CSSR, September 1–6, 1986.     

Fluctuations in the Curie-Weiss version of the random field Ising model

The fluctuations of the order parameter in the Curie-Weiss version of the Ising model with random magnetic field are computed. Away from criticality or at first-order critical points they have a Gaussian distribution with random (i. e.,sample-dependent) mean, thermal fluctuations contributing in same order as the fluctuations of the field; at second- or higher-order critical points, non-Gaussian sample-dependent distributions appear, and the fluctuations of the fields are enhanced, dominating over the thermal ones.     

Mean field study of the mixed Ising model in a random crystal field

The magnetic properties of a mixed Ising ferrimagnetic system, in which the two interacting sublattices have spins σ, (±1/2) and spins S, (±1,0) in the presence of a random crystal field, have been studied with the mean field approach. The obtained results show the existence of some interesting phenomena, such as the appearance of a new ferrimagnetic phase, namely, partly ferrimagnetic phase and consequently the existence of four topologically different types of phase diagrams. Furthermore, compensation behaviour and re-entrant phenomenon are found for appropriate ranges of crystal field. Thermal magnetization behaviour and phase diagrams have been discussed in detail.     

Ground states and thermal states of the random field Ising model

The random field Ising model is studied numerically at both zero and positive temperature. Ground states are mapped out in a region of random field and external field strength. Thermal states and thermodynamic properties are obtained for all temperatures using the Wang-Landau algorithm. The specific heat and susceptibility typically display sharp peaks in the critical region for large systems and strong disorder. These sharp peaks result from large domains flipping. For a given realization of disorder, ground states and thermal states near the critical line are found to be strongly correlated--a concrete manifestation of the zero temperature fixed point scenario.     

Supersymmetry and its spontaneous breaking in the random field Ising model

We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) ? 5.1 and leads to a breakdown of the "dimensional reduction" property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.     

Fast vectorized algorithm for the Monte Carlo simulation of the random field Ising model

An algorithm for the simulation of the 3-dimensional random field Ising model with a binary distribution of the random fields is presented. It uses multi-spin coding and simulates 64 physically different systems simultaneously. On one processor of a Cray YMP it reaches a speed of 184 million spin updates per second. For smaller field strength we present a version of the algorithm that can perform 242 million spin updates per second on the same machine.     

Analysis of a long-range random field quantum antiferromagnetic Ising model

We introduce a solvable quantum antiferromagnetic model. The model, with Ising spins in a transverse field, has infinite range antiferromagnetic interactions and random fields on each site following an arbitrary distribution. As is well-known, frustration in the random field Ising model gives rise to a many valley structure in the spin-configuration space. In addition, the antiferromagnetism also induces a regular frustration even for the ground state. In this paper, we investigate analytically the critical phenomena in the model, having both randomness and frustration and we report some analytical results for it.