Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system in the presence of an applied magnetic field

The mixed spin-1/2 and spin-5/2 Ising model is investigated on the Bethe lattice in the presence of a magnetic field $h$ via the recursion relations method. A ground-state phase diagram is constructed which may be needful to explore important regions of the temperature phase diagrams of a model. The order-parameters, the corresponding response functions and internal energy are thoroughly investigated in order to typify the nature of the phase transition and to get the corresponding temperatures. So, in the absence of the magnetic field, the temperature phase diagrams are displayed in the case of an equal crystal-field on the $(k_{\textrm{B}}T/|J|, \,D/|J|)$ plane when $q=3,4$, $5$ and $6$. The model only exhibits the second-order phase transition for appropriate values of physical parameters of a model.


Introduction
Over the last five decades, the Ising model has been one of the most largely used models to describe critical behaviors of several systems in nature. Lately, numerous extensions have been made in the spin- 1 2 Ising model to describe a wide variety of systems. For example, the models consisting of mixed spins with different magnitudes are interesting extensions, forming the so-called mixed-spin Ising class [1,2]. Beyond that, magnetic materials have several important technological applications: they find wide use in information storage devices, microwaves communication systems, electric power transformers and dynamo, and high-fidelity speakers [3][4][5][6]. Thus, in response to an increasing demand placed on the performance of magnetic solids, there has been a surge of interest in molecular-based magnetic materials [7][8][9][10]. Indeed, the discovery of these materials [11] has been one of the advances in modern magnetism. Many magnetic materials have two types of magnetic atoms regularly alternating which exhibit ferrimagnetism. In this context, a good description of their physical properties is given by means of mixed-spin configurations. The interest in studying magnetic properties of these materials is due to their reduced translational symmetry rather than to their single-spin counterparts, since they consist of two interpenetrating sublattices. Thus, ferrimagnetic materials are of great interest due to their possible technological applications and from a fundamental point of view. These materials are modelled using mixed-spin Ising models that can be built up by infinite combinations of different spins.
In literature there exist many studies on mixed-spin Ising systems which intend to clarify the magnetic properties of magnetic systems. In this regard, there has been great interest in the study of magnetic properties of the systems formed by two sublattices with different spins and crystal-field interactions [12]. Theoretically, such systems have been widely analysed using several numerical approaches, e.g., effective-field theory [13][14][15][16][17][18], mean-field approximation [19][20][21][22][23], renormalization-group technique [24,25], numerical simulations based on Monte-Carlo [26][27][28][29][30] and exact recursion equations [31][32][33][34][35][36][37][38]. A somewhat newer interest is to extend such investigations into a more general mixed-spin Ising model with one constituent spin- 1 2 and the other constituent spin- 5 2 . To this end, Deriven et al. [18] used an effective-field theory with correlation to study the same model and got interesting results. Recently, Guo et al. [17] studied the thermal entanglement of the same model by means of the concept of negativity and also got interesting results concerning the effects of the magnetic field on the entanglement.
In the present paper, we use the exact recursion equations technique to examine the magnetic properties of the mixed spin-1 2 and spin-5 2 Ising model with equal crystal-field on the Bethe lattice in the presence of a longitudinal magnetic field. The aim of this work is to investigate the effect of the crystal-field and the magnetic field on the physical magnetic properties of the model. The remainder of this paper is arranged as follows. In section 2, the description of the model on the Bethe lattice is clarified. Furthermore, the order-parameters, the corresponding response functions and the internal energy are expressed in terms of recursion relations. In the next section, some definitions of the critical temperature of the model are explained. In section 4, we present some illustrations and discuss in detail the numerical results. We finally conclude in the last section .

Description of the model on the Bethe lattice
The mixed spins system on the Bethe lattice is shown in figure 1. We consider the mixed spin-1 2 and spin- 5 2 system consisting of two sublattices A and B. The sites of sublattice A are occupied by atoms of spins Those of the sublattice B are occupied by atoms of spins σ j , where σ j = ± 5 2 , ± 3 2 , ± 1 2 . In our case, the Bethe lattice is arranged in such a way that the central spin is spin-1 2 and the next generation spin is spin- 5 2 and so on to infinity. Thus, the Ising Hamiltonian of such a model on the Bethe lattice may be written as: J < 0 is the bilinear exchange coupling interaction strength. D and h are, respectively, the single-ion anisotropy or the crystal-field and the longitudinal magnetic field acting on the spins of the model.
In order to formulate the problem on the Bethe lattice, the partition function is the main ingredient

