Critical Behavior of the Blume-Emery-Griffiths Model for a Simple Cubic Lattice on the Cellular Automaton

The spin-1 Ising model with the nearest-neighbour bilinear and biquadratic interactions and single-ion anisotropy is simulated on a cellular automaton which improved from the Creutz cellular automaton (CCA) for a simple cubic lattice. The simulations have been made for several k=K/J and d=D/J in the 0≤d<3 and −2≤k≤0 parameter regions. We confirm the existence of the re-entrant and the successive re-entrant phase transitions near the phase boundary. The phase diagrams characterizing phase transitions are presented for comparison with those obtained from other calculations. The static critical exponents are estimated within the framework of the finite-size scaling theory at d=0, 1 and 2 in the interval −2≤k≤0. The results are compatible with the universal Ising critical behavior.     

Non-equilibrium relaxation study of the ±J Ising model with biquadratic exchange interaction

The spin-1 ±J Ising model with uniform biquadratic couplings on a simple cubic lattice is studied by the Monte Carlo simulation using the non-equilibrium relaxation method. The reentrant phase transition induced by competition between the bilinear and biquadratic couplings is eliminated gradually with increasing randomness of bilinear couplings and disappears entirely in the strong random system. The dynamic exponent of ferromagnetic transition shows non-universal behavior with changing randomness, while this behavior is not observed in the case of staggered quadrupolar transition.     

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 magnetic dipole and magnetoelastic interactions on the phase states of 2D non-Heisenberg ferromagnetic with complex exchange interactions

The phase states of the 2D non-Heisenberg ferromagnetic with anisotropic bilinear and biquadratic exchange interactions are investigated. The limiting cases of the system under consideration are the two-dimensional XY-model with biquadratic exchange interaction and the isotropic Heisenberg ferromagnetic. The account of the magnetic dipole interaction leads to the realization of spatially inhomogeneous quadrupolar phase. The stability regions of various phase transitions for different values of the material parameters are studied. The phase diagram is built. Besides, the temperature phase transitions are investigated. The influence of the magnetoelastic interaction on the formation of the long-range quadrupolar order is determined.     

Staggered Quadrupolar Phase and Bicritical Point of Spin-1 Bond and Anisotropy Dilution Blume-Emery-Griffiths Model

Within the effective field theory (EFT), staggered quadrupolar phase and bicritical point of spin-1 bond and anisotropy dilution Blume-Emery-Griffiths model is studied on simple cubic lattice in the restricted range of biquadratic interaction and bilinear interaction ratio α≤-1. The phase diagrams present a line of staggered quadrupolar-paramagnetic (SQ-P) phase and a line of ferromagnetic-paramagnetic (F-P) phase, separated by a bicritical point (BCP). A large negative ratio and two different dilution factors magnify the range of the SQ phase and reduce range of ferromagnetic phase in T-α or T-D plane. These parameters can assist the reentrant behavior of the SQ-P line and suppress that of the F-P line. The influence of bond dilution on the BCP is dissimilar to that of anisotropy dilution. There is a degenerate pattern at ground state in T-D plane. Some results obtained by the pure BEG model is in qualitative agreement with the results of Monte Carlo simulations.     

Equilibrium properties of a spin-1 Ising system with bilinear,biquadratic and odd interactions

The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, α = J/K; therefore, the dependence of the nature of the phase transition on α is investigated extensively and it is found that for α ⩽ 1 and α ⩾ 2000 a second-order phase transition occurs, and for 1 < α < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of α calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.     

The lock-in and reorientational phase transitions in metallic superlattices

We show that in the metallic magnetic superlattices, along with the bilinear RKKY-like exchange, there is significant biquadratic contribution arising from intrinsic and fluctuational mechanisms. Within the phenomenological approach, we postulate a particular form of free energy density functional and study the thermodynamic behaviour of the system under consideration. The model permits a transparent interpretation of the phase transitions, observed in metallic multilayers, with multiple oscillation periods and biquadratic exchange effects taken into account.     

Surface-induced ordering in thin uniaxial liquid crystal films

The interface localization transition in thin uniaxial liquid crystal films with competing surface fields has been studied using Metropolis Monte Carlo simulations. The model is constructed from a lattice of continuously orientable interacting spins, and the Hamiltonian contains both bilinear and biquadratic contributions. The biquadratic contribution to the Hamiltonian is familiar from the Lebwohl-Lasher model, and accounts for the particle anisotropy in a liquid crystal. The head-tail asymmetry of the molecules in a uniaxial liquid crystal is taken into account through a bilinear contribution familiar from the classical ferromagnetic Heisenberg model with exchange anisotropy Lambda. The critical temperature T(c), characterizing the interface localization transition within the uniaxial liquid crystal film, depends strongly on the relative magnitudes of the bilinear and biquadratic interactions between the spins. For systems dominated by the biquadratic interaction, T(c) is found to be close to the bulk critical temperature of the system. But as the biquadratic interaction strength is reduced, T(c) departs markedly from the bulk critical temperature of the system.     

