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© 2021 Elsevier B.V.The Blume-Emery-Griffiths (BEG) model is considered on the Bethe lattice (BL) in terms of exact recursion relations (ERR) under the effect of a crystal field (D) which was either turned on or off randomly for a given probability. The repulsive case of biquadratic exchange interaction ((K < 0)) between the nearest-neighbor (NN) spins is assumed and the effects of changing the coordination number are also investigated. The thermal variation of the order-parameters, i.e. dipole and quadrupole moments, is examined to obtain the phase diagrams. Very rich phase diagrams with the second- and first-order phase transition lines, tricritical, bicritical and critical end points and the occurrence of reentrant behavior are observed. Different phase regions were explored which include the ferromagnetic (F), paramagnetic (P), staggered quadrupolar (SQ) and ferrimagnetic (FI) phases.

The magnetic behaviors of bilayer with spin-1 and 2 Ising model on the Bethe lattice are investigated using the Monte Carlo simulations. The thermal magnetizations, the magnetic susceptibilities and the transition temperature of the bilayer spin-1 and 2 on the Bethe lattice are studied for different values of crystal field and intralayer coupling constants of the two layers and interlayer coupling constant between the layers. The thermal and magnetic hysteresis cycles are given for different values of the crystal field, for different temperatures and for different exchange interactions.

The magnetic properties of alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice have been studied by using the Monte Carlo simulations. The ground state phase diagrams of alternate mixed spin-5/2 and spin-2 Ising model on the Bethe lattice has been obtained. The thermal total magnetization and magnetization of spins-5/2 and spin-2 with the different exchange interactions, external magnetic field and temperatures have been studied. The critical temperature have been deduced. The magnetic hysteresis cycle on the Bethe lattice has been deduced for different values of exchange interactions, for different values of crystal field and for different sizes. The magnetic coercive field has been deduced.

The magnetic properties of spins -S and σ Ising model on the Bethe lattice have been investigated by using the Monte Carlo simulation. The thermal total magnetization and magnetization of spins S and σ with the different exchange interactions, different external magnetic field and different temperatures have been studied. The critical temperature and compensation temperature have been deduced. The magnetic hysteresis cycle of Ising ferrimagnetic system on the Bethe lattice has been deduced for different values of exchange interactions between the spins S and σ , for different values of crystal field and for different sizes. The magnetic coercive filed has been deduced.

The biquadratic exchange interaction is randomized in a bimodal form with probabilities (p) and (1 - p) for the cases with K > 0 (attractive case) and K < 0 (repulsive case), respectively, and its effects on the phase diagrams of the spin-1 Blume-Emery-Griffiths model are studied on the Bethe lattice by using the recursion relations. It was found that the critical behaviors of the model change drastically. (C) 2015 Elsevier B.V. All rights reserved.

+/- J model was exploited on the Bethe lattice for the spin-1 Blume-Capel model. Either ferromagnetic (J > 0) or antiferromagnetic (J < 0) interactions are assumed between the nearest-neighbor (NN) spins on the Bethe lattice for given probabilities p or 1 - p, respectively. The phase diagrams of the model were calculated exactly on the temperature (T)-probability (p) or crystal field (D) planes for given coordination numbers q = 3, 4 and 6. In addition to the first- and second-order phase transitions, another kind phase transition with zero magnetization and quadrupolar order-parameter presenting a jump was obtained above the second-order phase transition temperature which seems to be indicating the spin glass phase. When the lines of these three phases combine is a multicritical point and the first two combine is a tricritical point. In the (T,p) planes, the lower critical dimensions (the value of q) of obtaining any phase transition increase with increasing negative D values. (C) 2013 Elsevier B.V. All rights reserved,Spin-1 Bethe lattice Bilinear interaction Blume-Capel Spin glass

The effects of crystal field (Delta) 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, Delta 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 disappearas p decreases. The critical lines do not present any qualitative differences with z, but they are seen at higher temperatures for higher z. (C) 2011 Elsevier B.V. All rights reserved.

The ground-state and finite-temperature behavior of the mixed spin-1 and spin-1/2 Ising–Heisenberg model on the diamond-like decorated Bethe lattice is investigated within the framework of two rigorous methods: the decoration-iteration transformation and exact recursion relations. The model under consideration describes a hybrid classical-quantum system consisting of the Ising and Heisenberg spins, which interact among themselves either through the Ising or XXZ Heisenberg nearest-neighbor interaction. Both sublattice magnetizations of the Ising and Heisenberg spins are exactly calculated with the aim to examine phase diagrams, thermal variations of the total and sublattice magnetizations. The finite-temperature phase diagrams form continuous (second-order) phase transition lines only, which exhibit a small reentrant region if the diamond-like decorated Bethe lattice with a sufficiently high coordination number is considered.

