An effective field theory with correlations has been used to study the critical behavior of the spin-1 Blume–Capel bilayer Ising model on a square lattice. The effects of the Hamiltonian parameters on thermodynamic quantities and phase diagrams are investigated in detail. We found that the system exhibits the first and the second order transitions as well as tricritical point. Furthermore, we have observed that the change of tricritical point values depends on interaction parameters.
An effective field theory with correlations has been used to study the critical behavior of the spin-1 Blume–Capel bilayer Ising model on a square lattice. The effects of the Hamiltonian parameters on thermodynamic quantities and phase diagrams are investigated in detail. We found that the system exhibits the first and the second order transitions as well as tricritical point. Furthermore, we have observed that the change of tricritical point values depends on interaction parameters.
Using a 1/z expansion we study the correlation of fluctuations in a spin-1 Ising model with anisotropy. Previously, the 1/z expansion has been applied only to the second-order phase transitions of ferromagnets and granular superconductors. By imposing conditions on the free energy F and the parameter x=1-〈S1z2〉, we use the 1/z technique to examine both the first- and second-order parts of the Blume-Capel phase diagram. In agreement with many numerical simulations and experiments on $^{3}\mathrm{\ensuremath{-}}^{4}$He mixtures, the value of x at the tricritical point is not shifted from its mean-field value of 2/3 to order 1/z. On the other hand, the tricritical values of the anisotropy parameter A and of the transition temperature Tc are altered by the correlation of fluctuations. At least to order 1/z, all the derivatives of Tc are continuous across the tricritical point.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.
In this work, we investigate the thermodynamic properties of the quantum Blume-Capel model with spin S=5/2 in the presence of transverse and random crystalline fields. The system is described by a Hamiltonian that includes ferromagnetic exchange interactions between nearest neighbors, a longitudinal single-ion anisotropy, and a transverse single-ion anisotropy. Using a mean-field approach based on Bogoliubov's inequality for the Gibbs free energy, we derive the fundamental thermodynamic potential and the equation of state for the magnetization. The influence of the longitudinal and transverse anisotropy parameters on the magnetic ordering and phase transitions is analyzed in detail. We present magnetization versus temperature diagrams for various combinations of the anisotropies, exploring both positive and negative values. Our results reveal that the system exhibits standard second-order phase transitions for most parameter ranges, with no evidence of tricritical behavior. However, for certain positive values of the anisotropies, the model displays a first-order phase transition within the ordered phase, characterized by a jump from a higher-spin ordered state to a lower-spin ordered state. The critical temperatures are shown to be sensitive to the magnitude and sign of the anisotropy parameters. In particular, negative transverse anisotropies favor magnetic order, raising the critical temperature, while positive anisotropies promote disorder, lowering the critical temperature. This study provides a comprehensive analysis of the phase diagram of the S=5/2 quantum Blume-Capel model and highlights the role of transverse fields in modifying the critical behavior.
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.
Purpose The authors investigate the magnetic properties of a mixed spin-3/2 and spin-2 Blume-Capel model on square and cubic lattices with two different single-ion anisotropies. Design/methodology/approach To study the critical behavior of a mixed spin-3/2 and spin-2 system, the authors have used a real space renormalization group approximation and specifically the Migdal-Kadanoff technique. The authors give the phase diagrams for two different cases: (1) on the (Δ/|J|, 1/|J|) plane with ΔA = ΔB = Δ, and (2) on the (ΔA/|J|, 1/|J|) and (ΔB/|J|, 1/|J|) planes for selected values of ΔB/|J| and ΔA/|J|, respectively. Findings The phase diagrams obtained show that the system exhibits both second- and first-order phase transitions as well as tricritical points for some values of the anisotropies. Moreover, using the variation of the free energy and its derivative at low temperatures, the authors have seen the appearance of first-order transitions at very low temperatures. Originality/value Few investigations of mixed spin-3/2 and spin-2 systems with crystal field have been realized. For this reason, the authors use the renormalization group approach to complete the work done on these systems. In absence of an exact solution, this contributes to the synthesis of the approximation results on mixed spins models.
We investigate the phase diagram for the spin-3/2 ferromagnetic Blume-Capel model in a transverse crystal field using the standard mean-field approximation within the framework of Bogoliubov inequality for free energy. We draw a very rich phase diagram with first- and second-order transition lines; tricritical and tetracritical points; critical endpoint of order 2 and double critical endpoint. Additionally, the behaviour of magnetisation as a function of temperature over a wide range of values of both longitudinal and transverse crystal fields is also analysed. To the best of our knowledge, this quantum spin model has only been studied employing an effective field theory, which in turn was not able to characterise completely the multicritical phenomena in its phase diagram, because that procedure is not based on free energy. Thus, our findings on the phase diagram for the present model are novel as they have been not previously reported.