Two-layer Bethe Lattice Research Articles

Spin-1/2 Ising model on a AFM/FM two-layer Bethe lattice in a staggered magnetic field

A bilayer spin-1/2 Ising model consisting of two superposed Bethe lattices with antiferromagnetic/ferromagnetic interactions is studied by the use of exact recursion relations in a pairwise approach in the presence of an external staggered magnetic field. Besides the ground state phase diagrams calculated in different possible planes of the model parameters space, the thermal variations of the order-parameters and the free energy are investigated to obtain the temperature-dependent phase diagrams of the model for different values of the coordination numbers q. Our calculations reveal that depending on the strength of the model parameters, the model exhibits a variety of interesting phase transitions and therefore phase diagrams.

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

Abstract 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.

The temperature-dependent phase diagrams of the spin-3/2 Ising model on a FM/AFM two-layer lattice with a crystal field

The temperature-dependent phase diagrams of the spin-3/2 Ising model on a two-layer Bethe lattice with ferromagnetic (FM)/antiferromagnetic (AFM) intra-layer and either FM or AFM type inter-layer interactions are investigated under a constant magnetic field ( H ) and in the presence of a crystal field ( D ) by using exact recursion equations in a pairwise approach for coordination numbers q = 3 , 4 and 6, in detail. In the light of the ground-state (GS) phase diagrams, the temperature-dependent phase diagrams of the model are obtained by studying the thermal variations of the order parameters, response functions and free energy. Then, they are illustrated on the ( k T / J 1 , J 3 / J 1 ) and ( k T / J 1 , J 2 / J 1 ) planes for the given system parameters. It is observed that the system exhibits first- and second-order phase transitions for all q values, and hence, in some cases, tricritical points. The existence of critical-end points and that of isolated points are also observed. The re-entrant behavior owes its presence to the two Néel temperatures, T N , that are present for all q .

The Crystal Field Effects on Spin-1 Ising model with FM/AFM interactions on a two-layer Bethe lattice

The Crystal Field Effects on Spin-1 Ising model with FM/AFM interactions on a two-layer Bethe lattice

Spin-3/2 Ising model AFM/AFM two-layer lattice with crystal field

The spin-3/2 Ising model is investigated for the case of antiferromagnetic (AFM/AFM) interactions on the two-layer Bethe lattice by using the exact recursion relations in the pairwise approach for given coordination numbers q = 3, 4 and 6 when the layers are under the influences of equal external magnetic and equal crystal fields. The ground state (GS) phase diagrams are obtained on the different planes in detail and then the temperature-dependent phase diagrams of the system are calculated accordingly. It is observed that the system presents both second- and first-order phase transitions for all q, therefore, tricritical points. It is also found that the system exhibits double-critical end points and isolated points. The model also presents two Néel temperatures, TN, and the existence of which leads to the reentrant behaviour.

The Phase Diagrams of Spin-1/2 Ising Model on a Two-Layer Bethe Lattice with AFM/AFM Interactions

The phase diagrams of spin-1/2 Ising model on a two-layer Bethe lattice with antiferromagnetic interactions for each layer and either antiferromagnetic or ferromagnetic interaction between the layers are investigated by using the pairwise approach for given values of coordination number q. The exact expressions of the order-parameters, response functions and free energy are obtained in terms of the recursion relations. The ground-state phase diagrams are calculated for given system parameters of the model. In the guidance of the ground-state phase diagrams, the temperature dependent phase diagrams of the model are also studied in detail for given coordination numbers q = 3, 4 and 6. It was found that the system presents only second-order phase transitions with different thermal behaviors for all values of q. In addition, two Neel temperatures, TN, are found for q = 6 only.

The hysteresis loops of FM/AFM two-layer Bethe lattice

The distinct hysteresis loops (HLs) of ferromagnetic/antiferromagnetic (FM/AFM) two-layer Bethe lattice with the Ising spins of the top layer having only FM interactions and the bottom ones having only AFM interactions with the interlayer coupling is either FM or AFM type are studied by using a pairwise approach. The sublattice magnetizations are studied by increasing and decreasing the external magnetic field (H) to obtain the HLs. The shapes of the HLs are strongly dependent on the competitions among the system parameters and on the phase configurations. The HLs are formed only when the AFM-type interactions are involved. The small loops of hysteresis are also formed because of the reentrant behavior in the FM region.

