Blume-Emery-Griffiths Research Articles

PHASE DIAGRAMS OF THE SPIN-1 BLUME–EMERY–GRIFFITHS MODEL IN A RANDOM TRANSVERSE FIELD

The effect of the random quantum transverse field Ω on the tricritical behavior of the spin-1 Blume–Emery–Griffiths (BEG) model is studied using an effective field theory. It is found that the tricritical behavior depends on both the biquadratic interaction K, single-ion anisotropy Δ and the concentration p of the disorder of Ω. Indeed, there exists a special value p1 of the probability p below which the tricritical behavior disappears. In addition, at sufficiently low temperatures, the system exhibits long-range order accompanied by the tricritical behavior below a special value p2 of the probability p.

EQUILIBRIUM AND NONEQUILIBRIUM BEHAVIOR OF THE BLUME–EMERY–GRIFFITHS MODEL USING THE PAIR APPROXIMATION AND PATH PROBABILITY METHOD WITH THE PAIR DISTRIBUTION

Equilibrium properties of Blume–Emery–Griffiths (BEG) model with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) are studied on a body centered cubic lattice by using the pair approximation of the cluster variation method, which is identical to the Bethe approximation. The thermal variation of the stable, metastable and unstable dipole and quadrupolar moment order parameters are found for various values of the two different coupling parameters J/K and D/K. The phase transitions of the stable, metastable and unstable branches of the order parameters are investigated extensively. The critical temperatures in the case of a second-order phase transition are obtained for different values of J/K and D/K by the Hessian determinant. The first-order phase transition temperature is determined by matching the values of the two branches of the free energy values. On the other hand, nonequilibrium behavior of the model is studied by using the path probability method with the pair distribution and the set of nonlinear differential equations, which is also called the dynamic or rate equations, is obtained. The solution of the dynamic equations is expressed by means of flow diagrams. The stable, metastable and unstable solutions are shown and the "overshooting" phenomenon is seen in the flow diagrams, explicitly. The role of the unstable points, as separators between the stable and the metastable points, is described and how a system freezes in a metastable state is also investigated, extensively. Moreover, the dynamic equations are also solved by using the Runge–Kutta method in order to study the relaxation of order parameters. In this investigation, the "flatness" property of metastable states and the "overshooting" phenomenon are seen explicitly.

Monte Carlo study of the Blume-Emery-Griffiths model at the ferromagnetic-antiquadrupolar-disordered phase interface

In this paper, we give high precision Monte Carlo (MC) results for the Blume-Emery-Griffiths (BEG) model at the ferromagnetic-antiquadrupolar-disordered phase interface. The data are analyzed by the multiple histogram technique. We show that this point of the phase diagram presents a highly degenerate ground-state with a residual entropy at temperature T=0 taking the values S ∞ = 0.7670′0.0005 and 0.7048′0.0008 for the two and the three-dimensional systems, respectively. We prove that the system with dimension d=2 does not present any re-entrant behavior, while a re-entrance occurs in the system d=3 with a phase transition at T c =1.036′0.007. Results for a multilayer system are also presented. They show that the re-entrance begins to appear for M≥5, where M is the number of layers. Finally, the staggered quadrupolar phase is shown to be unstable for d = 2 and d = 3 as well.

Thermodynamic versus statistical nonequivalence of ensembles for the mean-field Blume–Emery–Griffiths model

We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume–Emery–Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canonical sets of equilibrium distributions of states of the BEG model, we show that for values of the mean energy where the microcanonical entropy is nonconcave, the microcanonical distributions of states are nowhere realized in the canonical ensemble. Moreover, we show that for values of the mean energy where the microcanonical entropy is strictly concave, the equilibrium microcanonical distributions of states can be put in one-to-one correspondence with equivalent canonical equilibrium distributions of states. Our numerical computations illustrate general results relating thermodynamic and statistical equivalence and nonequivalence of ensembles proved by Ellis et al. (J. Stat. Phys. 101 (2000) 999).

