Dynamic phase transition in the kinetic spin-5/2 Blume–Emery–Griffiths model in an oscillating external magnetic field

Employing a mean-field approach, we study the stationary states of the kinetic spin-5/2 Blume–Emery–Griffiths (BEG) model under the presence of a time-varying (sinusoidal) magnetic field by using the Glauber-type stochastic dynamics. We employ the Glauber transition rates to construct the set of dynamic mean-field equations. We investigate the time variation in average order parameters to find the phases in the system, and the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the dynamic phase transtions and to the dynamic phase transition temperature. The dynamic phase diagrams are presented in three different planes. The phase diagrams contain the ferromagnetic-5/2, the ferromagnetic-3/2, the ferromagnetic-1/2, the ferroquadrupolar, and disordered fundamental phases. They also include the nine coexisting or mixed phases composed of binary and ternary combinations of fundamental phases that strongly depend on the interaction parameters. The phase diagrams display the critical end point, double critical end point, triple point, quadruple point, and one, two, or three special points and the dynamic tricritical point that depends on the interaction parameters.     

Multicritical dynamical phase diagrams of the kinetic Blume–Emery–Griffiths model with repulsive biquadratic coupling in an oscillating field

We study, within a mean-field approach, the stationary states of the kinetic Blume–Emery–Griffiths model with repulsive biquadratic coupling under the presence of a time-varying (sinusoidal) magnetic field. We employ the Glauber-type stochastic dynamics to construct set of dynamic equations of motion. The behavior of the time dependence of the order parameters and the behavior of the average order parameters in a period, which is also called the dynamic order parameters, as functions of the reduced temperature are investigated. The dynamic phase transition points are calculated and phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane. The dynamical transition from one regime to the other can be of first- or second order depending on the region in the phase diagram. According to the values of the crystal field interaction or single-ion anisotropy constant and biquadratic exchange constant, we find 20 fundamental types of phase diagrams which exhibit many dynamic critical points, such as tricritical points, zero-temperature critical points, double critical end points, critical end point, triple point and multicritical point. Moreover, besides a disordered and ordered phases, seven coexistence phase regions exist in the system.     

Dynamic phase transition and dynamic phase diagrams in the spin-5/2 Blume–Capel model under a time-dependent oscillating external field

We calculate the dynamic phase transition (DPT) temperatures and present the dynamic phase diagrams in the kinetic spin-5/2 Blume–Capel model under the presence of a time-dependent oscillating external magnetic field. First, we employ the Glauber transition rates to construct the mean-field dynamic equation. Then, we study the time variation of the average magnetization to find the phases in the system. We also investigate the behaviour of the dynamic magnetization to characterize the nature (continuous and discontinuous) of transition and to obtain the DPT points. We present the dynamic phase diagrams in two different planes. The phase diagrams include the ferromagnetic-5/2 (f_5/2), the ferromagnetic-1/2 (f_1/2) and paramagnetic (p) fundamental phases. In addition to these fundamental phases, we find 10 mixed phases, depending on the interaction parameters. The phase diagrams display many special points, such as a dynamic tricritical point, a double critical end point, a triple point and a quadruple point.     

Dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model in an oscillating field: the effective-field theory based on the Glauber-type stochastic dynamics

Using the effective-field theory based on the Glauber-type stochastic dynamics (DEFT), we investigate dynamic phase transitions and dynamic phase diagrams of the Blume–Emery–Griffiths model under an oscillating magnetic field. We presented the dynamic phase diagrams in (T/J, h₀/J), (D/J, T/J) and (K/J, T/J) planes, where T, h₀, D, K and z are the temperature, magnetic field amplitude, crystal–field interaction, biquadratic interaction and the coordination number. The dynamic phase diagrams exhibit several ordered phases, coexistence phase regions and special critical points, as well as re-entrant behavior depending on interaction parameters. We also compare and discuss the results with the results of the same system within the mean-field theory based on the Glauber-type stochastic dynamics and find that some of the dynamic first-order phase lines and special dynamic critical points disappeared in the DEFT calculation.     

Rates of Convergence in the Blume–Emery–Griffiths Model

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature $\beta $ and the interaction strength $K$ . The rates of convergence results are obtained as $(\beta ,K)$ converges along appropriate sequences $(\beta _n,K_n)$ to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein’s method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry–Esseen quality, on approximation error.     

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|)$(H/|J|,\mathit{kT}/|J|)$ 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|$K/\left|J\right|$, D/|J|$D/\left|J\right|$ 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$q=4$ and 6.     

Nonequilibrium behavior of the kinetic metamagnetic spin-5/2 Blume–Capel model

The dynamic magnetic behavior of the kinetic metamagnetic spin-5/2 Blume–Capel model is examined, within a mean-field approach, under a time-dependent oscillating magnetic field. To describe the kinetics of the system, Glaubertype stochastic dynamics has been utilized. The mean-field dynamic equations of the model are obtained from the Master equation. Firstly, these dynamic equations are solved to find the phases in the system. Then, the dynamic phase transition temperatures are obtained by investigating the thermal behavior of dynamic sublattice magnetizations. Moreover, from this investigation, the nature of the phase transitions(first- or second-order) is characterized. Finally, the dynamic phase diagrams are plotted in five different planes. It is found that the dynamic phase diagrams contain the paramagnetic(P),antiferromagnetic(AF5/2, AF3/2, AF1/2) phases and five different mixed phases. The phase diagrams also display many dynamic critical points, such as tricritical point, triple point, quadruple point, double critical end point and separating point.     

