Ising Square Research Articles

Hysteresis for anisotropic ± J Ising square lattices

It is known that ± J square Ising lattices exhibit hysteresis in the isotropic case emulating some general behavior of spin glasses. On the other hand, some experimental curves of hysteresis in real spin glasses show additional features not explained by simulation on isotropic systems. In the present work, we explore the possibility of explaining some of these features by introducing anisotropic ± J exchange interactions so they are stronger along one direction by an anisotropy factor f (square symmetry is broken), which is varied in the interval 1–8. For each sample, hysteresis cycles are found varying temperature by means of Monte Carlo method. The analysis is extended over many samples for each size searching for average magnetization values under different realizations. Two interesting properties are established in this paper: (a) Loops break into sub loops; (b) Energy loss per cycle (area within the hysteresis loop) tends to increase with f, which is an indication for a more stable spin-glass phase in the presence of anisotropy. Comments comparing these results with experiment are also included.

Cluster variation method and Monte Carlo simulations in Ising square antiferromagnets

Computer simulations in Ising square antiferromagnets with first- ( J 1) and second- ( J 2) neighbor interactions within the Cluster Variation Method are reported. The results are compared with those obtained by Monte Carlo (MC) simulations. The good agreement between both methods confirms the reliability of the Cluster Variation Method (CVM) simulation to describe the excitations of systems with first- and second-neighbor interactions. The phase transitions occurring in the neighborhood of V=( J 2/ J 1)>0.5, are analyzed in detail.

Weakly(x=0)and randomly(x=0.033)coupled Ising antiferromagnetic planes in(Li1−3xFex)NiPO4compounds

Neutron diffraction and magnetic susceptibility studies of $({\mathrm{Li}}_{1\ensuremath{-}3x}{\mathrm{Fe}}_{x}){\mathrm{NiPO}}_{4} (x=0$ and $x=0.033)$ compounds reveal remarkable differences between the magnetic properties of pure ${\mathrm{LiNiPO}}_{4}$ and those of its lightly iron-doped derivative. The spin system associated with the ${\mathrm{Ni}}^{2+}$ ions $(S=1)$ in the pure compound undergoes a collinear antiferromagnetic ordering at ${T}_{N}=19.1\ifmmode\pm\else\textpm\fi{}0.5 \mathrm{K},$ with the characteristics of weakly coupled two-dimensional (2D) Ising square planes. By contrast, randomly intercalated iron spins (in Li sites) between ${\mathrm{Ni}}^{2+}$ planes comprise a spin-glass-like subsystem which, despite their minute amount, drives the antiferromagnetic transition to higher temperatures ${T}_{N}=25.2\ifmmode\pm\else\textpm\fi{}0.5 \mathrm{K},$ and significantly modifies the critical behavior of the 2D ${\mathrm{Ni}}^{2+}$ system. It is argued that the doped compound can serve as a model system for studying the randomly coupled planar Ising model.

Hysteresis in ±J Ising square lattices

The magnetic hysteresis of $\ifmmode\pm\else\textpm\fi{}J$ Ising lattices is analyzed performing a zero-temperature random-walk minimizing energy. A steplike structure presenting a loop divided in four sections is observed. It is shown by Monte Carlo calculations that this structure is rounded off as temperature increases until a thin S shape is obtained, which is in general agreement with experimental results. A simple explanation for this form of hysteresis is given supporting universality and size independence.

2D Ising model for hydrated protein surfaces

To understand the nature of the glassy dielectric relaxation recently observed in hydrated protein powders, we model the protein surface as a 2D Ising square net, and identify frustration at the molecular level in the competing interactions of migrating protons with bound water.

Cluster variation approach to the Ising square lattice with two- and four-spin interactions

The phase diagram of the Ising square ferromagnet with nearest-neighbor, next-nearest-neighbor, and four-spin (plaquette) interactions is investigated using the cluster variation method in the square approximation. A complete analysis in temperature is performed for the ferromagnetic, superantiferromagnetic, and frustrated region of the ground state. Particular attention is devoted to the role of the four-spin interaction. The most remarkable effect is the occurrence, in several regions of the interaction parameter space, of first-order transitions and sometimes reentrance phenomena. Critical and tricritical points are almost analytically determined.