Relaxation phenomena in the Blume–Emery–Griffiths model near the critical and multicritical points

AbstractThe relaxation behaviour of the Blume–Emery–Griffiths (BEG) model Hamiltonian with bilinear and biquadratic nearest‐neighbour ferromagnetic exchange interaction and crystal field interactions is studied by using the molecular field approximation (MFA) and Onsager's theory of irreversible thermodynamics. First the equilibrium behaviour and the phase diagrams of the model in MFA are given briefly in order to understand the relaxation behaviour. Then, Onsager theory is applied to the model and the set of linear differential equations, which is also called the kinetic or rate equations, is obtained. By solving these equations a set of relaxation times is calculated and examined for temperatures near the critical and multicritical points. It is found that one of the relaxation times (τ1) is sharply cusped at the triple and critical points, whereas it approaches infinity near the critical‐end point. On the other hand, the other relaxation time (τ2) displays a jump‐discontinuity behaviour at the triple and critical points but it has a cusp at the critical‐end point. A maximum of τ2 has also been observed above the critical‐end point. Moreover, similar calculations are performed for non‐zero magnetic field (H) to investigate the effect of H on relaxation times. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Blume–Emery–Griffiths model at an infinitely many ground states interface and exponential decay of correlations at all non-zero temperatures

In this paper we study the spin–spin correlation function decay properties of the Blume–Emery–Griffiths (BEG) model with Hamiltonian located on the interface between the disordered and the anti-quadrupolar phases. On this interface, the BEG model has infinitely many ground state configurations. We show that, for any dimension d, there exists a parameter value, yd, below which the spin–spin correlation function with zero boundary condition decays exponentially fast at all non-zero temperatures. This result suggests that reentrant behaviour predicted by mean-field and numerical calculations may be absent for those values of parameters.

Layering sublimation transitions of the spin-1 Blume–Emery–Griffiths model in a transverse field

The effect of the transverse field Ω on bulk melting and layering sublimation transitions of a spin-1 Blume–Emery–Griffiths (BEG) model is studied using the mean field theory. It is found that there exists a critical value of the transverse field Ω l above which a sequence of layering sublimation occurs before melting bulk transition. Ω l depends on the value of temperature T and chemical potential Δ. Indeed, Ω l decreases with increasing temperature and/or decreasing chemical potential. Phase diagrams in the ( Ω–T) and ( Ω– Δ) planes are established. However, the increase of the transverse field favours the layering sublimation and bulk melting transitions. Moreover, the ( Ω–T) phase diagrams show that the sequence of the critical end points go to the roughness transverse field Ω R and the roughness temperature T R which depend on the value of the chemical potential. Hence, Ω R increases and T R decreases when Δ increases. The profiles of densities and magnetizations as a function of the chemical potential, transverse field and the depth of the film are calculated.

THE DYNAMICS OF THE BLUME–EMERY–GRIFFITHS MODEL

The Fock-space method is presented to analyze the dynamics of the Blume–Emery–Griffiths (BEG) model including both spin-flip and spin-exchange processes with weighted transition rates. In this connection, the master equation on a lattice is formulated in a Quantum–Hamiltonian technique with second quantized operators. Introducing raising and lowering Para-Fermi operators for spin S = 1 the master equation is established for a three-state model where two states represent two different kinds of particles and the remaining state corresponds to an empty state. The coupled dynamical equations for the particle density and the local relative composition are derived including fluctuation corrections in lowest order of a gradient expansion. Although the underlying dynamics are subjected to the exclusion principle the resulting equations of motion may be classified according to the scheme due to Halperin and Hohenberg where the field-dependent kinetic coefficients are given explicitly. The homogeneous stationary solutions of these dynamical equations correspond to the mean-field approximation of the BEG Hamiltonian. For a special case, the diluted kinetic Ising model, phase separation is observed below a characteristic temperature. Furthermore, the crossover between thermal and non-thermal driven processes is discussed.

Ferrimagnetic phases of the Blume-Emery-Griffiths model and the Potts model on the diamond lattice

The phase transitions in the Blume-Emery-Griffiths (BEG) model and antiferromagnetic Potts model on the diamond lattice are investigated using the cluster-variation method in the pair approximation. The ferrimagnetic phases are found to be different from those on the simple-cubic lattice. The phase diagrams of the BEG model are also calculated. In the vicinity of the parameter line where the BEG model reduces to the three-state antiferromagnetic Potts model, new types of phase diagram are obtained. The results are different from those of the mean-field theory, which is a good approximation only for large coordination number of the lattice.