33003-2
which is given as: If the Bethe lattice is cut at the central spin S 0 , it splits into q disconnected pieces. Thus, the partition function on the Bethe lattice can be written as: where S 0 is the central spin value of the lattice, g n (S 0 ) is the partition function of an individual branch and the suffix n represents the fact that the sub-tree has n shells, i.e., n steps from the root to the boundary sites. If we continue to cut the Bethe lattice on the sites σ 1 and S 2 which are respectively the nearest and the next-nearest of the central spin S 0 , we can obtain the recurrence relations for g n (S 0 ) and g n−1 (σ 1 ) as: Now, we explicitly calculate some g n (S 0 ) and g n−1 (σ 1 ) as follows: After calculating all the g n (S 0 ) and g n−1 (σ 1 ), we can define the recursion relations for the spin-1 2 as:

33003-3
and for the spin-5 2 as: To investigate our model, we define two order-parameters, the magnetization M and the corresponding quadrupolar moment Q. For the sublattice A, the sublattice magnetization M A is defined by: After some mathematical manipulations, the sublattice magnetization M A is explicitly given by: In the same way, we also calculate the two order-parameters for the sublattice B as follows: where: In order to determine the compensation temperature, one has to define the global magnetization M T of the model which is given by: (2.17) To really study the model in detail and single out the effect of the crystal-field and of the applied magnetic field on the magnetic properties of the model, we have also examined the thermal variations of the response functions, i.e., the susceptibilities, the specific heat and the internal energy defined respectively by: where F is the free energy of the model.

Definition of the critical temperature
The Curie temperature or the second-order transition temperature T c is the temperature at which both sublattice magnetizations and the global magnetization continuously go to zero. T c separates the ordered ferrimagnetic phase (F) from the disordered paramagnetic phase (P). At T c , one can obtain explicit expressions of the recursion relations as follows: for the spin-1 2 ,  In addition to the thermal variations of the order-parameters and the global magnetization of the model, we also calculate and analyze the free energy F of the model in order to identify the first-order transition temperature T t . Thus, using the definition of the free energy F = −k B T ln(Z ) in the thermodynamic limit (n → ∞) and in order to introduce the recursion relations, we can rewrite the free energy as: After some mathematical manipulations, the free energy expression in terms of recursion relations is explicitly given by: (3.4)

33003-5
We also investigate the compensation temperature T comp at which the global magnetization vanishes while both sublattice magnetizations cancel each other. T comp is found by locating the crossing point between the absolute values of sublattice magnetizations, i.e., Considering different definitions of the critical temperature, we can now investigate the thermal variations of the calculated thermodynamical quantities of interest and display the temperature phase diagrams of the model for q = 3, 4, 5 and 6.

Numerical results and discussions
In this section, we present and discuss the results we obtained for the thermal variations of the order-parameters, the response functions, the internal energy and the temperature phase diagrams of the model. We begin discussions with the ground-state phase diagram which is necessary for understanding the obtained temperature phase diagrams.

Ground-state phase diagram
Before presenting the numerical results for the temperature dependence of magnetic properties of the model, we first investigate the ground-state phase diagram. The ground-state structure of the model can be represented by comparing the values of the energy H 0 for different spin configurations which can be expressed as:  (B 1 , B 2 , · · · , B 8 ) and coexistence lines where the spin pair energy of some phases is the same. In the absence of the magnetic field, for all values of q and D/q|J |, M B shows five saturation values whereas for M A , ± 1 2 is the only saturation value. Thus, we get the ferrimagnetic phases F(± 1 2 , ∓ 5 2 ), F(± 1 2 , ∓ 3 2 ), F(± 1 2 , ∓ 1 2 ) and at the borders of these phases, two hybrid phases: F(± 1 2 , ∓1), F(± 1 2 , ∓2) at the multicritical points B 4 and B 5 , respectively. These hybrid phases should correspond to cases where the sublattice B is half-half covered by spins of the two neighboring phases. It is important to indicate that the groundstate phase diagram is very useful because it helps to check the reliability of the theoretical results and to classify different phase domains of the model for the temperature dependence phase diagrams.

Thermal variations of the order-parameters, the response functions and the internal energy
As it is explained above, thermal variations of the order-parameters, the response functions and the internal energy for the present model were calculated in terms of recursion relations. Thermal variations of the order-parameters are crucial in obtaining the temperature dependence phase diagrams of the model. Thus, when the magnetization curves go to zero continuously separating the ferrimagnetic phase from the paramagnetic phase, one gets the second-order phase transition or Curie temperature, i.e., the temperature at which magnetizations become zero. In the case of a jump in the magnetization curves

33003-7
followed by a discontinuity of the first derivative of the free-energy F , one gets a first-order transition temperature. Besides these two temperatures, there is another temperature referred to as compensation temperature defined as the temperature where the global magnetization becomes zero prior to the critical temperature. Therefore, in order to identify transitions and compensation lines, one should study the thermal behaviors of the considered thermodynamical quantities of the model. Now, we present some results on the thermal behaviors of the order-parameters, the response functions and the internal energy in the the absence of the magnetic field h when q = 3, 4 and 6.

