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Phase Transitions

The adjustable bimodal random exchange interaction effects on the mixed spin-(12,1) Blume-Capel Quantum model with biaxial crystal field: A mean-field approximation calculations

M. Kake, G. Seto, Toussaint Djidjoho Oke, R. A. A. Yessoufou, Erhan Albayrak, A. V. Kpadonou

Published Mar 4, 2025• 0 citations
Publisher

Abstract

The magnetic and thermodynamic properties of the Blume-Capel model are investigated within an approach combined with quantum mechanics and mean-field approximation for the spin-1/2 and spin-1 with coordination number z = 3. The spin-1 sites are subjected to transverse crystal fields Dx and Dy. In addition, the bilinear interaction parameter J is randomly distributed with an adjustable parameter α>0 such that certain sites are under the influence of αJ with probability p while the rest are subject to J with probability 1−p. The thermal behaviors of the order parameters have been studied to determine the nature of phase transitions and to calculate phase diagrams in the (Dx,kT), (Dy,kT), (α,kT), (|J|,kT), and (p,kT) planes. The obtained results revealed a rich variety of critical phenomena including first- and second-order phase transitions, tricritical points, and compensation temperatures. Under an external magnetic field, the system exhibits both simple and multi-hysteresis behaviors. Furthermore, the coercive field and remanent magnetization decrease with increasing temperature.

DOI
10.1080/01411594.2025.2481483

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The magnetic and thermodynamic properties of the Blume-Capel model are investigated within an approach combined with quantum mechanics and mean-field approximation for the spin-1/2 and spin-1 with coordination number z = 3. The spin-1 sites are subjected to transverse crystal fields Dx and Dy. In addition, the bilinear interaction parameter J is randomly distributed with an adjustable parameter α>0 such that certain sites are under the influence of αJ with probability p while the rest are subject to J with probability 1−p. The thermal behaviors of the order parameters have been studied to determine the nature of phase transitions and to calculate phase diagrams in the (Dx,kT), (Dy,kT), (α,kT), (|J|,kT), and (p,kT) planes. The obtained results revealed a rich variety of critical phenomena including first- and second-order phase transitions, tricritical points, and compensation temperatures. Under an external magnetic field, the system exhibits both simple and multi-hysteresis behaviors. Furthermore, the coercive field and remanent magnetization decrease with increasing temperature.
In this work, we investigate the thermodynamic properties of the quantum Blume-Capel model with spin S=5/2 S = 5/2 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 S = 5/2 S=5/2 quantum Blume-Capel model and highlights the role of transverse fields in modifying the critical behavior.
We investigated the critical behavior of the S=1 Blume Capel model in a scenario where the system is subject to a transverse crystalline field (D), which was randomly distributed through a bimodal probability distribution where p (1−p) corresponds to a fraction of sites that are (are not) under the action of the crystalline field. Using mean field theory based on Bogoulibov inequality for free energy, we obtain phase diagrams in the planes of the crystalline field versus temperature (D−τ) and magnetization versus temperature and (m−τ). We show that the topology of the phase diagrams depends significantly on the model parameters. The competition between temperature (τ), exchange interaction, and the random distribution of the crystalline field are responsible for revealing essential details of the critical behavior observed in the phase diagrams obtained as tricritical points (TCP) that separate the phase transition lines first and second order.
The magnetization and phase transition of mixed spin-1/2 and spin-1 Blume-Capel model on a cylindrical Ising nanotube with bimodal random crystal fields were investigated by using the effective field theory. Employing numerical calculation, the phase diagrams and the magnetization which depends on the temperature and the parameters of random crystal fields were obtained. The results show that：(1) Changing the probability and the ratio of the crystal fields, bimodal random crystal fields may describe different doped atoms acting on spins. (2) The critical points do exist for certain values of the probability, the negative or positive crystal field and the ratio of the crystal fields. (3) The system shows several phase transition temperatures, first-order and second-order phase transitions.
The spin-5/2 Blume–Capel model was studied using the mean-field approximation and the Migdal–Kadanoff renormalization group method for two- and three-dimensional systems. We determined the phase diagrams in the (crystal field, temperature) plane where the system exhibited first- and second-order phase transitions as well as isolated critical, bicritical and triple points. In order to show first-order transitions at low temperature, we presented the total magnetization per site and the derivative of the free energy as a function of the crystal field. Moreover, the critical exponents of the system were calculated by linearizing the renormalization transformation at the vicinity of the second-order fixed points.
The mean-field approximation is used to calculate the phase diagrams of the spin-3/2 random transverse crystal field (RTCF) model. A bimodal distribution of the RTCF is employed in which one of the modes is in competition with the other mode whose strength is adjusted with a controllable parameter α. Thus, some sites in the lattice are under the effect of RTCF Dx with probability p and the rest of the sites are under the effect of αDx with 1 − p. The phase diagrams of the model are obtained on the RTCF-temperature planes by varying p on square lattices. Both second- and first-order phase transitions are displayed and thus tricritical points are also observed. The critical end points and end points for the first-order phase transition lines were also found.
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.