,
Erna Laeticia Tchinda Ngounou,
Bertrand Sitamtze Youmbi,
Georges Collince Fouokeng*,
Aurelien Kenfack Jiotsa,
Nirina Randrianantoandro
In the present work, we report on a theoretical investigation of the magnetoelectric parameters of the helical Heisenberg multiferroic spin chain model. The spin-wave approximation in the bosonization process is used to evaluate the energy spectrum of the quadratic form of the model as that of regular quantum gases. The quantized form of the model is conveniently treated using the canonical ensemble in terms of the free energy developed by the use of Landau theory. Following the Fermi-Dirac statistics of quantum gases, the joint effect of Dzyaloshinskii -Mriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions on the magnetoelectric properties of the helical multiferroic spin structure controlled by a static electric field in the y-direction and a magnetic field in the z-direction is quantified. The magnetization, susceptibility, and electric polarization are used as measurable parameters. The work convincingly establishes that the juxtaposition of KSEA interactions to DM interactions is worthy since, on the one hand, it amplifies the magnetic property of the system, and on the other hand makes it possible to control the phase transition dynamics induced in such materials by the symmetric inversion due to the DM interactions. At low temperature, the measurable parameters are closely related to ferroelectricity and ferromagnetism in multiferroic materials, where all effects favor the symmetrization mechanism and hence magnetoelectric properties. These properties are critical in spintronics and information storage control.
| Published in | American Journal of Modern Physics (Volume 15, Issue 2) |
| DOI | 10.11648/j.ajmp.20261502.11 |
| Page(s) | 13-23 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Helical Spin, Isotropic Spin, Magnetoelectric, DM Interaction, KSEA Interaction, Quantum Phase Transition, Spin-orbit Coupling
can also be obtained. However, Moriya 
(1) 
(2) 
, 
, and, 
, 
are the interaction parameters in the x- and y- directions, respectively, i.e., the intrachain Heisenberg interaction parameter with nearest and NNN responsible for the helical structure in the spin system, 
(3) 
the transverse magnetic field, 
(4) 
, and, 
being the magnetocrystalline anisotropy constants 
(5) 
a net electric polarization well described by the spin-current model and found under spiral and conical spin structures 
is the propagation direction of the spin spiral and 
the moments on neighboring spins. 
is the external electric field. The sum is over the sites 
, with 
the number of spins per unit volume, 
(6) 
, the DM factor taken along the z-direction), 
(7) 
. 
) in charge of SU (2) symmetry breaking on its own, which includes NN spins and NNN, has been shown to be suitable to explain meta-magnetoelectric excitations due to its significant polarization in the ferroelectric phase and strong ME coupling 
( 
the Pauli matrix defined at site i in the three space directions 
) into the solvable model, the spin wave theory, based successively on the Jordan-Wigner (JW) transformations, Fourier transformations and thus Bogoliubov transformations, in the adiabatic limit, is used in this work as a diagonalization procedure. The NNN interactions, represented by a set of four fermions in JW representations, can result in 2l-fermions vertices for spin exchanges between NN of rank 1. The system deals with non-vanishing four-fermion order terms implying that the NNN term turns the model into deep paramagnetic phase for all positive value of 
where all spins are oriented in the direction of the transverse field such that 
. 
(8) 
(9) 
is the Bogoliubov quasiparticle expressed in terms of the Fourier transform operator 
and be the wave numbers in the range 
. 
and magnetization 
are related to the electric 
and magnetic fields 
by the following relations 
(10) 
(11) 
and 
are the spontaneous polarization and magnetization components, respectively, 
and 
are the electrical permittivity and magnetic permeability constants in the vacuum, 
and 
are the second-order tensors of electric and magnetic susceptibilities of the substance. 
and 
are the electric field and magnetic field components in the medium. 
are the tensors used to determine the magnitude of the linear ME effect in the material. 
of the system with applied fields (magnetic and electric), can be expressed in Landau theory as 
(12) 
is the ground state free energy. 
(13) 
(14) 
( 
is the absolute temperature and 
is the Boltzmann constant). 
(15) 
(16) 
(17) 
(18) 
and 
are respectively a dimensionless energy and the number of particles with energy 
defined according to Fermi-Dirac statistics at thermal equilibrium. 
the phase factor linking 
to 
. The coefficients 
, 
, 
, and 
are respectively the magnetization, magnetic susceptibility, electric polarization and the electric susceptibility of the system. 
and 
. 
measured under the joint effect of DM and KSEA interaction, with 
in (M1a) and 
in (M1b). The other parameters are
and 
.
measured under the joint effect of DM and KSEA interaction, with in (M2a) and in (M2b). The other parameters are 
and .
and electric polarization 
are comparable concepts used to quantify magnetic 
and electric field 
responses, respectively, indicating magnetic dipole moment density and electric dipole moment density, both of which align spin moment. Magnetic/electric susceptibility are characteristic parameters that indicate how a quantum system reacts to external fields, resulting in quantum/dynamical quantum phase transitions. In ME multiferroics, ME coupling is a direct result of applied magnetic and electric fields. It is the signature that controls the ME parameters by the action of applied external fields. All influences on the ME coupling, such as spin-orbit interactions investigated in this study, are critical for creating and understanding novel material states devices. Modifying the ME coupling factor affects the electric structure and thus the energy levels, spin dynamics, and even the material's topology 
, 
and in Figures 5 and 6 the polarization behavior at 
(for M1a, M2a, P1a and P2.a) and 
(for M1b, M2b, P1.b and P2.b) alongside fits obtained under different values of 
and 
. 
measured under the joint effect of DM and KSEA interaction, with 
in (S1a) and 
in (S1b). The other parameters are 
and .
measured under the joint effect of DM and KSEA interaction, with in (S2a) and in (S2b). The other parameters are: 
and .
measured under the joint effect of DM and KSEA interaction, with in (P1a) and in (P1b). The other parameters are and .
measured under the joint effect of DM and KSEA interaction, with in (P2a) and in (P2b). The other parameters areand .