33003-8
Thus, on the one hand, as shown in figure 4, we have displayed the thermal behaviors of the sublattice magnetizations and corresponding susceptibilities when q = 3, 4, 6 and D/|J | = 1. In this figure, one can notice that the model only exhibits the second-order phase transition, and the transition temperature T c at which the transition occurs increases with an increasing coordination number q. Here, T c separates the ferrimagnetic phase F(± 1 2 , ∓ 5 2 ) from the paramagnetic phase (P) and T c /|J | = 2.1724 (respectively T c /|J | = 3.2575 and 5.1398) for q = 3 (respectively for q = 4 and 6). Also, one remarks that for T → T c , χ 1/2 → −∞ whereas χ 5/2 → +∞. For T > T c , the susceptibility χ 1/2 rapidly increases whereas the susceptibility χ 5/2 rapidly decreases when the temperature increases and is very far from the Curie temperature T c , χ 1/2 → 0 and χ 5/2 → 0.

33003-9
Let us now discuss the thermal variations of sublattice magnetizations, the corresponding response functions and the internal energy of the system in the presence of a longitudinal magnetic field h. Figure 6 expresses the effects of an applied magnetic field h on the magnetic properties of the model when q = 3 and D/|J | = 1.0 for selected values of h/|J |. From panel (a) to panel (b), the sublattice magnetizations and the global magnetization continuously fall from their saturation values to non-zero values when the temperature increases. The remaining values of magnetizations are more important when the value of the applied magnetic field h/|J | increases. Thus, one can observe that the system does not present any transition when h/|J | 0. It is important to indicate that in the case of h/|J | = 0, the model exhibits the second-order transition at a Curie temperature T c /|J | = 2.1724, where the two sublattice magnetizations and the global magnetization continuously go to zero after falling from their saturation values at T = 0. In panels (c)-(e), we have displayed the temperature dependence of the total susceptibility χ T , the specific heat C and the internal energy U . One can see from these panels that the response functions and the internal energy also indicate a second-order transition which occurs at the same T c /|J | as in the case   compensation types of Ising model. Figure 9 shows temperature dependencies of the global magnetization M T for selected values of the crystal-field when q = 3. As seen in figure 9, the model exhibits five types of compensation behaviors, namely Q-, R-, S-, L-and P-type compensation behaviors as classified in the extended Néel nomenclature [39].

Finite-temperature phase diagrams
Considering all the above calculations, we can illustrate the temperature phase diagrams of the model.

33003-12
Bethe approach study of the mixed spin-1/2 and spin-5/2 Ising system It is important to mention that figure 10 presents some resemblances concerning the second-order transition lines with figure 3 of [18]. Moreover, by increasing the value of the coordination number q, the ferrimagnetic domain F becomes important.

Conclusion
In this work, the study of the mixed spin-1 2 and spin- 5 2 Ising ferrimagnetic model on the Bethe lattice in the presence of a longitudinal magnetic field is undertaken via exact recursion equations method. All the thermodynamical quantities of interest are calculated as functions of recursion relations.
The ground-state phase diagram of the model is displayed as shown in figure 2. From this phase diagram, we have found eleven existing and stable phases and along the D/q|J |-axis, two particular hybrid phases appear at the two multicritical points B 4 and B 5 . The ground-state phase diagram is considered and used as a guide in obtaining different temperature phase diagrams. In the presence and in the absence of a longitudinal magnetic field h, we investigated thermal variations of sublattice magnetizations, global magnetization, the corresponding response functions and the internal energy as seen in figures 3-9. From these figures, the order-parameters in most cases showed a usual decay with thermal fluctuations. By using thermal behaviors of the considered order-parameters, and by analysing the corresponding response functions and the internal energy, the nature of different phase transitions encountered is identified. This enables us to construct and to discuss in detail different temperature dependence phase diagrams in the case of equal crystal-field interactions as shown in figure 10. The model shows rich physical properties, namely the second-order transition and multicritical points for all values of the crystal-field interactions and for all values of the coordination number q. Here, the model only presents second-order transition for all values of q. The multicritical points B 4 and B 5 which respectively indicate the positions of the hybrid phases F(± 1 2 , ∓1) and F(± 1 2 , ∓2), respectively separate the ferrimagnetic phases F(± 1 2 , ∓ 1 2 ), F(± 1 2 , ∓ 3 2 ) and F(± 1 2 , ∓ 5 2 ).