The phase diagram of mixed spin-3/2 and spin-2 Ising ferromagnetic model with different single-ion anisotropies is investigated by the use of a mean-field theory based on the Bogoliubov inequality for the Gibbs free energy. Global phase diagrams are obtained in the temperature-anisotropy plane. In particular, by changing values of the single-ion anisotropies, several different types of phase diagrams of first-order transition between two ordered phases, are studied in detail. A variety of multicritical points such as tricritical points, isolated critical points, and triple points are obtained.

The spin-1 Ising model is studied for the case of antiferromagnetic (AFM)/AFM interactions on the bilayer Bethe lattice by using the pairwise approach for several values of the coordination numbers q = 3 , 4 and 6 when the layers are linked with the external magnetic fields. The ground state (GS) phase diagrams, thermal variations of the order-parameters and the response functions are studied in detail to obtain the phase diagrams of the model. It was found that the system gives only one Neel temperature, T N , for q = 3 . Two T N 's and first-order phase transition temperature, T t , thus tricritical point, are found and the existence of two T N 's leads to the reentrant behavior when q = 4 and 6 for some given system parameters.

The totally antiferromagnetic Ising model is analyzed on a bilayer Bethe lattice in detail by studying the order-parameters, response functions, i.e. susceptibility and specific heat, and free energy by using the recursion relations in a pairwise approach. The ground state phase diagrams of the model are also obtained on the (J(2)/vertical bar J(1)vertical bar,J(3)/q vertical bar J(1)vertical bar) plane for given values of H/q vertical bar J(1)vertical bar and on the (H/q vertical bar J(1)vertical bar,J(3)/q vertical bar J(1)vertical bar) plane for given J(2)/vertical bar J(1)vertical bar. As a result, we have obtained the temperature-dependent phase diagrams for various values of the coordination number q on the (J(3)/vertical bar J(1)vertical bar,kT/vertical bar J(1)vertical bar) and (H/vertical bar J(1)vertical bar,kT/vertical bar J(1)vertical bar) planes for given values of the rest of the system parameters. (C) 2008 Elsevier B.V. All rights reserved.

The low-temperature magnetization plateaus of the general spin Blume–Capel model on the Bethe lattice in an external magnetic field and single-ion anisotropy are studied by using iteration technique and restricting ourselves to the case of antiferromagnetic exchange interactions. Exact expressions for the local magnetizations are obtained. Low-temperature magnetization shows step-like plateau characterized by several critical magnetic fields which reflects the competing effect of magnetic interactions and thermal fluctuations. The effects of the external magnetic field and single-ion anisotropy parameters on the magnetization plateaus are discussed in detail.

A bilayer Ising model consisting of two Bethe lattices each with a branching ratio of q Ising spins with one of the layers having only spin- 3 2 and the other having only spin- 1 2 is laid over the top of the other and the two layers are tied together via an interaction between the vertically aligned spins. The problem was studied by using the exact recursion relations in a pairwise approach in terms of the intralayer bilinear interactions J 1 and J 2 of the upper and lower layers, respectively, and the interlayer bilinear interaction J 3 . After obtaining the ground state phase diagrams on the ( J 2 / | J 1 | , J 3 / q | J 1 | ) plane with either J 1 > 0 or J 1 0 , the variations of the order-parameters and the free energy were analyzed to obtain the temperature dependent phase diagrams. They were calculated for only the ferromagnetic ordering in each of the layers and ferromagnetic or antiferromagnetic ordering of the adjacent nearest-neighbor (NN) spins of the layers. It was found that the system presents both second- and first-order phase transitions, besides the isolated critical and triple points. The model also presents compensation temperatures when J 2 of spin- 1 2 layer can compete with J 1 of spin- 3 2 layer.

The spin- 3 2 Ising model on the bilayer Bethe lattice is studied in terms of the exact recursion relations with the intralayer bilinear interactions J 11 and J 22 of the two layers with ferromagnetic coupling and interlayer bilinear interaction J 12 between the layers with ferromagnetic or antiferromagnetic coupling. We first have obtained the ground state configurations of the model on the ( J 22 / | J 11 | , J 12 / q | J 11 | ) planes for J 11 0 and J 11 > 0 . Then the thermal behaviors of the order-parameters, 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, are studied to obtain the phase diagrams of the model on the ( kT / J 11 , J 12 / J 11 ) plane for given values of J 22 / J 11 and the coordination number q . As a result, a few types of the critical temperatures of the model are obtained.

