Two-dimensional Ising Ferromagnet Research Articles

Specific Heat Singularity in Two-Dimensional Random Ising Ferromagnets

We use transfer-matrix methods to calculate free energies and specific heats on long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings. By implementation of our code on a highly parallel computer, we have managed to generate high-quality data for strip widths up to L = 18 sites. An unequivocal trend towards a divergency of the specific heat in the thermodynamic limit can be discerned. Finite-size data appear to behave halfway between a single- ( ln L) and double- ( ln ln L) logarithmic dependence on strip width. This is in line with a crossover within the Dotsenko-Shalaev picture of logarithmic corrections to pure-system singularities.

Universality and logarithmic corrections in two-dimensional random Ising ferromagnets

We address the question of weak versus strong universality scenarios for the random-bond Ising model in two dimensions. A finite-size scaling theory is proposed, which explicitly incorporates $\ln L$ corrections ($L$ is the linear finite size of the system) to the temperature derivative of the correlation length. The predictions are tested by considering long, finite-width strips of Ising spins with randomly distributed ferromagnetic couplings, along which free energy, spin-spin correlation functions and specific heats are calculated by transfer-matrix methods. The ratio $\gamma/\nu$ is calculated and has the same value as in the pure case; consequently conformal invariance predictions remain valid for this type of disorder. Semilogarithmic plots of correlation functions against distance yield average correlation lengths $\xi^{av}$, whose size dependence agrees very well with the proposed theory. We also examine the size dependence of the specific heat, which clearly suggestsa divergency in the thermodynamic limit. Thus our data consistently favour the Dotsenko-Shalaev picture of logarithmic corrections (enhancements) to pure system singularities, as opposed to the weak universality scenario.

The two-dimensional site-diluted Ising ferromagnet a damage-spreading analysis

The quenched site-diluted Ising ferromagnet on a square lattice, for which each site is occupied or empty with probabilities p and 1 − p, respectively, is studied numerically through damage-spreading procedures. By making use of the Glauber dynamics, the percolation threshold p c is estimated. Within the heat-bath dynamics, the damage-spreading temperatures T d ( p) (for several values of p> p c ) are computed, indicating a strong correlation with the corresponding critical temperatures T c ( p). A procedure for estimating the fractal dimensions of clusters of damaged sites, at low temperatures, is presented; as p → p c , our estimate is very close to 91/48, which is the fractal dimension of the infinite cluster at p = p c in two-dimensional site percolation. Whenever possible to compare, our results are in good agreement with the best estimates available from other techniques, in spite of a modest computational effort.

Correlation functions in a corner-shaped Ising model.

We obtain exact results at all temperatures for the two-point function of surface spins near the corner of a two-dimensional rectangular Ising ferromagnet; at the critical point, the functional form predicted by conformal theory is recovered. We are also able to calculate exactly a large class of correlation functions involving spins off the surface, and to interpret the results geometrically.

Geometrical string and dual spin systems

We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual hamiltonian represents a new type of spin system with local gauge invariance. At each vertex ξ there are d ( d − 1) /2 Ising spins ∧ μ, η = ∧ η, μ N . ≠ P = 1, … , d and one Ising spin Γ on every link ξ ξ + e,). For the frozen spin Γ  1 the dual hamiltonian factorizes into d ( d − 1) /2 two-dimensional Ising ferromagnets and into antiferromagnets in the case Γ  −1. For fluctuating F it is a sort of spin-glass system with local gauge invariance. The generalization to p-membranes is given.

Aging in two-dimensional random-exchange Ising ferromagnets

Domain growth in the two-dimensional random-exchange Ising ferromagnet Rb2Cu0.89Co0.11F4 is studied using ac and dc magnetization measurements. From a scaling analysis of the time and frequency dependences of the nonequilibrium part of the ac susceptibility the ferromagnetic domain size R(t) is found to grow according to R(t) α ⦜ln t]1/gy, with ψ = 14.