Magnetic properties of the bond and crystal field dilution spin-3/2 Blume–Capel model in an external magnetic field

Magnetic properties of the bond and crystal field dilution spin-3/2 Blume–Capel model in an external magnetic field (h)$(h)$ on simple cubic lattice are studied by using the effective field theory. In the m−T$m-T$ plane, the degeneracy of the magnetization (m)$(m)$ is affected by the concentration of bond or crystal field dilution at low temperature (T)$(T)$. The magnetization curves can appear to fluctuate in certain regions of negative crystal field. In the m−h$m-h$ plane, the initial magnetization curve has an irregular behavior due to the introduction of bond dilution. The crystal field dilution has the influence on the process of magnetic domain displacement. In the χ−h$\chi -h$ plane, there exists one susceptibility (χ)$(\chi )$ shoulder and one step for different negative crystal field. The susceptibility curve takes on the feature of multi-peaks distribution under bond and crystal field dilution conditions.     

Study of the first-order transition in the spin-1 Blume–Capel model by using effective-field theory

The spin-1 Blume–Capel model on a square lattice is studied by using an effective-field theory (EFT) with correlation. We propose an expression for the free energy within the EFT. The phase diagram is constructed in the temperature (T) and single-ion anisotropy amplitude (D) plane. The first-order transition line is obtained by Maxwell construction (comparison between free energies). Our results predict first-order transitions at low temperatures and large anisotropy strengths, which correspond in the phase diagram to the existence of a tricritical point (TCP). We compare our results with mean-field approximation (MFA), that show a qualitative correct behavior for the phase diagram.     

Metastability in the Spin-1 Blume–Emery–Griffiths Model within Constant Coupling Approximation

In this paper, the equilibrium properties of spin-1 Blume–Emery–Griffiths model are studied by using constant-coupling approximation. The dipolar and quadrupolar order parameters, the stable, metastable and unstable states and free energy of the model are investigated. The states are defined in terms of local minima of the free energy of system. The numerical calculations are presented for several values of exchange interactions on the simple cubic lattice with q = 6.     

Blume–Emery–Griffiths纳米管的热力学与相变性质

利用有效场理论研究了圆柱形纳米管上Blume-Emery-Griffiths系统的热力学与相变性质,得到了系统的磁化强度、磁化率、比热和相图.讨论了四次交换作用与二次交换作用的比值 与晶格场对系统热力学量和相图的影响.研究发现:系统存在三临界点,且三临界点由参数 和晶格场共同决定,即若确定了参数 ,则三临界点所对应的晶格场也能确定.随着参数 的增加,系统出现三临界点时所对应的温度和晶格场也相应增大.     

Influence of transverse field on the spin-3/2 Blume–Capel model on rectangular lattice

Transverse field effect on thermodynamic properties of the spin-3/2 Blume–Capel model on rectangular lattice in which the interactions in perpendicular directions differ in signs is studied within the mean field approximation. Phase diagrams in the (transverse field, temperature) plane are constructed for various values of single-ion anisotropy.     

The Blume–Emery–Griffiths neural network: dynamics for arbitrary temperature

The parallel dynamics of the fully connected Blume–Emery–Griffiths neural network model is studied for arbitrary temperature. By employing a probabilistic signal-to-noise approach, a recursive scheme is found determining the time evolution of the distribution of the local fields and, hence, the evolution of the order parameters. A comparison of this approach is made with the generating functional method, allowing to calculate any physical relevant quantity as a function of time. Explicit analytic formula are given in both methods for the first few time steps of the dynamics. Up to the third time step the results are identical. Some arguments are presented why beyond the third time step the results differ for certain values of the model parameters. Furthermore, fixed-point equations are derived in the stationary limit. Numerical simulations confirm our theoretical findings.     

Multilayer transition in a spin-1 Blume–Capel model with RKKY interaction and quantum transverse anisotropy

Using mean-field theory, we have studied the effect of quantum transverse anisotropies with RKKY interaction on the multi-layer transition and magnetic properties of the spin-1 Blume--Capel model of a system formed by two magnetic multi-layer materials, of different thicknesses, separated by a non-magnetic spacer of thickness M. It is found that the multilayer magnetic order--disorder transition temperature depends strongly on the value of the transverse anisotropy. The multilayer transition temperature decreases when increasing the transverse anisotropy. Furthermore, there exists a critical quantum transverse anisotropy Δ_xL beyond which the separate transitions occur in the two magnetic layers. The critical transverse anisotropy Δ_xL decreases (increases) on increasing the non-magnetic spacer of thickness M (on increasing the crystal field), and Δ_xL undergoes oscillations as a function of the Fermi level.     

Ferromagnetic phases due to competing short- and long-range interactions in the spin-5/2 and spin-2 Blume–Capel model

We broaden the study of the statistical physics of the spin-S Blume–Capel model with ferromagnetic mean-field interactions J in competition with short-range antiferromagnetic interactions K in a linear chain in the thermodynamic limit. This work is dedicated to the case when S takes the half-integer spin $S=5/2$ and when S assumes the integer value $S=2$. In both cases the phase diagrams exhibit new ferromagnetic phases (for certain values of K) enclosed by branches emerging from the first-order frontiers of the pure ferromagnetic model. For finite temperatures the complex topologies were obtained by numerical minimization of the free energy and some results were confirmed by Monte Carlo simulations.     

Random-bond and random-anisotropy effects in the phase diagram of the Blume–Capel model

We report on Monte Carlo studies of the influence of quenched randomness on the phase diagram of the three-dimensional (3D) Blume–Capel model. The randomness is supposed to act either on the exchange coupling constants (bond randomness) or on the anisotropy distribution. With increasing disorder, first-order phase transitions are shown to change into second-order phase transitions. The trajectory of the tricritical point in the phase space as a function of disorder is presented. We have also calculated critical exponents at some points in the second-order phase region which show a change of universality class in agreement with the Harris criterion.