Numerical study of the mixed spin-1 and spin-5/2 BEG model on the Bethe lattice

The mixed spin-1 and spin- \frac52\frac{5}{2} ferromagnetic Ising model with bilinear (J) and biquadratic (K) nearest-neighbor exchange interactions and a single-ion potential or crystal-field interaction (D) is studied on the Bethe lattice by means of exact recursion equations. First, the phase diagram of the system at zero temperature is obtained in the (D/Jq, K/Jq) plane, where q denotes the coordination number of the lattice. Second, the sublattice magnetizations as functions of the temperature, the crystal-field and the biquadratic interaction strengths are thoroughly investigated. For q = 3, the resulting phase diagrams show first and second order phase transitions as well as compensation points where the net magnetization of the whole lattice should vanish in the antiferromagnetic version of the model. One interesting feature of the model concerns the presence of tricritical points. Our calculations show that at non-zero temperature, none of the sublattice can order separately. However, under an external magnetic field, some interesting phase diagrams with partially ordered phases arise.     

The ferrimagnetic phase of the Blume-Emery-Griffiths model on the cellular automaton

The Blume-Emery-Griffiths model is simulated using the cooling algorithm which is improved from the Creutz cellular automaton (CCA) under periodic boundary conditions. The simulations are carried out on a simple cubic lattice at K/J = −1.5 in the range of −3.5 < D/J < 0.5, with J and K representing the nearestneighbour bilinear and biquadratic interactions, D being the single-ion anisotropy parameter. The phase diagram characterizing phase transition of the model is obtained. We found different kinds of phase transitions between the ferromagnetic, quadrupolar, staggered quadrupolar and ferrimagnetic phases for K/J = −1.5. In particular, the region of the phase diagram containing a ferrimagnetic phase is explored and compared to those obtained by other methods. The simulations confirm that the ferrimagnetic phase occurs in the narrow interval −3.006 ≤ D/J < −3. This result is in a good agreement with Monte Carlo renormalization group and closer to the cluster variation method result than the mean field approximation result.      

The Glauber dynamics for a spin-1 metamagnetic Ising system with bilinear and biquadratic interactions

We present a study, within a mean-field approximation, of the dynamics of a spin-1 metamagnetic Ising system with bilinear and biquadratic interactions in the presence of a time-dependent oscillating external magnetic field. First, we employ the Glauber transition rates to construct the set of mean-field dynamic equations. Then, we study the time variation of the average order parameters to find the phases in the system. We also investigate the thermal behavior of dynamic order parameters to characterize the nature (first- or second-order) of the dynamic transitions. The dynamic phase transitions are obtained and the phase diagrams are constructed in two different the planes. The phase diagrams contain a disordered and ordered phases, and four different mixed phases that strongly depend on interaction parameters. Phase diagrams also display one or two dynamic tricritical points, a dynamic double critical end and dynamic quadruple points. A comparison is made with the results of the other metamagnetic Ising systems.     

A Two-Sublattice Non-Heisenberg Magnet with <Emphasis Type="Italic">S</Emphasis> = 1 and Complex Interion Anisotropy

Static and dynamic properties of a two-sublattice non-Heisenberg magnet with complex interion anisotropy of both bilinear and biquadratic exchange interactions are investigated. Existence conditions are found for phases with the dipole order parameter (AFM phases) and tensor parameters (OQU phases), as well as for intermediate states characterized by both vector and tensor parameters. Conditions for phase transitions and their types are determined.     

Phase transitions as a function of material constants and temperature in intermetallic compounds of the terfenol-D type

A model of magnetic and magnetoelastic properties of intermetallic compounds has been considered with the inclusion of the influence of the “giant” magnetoelastic coupling and the biquadratic exchange interaction. The phase transitions as a function of material constants and temperature have been investigated in the framework of the proposed model. It has been demonstrated that the ferromagnetic and quadrupole phases can be formed in the system under consideration. In this case, the phase transition between these phases is a first-order transition and occurs through the intermediate, i.e., quadrupole-ferromagnetic, state. The dependences of the phase transition temperature on the Heisenberg and biquadratic exchange interaction constants have been obtained for compounds of the terfenol-D type.     

Dynamic Behavior of a Spin- 1 Ising Model. I. Relaxation of Order Parameters and the “Flatness” Property of Metastable States

The dynamic behavior of a spin-1 Ising system with arbitrary bilinear and biquadratic pair interactions is studied by using the path probability method, and approaches of the system toward the stable or metastable equilibrium states according to the ratio of interaction parameters and rate constants are presented. In particular, we investigate the relaxation of the order parameters for temperatures less than, equal to, and greater than the second-order and first-order phase transitions. From this investigation, the “flatness” property of metastable states is seen explicitly. We also show how a system freezes in a metastable state as well as how it escapes from one metastable state to the other.     

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.     