Heterochirality in Langmuir monolayers and antiferromagnetic Blume–Emery–Griffiths model

Chirality is quite a common property in organic molecules. Chiral molecules exist in two forms (called enantiomers) which cannot be superimposed by rotations and translations and may have very different biochemical properties. Experiments on Langmuir monolayers made up of chiral amphiphiles have shown that, in most cases, heterochiral interactions dominate, implying the formation of a racemic compound. It has been shown that a monolayer mixture of two enantiomers can be simply described by a two-dimensional (2-D) spin-1 lattice gas [Blume–Emery–Griffiths (BEG) model], where the heterochiral preference is represented by an effective antiferromagnetic coupling. By now just mean field calculations have been performed on this model. Here we present a revisitation of the tripod amphiphile model, proposed by Andelman and de Gennes [D. Andelman and P.-G. de Gennes, Compt. Rend. Acad. Sci. (Paris) 307, 233 (1988)], together with a rigorous proof of the heterochiral preference shown by the model in the hypothesis of van der Waals interactions. Moreover, a cluster variation analysis of the antiferromagnetic BEG model on a triangular lattice is performed and possible interpretations in terms of surface pressure–concentration phase diagrams for monolayer mixtures of enantiomers are discussed. The choice of a triangular lattice has been suggested by the triangularlike structure of condensed phases of Langmuir monolayers, shown by x-ray diffraction experiments.

Equivalence of thep-degenerate and ordinary Blume-Emery-Griffiths models

The p-degenerate Blume-Emery-Griffiths (BEG) model, introduced by Vives, Cast\'an, and Lindg\aa{}rd in the context of martensitic transitions, is equivalent to the ordinary BEG model with the crystal field \ensuremath{\Delta} shifted by $\mathrm{kT}\mathrm{ln}p.$

Stable, metastable and unstable solutions of the Blume–Emery–Griffiths model

The temperature dependence of the magnetization and quadrupole order parameters of the Blume–Emery–Griffiths (BEG) model Hamiltonian with the nearest-neighbor ferromagnetic exchange interactions [both bilinear ( J) and biquadratic ( K)] and crystal field interaction ( D) is studied using the lowest approximation of the cluster variation method. Besides the stable solutions, metastable and unstable solutions of the order parameters are found for various values of the two different coupling parameters, α= J/ K and γ= D/ K. These solutions are classified using the free energy surfaces in the form of a contour map. The phase transitions of the stable, metastable and unstable branches of the order parameters are investigated extensively. The critical temperatures in the case of a second-order phase transition are obtained for different values of α and γ calculated by the Hessian determinant. The first-order phase transition temperatures are found using the free energy values while increasing and decreasing the temperature. The temperature where both the free energies equal each other is the first-order phase transition temperature. Finally, the results are also discussed for the Blume–Capel model which is the special case of the BEG model.

Mean-field study of the degenerate Blume–Emery–Griffiths model in a random crystal field

The degenerate Blume–Emery–Griffiths (DBEG) model has recently been introduced in the study of martensitic transformation problems. This model has the same Hamiltonian as the standard Blume–Emery–Griffiths (BEG) model but, to take into account vibrational effects on the martensitic transition, it is assumed that the states S=0 have a degeneracy p ( p=1 corresponds to the usual BEG model). This model was studied by E. Vives et al. for a particular value of Δ, through a mean-field approximation and numerical simulation. When the parameter p increases, the ferromagnetic phase shrinks and the region where the transition is of first order increases. In some materials, however, the transition would be better described by a disordered DBEG model; further, the inclusion of disorder in the DBEG model may be relevant in the study of shape memory alloys. From the theoretical point of view, it would be interesting to study the consequence of conflicting effects: the parameter p, which increases the first-order phase-transition region, and disorder in the crystal field, which tends to diminish this region in three dimensions. In order to study this competition in high-dimensional systems, we apply a mean-field approximation: it is then possible to determine the critical behavior of the random DBEG model for any value of the interaction parameters. Finally, we comment on (preliminary) results obtained for a two-dimensional system, where the randomness in the crystal field has a more drastic effect, when compared to the three-dimensional model.

