Random crystal field effects for spin-3/2 Blume-Capel model

The effects of crystal field (Δ) which is randomly turned on with probability p or turned off with probability 1−p on the sites of the Bethe lattice are investigated to study the critical behavior of the spin-3/2 Blume-Capel model. The order-parameters are obtained in terms of the recursion relations and then the possible phase diagrams are calculated at finite temperatures for given system parameters p, Δ and coordination numbers z=3, 4, 5 and 6. It was found that the phase diagrams change drastically for given probability and randomness. The first-order phase transitions are only found for higher p values and the lines of which get shorter and eventually disappear as p decreases. The critical lines do not present any qualitative differences with z, but they are seen at higher temperatures for higher z.     

Random Crystal Field Effects on the Mixed-Spin 1/2 and 5/2 Blume-Capel Model

The Bethe lattice approach is used to examine the effect of a random crystal field (RCF) on the mixed spin-1/2 and 5/2 Blume-Capel model. A bimodal form of RCF is considered which either turns the crystal field on or off for given probabilities p and 1 ? p, respectively, for the sites with spin-5/2. The exact recursion relations are employed to obtain all the characteristics of the model for given coordination numbers q =?3,4,5and 6. The thermal variations of the order-parameters are studied and net magnetizations are classified according to the usual letter coding. The phase diagrams are obtained on the probability-temperature and crystal field-temperature planes. It is found that the model yields only one compensation temperature for appropriate values of system parameters.     

The mixed-spins 1/2 and 3/2 Blume—Capel model with a random crystal field

The random crystal field(RCF) effects are investigated on the phase diagrams of the mixed-spins 1/2 and 3/2 Blume-Capel(BC) model on the Bethe lattice.The bimodal random crystal field is assumed and the recursion relations are employed for the solution of the model.The system gives only the second-order phase transitions for all values of the crystal fields in the non-random bimodal distribution for given probability.The randomness does not change the order of the phase transitions for higher crystal field values,i.e.,it is always second-order,but it may introduce first-order phase transitions at lower negative crystal field values for the probability in the range about 0.20 and 0.45,which is only the second-order for the non-random case in this range.Thus our work claims that randomness may be used to induce first-order phase transitions at lower negative crystal field values at lower probabilities.     

Mixed spin transverse Ising model with longitudinal random crystal field interactions

The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory. In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators. We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice (N=3) and a simple-cubic lattice (N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features resulting from the fluctuation of the crystal field interactions. Received: 13 February 1998 / Accepted: 17 March 1998     

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.     

Random crystal-field effects in a mixed spin-1/2 and spin-2 ferrimagnetic Ising system

The magnetic properties of a mixed spin ferrimagnetic system σ?=?1/2 and S?=?2 at the sites of a square lattice with a random crystal field are studied by the mean field approximation. The ground state of the system is determined and the total magnetization is plotted according to the model parameters. Different behaviors for the thermal dependence of the magnetization are highlighted. In particular, one sees the appearance of one or two compensating points and many types of phase diagrams with first and second order phase transition lines as well as isolated critical and tricritical points.     

The spin-1 Blume–Capel model on the Bethe lattice in ± J distribution with an adjustable parameter between FM and AFM phases

The spin-1 Blume–Capel (BC) model is studied on the Bethe lattice (BL) for the ?±? J distribution with a competing adjustable parameter α which alters the strength of bilinear exchange interaction parameter for the ferromagnetic phase (J?>?0) with respect to antiferromagnetic phase (J?<?0). The J?>?0 and αJ?<?0 values are also distributed throughout the BL with probabilities p and $1?p,$ respectively. The order-parameters are obtained on the BL in terms of exact recursion relations (ERR’s) and their temperature (T) variations are studied to calculate the phase diagrams on the (α, T) planes for given values of p, crystal field (D) and coordination number $q=3$ corresponding to honeycomb lattice. It is found that the model gives both first- and second-order phase transitions and also tricritical points. In addition to the well known ordinary phases and TCP’s, the spin glass phase and two more special points are also observed.     

Phase transition properties of the spin-1 Blume-Emery-Griffiths model with random transverse crystal field

Phase transition properties of a spin-1 Blume-Emery-Griffiths model (BEGM) with random transverse crystal field is studied by the effective field theory for a simple cubic lattice. In T−D_x space, we obtain the phase diagrams with the ratio α between the biquadratic interaction and the exchange interaction as well as a tunable parameter l of the transverse crystal field. The tricritical point (TCP) appears at α<0, which undergoes a crossover from positive to negative direction of the transverse crystal field when l<0. The TCP cannot be observed for α>0. The maximum critical temperature increases with the increase of α. The position of the peak value tends to the drift of negative or positive direction for a different magnitude or an imperfect (±) transverse crystal field distribution. In T−α space, the range of ordered phase is magnified when the ratio is changed from α<0 to α>0. The random transverse crystal field obviously affects the TCP.     

