Blume Emery Griffiths Model Research Articles

Multicritical behavior of the antiferromagnetic [formula omitted] Blume–Emery–Griffiths model

The multicritical behavior of the antiferromagnetic spin- 3 2 Blume–Emery–Griffiths model in an external magnetic field is studied within the lowest approximation of the cluster variation method. We have investigated the thermal variations of order parameters for different values of interaction parameters and external magnetic field and constructed the resulting phase diagrams. The model exhibits distinct critical regions, including the first-order, second-order tricritical point, double critical point and zero critical point. Comparison of the phase diagrams with closely related systems is made.

Analysis of phase transitions in the mean-field Blume–Emery–Griffiths model

In this paper we give a complete analysis of the phase transitions in the mean-field Blume-Emery-Griffiths lattice-spin model with respect to the canonical ensemble, showing both a second-order, continuous phase transition and a first-order, discontinuous phase transition for appropriate values of the thermodynamic parameters that define the model. These phase transitions are analyzed both in terms of the empirical measure and the spin per site by studying bifurcation phenomena of the corresponding sets of canonical equilibrium macrostates, which are defined via large deviation principles. Analogous phase transitions with respect to the microcanonical ensemble are also studied via a combination of rigorous analysis and numerical calculations. Finally, probabilistic limit theorems for appropriately scaled values of the total spin are proved with respect to the canonical ensemble. These limit theorems include both central-limit-type theorems when the thermodynamic parameters are not equal to critical values and non-central-limit-type theorems when these parameters equal critical values.

Multicritical behavior of the antiferromagnetic Blume–Emery–Griffiths model with the repulsive biquadratic coupling in an external magnetic field

We have studied the antiferromagnetic Blume–Emery–Griffiths model with the repulsive biquadratic coupling in an external magnetic field using the lowest approximation of the cluster variation method which is identical to the mean-field approximation. First, we have investigated the thermal variations of the sublattice magnetizations and obtained four different main topological types. Then, we have calculated the phase diagrams and five main different phase diagram topologies are found. Finally, the discussion and comparison of the phase diagrams are made.

THE METASTABLE PHASE DIAGRAM OF THE BLUME–EMERY–GRIFFITHS MODEL IN ADDITION TO THE EQUILIBRIUM PHASE DIAGRAM USING THE PAIR APPROXIMATION

As a continuation of our previously published work, the metastable phase diagram of the Blume–Emery–Griffiths model with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) is presented in addition to the equilibrium phase diagram in (T/K, J/K) and (T/K, D/K) plane by using the pair approximation of the cluster variation method on a body centered cubic lattice. We also calculate the phase transitions for the unstable branches of order parameters. The calculated first- and second-order phase boundaries of the unstable branches of the order parameters are superimposed on the equilibrium phase diagram and metastable phase diagram. It is found that the metastable phase diagram and the first- and second-order phase boundaries for unstable branches of order parameters always exist at low temperatures, which are consistent with the experimental and theoretical works.

The storage capacity of the Blume–Emery–Griffiths neural network

We analyse the so-called Blume–Emery–Griffiths (BEG) neural network at zero temperature. An upper bound on its storage capacity is given if we want the stored patterns to be fixed points of the retrieval dynamics. Besides, we discuss a more liberal notion of storage capacity introduced by Newman (1988 Neural Netw. 1 223–38) in the context of the Hopfield model (Hopfield 1982 Proc. Natl Acad. Sci. USA 79 2554–8). We show that, similar to the findings in the neural networks literature, the BEG model with this notion of storage capacity can store a number of patterns proportional to the number of neurons in the model.

Multicritical phase diagrams of the Blume–Emery–Griffiths model with repulsive biquadratic coupling including metastable phases: the pair approximation and the path probability method with pair distribution

As a continuation of the previously published work, the pair approximation of the cluster variation method is applied to study the temperature dependences of the order parameters of the Blume–Emery–Griffiths model with repulsive biquadratic coupling on a body centered cubic lattice. We obtain metastable and unstable branches of the order parameters besides the stable branches and phase transitions of these branches are investigated extensively. We study the dynamics of the model by the path probability method with pair distribution in order to make sure that we find and define the metastable and unstable branches of the order parameters completely and correctly. We present the metastable phase diagram in addition to the equilibrium phase diagram and also the first-order phase transition line for the unstable branches of the quadrupole order parameter is superimposed on the phase diagrams. It is found that the metastable phase diagram and the first-order phase boundary for the unstable quadrupole order parameter always exist at the low temperatures which are consistent with experimental and theoretical works.