The mixed spin-2 and spin- 5 2 Blume–Capel Ising ferrimagnetic system, i.e. negative bilinear interaction of the nearest-neighbor spins of the sublattices, with two crystal–field interactions, D A for spin-2 and D B for spin- 5 2 , is investigated on the Bethe lattice by using the exact recursion equations. The exact expressions for the order-parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. Then, the phase diagrams on the ( kT / | J | , D A / | J | ) and ( kT / | J | , D B / | J | ) planes are studied for various values of D B / | J | and D A / | J | , respectively, when the coordination number q = 3 . It is observed that the system presents both second- and first-order phase transitions, therefore, tricritical points, besides the compensation temperatures for the sublattice magnetizations. In addition to this, we have also found that the sublattice quadrupolar moments also compensate each other meaning that their values are equal, whenever the sublattice magnetizations compensate, but at different temperatures, which is a type of compensation not mentioned in the literature as far as in our knowledge.

We give an exact formulation of a mixed spin-1 and spin-3/2 Ising model on the Bethe lattice, which shows ferrimagnetism and compensation points. The model incorporates antiferromagnetic nearest-neighbor interaction which is relevant to describe ferrimagnetism. The influence of two sublattice crystal fields, D A and D B , on compensation points is studied in detail. For certain crystal-field values, the single or double compensation temperature may occur in the present system.

The magnetic properties of a mixed Ising ferro and ferrimagnetic system on a Bethe lattice, in which the two interpenetrating sublattices have spins that can take two values, σ = ± 1 2 , alternated with spins that can take four values, S = ± 3 2 , ± 1 2 , are studied. On the Bethe lattice, we obtain closed expressions for sublattice and total magnetizations. The thermal behaviors of magnetizations are examined in detail. We find some interesting phenomena in these quantities, due to applied longitudinal magnetic field.

The antiferromagnetic spin- 3 2 Blume–Capel model on the Bethe lattice in an external magnetic field is investigated and phase diagrams are obtained. Using the recursion method we obtain the plots of magnetization versus external field for different temperatures and construct the resulting phase diagrams. The model exhibits distinct critical regions, including the first-order, second-order and tricritical point.

A theory of disordered binary alloys A x B 1− x ( A = Ni , Co ; B = Fe ; x ∼0.06) is used to determine the changes in the electronic structure and magnetic properties of body centered cubic (BCC) iron induced by doping with nickel and cobalt impurities. This approximation is an extension of the cluster-Bethe lattice method, in which we incorporate electronic correlations, itinerant and localized nature of electrons 3d, and both long-range and short-range chemical correlations. The magnetism is described by means of a Hubbard Hamiltonian that in conjunction with Green's functions techniques is used to calculate local densities of electronic states. For it we take an atom in the real lattice and it is joined to a Bethe's lattice with like coordination number. The magnetic moments on sites occupied for A and B atoms are obtained self-consistently. Nickel and cobalt impurities in BCC iron can provide crucial information on the modification of the electronic band structure and magnetic moments from pure Fe. The results obtained are compared with those of both pure Fe and binary alloys of Co–Fe and Ni–Fe, which have been obtained by other authors using methods such as: first-principles electronic structure calculations using the layer Korringa–Kohn–Rostoker (KKR), the full-potential linearized augmented plane wave method, the KKR coherent potential approximation combined with the local-density functional method and by the tight-binding linear-muffin-tin orbitals method, obtained good agree. These results and other that recently we have published indicate to us that our methodology can be a new alternative for calculations of the electronic structure and magnetic properties of impurities and alloys of ferromagnetic transition metals.

The order–disorder phase transitions in binary alloys with magnetic interactions are studied. We introduced an electronic theory for magnetic binary alloys, which is an extension of the cluster-Bethe lattice method. In this approximation we incorporate electronic correlations, itinerant and localize nature of electrons 3d, and both long-range and short-range chemical correlations. The magnetism is described by means of a Hubbard Hamiltonian that in conjunction with Green's functions techniques is used to calculate local densities of electronic states. For it we take an atom in the real lattice and it is joined to a Bethe's lattice with the same coordination number. The diagram for order–disorder phase transition is obtained. It is achieved by coupling the electronic theory introduced here, to combinatorial theory developed by Kikuchi. The first is used for calculating the alloy internal energy so long as the second is used for obtain the configurational entropy. An application to order–disorder transition of CoxFe1−x is studied and good agreement with experiment is obtained.

