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SPIN

The Ising Model on the Bethe Lattice with Two Atoms Per Site

Erhan Albayrak

Published Jul 18, 2024• 3 citations
Publisher

Abstract

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.

DOI
10.1142/s2010324724500140

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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.

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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.

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The mixed spin-1/2 and spin-1 model with alternating coordination number

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The mixed spin-1/2 and spin-1 Blume–Capel model is studied with randomly alternated coordination numbers (CN) on the Bethe lattice (BL) by utilizing the exact recursion relations. Two different CNs are randomly distributed on the BL by using the standard–random (SR) approach. It is observed that this model presents first-order phase transitions and tricritical points for variations of CNs 3 and 4, even if these behaviors are not displayed for the regular mixed-spin on the BL. The phase diagrams are mapped by obtaining the phase transition temperatures of the first- and second-order on several planes.

International Journal of Modern Physics B•Apr 29, 2019•4 citations
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.
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.
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.
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.
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.
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.
The mixed spin-1/2 and spin-1 Blume–Capel model is studied with randomly alternated coordination numbers (CN) on the Bethe lattice (BL) by utilizing the exact recursion relations. Two different CNs are randomly distributed on the BL by using the standard–random (SR) approach. It is observed that this model presents first-order phase transitions and tricritical points for variations of CNs 3 and 4, even if these behaviors are not displayed for the regular mixed-spin on the BL. The phase diagrams are mapped by obtaining the phase transition temperatures of the first- and second-order on several planes.