Monte Carlo algorithms with absorbing Markov chains: Fast local algorithms for slow dynamics.

A class of Monte Carlo algorithms which incorporate absorbing Markov chains is presented. In a particular limit, the lowest-order of these algorithms reduces to the $n$-fold way algorithm. These algorithms are applied to study the escape from the metastable state in the two-dimensional square-lattice nearest-neighbor Ising ferromagnet in an unfavorable applied field, and the agreement with theoretical predictions is very good. It is demonstrated that the higher-order algorithms can be many orders of magnitude faster than either the traditional Monte Carlo or $n$-fold way algorithms.

Application of a constrained-transfer-matrix method to metastability in the d = 2 Ising ferromagnet

Applying a numerical transfer-matrix formalism, we obtain complex-valued constrained free energies for the two-dimensional square-lattice nearest-neighbor Ising ferromagnet below its critical temperature and in an external magnetic field. In particular, we study the imaginary part of the constrained free-energy branch that corresponds to the metastable phase. Although droplets are not introduced explicitly, the metastable free energy is obtained in excellent agreement with field-theoretical droplet-model predictions. The finite-size scaling properties are different in the weak-field and intermediate-field regimes, and we identify the corresponding different critical-droplet shapes. For intermediate fields, we show that the surface free energy of the critical droplet is given by a Wulff construction with the equilibrium surface tension. We also find a prefactor exponent in complete agreement with the field-theoretical droplet model. Our results extend the region of validity for known results of this field-theoretical droplet model, and they indicate that this transfer-matrix approach provides a nonperturbative numerical continuation of the equilibrium free energy into the metastable phase.

What is the dimension from scaling of finite systems?

Finite-size scaling allows the extraction of a dimension from scaling [ital d][sub [ital s]], as well as critical exponents for systems with a second-order phase transition. The calculation of [ital d][sub [ital s]] allows comparisons to be made with other expansions in dimension, in which [ital d][sub [ital s]] plays the role of the physical dimension. This is demonstrated by a numerical transfer-matrix study of the pure ferromagnetic Ising model in two and three dimensions, as well as for quenched random bonds in the two-dimensional Ising ferromagnet.

Distributions of magnetization in the two-dimensional random Ising model

The two-dimensional random Ising ferromagnet is investigated numerically by the Monte Carlo method. The integrated distribution function of magnetization is calculated above and below the Griffiths temperature which corresponds to the critical temperature of the pure system with smaller interaction. It is found that this function shows different behavior above or below this temperature.

Order parameter distribution of the random bond Ising ferromagnet with antiperiodic boundary conditions

The two-dimensional random bond Ising ferromagnet with antiperiodic boundaries in one direction has been studied by Monte Carlo simulations. From the overlap distribution function P(q), the scaling behavior of P(q) near q=0 with linear dimension L has been obtained. In contrast to recent simulations of the four-dimensional Ising spin glass, our results are consistent with the predictions of the droplet model.

Metastability and exponential approach to equilibrium for low-temperature stochastic Ising models

We present a proof of the exponential convergence to equilibrium of single-spin-flip stochastic dynamics for the two-dimensional Ising ferromagnet in the low-temperature case with not too small external magnetic fieldh uniformly in the volume and in the boundary conditions.

Application of the Monte Carlo coherent-anomaly method to two-dimensional lattice-gas systems with further-neighbor interactions.

A Monte Carlo version of the coherent-anomaly method has been used to determine critical properties of a two-dimensional Ising ferromagnet with nearest- and next-nearest-neighbor interactions and of a series of two-dimensional lattice-gas systems of particles interacting via 6-12 Lennard-Jones potential. It has demonstrated that the method leads to quite accurate determination of critical temperature but is less successful when used to determine the values of critical exponents \ensuremath{\gamma} and \ensuremath{\nu}.

