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.     

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.     

Diagrammatic expansion and metastability in the random-field Ising model

Within the perturbation diagrammatic expansion we discuss the origin of differences in determinations of the lower critical dimension of the random-field Ising model and show that below four dimensions metastability and hysteresis occur. We also explain the occurrence of a quasicritical d=2 behavior at weak random fields, which is responsible for local stability of the ordered state above two dimensions.     

The Ising model on the layered J1-J2 square lattice

We investigate the phase transitions in the Ising model on a layered square lattice with first-($J_1$) and second-($J_2$) neighbor intralayer interactions and interlayer couplings (J). The thermodynamics of the system is evaluated within a cluster mean-field approximation, which allows us to identify the nature of the thermally driven phase transitions hosted by the model. As a result, we find that interlayer couplings reduce the region of first-order phase transitions between paramagnetic and superantiferromagnetic states. We also find that the interlayer couplings reduce the frustration effects by reducing the entropy content of the low-temperature phases. Our results suggest that tricriticality is present in the special case $J=J_1$, which is in qualitative agreement with recent Monte Carlo simulations for the model.     

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.     

Longitudinal-Random-Field Mixed Ising Model with Arbitrary Spins

The longitudinal-random-fieM mixed Ising model consisting of arbitrary spin values has been studied by the use of an effective field theory with correlations （EFT）. The phase diagrams of systems with mixed spins： σ = 1/2, S = 1; σ = 1/2, S = 3/2 are plotted. Not only the discontinuity at T = 0 K, is found when both longitudinal fields are trimodal distributed, but also the trieritical behavior is observed in these phase diagrams between the bimodal and trimodal distributions of longitudinal fields, which is different from the single-spin one. The appearance of tricritical point is independent of the coordination number and spin values.     

The transverse spin-1 Ising model with random interactions

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(J_ij)=pδ(J_ij−J)+(1−p)δ(J_ij−αJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter α.     

Phase diagram of the 3D bimodal random-field Ising model

The one-parametric Wang-Landau (WL) method is implemented together with an extrapolation scheme to yield approximations of the two-dimensional (exchange-energy, field-energy) density of states (DOS) of the 3D bimodal random-field Ising model (RFIM). The present approach generalizes our earlier WL implementations, by handling the final stage of the WL process as an entropic sampling scheme, appropriate for the recording of the required two-parametric histograms. We test the accuracy of the proposed extrapolation scheme and then apply it to study the size-shift behavior of the phase diagram of the 3D bimodal RFIM. We present a finite-size converging approach and a well-behaved sequence of estimates for the critical disorder strength. Their asymptotic shift-behavior yields the critical disorder strength and the associated correlation length's exponent, in agreement with previous estimates from ground-state studies of the model.     

Critical and tricritical behavior of a selectively diluted triangular Ising antiferromagnet in a field

We study a geometrically frustrated triangular Ising antiferromagnet in an external magnetic field which is selectively diluted with nonmagnetic impurities employing an effective-field theory with correlations and Monte Carlo simulations. We focus on the frustration-relieving effects of such a selective dilution on the phase diagram and find that it can lead to rather intricate phase diagrams in the dilution-field parameters space. In particular, in a highly (weakly) diluted system the frustration is greatly (little) relieved and such a system is found to display only the second(first)-order phase transitions at any field. On the other hand, for a wide interval of intermediate dilution values the transition remains second-order at low fields but it changes to first-order at higher fields and the system displays a tricritical behavior. The existence of the first-order transition in the region of intermediate dilution and high fields is verified by Monte Carlo simulations.     

Chain length probability distribution — equivalence of ASYNNNI and 1d Ising model

An expression for the chain length probability distribution p(l) of a one dimensional Ising chain was derived using the cluster variation method formalism, the p(l) being expressed through the pair cluster probabilities. It was shown numerically that the same expression also applies in the case of one dimensional chains formed along one of the next-nearest neighbor interactions included in the two dimensional ASYNNNI (Asymmetric Next-Nearest Neighbor Ising) model, widely used to describe the statistics of oxygen ordering in the basal CuO _x planes of the YBa₂Cu₃O_6+x type high-T _c superconducting materials. Equivalency between ASYNNNI and 1d Ising model is discussed.      

具有Dzyaloshinskii-Moriya作用的Blume-Capel模型的临界性质

应用二自旋集团平均场近似的方法,研究了蜂窝晶格和正方晶格上具有Dzyaloshinskii-Moriya(DM)作用的Blume-Capel模型的临界性质,得到了该系统的相图。结果表明,所研究系统存在三临界点,并且三临界温度不随DM作用参量单调变化,三临界温度有最小值。系统的这种临界性质是交换耦合作用、晶体场作用和DM作用三者相互竞争的结果。     

Generating functionals and the Ising model

The Ising model is studied by the generating functional approach in order to provide a better understanding of that method. It is shown how to derive a general solution of a functional equation in terms of infinite-dimensional integrals. This solution is not unique; the different possibilities are characterized by different paths of integration. Further, the saddle point approximation is used for the integrals in order to obtain second-order correlation functions. It is shown that besides the normal solution, one obtains several anomalous ones, which correspond directly to the nonphysical solutions of the transfer matrix method for treating the partition function. It is also shown that only the correct solution can give a realistic behavior of the correlation function at large distances. The relevance of the saddle point methods for describing phase transitions is also discussed.     

Approximate calculations for the two-dimensional Ising model

A self-consistent molecular field approximation for the two-dimensional, square-lattice Ising model is used to calculate the energy and magnetization. Agreement with the exact calculations is good except near the critical temperature, which differs from the exact critical temperature by 4%. The specific heat has no anomalous behavior asT approachesT _c from above, and the magnetization follows the incorrect Weiss (T _c-T)^1/2 law asT approachesT _c from below.     

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.     

Droplet dynamics for asymmetric Ising model

Nucleation from a metastable state is studied for an anisotropic Ising model at very low temperatures. It turns out that the critical nucleus as well as configurations on a typical path to it differ from the Wulff shape of an equilibrium droplet.     

The shapes of bowed interfaces in the two-dimensional Ising model

We show that properly normalized net energy fluctuations associated with interfaces in two-dimensional Ising models are described, asymptotically, by random walk partition functions. Two examples are investigated: one is a droplet on a wall, and the other is two nearby, ideally parallel interfaces; the mean shapes of the interfaces in both cases prove to be elliptic, bowed outward from the wall or from each other, the semiminor axis of the latter ellipse being 1/2 that of the former, in accord with random walk results.