Phase diagrams of a nonequilibrium mixed spin-3/2 and spin-2 Ising system in an oscillating magnetic field

The phase diagrams of the nonequilibrium mixed spin-3/2 and spin-2 Ising ferrimagnetic system on square lattice under a time-dependent external magnetic field are presented by using the Glauber-type stochastic dynamics. The model system consists of two interpenetrating sublattices of spins σ=3/2 and S=2, and we take only nearest-neighbor interactions between pairs of spins. The system is in contact with a heat bath at absolute temperature T_abs and the exchange of energy with the heat bath occurs via one-spin flip of the Glauber dynamics. First, we investigate the time variations of average order parameters to find the phases in the system and then the thermal behavior of the dynamic order parameters to obtain the dynamic phase transition (DPT) points as well as to characterize the nature (first- or second-order) phase transitions. The dynamic phase diagrams are presented in two different planes. Phase diagrams contain paramagnetic (p), ferrimagnetic (i₁, i₂, i₃) phases, and three coexistence or mixed phase regions, namely i₁+p, i₂+p and i₃+p mixed phases that strongly depend on interaction parameters.     

Critical properties of a random transverse crystal field Ising model with bond dilution

Within effective field theory (EFT), the critical properties of a random transverse crystal field Ising model with bond dilution are studied on a square lattice. Under both weak and strong bond dilution conditions, we consider three cases (α = 0,±0.5) of a transverse crystal field ratio, obtaining global phase diagrams in T−D _x space for changes in the random transverse crystal field concentration. The phase diagrams obtained for a weak bond dilution are very similar in shape to those of pure bond but with decreases in corresponding ordered phases and critical values. However, the phase diagrams for a strong bond dilution exhibit varieties, including a change in reentrant phenomenon, the occurrence of transverse crystal field degeneration, and the opposite direction crossover of temperature peak value.     

The random distribution of the coordination numbers in the mixed spin-1/2 and spin-2 Blume-Capel model

The effects of random distribution of coordination numbers are investigated for the mixed spin-1/2 and spin-2 Blume-Capel model on the Bethe lattice in terms of exact recursion relations. The usual coordination numbers, i.e. $q=3,4$ and 6, corresponding to the honeycomb, square and simple cubic lattices, respectively, are considered. Two different q values are taken as couples and are varied randomly in terms of a standard-random approach on the shells of the Bethe lattice with some probabilities. It is found that the model gives either first- or second-order phase transitions for appropriate values of probability (p) and single ion anisotropy (d). One or two tricritical points are also observed depending on the given values of p and d, respectively.     

The transverse spin-1 Ising model with random interactions

The phase diagrams of the transverse spin-1 Ising model with random interactions are investigated using a new technique in the effective field theory that employs a probability distribution within the framework of the single-site cluster theory based on the use of exact Ising spin identities. A model is adopted in which the nearest-neighbor exchange couplings are independent random variables distributed according to the law P(J_ij)=pδ(J_ij−J)+(1−p)δ(J_ij−αJ). General formulae, applicable to lattices with coordination number N, are given. Numerical results are presented for a simple cubic lattice. The possible reentrant phenomenon displayed by the system due to the competitive effects between exchange interactions occurs for the appropriate range of the parameter α.     

Effects of the bond dilution on the phase diagrams of a spin-1 transverse Ising model with crystal field interaction on a honeycomb lattice

A bond diluted spin-1 Ising model with crystal and transverse field interactions is examined for honeycomb lattice by introducing an effective field approximation that takes into account the correlations between different spins that emerge when expanding the identities. The effects of the crystal field as well as the transverse field and dilution on the phase diagrams and order parameter m_z are discussed in detail. A number of interesting and unusual phenomena such as reentrant phenomena and three successive phase transitions originating from the crystal field as well as the transverse field and bond dilution have been found.     

The mixed spin-1/2 and spin-1 Ising system on a two-layer Bethe lattice

Two layered magnetic Bethe lattice with varying coordination number q is introduced and numerically studied via exact recursion relations within a pairwise approach. The system is influenced by competing interlayer and intralayer nearest-neighbour (NN) coupling interactions and also by the crystal and external magnetic fields. Cases where both layers are ferromagnetic or one is ferro and the other antiferromagnetic are considered. System configurations’ energy calculations are used to devise some ground state phase diagrams that have proven useful for the investigation of the very low temperature behaviour of the model. Analysis of the thermal behaviours of the total magnetization within the model parameters’ space yield interesting phase diagrams which display fascinating properties, in particular the presence of tricritical points. Increasing negative values of the crystal field strength stabilizes the disordered paramagnetic phase and sometimes gives rise to wavy transition lines.     

