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.     

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.     

Mixed spin Ising model with four-spin interaction and random crystal field

The effects of fluctuations of the crystal field on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions are investigated within the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the two-dimensional square lattice. It has been found that the system exhibits a variety of interesting features resulting from the fluctuation of the crystal field interactions. In particular, for low mean value D of the crystal field, the critical temperature is not very sensitive to fluctuations and all transitions are of second order for any value of the four-spin interactions. But for relatively high D, the transition temperature depends on the fluctuation of the crystal field, and the system undergoes tricritical behaviour for any strength of the four-spin interactions. We have also found that the model may exhibit reentrance for appropriate values of the system parameters.     

Thermal fluctuations in some random field models

Exact identities are derived for a family of models including (a) a domain wall in a random field Ising model (RFIM), and (b) the random anisotropyXY model in the no-vortex approximation. In particular, the second moment of thermal fluctuations is not affected by frozen randomness. It is checked in a one-dimensional model that higher moments are on the contrary strongly enhanced. Thus, thermal fluctuations are strongly non-Gaussian. This reflects excursions between remote potential wells in the phase space. It is shown exactly that the Imry-Ma argument yields a correct evaluation of the field-induced fluctuations for the one-dimensional model.     

A random field model for anomalous diffusion in heterogeneous porous media

Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.     

Absence of tricritical behavior of the random field Ising model in a honyecomb lattice

In this letter, we study the behavior of the random field Ising model on a honeycomb lattice by means of the effective field theory. We obtain the phase diagram in the T$T$–H$H$ plane for clusters with one spin in a finite size cluster scheme and it is observed the absence of a tricritical point.     

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.     

Phase transitions in diluted magnets: Critical behavior,percolation, and random fields

Transition metal halides provide realizations of Ising,XY, and Heisenberg antiferromagnets in one, two, and three dimensions. The interactions, which are of short range, are generally well understood. By dilution with nonmagnetic species such as Zn⁺⁺ or Mg⁺⁺ one is able to prepare site-random alloys which correspond to random systems of particular interest in statistical mechanics. By mixing two magnetic ions such as Fe⁺⁺ and Co⁺⁺ one can produce magnetic crystals with competing interactions-either in the form of competing anisotropies or competing ferromagnetic and antiferromagnetic interactions. In this paper the results of a series of neutron scattering experiments on these systems carried out at Brookhaven over the past several years are briefly reviewed. First the critical behavior in Rb₂Mn_0.5Ni_0.5F₄ and Fe_cZn_1–cF₂ which correspond to two-dimensional and three-dimensional random Ising systems, respectively, are discussed. Percolation phenomena have been studied in Rb₂Mn_cMg_l–cF₄, Rb₂Co_cMg_l–cF₄, KMn_cZ_l-cF₃, and Mn_cZn_l–cF₂ which correspond to two-and three-dimensional Heisenberg and Ising models, respectively. In these casesc is chosen to be in the neighborhood of the nearest-neighbor percolation concentration. Application of a uniform field to the above systems generates a random staggered magnetic field; this has facilitated a systematic study of the random field problem. As we shall discuss in detail, a variety of novel, unexpected phenomena have been observed.     

Large-n limit of the Heisenberg model: Random external field and random uniaxial anisotropy

The thermodynamic equivalence of the large-n limit of then-vector model in a random external field and the corresponding disordered spherical model is proved. An analytic expression for the free energy and a phase diagram of the large-n limit of then-vector model with random uniaxial anisotropy are obtained by rigorous argument. The ferromagnetic order in the large-n limit is proved to be stable against the switching on of an arbitrarily small random anisotropy.     

横向随机晶场Ising模型的相图和磁化行为研究

在有效场理论和切断近似框架内，选择自旋S=1的二维方格子，研究横向随机晶场Ising模型的相图和磁化行为，重点是横向随机晶场浓度和晶场比率对相图和磁化的影响.给出了i>T-D_x空间的相图和m-T空间的磁化图.在晶场稀疏情况下，负晶场方向存在临界温度的峰值，正方向可出现重入现象.晶场比率取+0.5和-0.5时，磁有序相范围缩小，特别是晶场比率取-0.5时，随晶场浓度的降低，临界温度峰值从横向晶场负方向渡越到正方向.固定某一负晶场值，不同晶场比率的磁化行为有明显差异.同时与纵向稀疏晶场Ising模型结果进行有意义的比较. 关键词： 横向随机晶场Ising模型 相图 磁化行为     

Glassy phase and thermodynamics for random field Ising model on spherical lattice in magnetic field

