Dilute antiferromagnets: Imry-Ma argument,hierarchical model,and equivalence to random field Ising models

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.     

Ising model in an external field on a hierarchical lattice

The two-dimensional renormalization map of the diamond-hierarchical Ising model in an external field is given, and pictures of the distribution of zeros of the partition function in the complex plane of temperature for varying values of coupling constant and external field are shown. Critical exponents of the model are found, and results are different from those of the Ising model on a two- or three-dimensional regular lattice.     

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.     

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.     

Stability of hierarchical interfaces in a random field model

We study a hierarchical model for interfaces in a random-field ferromagnet. We prove that in dimensionD>3, at low temperatures and for weak disorder, such interfaces are rigid. Our proof uses renormalization group transformations for stochastic sequences.     

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.     

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.     

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.     

对《二维六角形晶格伊辛模型的重正化群解》一文的进一步计算

针对《二维六角形晶格伊辛模型的重正化群解》一文中有关〈V〉0的计算进行了修正,给出了新的重正化群的变换、重正化群的线性化变换矩阵以及临界指数.     

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.     

The random field method for the Ising model

The random field method is used for investigation of the Ising model. The generalization of the variational principle and a new representation for the free energy of the Ising model are proposed.     

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.     

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.     

A hierarchical model for random walks in random media

We show that the random walk generated by a hierarchical Laplacian in ^d has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions.     

二维六角形晶格伊辛模型的重正化群解

选取六角形晶格为Kadanoff集团,用重正化群方法求得临界点和与之相应的各临界指数,与三角形晶格重正化群解相比,其精度提高了15％。     

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.     

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.     

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.     

An exact limit for the Peierls distortion energy in an antiferromagnetic chain by means of a renormalization procedure

A renormalization procedure gives a rigorous upper bound for the ground-state energy per spin for a Peierls-distorted antiferromagnetic chain with Heisenberg interaction.     

Stability of interfaces in a random environment. A rigorous renormalization group analysis of a hierarchical model

We study a hierarchical model of domain walls in aD-dimensional bond disordered Ising model at low temperatures. Using a renormalization group method inspired by the work of Bricmont and Kupiainen for the random field Ising model, we prove the existence of rigid interfaces at low enough temperatures in dimensionsD>3.