Exact correlation functions of Bethe lattice spin models in external magnetic fields

We develop a transfer matrix method to compute exactly the spin-spin correlation functions $〈{s}_{0}{s}_{n}〉$ of Bethe lattice spin models in the external magnetic field $h$ and for any temperature $T.$ We first compute $〈{s}_{0}{s}_{n}〉$ for the most general spin-$S$ Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin-$S$ Ising model includes the spin-$\frac{1}{2}$ simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. From the spin-spin correlation functions, we obtain functions of correlation length $\ensuremath{\xi}(T,h)$ for the simple Ising model and BEG model, which show interesting scaling and divergent behavior as $\stackrel{\ensuremath{\rightarrow}}{h}0$ and $T$ approaches the critical temperature ${T}_{c}.$ Our method to compute exact spin-spin correlation functions may be applied to other Ising-type models on Bethe and Bethe-like lattices.

Exact spin–spin correlation functions of Bethe lattice Ising and BEG models in external fields

We develop a transfer matrix method to compute exactly the spin–spin correlation functions for Bethe lattice Ising model and Blume–Emery–Griffiths (BEG) model in the external magnetic field h and for any temperature T. The correlation length ξ( T, h) obtained from the spin–spin correlation function shows interesting scaling and divergent behavior as h→0 and T approaches the critical temperature T c .

An effective field study of the staggered quadrupolar phase in the BEG model

The so-called new effective field theory, based on a finite cluster theory that correctly incorporates the single-site kinematic relations, is applied to the Blume Emery Griffiths (BEG) model with positive biquadratic interactions in order to test its ability to deal with magnetic systems having competing interactions. In particular, the region of the phase diagram containing a staggered quadrupolar phase is explored and comparison with the results of Monte Carlo simulation made. The results are also compared to those obtained by a cluster variation method in pair approximation.

Study of Blume–Emery–Griffiths model by a modified Bethe-Peierls method

We present here a study of the Blume–Emery–Griffiths (BEG) model for general spins using a modified Bethe–Peierls (BP) method calling it the Bethe–Peierls–Lines (BPL) method, incorporating the well known Lines approximation. We compare our results of BPL as well as BP methods for the ferro-to para-magnetic transition temperatures Tc with those available from other theories such as the Correlated-Effective-Field Theory (CEFT), Cluster Variation Method (CVM) and the Monte-Carlo (MC) method. We find that our BPL method gives results for variation of Tc with the single-ion uniaxial anisotropy (D) and biquadratic interaction (J') coefficients close to those obtained by CVM method for spin S = 1 on a honeycomb lattice. We also find that the BPL results are in good agreement for the cubic lattice for S = 3/2 and J' = 0 with those of the MC results, and for S = 1 and J' = −0.5J with those of the CVM results. For these three dimensional lattices our BPL method gives the correct reentrant behaviour for negative values of J'. We also compare our BP and BPL results for magnetic susceptibility with those of the CEFT for S = 3/2 and 3 nearest neighbours and find fairly good agreement for D > 0.

Phase Diagrams of a Double Layer with Spin 1

The anisotropic Blume-Emery-Griffiths (B-E-G) model with single-ion anisotropy and biquadratic interactions has been applied for the description of very thin magnetic films with S = 1. According to the Bogolyubov minimisation principle, the expression for the Gibbs free energy has been derived in the frame of the Molecular Field Approximation (MFA), as a basis from which all thermodynamic properties can be studied. In particular, the method has been employed to study the phase diagrams of a double layer. The numerical results are presented in figures and a discussion is given.

Tri-critical behaviour of the BEG model on a diamond lattice

The critical behaviour of the Blume Emery Griffiths (BEG) spin 1 Ising model on a diamond lattice is studied within the framework of a finite cluster theory. An expansion technique for the cluster identities that allows the kinematic properties of the spin operators to be accounted for in a systematic way is employed. The location of the tri-critical points in temperature versus single-ion anisotropy phase space is determined for a range of biquadratic exchange interaction strengths.

Reduction of a Z(3) gauge theory on the flat lattices to the spin-1 BEG model

The Z(3) gauge model with double plaquette representation of the action on the flat triangular and square lattices is constructed. It is reduced to the spin-1 Blume-Emery-Griffiths (BEG) model. An Ising-type critical line of a second-order phase transition is found.