Neutron Scattering Study of Two-Dimensional Randomly Mixed Magnets with Competing Anisotropies K2Cu1-xCoxF4

The random mixture of a quasi-two-dimensional magnet of K 2 Cu 1- x Co x F 4 is studied by neutron scattering in the ferromagnetic composition range, 0.2> x >0. K 2 CuF 4 is the XY-like ferromagnet with the distorted K 2 NiF 4 -type structure. The substitution of Ising spins of Co for XY-like spins of Cu by a few percent brings about a rotation of the easy axis. Two critical lines of three-dimensional XY and two-dimensional Ising characters, intersecting at a tetracritical point, are observed as is commonly seen in randomly mixed magnets with orthogonal anisotropies. It was also found that K 2 Cu 1- x Co x F 4 with 0.2> x >0.05 is a model system of the two-dimensional Ising ferromagnet.

Studies of novel magnetic systems by electron capture spectroscopy (ECS)

The electron spin polarization (ESP) at surfaces of bulk and artificially structured, thin film magnetic materials is studied using electron capture spectroscopy (ECS) and ion-induced emission of spin-polarized electron. For V and Tb surfaces, it is found that the temperature dependence of the surface magetic order is quite different from that of the bulk. For ultra-thin magnetic superlattices, V(100)/Ag(100) and Tb (0001)/W(110), we find novel magnetic surface anisotropies and critical behavior. At surfaces of 5 monolayers V(100) on Ag(100), the critical behavior of the surface-ESP is found to be identical to that of the two-dimensional Ising ferromagnet. At surfaces of PtMnSb and Fe80B20, we findnonzero ESP due to long-ranged or short-ranged ferromagnetic order. Thesign of the ESP at surfaces of the Heusler alloy PtMnSb is in agreement with recent band structure calculations. At surfaces of the spin glass Fe80B20, we find, in agreement with ECS data, that the ESP ofion-induced emitted spin-polarized electrons is extremely sensitive to the cleanness of thetopmost surface layer.

Contact angle for two-dimensional Ising ferromagnets.

The contact angle of a sessile drop on a wall is considered within a modified planar Ising model. The thermal variation of this contact angle is computed exactly, and it is shown how a simple thermalization of the boundaries (wall) may lead to an exact treatment of multiple wetting transitions.

Two-dimensional ferromagnetism of ultra-thin artificial vanadium films

Artificial ultra-thin (1–7 monolayers) films of V (100)p(1 × 1) on Ag(100) are studied by electron capture spectroscopy and found to be ferromagnetic, with a surface Curie temperature T Cs depending on film thickness. The critical behavior of the magnetization of the topmost layer is found to be identical to that of the two-dimensional Ising ferromagnet.

Instability of two-dimensional Ising ferromagnets with dipole interactions

Two-dimensional Ising ferromagnets with spins perpendicular to the plane are destabilised by dipole interactions. Ferromagnetism is destroyed by domain walls which, at low temperatures, form a square network on a square lattice. At higher temperatures, domain walls form a floating solid, the period of which decreases continuously with increasing temperature.

On limit theorems for the bivariate (magnetization, energy) variable at the critical point

The limiting (magnetization, energy) bivariate variable is studied for Ising ferromagnets at the critical point. The factorization property of the limiting bivariate moment generating function is shown to be intimately connected to critical point exponent inequalities and to the behaviour of the scaling limit near and at the critical point. The validity of this can be deduced from the study of the second and the fourth magnetization cumulants at zero external field. The limiting bivariate variable is exactly calculated at the critical point for the Curie-Weiss model (MF) and for the edge of a two-dimensional Ising ferromagnet wrapped on a cylinder. It is shown that the mean field case leads to a non-Gaussian limiting distribution in contradistinction with the particular Ising model we consider for which we obtain a product of two Gaussian probability distributions.

Erratum: “The Curie Temperature of the Two-Dimensional Quadratic Ising Ferromagnet with Mixed Spins of S=1/2 and S=1”

Erratum: “The Curie Temperature of the Two-Dimensional Quadratic Ising Ferromagnet with Mixed Spins of S=1/2 and S=1”