Phase diagram and thermodynamic parameters of the random field Ising model (RFIM) on spherical lattice are studied by using mean field theory. This lattice is placed in an external magnetic field (B). The random field (hi) is assumed to be Gaussian distributed with zero mean and a variance     

The random field Ising model with an asymmetric and anisotropic trimodal probability distribution

The Ising model in the presence of a random field, drawn from the asymmetric and anisotropic trimodal probability distribution P(_hi)=pδ(_hi−_h0)+qδ(_hi+λ∗_h0)+rδ(_hi)$P\left(h_i\right)=p\delta (h_i-h_0)+q\delta (h_i+\lambda \ast h_0)+r\delta \left(h_i\right)$, is investigated. The partial probabilities p,q,r$p,q,r$ take on values within the interval [0,1]$[0,1]$ consistent with the constraint p+q+r=1$p+q+r=1$; asymmetric distribution, _hi$h_i$ is the random field variable with basic absolute value _h0$h_0$ (strength); λ$\lambda$ is the competition parameter, which is the ratio between the respective strength of the random magnetic field in the two principal directions (+z)$(+z)$ and (−z)$(-z)$ and is positive so that the random fields are competing, anisotropic distribution. 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 mainly second order phase transitions, which, for some values of p,q$p,q$ and _h0$h_0$, are followed by first order phase transitions joined smoothly by a tricritical point; occasionally, two tricritical points appear implying another second order phase transition. In addition to these points, re-entrant phenomena can be seen for appropriate ranges of the temperature and random field for specific values of λ$\lambda$, p$p$ and q$q$. Using the variational principle, we write down the equilibrium equation for the magnetization and solve it for both phase transitions and at the tricritical point in order to determine the magnetization profile with respect to _h0$h_0$, considered as an independent variable in addition to the temperature.     

Kinklike structures in scalar field theories: From one-field to two-field models

In this Letter we study the possibility of constructing two-field models from one-field models. The idea is to start with a given one-field model and use the deformation procedure to generate another one-field model, and then couple the two one-field models nontrivially, to get to a two-field model, together with some explicit topological solutions. We show with several distinct examples that the procedure works nicely and can be used generically.     

Analysis of numerical limits in 2-D simulations of a ferromagnetic/antiferromagnetic interface using the random field Ising model

This paper explains the Random Field Ising Model simulations of a two-dimensional ferromagnetic/antiferromagnetic interface, influenced by the exchange-bias interaction. Exchange-biased shifts, coercivity fields, the number of unreversed spins as well as the numerical errors are provided. These were tested for different structure dimensions and boundary conditions in order to find limitations of the method. The algorithm developed is simple, very effective, and provides deeper insight into the nature of the exchange-bias phenomenon.     

Parity effects in eigenvalue correlators, parametric and crossover correlators in random matrix models: Application to mesoscopic systems

This paper summarizes some work that I have been doing on eigenvalue correlators of random matrix models which show some interesting behavior. First we consider matrix models with gaps in their spectrum or density of eigenvalues. The density-density correlators of these models depend on whether N, where N is the size of the matrix, takes even or odd values. The fact that this dependence persists in the large N thermodynamic limit is an unusual property and may have consequences in the study of one electron effects in mesoscopic systems. Secondly, we study the parametric and cross correlators of the Harish Chandra-Itzykson-Zuber matrix model. The analytic expressions determine how the correlators change as a parameter (e.g. the strength of a perturbation in the Hamiltonian of the chaotic system or external magnetic field on a sample of material) is varied. The results are relevant for the conductance fluctuations in disordered mesoscopic systems.     

Liouville field theory coupled to a critical Ising model: non-perturbative analysis,duality and applications

Two different kinds of interactions between a $\text{Z}{}_{\mathrm{n}}$-parafermionic and a Liouville field theory are considered. For generic values of n, the effective central charges describing the UV behavior of both models are calculated in the Neveu–Schwarz sector. For n=2 exact vacuum expectation values of primary fields of the Liouville field theory, as well as the first descendent fields are proposed. For n=1, known results for sinh-Gordon and Bullough–Dodd models are recovered whereas for n=2, exact results for these two integrable coupled Ising–Liouville models are shown to exchange under a weak–strong coupling duality relation. In particular, exact relations between the parameters in the actions and the mass of the particles are obtained. At specific imaginary values of the coupling and n=2, we use previous results to obtain exact information about: (a) integrable coupled models like Ising–$\text{M}{}_{\mathrm{p/p\prime }}$, homogeneous sine-Gordon model SU(3)₂ or the Ising–XY model, (b) Neveu–Schwarz sector of the Φ₁₃ integrable perturbation of N=1 supersymmetric minimal models. Several non-perturbative checks are done, which support the exact results.