Phase diagrams of the spin-2 Ising model in the presence of a quenched diluted crystal field distribution

We have investigated the random crystal field effects on the phase diagrams of the spin-2 Blume-Capel model for a honeycomb lattice using the effective-field theory with correlations.To do so,the thermal variations of magnetization are studied via calculating the phase diagrams of the model.We have found that the model displays both second-order and first-order phase transitions in addition to the tricritical and isolated points.Reentrant behavior is also observed for some appropriate values of certain system parameters.Besides the usual ground-state phases of the spin-2 model including ±2,±1,and 0,we have also observed the phases ±3/2 and ±1/2,which are unusual for the spin-2 case.     

Dynamic phase transitions in the kinetic spin-1 Blume-Capel model on the Bethe lattice

The stationary states of the kinetic spin-1 Blume-Capel (BC) model on the Bethe lattice are analyzed in detail in terms of recursion relations. The model is described using a Glauber-type stochastic dynamics in the presence of a time-dependent oscillating external magnetic field (h) and crystal field (D) interactions. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. It is found that the magnetization oscillates around nonzero values at low temperatures (T) for the ferromagnetic (F) phase while it only oscillates around zero values at high temperatures for the paramagnetic (P) phase. There are regions of the phase space where the two solutions coexist. The dynamic phase diagrams are obtained on the (kT/J,h/J) and (kT/J,D/J) planes for the coordination number q=4. In addition to second-order and first-order phase transitions, dynamical tricritical points and triple points are also observed.     

Phase transition features of the bond-dilution transverse ferromagnetic Ising spin system with random crystal field

The bond dilution spin-1 transverse ferromagnetic Ising model with random crystal field is investigated in the framework of effective field theory (EFT). The general expressions of magnetization are derived for lattice with any coordination number z. In particular, we calculate the phase diagrams of a square lattice. The second-order phase transition lines and the tricritical points are presented in different conditions for the square lattice. Special emphasis is placed on the influence of bond dilution, random crystal field and the transverse field on the phase diagrams. We discover some unusual phenomena. New results obtained in this paper are discussed in detail.     

Phase diagrams of diluted mixed spin XY model

Phase diagrams of diluted mixed spin XY magnetic systems on 3D cubic lattice are investigated by the use of the variational principle and the replica method. The results are given with the consideration of the different nearest-neighbour interactions between randomly occupied magnetic sites.Dilution and competition are found to be responsible of the spin glass phase. The range of dilution in which this phase may appear depends on the spin number, the sign and strength of magnetic interactions.The experimental phase diagrams of the particular 3D hexagonal Fe_pMn_1−pTiO₃ and the planar system K₂Cu_pMn_1−pF₄ are well represented by the obtained results.     

Phase diagrams of competing quadruple and binary interactions on Cayley tree-like lattice: Triangular Chandelier

We study the phase diagrams for the Ising model on a Cayley tree-like lattice, called Triangular Chandelier, with competing nearest-neighbour interactions J₁, prolonged next-nearest-neighbour interactions J_p and one-level next-nearest-neighbour quadruple interactions J_l1. The phase diagrams display the multicritical points (the Lifshitz points) that are at nonzero temperature and many modulated phases. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is established; it recovers, as particular case, previous work of Vannimenus extension result given by Ganikhodjaev and U?uz for k=3. At vanishing temperature, the phase diagram is fully determined for all values and signs of J₁,J_p and J_l1. At finite temperatures several interesting features are exhibited for typical values of J_l1/J₁ and −J_p/J₁.     

Trilayer Bethe lattices in the form of spin-(1/2,3/2,1/2)

The critical behaviors of the trilayer Bethe lattice in the form of spin-(1/2,3/2,1/2) with the exterior two layers consisting of spin-1/2 and the interior one having only spin-3/2 Ising spins are studied in terms of the recursion relations with either ferromagnetic or antiferromagnetic bilinear exchange interactions between the nearest-neighbor spins. The ground-state phase diagrams are calculated and it is found that the model presents six different ground state phase configurations. The thermal variations of the order-parameter and free energy are investigated, therefore, topologically different phase diagrams of the model are obtained. It is found that the model exhibits second- and first-order phase transitions, tricritical and bicritical points for the values of the coordination numbers q=3, 4 and 6 and, the reentrant behavior for q=4 and 6, respectively.