Critical behavior of the 3D Ising model on a poissonian random lattice

The single-cluster Monte Carlo algorithm and the reweighting technique are used to simulate the 3D ferromagnetic Ising model on 3D Voronoi-Delauney lattices. It is assumed that the coupling factor J varies with the distance r between the first neighbors as J(r)∝e^−ar, with a≥0. The critical exponents γ/ν, β/ν, and ν are calculated, and according to the present estimates for the critical exponents, we argue that this random system belongs to the same universality class of the pure 3D ferromagnetic Ising model.     

Fock space for fermion-like lattices and the linear Glauber model

The concept of Fock space representation is developed to deal with stochastic spin lattices written in terms of fermion operators. A density operator is introduced in order to follow in parallel the developments of the case of bosons in the literature. Some general conceptual quantities for spin lattices are then derived, including the notion of generating function and path integral via Grassmann variables. The formalism is used to derive the Liouvillian of the d-dimensional Linear Glauber dynamics in the Fock-space representation. Then the time evolution equations for the magnetization and the two-point correlation function are derived in terms of the number operator.     

Local analysis of frustration based on Kagomé lattices

Magnetic frustration in the framework of the Edwards-Anderson model is studied for the ground level (T=0) of the Kagomé lattice (KL). A sample consists of a realization of a random distribution of ferromagnetic (F) and antiferromagnetic (AF) interactions of the same strength along lines connecting nearest neighbors. Our goal is to compare two methods to calculate the following parameters: ground state energy, frustration and average frustration segment as functions of x, the concentration of F bonds. In doing so we make use of topological concepts such as plaquettes and frustration segments. The probability of a plaquette being unfrustrated (or flat) is ℘_p(x), while the probability of a plaquette being frustrated (or curved) is ℘_c(x). The analysis is done locally on a representative portion of the lattice which is called cell; cells of two different sizes are used in the present work. One method (which is simpler) is based on the probability function of any plaquette configuration ?(℘_p,℘_c). The other method is more exact but also more complex and increasingly difficult to use for large cells; it is based on the probability of bond configurations ψ(x). These methods are compared between themselves noting that the simpler method can be enough for most of the range for x. In addition, numerical simulations for many random samples at different concentrations x for a size given by 75 spins with periodic boundary conditions were done. This provides reference lines to compare with the properties under study. The local frustration analysis to obtain both ?(℘_p,℘_c) and ψ(x) is done over two cells of different size. Robustness of the criteria used in the local frustration analysis is also investigated.     

Field theory of finite-size effects in Ising-like systems

We propose a new perturbation approach to finitesize effects within the ⁴ field theory for a one-component order parameter with periodic boundary conditions. Our approach is applicable both above and belowT _c . Renormalization-group calculations of finite-size scaling functions in three dimensions are compared with new Monte Carlo data for theL×L×L Ising model withL=8, 16, 32. The field-theoretic predictions are in good overall agreement with the Monte Carlo data.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

Phase diagram of uniaxial antiferromagnetic particles: Field perpendicular to the easy axis

We consider a spherical uniaxial antiferromagnetic particle in the presence of an external magnetic field perpendicular to its easy axis. The model is described by a classical Heisenberg Hamiltonian including a single-ion uniaxial anisotropy, where the magnetic moments of the particle are represented by continuous spin vectors. We employ mean-field calculations and Monte Carlo simulations to determine the phase diagram of the system. The phase diagram in the plane field versus temperature is obtained for particles with radii ranging from three up to twelve spacing lattice units. We have seen that a particle with more than nine shells behaves as a true thermodynamic system. We find the explicit dependence of the zero temperature critical field and the Néel temperature on the diameter of the particle. At low temperatures, we have also shown that, for particles with three or more shells, the critical field follows a T² law, which is in agreement with the predictions of the spin-wave theory, when the field is perpendicular to the easy axis.     

Critical exponents of the Ising model on low-dimensional fractal media

The critical behavior of the Ising model on fractal substrates with noninteger Hausdorff dimension d_H<2 and infinite ramification order is studied by means of the short-time critical dynamic scaling approach. Our determinations of the critical temperatures and critical exponents β, γ, and ν are compared to the predictions of the Wilson-Fisher expansion, the Wallace-Zia expansion, the transfer matrix method, and more recent Monte Carlo simulations using finite-size scaling analysis. We also determined the effective dimension (d_ef), which plays the role of the Euclidean dimension in the formulation of the dynamic scaling and in the hyperscaling relationship d_ef=2β/ν+γ/ν. Furthermore, we obtained the dynamic exponent z of the nonequilibrium correlation length and the exponent θ that governs the initial increase of the magnetization. Our results are consistent with the convergence of the lower-critical dimension towards d=1 for fractal substrates and suggest that the Hausdorff dimension may be different from the effective dimension.     

