Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.30125

Quantum Spin-5/2 Blume-Capel Model in a Random Transverse-Crystalline Field Anisotropy

Claudio M. Salgado, Karollaine C. Leite, T.M. Tunes, M. F. Z. Arruda, J.L.B. Faria, Alberto S. de Arruda

Published May 28, 2026• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2605.30125

Related papers

Random transverse single-ion anisotropy in the spin−1 Blume–Capel quantum model

C.M. Salgado, N.L. de Carvalho, P. H. Z. de Arruda +4 more

We have studied the effects of the random transverse single-ion anisotropy in the phase diagram and in the thermodynamic properties of the spin−1 Blume–Capel quantum model, which has been used the approach the mean-field theory based on the Bogoliubov inequality for the Gibbs free energy. The phase diagrams were obtained in the temperature τ=kBT∕zJ versus single-ion anisotropy δx=Dx∕zJ plane for several values of δy=Dy∕zJ, p and q. The results show that there is a tricritical behavior only in the range of δy given by −0.048≤δy<0.5. This behavior is shown again by the curves of the magnetization m as a function of temperature τ with corresponding values of δy, p and q.

•Jan 1, 2019•0 citations

Thermodynamic properties of the spin S=3/2 quantum ferromagnetic Blume-Capel model in a transverse crystal field

D. C. Carvalho, J. A. Plascak

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.

Physical review. E•Mar 24, 2021•15 citations

Critical behavior of the S = 1 Blume Capel model with transverse and random anisotropy based on the Bogoliubov variational approach

Itacy José de Souza, J.L.B. Faria, Ricardo Ferreira Bento +4 more

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.

Phase Transitions•Aug 3, 2025•1 citation

Quantum spin-2 ferromagnetic Blume-Capel model in the presence of a transverse crystal field

Jordana Torrico, D.C. Carvalho, J. A. Plascak

The ferromagnetic spin-2 quantum Blume-Capel model in the presence of a transverse crystal field interaction is studied by employing a variational approach based on Bogoliubov inequality for the free energy and an exact mapping of the ground state. The thermodynamic properties have been obtained using a single spin cluster, which is equivalent to the usual mean-field approximation. The temperature dependence of the magnetization has been analyzed and the global phase diagram, as function of the model parameters, has been obtained for any values of the coordination number of the lattice. A variety of quantum phase transitions, multiple and multicritical points, are present in several topologies of the corresponding phase diagrams. The ground state properties of the system have also been obtained through an exact mapping of the spin-2 model onto another equivalent spin-1 Blume-Capel model in a transverse field. This mapping allows one to determine an exact asymptote formula of the phase diagram at zero temperature in the limit of large longitudinal crystal field for any lattice dimension.

Physical review. E•May 15, 2026•1 citation

Phase diagram for the spin-3/2 quantum ferromagnetic Blume-Capel model in a transverse crystal field: an approximation at the mean-field level

Carvalho, D. C., J. A. Plascak

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.

arXiv:2007.07763•Jul 15, 2020•0 citations

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 +3 more

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.

Phase Transitions•Mar 4, 2025•0 citations
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.
We have studied the effects of the random transverse single-ion anisotropy in the phase diagram and in the thermodynamic properties of the spin−1 Blume–Capel quantum model, which has been used the approach the mean-field theory based on the Bogoliubov inequality for the Gibbs free energy. The phase diagrams were obtained in the temperature τ=kBT∕zJ versus single-ion anisotropy δx=Dx∕zJ plane for several values of δy=Dy∕zJ, p and q. The results show that there is a tricritical behavior only in the range of δy given by −0.048≤δy<0.5. This behavior is shown again by the curves of the magnetization m as a function of temperature τ with corresponding values of δy, p and q.
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.
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 ferromagnetic spin-2 quantum Blume-Capel model in the presence of a transverse crystal field interaction is studied by employing a variational approach based on Bogoliubov inequality for the free energy and an exact mapping of the ground state. The thermodynamic properties have been obtained using a single spin cluster, which is equivalent to the usual mean-field approximation. The temperature dependence of the magnetization has been analyzed and the global phase diagram, as function of the model parameters, has been obtained for any values of the coordination number of the lattice. A variety of quantum phase transitions, multiple and multicritical points, are present in several topologies of the corresponding phase diagrams. The ground state properties of the system have also been obtained through an exact mapping of the spin-2 model onto another equivalent spin-1 Blume-Capel model in a transverse field. This mapping allows one to determine an exact asymptote formula of the phase diagram at zero temperature in the limit of large longitudinal crystal field for any lattice dimension.
We investigate the phase diagram for the spin-3/23/23/2 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.
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.