Griffiths Model Research Articles

Dynamic phase transition in the kineticspin-3/2 Blume–Emery–Griffiths model in an oscillating field

The dynamic phase transitions are studied, within a mean-field approach, inthe kinetic Blume–Emery–Griffiths model under the presence of a time varying(sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. Thebehaviour of the time-dependence of the order parameters and the behaviour of theaverage order parameters in a period, which is also called the dynamic orderparameters, as a function of reduced temperature, are investigated. The nature(continuous and discontinuous) of transition is characterized by studying theaverage order parameters in a period. The dynamic phase transition points areobtained and the phase diagrams are presented in the reduced magnetic fieldamplitude and reduced temperature plane. The phase diagrams exhibit one, two, orthree dynamic tricritical points and a dynamic double critical end point, andbesides a disordered and two ordered phases, seven coexistence phase regions exist,which strongly depend on interaction parameters. We also calculate the Liapunovexponent to verify the stability of solutions and the dynamic phase transition points.

Critical behaviors of bond and crystal field dilution Blume–Emery–Griffiths model

The critical behavior of the spin-1 bond and crystal field dilution Blume–Emery–Griffiths model have been investigated on simple-cubic lattice within the framework of the effective field theory. In particular, both bond dilution and random crystal field are considered at the same time. The interplay between bond and crystal field dilution constructs rich and interesting phase diagrams. Significant distinctions are exhibited. When positive ratio α changes in a certain range, there exist double tricritical points in phase-transition lines in T– D plane. Moreover, this first-order phase transition is enlarged with increasing of ratio α at a fixed crystal field dilution concentration, while this first-order phase transition will shrink when bond dilution concentration is fixed. In addition, we observe that there exist two bond percolation thresholds for negative crystal field and α>0 in T– P plane.

Ferrimagnetism Caused by Charge Ordering in Doped Antiferromagnetic Ising Spins

The emergence of ferrimagnetism caused by the charge ordering of commensurately doped charged Ising spins is proposed in the extended Ising model, equivalent to the Blume–Emery–Griffiths model. The stability of the ferrimagnetic state is discussed on the basis of the energy at zero temperature, the sublattice magnetizations from mean field approximation, and Monte Carlo simulation, as examples on triangular and square lattices. The magnitude of the ferrimagnetic moment is suggested in connection with the dimensionality of the lattice.

Multicritical behaviors of the antiferromagnetic Blume–Emery–Griffiths model with the external magnetic field on the Bethe lattice

The antiferromagnetic Blume–Emery–Griffiths model in an external magnetic field is studied by using the exact recursion relations on the Bethe lattice for arbitrary values of biquadratic and for negative values of bilinear interactions. We have studied the thermal variations of two-sublattice magnetizations belonging to spin-1 BEG model to obtain the phase diagrams on the ( H / | J | , kT / | J | ) plane. As a result, we have found that the system presents second- and first-order phase transitions, therefore, tricritical points for appropriate values of K / | J | , D / | J | and q. We have also found that the second-order phase transition lines exhibit reentrant phenomena in temperature, besides it also shows reentrant phenomena for the first-order phase lines in external magnetic field for q = 4 and 6.

APPLICATION OF THE GENETIC ALGORITHM TO BLUME–EMERY–GRIFFITHS MODEL: TEST CASES

The equilibrium properties of the Blume–Emery–Griffiths (BEG) model Hamiltonian with the arbitrary bilinear (J), biquadratic (K) and crystal field interaction (D) are studied using the genetic algorithm technique. Results are compared with lowest approximation of the cluster variation method (CVM), which is identical to the mean field approximation. We found the genetic algorithm to be very efficient for fast search at the average fraction of the spins, especially in the early stages as the system is far from the equilibrium state. A combination of the genetic algorithm followed by one of the well-tested simulation techniques seems to be an optimal approach. The curvature of the inverse magnetic susceptibility is also presented for the stable state of the BEG model.

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.

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.

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.

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.

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.