Critical behavior of sound attenuation in a metamagnetic Ising system

In this study, the sound attenuation coefficient of a spin- metamagnetic Ising system is calculated by the method of thermodynamics of irreversible processes. The behavior of sound attenuation near the phase transition temperatures is analyzed according to various values of phenomenological rate coefficients (γij). For all γm and γs values it is found that sound attenuation peaks occur below TN(H) and depend on frequency ω and the value of the off-diagonal rate coefficient γ. On the other hand, the critical behavior of the sound attenuation in the hydrodynamic regime is obtained analytically via the critical exponents. Moreover, the behavior of the sound attenuation as a function of frequency is also investigated and ω² dependence is observed for the attenuation coefficient. These results are in a good agreement with ultrasonic investigations of magnetic systems.     

Effective-field theory of the Ising model with three alternative layers on the honeycomb and square lattices

The Ising model with three alternative layers on the honeycomb and square lattices is studied by using the effective-field theory with correlations. We consider that the nearest-neighbor spins of each layer are coupled ferromagnetically and the adjacent spins of the nearest-neighbor layers are coupled either ferromagnetically or anti-ferromagnetically depending on the sign of the bilinear exchange interactions. We investigate the thermal variations of the magnetizations and present the phase diagrams. The phase diagrams contain the paramagnetic, ferromagnetic and anti-ferromagnetic phases, and the system also exhibits a tricritical behavior.     

Phase diagram of an Ising model with competitive interactions on a Husimi tree and its disordered counterpart

We consider an Ising competitive model defined over a triangular Husimi tree where loops, responsible for an explicit frustration, are even allowed. We first analyze the phase diagram of the model with fixed couplings in which a “gas of noninteracting dimers (or spin liquid) — ferro or antiferromagnetic ordered state” zero temperature transition is recognized in the frustrated regions. Then we introduce the disorder for studying the spin glass version of the model: the triangular ±J model. We find out that, for any finite value of the averaged couplings, the model exhibits always a finite temperature phase transition even in the frustrated regions, where the transition turns out to be a glassy transition. The analysis of the random model is done by applying a recently proposed method which allows us to derive the critical surface of a random model through a mapping with a corresponding nonrandom model.     

Absorbing phase transition in contact process on fractal lattices

We investigate the critical behavior of nonequilibrium phase transition from an active phase to an absorbing state on two selected fractal lattices, i.e., on a checkerboard fractal and on a Sierpinski carpet. The checkerboard fractal is finitely ramified with many dead ends, while the Sierpinski carpet is infinitely ramified. We measure various critical exponents of the contact process with a diffusion-reaction scheme A→AA and A→0, characterized by a spreading with a rate λ and an annihilation with a rate μ, and the results are confirmed by a crossover scaling and a finite-size scaling. The exponents, compared with the ?-expansion results assuming , being the fractal dimension of the underlying fractal lattice, exhibit significant deviations from the analytical results for both the checkerboard fractal and the Sierpinski carpet. On the other hand, the exponents on a checkerboard fractal agree well with the interpolated results of the regular lattice for the fractional dimensionality, while those on a Sierpinski carpet show marked deviations.     

Tethered Monte Carlo: Computing the effective potential without critical slowing down

We present Tethered Monte Carlo, a simple, general purpose method of computing the effective potential of the order parameter (Helmholtz free energy). This formalism is based on a new statistical ensemble, closely related to the micromagnetic one, but with an extended configuration space (through Creutz-like demons). Canonical averages for arbitrary values of the external magnetic field are computed without additional simulations. The method is put to work in the two-dimensional Ising model, where the existence of exact results enables us to perform high precision checks. A rather peculiar feature of our implementation, which employs a local Metropolis algorithm, is the total absence, within errors, of critical slowing down for magnetic observables. Indeed, high accuracy results are presented for lattices as large as L=1024$L=1024$.     

Magnetization and magnetic susceptibility of the Ising ferromagnetic/antiferromagnetic superlattice

A linear cluster mean-field approximation is used to study the magnetic properties of the Ising ferromagnetic/antiferromagnetic superlattice, which is composed of a spin-1/2 ferromagnetic monolayer and a spin-1 antiferromagnetic monolayer with a single-ion anisotropy alternatively. By using the transfer matrix method, we calculate the magnetization and the initial magnetic susceptibility as functions of temperature for different interlayer coupling, single-ion anisotropy. We summarize the changing behaviors of the spin structure in ferromagnetic and antiferromagnetic layers and the characteristics of the corresponding magnetic susceptibilities, give the transition temperature as a function of the interlayer exchange coupling for different single-ion anisotropy, and analyze the features of the magnetization and the magnetic susceptibility.     

