Magnetic properties in the hybrid model of Ising spin-1/2: A Mean-Field theory approach in the kagome lattice

doi:10.1016/j.physleta.2021.127551

Physics Letters A

Volume 411, 30 September 2021, 127551

https://doi.org/10.1016/j.physleta.2021.127551 Get rights and content

Highlights

•
A frustrated antiferromagnetic system in MFT approximations is studied.
•
Dependence of magnetization of the model of a cluster of the 12 sites is studied.
•
The formation of the magnetization plateaus for low temperatures is shown.
•
Appearance of magnetic charges in the Kagome lattice is found.

Abstract

In this work, we investigate the magnetic properties using the hybrid Hamiltonian (spin/charge) in the Ising spin-1/2 antiferromagnetic model in the kagome lattice, which presents geometric frustration. The study was carried out using the Mean-Field theory obtained from the generalization of Callen's identity to obtain expressions for the magnitude magnetization as a function of temperature and an external field. In the presence of an external field the model exhibits the characteristics of a frustrated system through the appearance of magnetization plateaus on the magnetization curves. As geometric frustration in magnetic materials can provide exotic states of matter, in our investigations it was possible to verify the appearance of magnetic charges in the kagome lattice through a hexagonal sublattice using the spin/charge hybrid model.

Introduction

The antiferromagnetic spin models in structures that present geometric frustrations have been studied by several theorists and experimental. This interest is due to the exotic properties that such systems exhibit. These properties arise due to the magnetic moments with antiferromagnetic couplings, showing frustration due to the geometric structure that the model presents and failing to minimize its ground state energy, thus inhibiting the formation of a magnetically ordered collinear state. Thus many different spin configurations with the same ground state energy appear, and the best known example is probably the two-dimensional Ising antiferromagnet in a triangular lattice. Wannier [1] showed that this model has a great degeneracy of fundamental states. Other geometries also exhibit the phenomenon of frustration, such as the antiferromagnetic systems in the kagome lattice [2], [3], [4], and also in the tetrahedral structure of the pyrochlore lattice [5], [6], [7]. The latter being modeled for the first time by Anderson [8].

Still, under certain conditions, these frustrated systems can lead to the formation of exotic states of matter, as shown in the description of the low temperature behavior of pyrochlore spin ice, proposed by Castelnovo et al. [9]. It involves the emergence of magnetic monopole quasi-particles, where each spin behaves like a magnetic dipole, and can be effectively replaced by a dumbbell of magnetic charges with opposite signs, to which individual charge contributions can be added for each tetrahedron, resulting in the so-called effective charge/tetrahedron. Comparable with the pyrochlore lattice, but in two dimensions, the structure kagome lattice is a natural generalization [10].

Although traditionally the typical properties of frustrated magnetic systems are related to the quantum nature of magnetism, many properties of these systems can be investigated using well-defined classical models of statistical mechanics. Among the exotic phenomena that emerge from these systems, it is possible to verify with the application of an external magnetic field, the appearance of magnetization plateaus. We can cite some works that used the Ising model and found these plateaus, for example, in Monte Carlo simulations [11], [12], [13], [14], [15], and also with the application of the effective-field theory [16], [17], [18], [19], approximation Husimi recursive lattice [20], [21] and recursive tetrahedron [22].

The objective of this paper was to study the magnetization behavior of the Ising spin-1/2 hybrid antiferromagnetic model in the kagome lattice [23] in the presence of an external field, using the Mean-Field theory (MFT) in the structure of a cluster with 12 sites (see Fig. 1). It takes into account the spin/charge ratio, and can be related to the phenomenon of magnetic monopole quasi-particles [9] that can occur as out-of-ground state excitations in a magnetic system. In section 2, we show the hybrid Hamiltonian used and describe the use of MFT in order to obtain expressions for the magnetization by spin $〈 S_{i} 〉$ and also the magnetization in the spin/charge correspondence, defined as the local sum of the magnetic poles and called in the magnetic charge study $〈 Q_{u} 〉$ . In Section 3, we analyze numerical results and diagrams. From the magnetization curves for two different coupling values, we report on the fundamental state phase diagram as a function of the external magnetic field, thus revealing the occurrence of magnetization plateaus, and also we calculate the dependence of magnetization by spin and magnetization by magnetic charge as a function of temperature. In Section 4 we present the concluding observations.