Phase coexistence in binary mixtures in thin films with symmetric walls: model calculations for two- and three-dimensional Ising lattices

Binary mixtures (A, B) that undergo phase separation in the bulk are considered in thin film geometry, assuming that one of the components is preferentially attracted to one of the walls. We discuss the average profile of the order parameter {volume fraction φ(z) of one of the components} in the z-direction (perpendicular to the surfaces), paying attention to the lateral inhomogeneity of the thin film when the average volume fraction corresponds to a state inside the coexistence curve. We consider the situation where due to (short range) surface forces a second-order wetting transition would occur in semi-infinite geometry: in the thin film geometry, this transition is rounded off and its only remnant is a smooth increase of the adsorbed mass in the surface-enriched layer in the transition region. Monte Carlo calculations for nearest neighbor Ising square and simple cubic lattices are used to derive typical concentration profiles. In the two-dimensional case, also the kinetics of domain formation after the quench from a disordered state is considered, and it is shown that the typical concentration oscillations perpendicular to the wall (“surface directed spinodal decomposition”) do not occur, due to strong lateral fluctuations of the local position of the interface between the enrichment layer at the surface and the neighboring depleted region. Finally, also phase-separated states in thin films with competing walls (where one surface prefers A and the other prefers B) are briefly treated, and experimental applications are discussed.

Phase transitions in Ising square antiferromagnets with first- and second-neighbour interactions

The phase transitions occurring in the Ising square antiferromagnet with first- (J1) and second- (J2) nearest-neighbour interactions are studied using several mean-field approximations and for a wide range of R=J2/J1. The largest approximation used corresponds to a nine-point cluster approximation of the cluster variation method. In this case, the transition temperatures as a function of R are found to be in excellent agreement with those obtained by other methods. The mean-field approximations predict a first-order transition in the range 0.5<R<or approximately=1.2, where the critical exponents associated with the paramagnetic to superantiferromagnetic transition have been reported to vary continuously with R. In that range of R, the mean-field approximations also predict a crossover between two distinct instability temperatures, or spinodals, taking place immediately below the first-order transition. Mean-field results are also given for the magnetization m, the specific heat Cnu , the magnetic susceptibility X, the staggered susceptibility Xs, and the pair correlation function sigma ij between i and j sites.

Ordering in Ising square lattices with nearest and next nearest neighbor repulsion

The phase diagram of an Ising square lattice with nearest and next nearest neighbor repulsion of equal strength is analyzed by means of Monte Carlo simulation. At low temperatures a multidomain and a (2 × 1) ordered lattice gas phase are found for θ ≈ 1 4 , 3 4 and θ ≈ 1 2 in contradiction to previous Monte Carlo studies. The multidomain phase consists of patches of (2 × 2) and c(4 × 2) lattice gas phases. From a careful inspection of diffraction pattern it is possible to distinguish between multidomain and (2 × 1) lattice gas phases. The phase diagram for the present lattice gas model indicates regions of increased disorder between multidomain and (2 × 1) phases.

Random decorated antiferromagnetic ising model with mixed spins

A new approach to studying random quenched decorated models is introduced, which after an approximation on the decoration transformation equation makes use of the exact solution of the partition function of the host lattice. The method is applied to study the phase diagram of an antiferromagnetic Ising square lattice randomly decorated with spins S1=±1;0 with probability p or S2=±1 with probability 1−p, in the isotropic case, in which all the bonds are decorated. Several phase diagrams are obtained spanning the various parameter spaces of the model.

Finite-size scaling and the crossover to mean-field critical behavior in the two-dimensional Ising model with medium-ranged interactions.

Critical amplitudes in finite-size scaling relations show a singular dependence on the range of the interactions, R. The respective power laws are predicted from phenomenological crossover scaling considerations. These predictions are tested by Monte Carlo simulations for medium-ranged Ising square lattices. It is speculated that some deviations between the simulation results and corresponding predictions may be due to logarithmic corrections.

Specific heat and the phase diagram of the Ising square lattice with nearest- and next-nearest interactions

The specific heat and the phase diagram of the Ising square lattice with nearest- (J1) and next-nearest-neighbour (J2) interactions are studied within the cluster variation method. The Ising Hamiltonian is solved in the square (Wannier-Kramer) approximation. The phase diagram is calculated and compared with other previous studies.

First-order phase transitions in the Ising square lattice with first- and second-neighbor interactions.