Critical phenomena at edges and corners

Using Monte-Carlo techniques, the critical behaviour at edges and corners of the three-dimensional Ising model is studied. In particular, the critical exponent of the local magnetization at edges formed by two intersecting free surfaces is estimated to be, as a function of the opening angle , for , for , and for . The critical exponent of the corner magnetization of a cube is found to be . The Monte-Carlo estimates are compared to results of mean field theory, renormalization group calculations and high temperature series expansions. Received: 29 January 1998 / Accepted: 17 March 1998     

Magnetization behavior of hard/soft-magnetic composite pillar

Hard/soft-magnetic composite pillar array medium is proposed for ultra-high-density recording media. Magnetization reversal process for a single hard/soft-magnetic composite pillar in the medium is calculated using the Landau–Lifshitz–Gilbert equation. Magnetization reversal of the soft-magnetic unit helps the magnetization reversal for the hard-magnetic unit, and the effective coercivity for the hard-magnetic unit is greatly reduced. Thereby saturation recording to the high-K_u-hard-magnetic material used for perpendicular magnetic recording will be realizable.     

Waves and patterns in ring lattices with delays

We investigate the existence and the stability of waves and phase locked states in rings of coupled oscillators with delayed interactions. Using center manifold reduction and the normal form method, we reduce the equation governing the dynamics of the whole network to an amplitude-phase model (i.e. a set of coupled ordinary differential equations describing the evolution of both the amplitudes and the phases of the oscillators). Then we prove the existence of traveling waves, in-phase and anti-phase locked oscillations, in both one-dimensional and two-dimensional lattices. The influence of the interaction strength and the number of oscillators is investigated, and the possible coexistence of waves and phase locked oscillations is shown.     

Phase diagram of the Ising antiferromagnet with nearest-neighbor and next-nearest-neighbor interactions on a square lattice

The phase diagram of the Ising model in the presence of nearest-neighbor (J₁) and next-nearest-neighbor (J₂) interactions on a square lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by effective-field theory in finite cluster (we have chosen N=4 spins). We have proposed a functional for the free energy (similar to Landau expansion) to obtain the phase diagram in the (T,α) space (α=J₂/J₁), where the transition line from the superantiferromagnetic (SAF) to the paramagnetic (P) phase is of first-order in the range 1/2<α<0.95 in contrast to previous study of CVM (Cluster Variational Method) that predict first-order transition for α=1.0. Our results for α=1.0 are in accordance with MC (Monte Carlo) simulations, that predict a second-order transition.     

Magnetic properties of the mixed ferrimagnetic ternary system with a single-ion anisotropy on the Bethe lattice

The magnetic properties of the ternary system ABC consisting of spins , S=1, and are investigated on the Bethe lattice by using the exact recursion relations. We consider both ferromagnetic and antiferromagnetic exchange interactions. The exact expressions for magnetizations and magnetic susceptibilities are found, and thermal behaviors of magnetizations and susceptibilities are studied. We construct the phase diagrams and find that the system exhibits one, two or even three compensation temperatures depending on the values of the interaction parameters in the Hamiltonian. Moreover, the system undergoes a second-order phase transition for the coordination number q?3 and a second- and first-order phase transitions for q>3; hence the system gives a tricritical point. The system also exhibits the reentrant behaviors.     

Antiferromagnetic model and magnetization plateaus on the zigzag ladder with two- and three-site exchanges

The Ising approximation is proposed for the Heisenberg model with two- and three-spin exchange interactions on a zigzag ladder placed in a strong magnetic field. Using the transfer matrix technique the magnetization is calculated and compared with the results obtained in the dynamical recursive approach. Magnetization plateaus are found at m=0 and with the two- and three-spin exchanges. The magnetic susceptibility in small and large zigzag ladders for various exchange parameters and temperatures is also analyzed. As temperature decreases the magnetic susceptibility versus magnetic field shows the formation of spin pseudogaps and transformation from double to four peak structure. Small clusters also exhibit characteristic features of large thermodynamic systems.     

On the linear relaxation of a one-dimensional bonded fluid model

The linear relaxation of a one-dimensional bonded fluid model is studied in the dynamical mean field approximation at the pair level. The results are compared with simulation data. Relaxation at low temperatures shows a cooperative character that can be understood in terms of a simple diffusion picture.     

Phase diagram of the two-dimensional Ising model with random competing interactions

An Ising model with ferromagnetic nearest-neighbor interactions J₁ (J₁>0) and random next-nearest-neighbor interactions [+J₂ with probability p and −J₂ with probability (1−p); J₂>0] is studied within the framework of an effective-field theory based on the differential-operator technique. The order parameters are calculated, considering finite clusters with n=1,2, and 4 spins, using the standard approximation of neglecting correlations. A phase diagram is obtained in the plane temperature versus p, for the particular case J₁=J₂, showing both superantiferromagnetic (low p) and ferromagnetic (higher values of p) orderings at low temperatures.