Crystal field effects on a spin-1 Ising model with AFM–AFM interactions on a two-layer Bethe lattice

Crystal field effects and the influence of changing the coordination numbers q on the phase diagrams of a spin-1 Ising model are investigated for the case of antiferromagnetic (AFM)–AFM interactions on a two-layer Bethe lattice by using the pairwise approach when the layers are linked with the external magnetic fields. The ground state (GS) phase diagrams are calculated on different planes and are used as a guide, together with the free energy and order parameters, to obtain temperature-dependent phase diagrams. It was found that the system gives only one Néel temperature, TN, for q=3, but two TN's and reentrant behavior are observed for q=4 and 6. The first-order phase transitions and tricritical points are also observed for all q with appropriate values of the system parameters. The existence of the crystal field has increased the number of GS configurations and also let the system present first-order phase transitions for q=3 in comparison with no crystal field.

Spin-3/2 Ising model on a two-layer Bethe lattice with FM/AFM interactions

Spin-3/2 Ising model on a two-layer Bethe lattice with FM/AFM interactions

The spin-1 Ising model on a two-layer Bethe lattice with FM/AFM interactions

Two-layer Bethe lattice with the Ising spins of the top layer having only ferromagnetic (FM) interactions and the bottom layer having only antiferromagnetic (AFM) interactions are allowed to interact with the interlayer interaction of either FM or AFM type. The model is studied by using the exact recursion relations in a pairwise approach for given coordination numbers q = 3 , 4 and 6 with equal external magnetic fields acting on the layers. The phase diagrams of the model are obtained on different planes for given system parameters by studying the ground state (GS) phase diagrams and the thermal variations of the order-parameters and the response functions, i.e. the susceptibility and the specific heat, in detail. The model presents second- and first-order phase transitions, and where their lines are combined is the tricritical point. The critical end points also exist. The reentrant behavior is also seen when the model presents two Néel temperatures.

The exact phase diagrams of spin-1 Ising model on a two-layer Bethe lattice

The two-layer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants J 1 and J 2 of the two layers, interlayer coupling constant J 3 between the layers and the external magnetic fields, which are coupled to the two layers and assumed to be different for each layer, for given values of the coordination number q by using the recursion relation scheme. The ground-state configurations of the system are obtained on the ( J 2 / | J 1 | , J 3 / q | J 1 | ) planes depending on J 1 < 0 or J 1 > 0 . Then, the phase diagram of the system is obtained on the ( kT / J 1 , J 3 / J 1 ) plane for given values of α = J 2 / J 1 and q in zero external magnetic fields. It was found that the system presents both first- and second-order phase transitions for all values of q. Besides, we also present the thermal change of the total and staggered magnetizations of the two layers and also the spin–spin correlation function between the nearest-neighbor spins of the adjacent layers.

Exact phase diagrams for an Ising model on a two-layer Bethe lattice.

Using an iteration technique, we obtain exact expressions for the free energy and the magnetization of an Ising model on a two-layer Bethe lattice with intralayer coupling constants J1 and J2 for the first and the second layer, respectively, and interlayer coupling constant J3 between the two layers; the Ising spins also couple with external magnetic fields, which are different in the two layers. We obtain exact phase diagrams for the system and find that when /J3/-->0, DeltaT(c) identical with[T(c)(J3)-T(c)(0)]/T(c)(0) approximately [J3]/J(1)/(1/psi), where T(c)(J3) is the phase-transition temperature for the system with interlayer coupling constant J3 and the shift exponent psi is 1 for J(1)=J(2) and is 0.5 for J1 not equal to J2. Such results are consistent with predictions of a scaling theory. We also derive equations for DeltaT(c) when /J3/ approaches infinity.