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physica status solidi (b)

The Ising model on a trilayer Bethe lattice

Erhan Albayrak

Published Oct 30, 2006• 10 citations
Publisher

Abstract

Abstract In this work, the Ising model on a trilayer Bethe lattice is studied in terms of recursion relations for which the nearest‐neighbor spins of each layer are coupled ferromagnetically and the adjacent spins of the nearest‐neighbor layers are coupled either ferromagnetically or antiferromagnetically depending on the sign of the bilinear interactions for given values of the coordination number q. Besides the ground‐state phase diagram, the phase diagrams of the model are also calculated for given values of the system parameters. It is found that the model presents both first‐ and second‐order phase transitions for all values of q. In order to construct the phase diagrams, we have also studied the thermal and interlayer coupling constant changes of the layer magnetizations, the spin–spin correlation functions between the nearest‐neighbor spins of the adjacent layers and the free energy of the model. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

DOI
10.1002/pssb.200642273

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Abstract The spin‐3/2 Ising model is studied on a two‐layer Bethe lattice with antiferromagnetic/antiferromagnetic (AFM/AFM) intralayer and either FM or AFM type of interlayer interactions by using the pairwise approach for the coordination numbers q = 3, 4, and 6. In the guidance of the ground‐state phase diagrams, the thermal variations of the magnetizations, susceptibility and specific heat are examined in detail to obtain the critical behaviors of the model. Thus, it was found that the model presents second‐ and first‐order phase transitions; where their lines are combined is a tricritical point. In addition, the model also presents two Néel temperatures, the existence of which leads to the re‐entrant behavior.

physica status solidi (b)•Jul 24, 2009•4 citations
Abstract In this work, the Ising model on a trilayer Bethe lattice is studied in terms of recursion relations for which the nearest‐neighbor spins of each layer are coupled ferromagnetically and the adjacent spins of the nearest‐neighbor layers are coupled either ferromagnetically or antiferromagnetically depending on the sign of the bilinear interactions for given values of the coordination number q. Besides the ground‐state phase diagram, the phase diagrams of the model are also calculated for given values of the system parameters. It is found that the model presents both first‐ and second‐order phase transitions for all values of q. In order to construct the phase diagrams, we have also studied the thermal and interlayer coupling constant changes of the layer magnetizations, the spin–spin correlation functions between the nearest‐neighbor spins of the adjacent layers and the free energy of the model. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
A spin-2 system consisting of two layers of Bethe lattices each with a branching ratio of q Ising spins was analyzed by the use of the exact recursion relations in a pairwise approach. The upper layer interacting with nearest-neighbor (NN) bilinear interaction J1 is laid over the top of the lower layer interacting with bilinear NN interaction J2, and the two layers are tied together via the bilinear interaction between the vertically aligned adjacent NN spins denoted as J3. The study of the ground state phase diagrams on the (J2/|J3|, J1/|J3|) plane with J3>0 and J3<0 and on the (J2/J1, J3/q J1) plane with J1>0 has yielded five distinct ground state configurations. The temperature dependent phase diagrams are obtained for the case with intralayer coupling constants of the two layers with ferromagnetic type J1 and J2>0, and the interlayer coupling constant of the layers with either ferromagnetic J3>0 or antiferromagnetic type J3<0 on the (kT/J1, J3/J1) planes for given values of the J2/J1 for various values of the coordination numbers. As a result, we have found that the model presents both second- and first-order phase transitions, therefore, tricritical points.
A bilayer Ising model consisting of two Bethe lattices, each of which is coupled with crystal fields of different strengths and each with a branching ratio of q Ising spins with one of the layers having only spin-1 and the other having only spin-3/2, is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. After obtaining the ground-state (GS) phase diagrams on different possible planes depending on the given system parameters, the changes in the order-parameters and the free energy are investigated by use of the exact recursion relations in a pairwise approach to calculate the phase diagrams of the model. The ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers are considered. The system presents both second- and first-order phase transitions. The lines of the first-order phase transitions end on either the stable or unstable tricritical points or at the isolated critical points. The model also displays one or two compensation temperatures when the bilinear interaction of the upper layer with spin-1 can compete with that of the lower layers with spin-3/2.
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 Néel temperatures, T N , are found for q = 6 only.
Two-layer Bethe lattice whose lattice sites are occupied with spin-3/2 atoms is solved exactly by using the 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 ferromagnetic (FM) and antiferromagnetic (AFM) interactions for the spins of the upper and lower layers, respectively, and either FM or AFM type interactions between the adjacent spins of the layers are assumed.The phase diagrams of the model are studied on different planes for given system parameters by obtaining 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.It was found that the model presents both second-and first-order phase transitions.The reentrant behavior is seen when the model presents two N éel temperatures for higher q values.The existence of the tricritical point and critical end points is also confirmed.
The influence of the crystal field on the phase diagrams of the bilayer spin-1 Ising model on the Bethe lattice is studied in terms of the intralayer coupling constants J1 and J2 of the two layers and interlayer coupling constant J3 between the layers for given values of the coordination number q by using the recursion relation scheme. The six distinct ground-state configurations of the model are obtained on the (J_{2}J_{1},J_{3}qJ_{1}) plane with J1>0 , the ferromagnetic coupling, for given values of the crystal field. Then, the phase diagram of the system is obtained on the (kTJ_{1},J_{3}J_{1}) plane for given values of the crystal field and alpha=J_{2}J_{1} with q=4 corresponding to the square lattice in real lattice systems. It was found that the system presents both first- and second-order phase transitions, therefore, tricritical points. The paramagnetic phase was also divided into two phases, P+ and P_{-} , by studying the thermal behavior of the quadrupolar moments of the two layers.
Abstract The spin‐3/2 Ising model is studied on a two‐layer Bethe lattice with antiferromagnetic/antiferromagnetic (AFM/AFM) intralayer and either FM or AFM type of interlayer interactions by using the pairwise approach for the coordination numbers q = 3, 4, and 6. In the guidance of the ground‐state phase diagrams, the thermal variations of the magnetizations, susceptibility and specific heat are examined in detail to obtain the critical behaviors of the model. Thus, it was found that the model presents second‐ and first‐order phase transitions; where their lines are combined is a tricritical point. In addition, the model also presents two Néel temperatures, the existence of which leads to the re‐entrant behavior.