The phase diagram of the Ising square lattice with first- and second-nearest-neighbor antiferromagnetic interactions (${\mathit{J}}_{1}$ and ${\mathit{J}}_{2}$) is calculated within the framework of the cluster-variation method. The Hamiltonian is solved within the nine-point-cluster approximation. In contrast to previous studies we find that the phase transition from the superantiferromagnetic to the paramagnetic state is of first order in the range 0.5R=${\mathit{J}}_{1}$/${\mathit{J}}_{2}$1.144. The non-Ising critical behavior observed in that range of interactions might be due to that phenomenon.

Response of Fully-Frustrated Ising Square Lattice to a Type of Symmetry Breaking Field

The fully-frustrated Ising system without the frozen spin state shows the diverging behavior with the critical index δ=3 in the response to a replica type of field at low temperature. The replica field provides a relevant external field for a specific Villain-Stephenson universality class treated recently by Horiguchi, Tanaka and Morita

Response of Fully-Frustrated Ising Square Lattice to a Type of Symmetry Breaking Field

Response of Fully-Frustrated Ising Square Lattice to a Type of Symmetry Breaking Field

Corner transfer matrices and corner magnetization for the Ising model

The authors consider the corner magnetization of an Ising square lattice with free edges along a diagonal direction. Using some remarkable properties of the corner transfer matrix, they give a unified treatment for various opening angles. The calculation of the magnetization is reduced to the solution of a matrix equation. In the thermodynamic limit they construct an explicit form of this equation and discuss its numerical and asymptotic solution and the results obtained thereby.

Critical wetting in the square Ising model with a boundary field

The Ising square lattice with nearest-neighbor exchangeJ>0 and a free surface at which a boundary magnetic fieldH1 acts has a second-order wetting transition. We study the surface excess magnetization and the susceptibility ofL×M lattices by Monte Carlo simulation and probe the critical behavior of this wetting transition, applying finite-size scaling methods. For the cases studied, the results are not consistent with the presumably exactly known values of the critical exponents, because the asymptotic critical region has not yet been reached. Implication of our results for critical wetting in three dimensions and for the application of the present model to adsorbed wetting layers at surface steps are briefly discussed.

The Ising square lattice in aL�M geometry: A model for the effect of surface steps on phase transitions in adsorbed monolayers

Critical phenomena in adsorbed monolayers on surfaces are influenced by limited substrate homogeneity, such as surface steps. We consider the resulting finite-size and boundary effects in the framework of a lattice gas system with nearest neighbor attraction in aL×M geometry, with two free boundaries of lengthM≫L, and periodic boundary conditions in the other direction (along the direction of the steps). This geometry thus models a “terrace” of the stepped surface, and adatoms adsorbed on neighboring terraces are assumed to be non-interacting. Also the effect of boundary “fields” is considered (describing the effects of missing neighbors and changed binding energy to the substrate near the boundary). Extensive Monte Carlo calculations on this model performed on a multi-transputer system are presented and analyzed in terms of phenomenological finite size scaling concepts. The fact that two scaling variables occur (ζ/L,L/M, with ζ being the correlation length in the bulk) is demonstrated explicitly. In the absence of boundary fields, the system forM≫L orders nearTc in a domain state, with domain walls running across the terrace, while at some temperature belowTc a transition to a monodomain state occurs. This domain state slightly belowTc is suppressed, however, by rather weak boundary fields. These results are interpreted in terms of exact theoretical predictions.

Magnetic phases in Ising square lattices with mixed bonds.

${\mathrm{CuO}}_{2}$ planes are modeled by means of a square lattice with antiferromagnetic coupling provided by the oxygen between two copper ions and ferromagnetic coupling when the oxygen is absent. The magnetic interaction is described using an Ising Hamiltonian with interacting first-nearest neighbors. The ferromagnetic exchange interaction is supposed to be larger than the antiferromagnetic one. A Monte Carlo routine is then defined in order to minimize the energy and to calculate physical parameters such as correlation to nearest neighbors, time correlation to the same site, and a schematic of a phase diagram.

Phase Diagram for the Simple Antiferromagnetic Ising Model Using Constant Coupling and Cluster Variation Approximations

AbstractAn explicit formula is derived for the critical curve in the H vs. T plane of the two‐dimensional antiferromagnetic Ising square, three‐dimensional antiferromagnetic simple cubic and body‐centered (b.c.c.) lattices and the critical data are evaluated by using constant coupling (CC) and cluster variation (CV) approximations. In the case of H = 0, Onsager's exact solution is obtained for the square lattice. In the case of H ≠ 0, the critical curve dividing antiferromagnetic and paramagnetic regions for each lattice is obtained and compared with other approximations in the literature.