We present the exact formulation for the mixed spin - 1 2 and spin - 3 2 Blume–Capel Ising ferrimagnetic system on the Bethe lattice by the use of exact recursion relations. The exact expressions for the magnetization, quadrupole moment, Curie temperature and free energy are found and the phase diagrams are illustrated on the Bethe lattice with the coordination numbers q = 3 , 4, 5 and 6. It is found that the phase diagram of this mixed spin system only presents second-order phase transitions. The thermal variation of the magnetization belonging to each sublattice and the net magnetization are also presented.

The multicritical behaviour of the spin- 3 2 Blume–Emery–Griffiths model with bilinear and biquadratic exchange interactions and single-ion crystal field is studied on the Bethe lattice by introducing two-sublattices A and B within the exact recursion equations. Exact expressions for the free energy, the Curie or second-order phase transition temperatures, as well as for the magnetization and quadrupolar moment order parameters are obtained. The general procedure of investigation of critical properties is discussed and phase diagrams are obtained, in particular, for negative biquadratic couplings. The phase diagrams of the model exhibits a rich variety of behaviours. Results are compared with other approximate methods.

The statistical physics of the spin- 3 2 Blume–Emery–Griffiths model on the Bethe lattice is studied by using the exact recursion equations. Exact expressions for the free energy, the Curie or the second-order phase transition temperatures as well as the magnetization and quadrupolar moment order parameters are found. The phase diagrams in ( kT / J , D / J ) and ( kT / J , K / J ) planes are presented in depth for various values of constants K / J and D / J , respectively. The phase diagrams are discussed and a comparison is made with the results of the other approximation methods.

A cluster variational method for spin 3 2 Ising models on regular lattices is presented that leads to results that are exact for Bethe lattices of the same coordination number. The method is applied to both the Blume–Capel (BC) and the isotropic Blume–Emery–Griffiths model (BEG). In particular, the first-order phase line separating the two low-temperature ferromagnetic phases in the BC model, and the ferrimagnetic phase boundary in the BEG model are studied. Results are compared with those of other theories whose qualitative predictions have been in conflict.

To investigate the magnetic properties of strongly correlated electron systems, the magnetization curve and the magnetic susceptibility of the infinite-dimensional Hubbard model are calculated using the quantum Monte Carlo method for solving the single-impurity part. For the Bethe lattice, it is found that the magnetization curve is similar to that obtained by the Gutzwiller approximation: the curve rises steeply towards the saturation value near the metal-insulator (MI) transition, in contrast with the result obtained by Georges and Krauth for the simple hypercubic lattice. The susceptibility is consistent with χ ∼ ( U c - U ) −1 except near the MI transition, and rather close to the Gutzwiller result.

A randomly decorated Ising model on a Bethe lattice is considered to study hole-hole correlations of CuO 2 -based superconducting sheets. Two intrinsically different types of pairing mechanism are observed: one, induced by magnetic fluctuations and the other as a consequence of a free-energy minimisation process due to frustration effects so intense that one might conjecture that frustration is an essential ingredient to the understanding of such high superconducting transition temperature displayed by those materials.

The free energy of the ±J model at the finite temperatures on the Bethe lattice was obtained to be βF = (1 − 1 2z ) ∫ −x x ( 1 2 x) cosech (πx/2β)[S(x)] z d x+ 1 2 z∫ −x x [ cos (J 0 x)/2x] cosech (πx/2β) {S(x)] z−1 dx , where S(x) is the Fourier transform of the distribution function g(h) of the single bond effective field h. A tentative self-consistent S(x) gives the spin glass transition temperature and the energy at T = 0 exactly.

The paper is an investigation of the magnetic susceptibilities of a random binary Bethe lattice with anisotropic Heisenberg exchange at high temperature by the adoption of the extended technique of the pair approximation.

Spin-wave spectra at the surface of binary alloys with either one or two ferromagnetic components are studied by means of a cluster-Bethe-lattice approximation and a Heisenberg Hamiltonian. The surface can be magnetically harder than the bulk over a range of temperatures. Applications to FeCo are presented.

The exact expressions for the Curie temperature, the spontaneous magnetization and the quadrupolar moment are reported for a spin-one Ising model on a Bethe lattice with the inclusion of single-ion anisotropy and biquadratic exchange. Detailed computations of these quantities are presented for a wide range of parameters.