The spin‐3/2 Blume–Emery–Griffiths model with repulsive biquadratic coupling in an external field

AbstractThe spin‐3/2 Blume–Emery–Griffiths model is solved on the Bethe lattice in the absence and presence of the external field using the iteration technique (recursion method). The phase diagrams are obtained for the coordination number q = 3, 4, 6 and 8. We confirm the existence of a staggered quadrupolar phase in addition to a ferrimagnetic and ferromagnetic ordered phases and analyze the occurence of re‐entrant boundaries inside the ordered region as a character of the coordination of the lattice. A comparison is made with the results of other approximate methods. (© 2004 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 nonzero temperatures

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

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.

Multicritical phase diagrams of the spin- [formula omitted] Blume–Emery–Griffiths model on the Bethe lattice using the recursion method

The multicritical behaviour of the spin- 3 2 Blume–Emery–Griffiths model with bilinear and biquadratic exchange interactions and single-ion crystal field is studied on the Bethe lattice by introducing two-sublattices A and B within the exact recursion equations. Exact expressions for the free energy, the Curie or second-order phase transition temperatures, as well as for the magnetization and quadrupolar moment order parameters are obtained. The general procedure of investigation of critical properties is discussed and phase diagrams are obtained, in particular, for negative biquadratic couplings. The phase diagrams of the model exhibits a rich variety of behaviours. Results are compared with other approximate methods.

Phase diagrams of the spin- [formula omitted] Blume–Emery–Griffiths model on the Bethe lattice using the recursion method

The statistical physics of the spin- 3 2 Blume–Emery–Griffiths model on the Bethe lattice is studied by using the exact recursion equations. Exact expressions for the free energy, the Curie or the second-order phase transition temperatures as well as the magnetization and quadrupolar moment order parameters are found. The phase diagrams in ( kT/ J, D/ J) and ( kT/ J, K/ J) planes are presented in depth for various values of constants K/ J and D/ J, respectively. The phase diagrams are discussed and a comparison is made with the results of the other approximation methods.

Phase diagrams of the Blume–Emery–Griffiths model with site dilution

Cluster variational method in pair approximation has been extended for the description of the site-diluted Blume–Emery–Griffths model with spin 1. The phase diagrams for the case of dilution are discussed in the presence of biquadratic interactions and single-ion anisotropy. In particular, the critical concentrations for the ferromagnetic and staggered quadrupolar phases, as well as the shift of tricritical point as a result of dilution have been studied. The numerical calculations are presented for a bilayer system with the coordination number z=5.

Mean-field critical behaviour for a fully frustrated Blume–Emery–Griffiths model

We present a mean-field analysis of a fully frustrated Ising spin model on an Ising lattice gas. This is equivalent to a degenerate Blume–Emery–Griffiths model with frustration, which we analyse for different values of the quadrupolar interaction. We find that the regular disorder shifts the tricritical point to lower temperatures.

The metastable phase diagram of the Blume–Emery–Griffiths model in addition to the equilibrium phase diagram

As a continuation of our previously published works, the metastable phase diagram of the Blume–Emery–Griffiths model is presented in addition to the equilibrium phase diagram by using the lowest approximation of the cluster variation method. We also calculate the phase transitions for the unstable branches of order parameters. The calculated first- and second-order phase boundaries of the unstable branches of the order parameters are superimposed on the equilibrium phase diagram and metastable phase diagram. It is found that the metastable phase diagram and the first- and second-order phase boundaries for the unstable branches of order parameters always exist at the low temperatures, which are consistent with the experimental and theoretical works.

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.

Cluster variational theory of spin [formula omitted] Ising models

A cluster variational method for spin 3 2 Ising models on regular lattices is presented that leads to results that are exact for Bethe lattices of the same coordination number. The method is applied to both the Blume–Capel (BC) and the isotropic Blume–Emery–Griffiths model (BEG). In particular, the first-order phase line separating the two low-temperature ferromagnetic phases in the BC model, and the ferrimagnetic phase boundary in the BEG model are studied. Results are compared with those of other theories whose qualitative predictions have been in conflict.