Section snippets

Model and formulation

In this section we will present the Mean-Field Theory applied to the hybrid spin model (spin-charge), in the structure of a kagome lattice, from the generalization of Callen's identity [25].

Let us introduce the Hamiltonian [23], treated in this work, in the following form: $H = - J \sum_{〈 i, j 〉} S_{i} S_{j} - h \sum_{i} S_{i} - K \sum_{〈 u, v 〉} Q_{u} Q_{v},$ where J represents the coupling between the nearest-neighbors of the $S_{α}$ spin variables, while the K constant corresponds to the coupling between the magnetic charges $Q_{θ}$ of the nearest-neighbor

Numerical results

In this section, we present the phase diagram ( $h \times T$ ) and the diagrams for the variables $m_{s}$ , $m_{q 1}$ , $m_{q 2}$ and $m_{q T}$ as a function of the external field h and temperature T, obtained at from the application of MFT in the Ising spin-1/2 hybrid anisotropic model in a kagome lattice. For simplicity we consider $k_{B} = 1$ . We show with the application of an external field that in the Ising spin-1/2 hybrid antiferromagnetic model it exhibits, besides the saturated ground state from a certain value of the external

Conclusion

We present a study aiming at analyzing the magnetization behavior of the Ising spin-1/2 hybrid antiferromagnetic model in the kagome lattice [23] in the presence of an external field using the Mean-Field theory in the structure of a cluster with 12 sites. The main objective was to study the spin/charge relationship that arises due to the geometric frustration that the model presents and can be related to the phenomenon of magnetic monopole quasi-particles [9]. The results showed that the

CRediT authorship contribution statement

S. Oliveira: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. J.P. Santos: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Supervision, Validation, Visualization, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

J.P. Santos would like to thank the Brazilian agencie CNPq (No. 306404/2019-2) for financial support.

References (30)

G.H. Wannier
Phys. Rev.
(1950)
H.O. Jeschke et al.
Phys. Rev. B
(2019)
K. Soldatov et al.
Phys. Lett. A
(2019)
N. Jiang et al.
Nat. Chem.
(2020)
E. Jurčišinová et al.
Physica A
(2020)
K. Soldatov et al.
Phys. Lett. A
(2017)
M. Mohylna et al.
Phys. Lett. A
(2019)
P.W. Anderson
Phys. Rev.
(1956)
C. Castelnovo et al.
Nature
(2008)
J.T. Chalker
Introduction to Frustrated Magnetism
(2011)

K. Zhao et al.

Science

(2020)

X. Yao

J. Magn. Magn. Mater.

(2010)

R.G. Melko et al.

J. Phys. Condens. Matter

(2004)

H. Fukazawa et al.

Phys. Rev. B

(2002)

L. Regeciová et al.

Eur. Phys. J. B

(2020)

Cited by (1)

Study of the magnetocaloric effect and magnetic properties in the spin-3/2 Blume–Capel model
2023, Journal of Magnetism and Magnetic Materials
In this study, the magnetic properties and the magnetocaloric effect of the spin-3/2 Blume–Capel model were explored through the Mean Field Theory, developed through a variational principle based on the Bogoliubov inequality to derive thermodynamic expressions such as free energy, magnetization and entropy. The analysis focused on magnetization, free energy and phase diagram with the presence of an external field greater than and equal to zero. The magnetocaloric effect was evaluated by observing the magnetic entropy changes due to different variations of the external magnetic field. Results were analyzed for both continuous and first-order phase transitions. To validate the magnetic entropy results, Maxwell’s relations were applied in the regions with continuous magnetization variations, and the Clausius–Clapeyron equation was used in the regions with magnetization discontinuity.

View full text

Magnetic properties in the hybrid model of Ising spin-1/2: A Mean-Field theory approach in the kagome lattice

Highlights

Abstract

Introduction

Section snippets

Model and formulation

Numerical results

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgements

Phys. Rev.

Phys. Rev. B

Phys. Lett. A

Nat. Chem.

Physica A

Phys. Lett. A

Phys. Lett. A

Phys. Rev.

Nature

Introduction to Frustrated Magnetism

Science

J. Magn. Magn. Mater.

J. Phys. Condens. Matter

Phys. Rev. B

Eur. Phys. J. B