Monte Carlo simulation of the three-dimensional XY model with bilinear–biquadratic exchange interaction

The three-dimensional XY model with bilinear–biquadratic exchange interactions J and J′, respectively, has been studied by Monte Carlo simulations. From the detailed analysis of the thermal variation of various physical quantities, as well as the order parameter and energy histogram analysis, the phase diagram including two different ordered phases has been determined. There is a single phase boundary from a paramagnetic to a dipole–quadrupole ordered phase, which is of second order in a high J/J′ ratio region, changing to a first-order one for 0.35⩽J/J′⩽0.5. Below J/J′=0.35 there are two separate transitions: the first one to the quadrupole long-range order (QLRO) phase at higher temperatures, followed by another one to the dipole–quadrupole long-range order (DLRO) phase at lower temperatures. The finite-size scaling analysis yields values of the critical exponents for both the DLRO and QLRO transitions close to the values for the conventional XY model which includes no biquadratic exchange.     

A renormalization-group study of an Ising spin-glass with annealed vacancies

Systems driven and characterized by fluctuations in density and magnetization can be realistically modeled using the Blume–Emery–Griffiths model; a spin-1 Ising model with bilinear, biquadratic, and crystal-field interactions. In this study, renormalization-group techniques are used on an exactly solvable system in which frustration is present due to competing ferromagnetic and antiferromagnetic interactions. Thus, this calculation models a spin-glass system with annealed vacancies. To determine the effects of these competing bilinear interactions, an exactly solvable frustrated hierarchical model has been constructed, similar to those introduced to study spin glasses [S.R. McKay, A.N. Berker, S. Kirkpatrick, Phys. Rev. Lett. 48 (1982) 767]. Phase diagrams have been calculated for a series of planes of constant biquadratic coupling while varying the temperature and concentration of annealed vacancies in the system. In addition, a phase diagram was produced for constant concentration of annealed vacancies as the biquadratic coupling (i.e. clustering bias) was varied. Each phase diagram reveals three qualitatively unique basins of attraction, each corresponding to a phase distinguished by a unique renormalization-group trajectory. The sink of each trajectory is interpreted to determine the nature of each phase: dense paramagnetic, dilute paramagnetic and spin-glass.     

A renormalization-group study into the effects of competing biquadratic and crystal-field interactions in the Blume–Emery–Griffiths model

The Blume–Emery–Griffiths model, a spin-1 Ising model with bilinear, biquadratic, and crystal field interactions, provides a general system for the analysis of systems driven by fluctuations in density and magnetization. In this study, we consider an exactly solvable system in which frustration is present due to competing biquadratic and crystal-field interactions. Thus, this calculation models a dilute ferromagnetic material with two types of nearest-neighbor site pairs, distinguished by whether or not simultaneous occupation is energetically favored. To determine the effects of this competition, we have constructed exactly solvable frustrated hierarchical models similar to those introduced to study spin glasses. The resulting phase diagrams reveal two distinct paramagnetic phases separated by a plane in parameter space in which the biquadratic interaction and crystal-field strength rescale chaotically. Each paramagnetic phase has a ferromagnetic complement in which the unique distribution of occupied sites possesses a net magnetization.     

Biquadratic exchange interactions in the Europium monochalcogenides

We have examined the biquadratic exchange interaction strengths in the Europium monochalcogenides EuO, EuS, EuSe and EuTe using magnetization data of the paramagnetic phase and elaborate the consequences this additional interaction mechanism has on the magnetic phase diagrams of EuSe and EuTe. It is shown that the cubic susceptibility χ₃ obeys a Curie-Weiss law at suffciently high temperatures and that the associated Curie-Weiss temperature θ₃ is a measure for the biquadratic interaction strength. For all these materials the biquadratic interaction is ferromagnetic (θ₃ > 0). This leads to a conflicting situation in the case of EuTe for which θ₁ < 0. We attribute the peculiar observation, that the MnO superstructure reflection intensities as observed with neutron scattering correspond only to 0.6 of that moment expected for perfect magnetic order, to the presence of biquadratic interactions. The critical field B_c follows a T² law in the spin-wave regime (T < 0.8 K) for EuTe and EuSe but for these two materials with an antiferromagnetic ground state the cubic susceptibility χ₃ diverges at a temperature T* which is 2.5 K and 1.2 K above the ordering temperature, respectively. In the temperature range T_c < T < T* the magnetization curves exhibit some weak but definite anomaly which might be interpreted as a field-induced transition into the ferromagnetic state. A new multicritical point has been identified along the critical field curve B_c of EuSe.     

Absence of re-entrant phase transition of the antiferromagnetic Ising model on the simple cubic lattice: Monte Carlo study of the hard-sphere lattice gas

We perform Monte Carlo simulations of the hard-sphere lattice gas on the simple cubic lattice with nearest neighbour exclusion. The critical activity is estimated, z_c = 1.0588 ± 0.0003. Using a relation between the hard-sphere lattice gas and the antiferromagnetic Ising model in an external magnetic field, we conclude that there is no re-entrant phase transition of the latter on the simple cubic lattice.