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International Journal of Modern Physics B

The spin-3/2 Blume–Capel model with two spins per site of the Bethe lattice

Erhan Albayrak

Published Jun 30, 2025• 2 citations
Publisher

Abstract

The exact recursion relations are employed to study the magnetization and phase transition properties by assuming that each site of a Bethe lattice is occupied by two spin-3/2 atoms forming diatomic molecules in the Blume–Capel model. The atoms interact ferromagnetically via three distinct bilinear coupling parameters: intra-site [Formula: see text], nearest-neighbor [Formula: see text] and diagonal inter-site [Formula: see text]. The crystal field (D) is also active at all sites of the Bethe lattice of coordination number three. The thermal variations of the magnetizations are examined at the core molecule to obtain its phase transition properties. The phase diagrams mapped on the [Formula: see text] and [Formula: see text] planes show that the model presents second- and first-order phase transitions. The obtained first-order phase transition line terminating at the isolated endpoints separating the phases ([Formula: see text]) from ([Formula: see text]) is the result of the usual BC phase diagram on the [Formula: see text] plane. The phase diagrams obtained on the [Formula: see text] planes give either only second-order lines, second- and first-order lines, or second- and two first-order lines for given parameters that are not seen in the usual spin-3/2 models.

DOI
10.1142/s0217979225502194

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The exact recursion relations are employed to study the magnetization and phase transition properties by assuming that each site of a Bethe lattice is occupied by two spin-3/2 atoms forming diatomic molecules in the Blume–Capel model. The atoms interact ferromagnetically via three distinct bilinear coupling parameters: intra-site [Formula: see text], nearest-neighbor [Formula: see text] and diagonal inter-site [Formula: see text]. The crystal field (D) is also active at all sites of the Bethe lattice of coordination number three. The thermal variations of the magnetizations are examined at the core molecule to obtain its phase transition properties. The phase diagrams mapped on the [Formula: see text] and [Formula: see text] planes show that the model presents second- and first-order phase transitions. The obtained first-order phase transition line terminating at the isolated endpoints separating the phases ([Formula: see text]) from ([Formula: see text]) is the result of the usual BC phase diagram on the [Formula: see text] plane. The phase diagrams obtained on the [Formula: see text] planes give either only second-order lines, second- and first-order lines, or second- and two first-order lines for given parameters that are not seen in the usual spin-3/2 models.
The molecular approach of a spin model is constructed on the Bethe lattice (BL), and then it is examined in terms of exact recursion relations. Rather than assuming that each BL site is inhabited by a single spin, each site is occupied by two spin-1/2 atoms A and B, forming a molecule. Each molecule is considered to contain two spin-1/2 atoms, as well as [Formula: see text] or 6 nearest-neighbor molecules. In addition to the internal interactions between the atoms of each molecule, the molecules interact via their atoms in terms of bilinear interaction parameters J. Atoms of a molecule interact with [Formula: see text], while the molecules interact via their atoms in terms of [Formula: see text] and [Formula: see text]. After obtaining the magnetizations of each atom in the central molecule of the BL, the average magnetization of the molecule is determined. It is found that the model presents first-and second-order and random phase transitions. The model also displays tricritical, bicritical and end points, in addition to reentrant behavior for appropriate J values.
Abstract The Bethe lattice is configured to create a diatomic molecule composed of one spin-1/2 ( σ ) atom and one spin-5/2 ( S ) atom. Each molecule is permitted to engage with its q nearest-neighbor molecules through diverse ferromagnetic (FM) bilinear interaction parameters. The crystal field ( D ) is incorporated into the model and is effective solely at spin-5/2 locations. Following an examination of magnetization behaviors, namely for spin-1/2, spin-5/2, and a molecular system, the phase diagrams are delineated for q = 3 and q = 4. As a result, four distinct phase regions were identified with the ground-state values of ( : FM 1 with , FM 2 with , FM 3 with , and FM 4 characterized by minimal M values. The model delineates first- and second-order phase transition temperatures as depicted on the specified ( D, T )-plane phase diagrams. Tricritical and end points are also observed.
The mixed spin-1/2 and spin-3/2 Blume–Capel (BC) model is considered on the Bethe lattice (BL) with randomly changing coordination numbers (CN) and examined in terms of exact recursion relations. A couple of two different CNs are changed randomly on the shells of the BL in terms of a standard–random approach to obtain the phase diagrams on possible planes of the system parameters. It is found from the thermal analysis of the order-parameters that the model only gives the second-order phase transitions as in the regular mixed case. As the probability of having larger CN increases, the temperatures of the critical lines also increase as expected.
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.
Phase diagrams and the thermal variations of the order‐parameters in the mixed spin‐1 and spin‐323 \over 223​ ising model on the Bethe lattice
Abstract The mixed spin‐1 and spin‐323 \over 223​ Blume‐Capel Ising ferromagnetic system (J > 0) with two crystal‐field interactions, DA for spin‐1 and DB for spin‐323 \over 223​, is studied on the Bethe lattice using the exact recursion equations for the central‐spin with spin‐323 \over 223​. The exact expressions for the order‐parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. The phase diagrams on the (kTcJ, DAJ)\bigg ({{kT_c}\over{J}}, \,{{D_A}\over{J}}\bigg )(JkTc​​,JDA​​) and (kTcJ, DBJ)\bigg ({{kT_c}\over{J}}, \,{{D_B}\over{J}}\bigg )(JkTc​​,JDB​​) planes are studied for selected values of DB/J and DA/J, respectively, for the coordination numbers q = 3, 4, and 6. Besides the second‐order phase transition lines, the lines of the first‐order phase transitions terminating either at a tricritical point or at an isolated critical point are found. The thermal variations of the order‐parameters are studied with two formulations, both obtained on the Bethe lattice for the mixed spin‐1 and spin‐323 \over 223​ Ising system using the exact recursion relations, but one with the central‐spin with spin‐1 and the other is for the central‐spin with spin‐323 \over 223​ for q = 3, thus, it is shown that the choice of the central‐spin on the Bethe lattice has no effect on the phase diagrams.
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.