, which is responsible for symmetry breaking, preserves the observed behavior in the evolution of magnetization and polarization with respect to the magnetic field (Figures 1 and 5), while breaking the symmetric behavior observed in their evolution when they depend on the electric field (Figures 2 and 6). However, the factor 
tends to resist the action needed by the interaction parameters 
, diminishing its influence and providing greater control over the magnetic and electric domains. 
measured under joint effect of DM and KSEA interaction, with in (Se1a) and in (Se1b). The other parameters are: and .
measured under joint effect of DM and KSEA interaction, with in (Se2a) and in (Se2b). The other parameters are: and .
shows a sudden change in concavity at the critical values of the applied field (Figures 3 and 4). These points correspond to the phase transition points. There are several such points depending on the electric field (Figure 4). In contrast, the electric susceptibility 
exhibits progression with a number of positive transitions at the critical points of the applied field (Figures 7 and 8). The multiplicity of critical points as a function of the electric field refers to the variation of the action of the transverse electric field with respect to that of the anisotropy 
and that of the magnetic field applied along the z-axis. The variation of the DM interaction generates quantum transition dynamics phase diagram.
et 
. With reference to the result obtained in Eq. (18), we can read the possibility of coupling the two susceptibilities by a non-linear factor whose profile is represented in Figure 9. DM | Dzyaloshinskii Moriya |
KSEA | Kaplan-Shekhtman-Entin-Wohlman-Aharony |
NN | Nearest Neighbor |
NNN | Next Nearest Neighbor |
ME | Magnetoelectric |
JW | Jordan-Wigner |
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APA Style
Ndakom, B., Ngounou, E. L. T., Youmbi, B. S., Nadjitonon, N., Fouokeng, G. C., et al. (2026). Magnetic Properties of One-dimensional Helical Spin System with Isotropic Spin Exchanges Under the Effect of DM and KSEA. American Journal of Modern Physics, 15(2), 13-23. https://doi.org/10.11648/j.ajmp.20261502.11
ACS Style
Ndakom, B.; Ngounou, E. L. T.; Youmbi, B. S.; Nadjitonon, N.; Fouokeng, G. C., et al. Magnetic Properties of One-dimensional Helical Spin System with Isotropic Spin Exchanges Under the Effect of DM and KSEA. Am. J. Mod. Phys. 2026, 15(2), 13-23. doi: 10.11648/j.ajmp.20261502.11
AMA Style
Ndakom B, Ngounou ELT, Youmbi BS, Nadjitonon N, Fouokeng GC, et al. Magnetic Properties of One-dimensional Helical Spin System with Isotropic Spin Exchanges Under the Effect of DM and KSEA. Am J Mod Phys. 2026;15(2):13-23. doi: 10.11648/j.ajmp.20261502.11
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Download@article{10.11648/j.ajmp.20261502.11,
author = {Blaise Ndakom and Erna Laeticia Tchinda Ngounou and Bertrand Sitamtze Youmbi and Ngarmaim Nadjitonon and Georges Collince Fouokeng and Aurelien Kenfack Jiotsa and Nirina Randrianantoandro},
title = {Magnetic Properties of One-dimensional Helical Spin System with Isotropic Spin Exchanges Under the Effect of DM and KSEA},
journal = {American Journal of Modern Physics},
volume = {15},
number = {2},
pages = {13-23},
doi = {10.11648/j.ajmp.20261502.11},
url = {https://doi.org/10.11648/j.ajmp.20261502.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmp.20261502.11},
abstract = {In the present work, we report on a theoretical investigation of the magnetoelectric parameters of the helical Heisenberg multiferroic spin chain model. The spin-wave approximation in the bosonization process is used to evaluate the energy spectrum of the quadratic form of the model as that of regular quantum gases. The quantized form of the model is conveniently treated using the canonical ensemble in terms of the free energy developed by the use of Landau theory. Following the Fermi-Dirac statistics of quantum gases, the joint effect of Dzyaloshinskii -Mriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions on the magnetoelectric properties of the helical multiferroic spin structure controlled by a static electric field in the y-direction and a magnetic field in the z-direction is quantified. The magnetization, susceptibility, and electric polarization are used as measurable parameters. The work convincingly establishes that the juxtaposition of KSEA interactions to DM interactions is worthy since, on the one hand, it amplifies the magnetic property of the system, and on the other hand makes it possible to control the phase transition dynamics induced in such materials by the symmetric inversion due to the DM interactions. At low temperature, the measurable parameters are closely related to ferroelectricity and ferromagnetism in multiferroic materials, where all effects favor the symmetrization mechanism and hence magnetoelectric properties. These properties are critical in spintronics and information storage control.},
year = {2026}
}
TY - JOUR T1 - Magnetic Properties of One-dimensional Helical Spin System with Isotropic Spin Exchanges Under the Effect of DM and KSEA AU - Blaise Ndakom AU - Erna Laeticia Tchinda Ngounou AU - Bertrand Sitamtze Youmbi AU - Ngarmaim Nadjitonon AU - Georges Collince Fouokeng AU - Aurelien Kenfack Jiotsa AU - Nirina Randrianantoandro Y1 - 2026/03/05 PY - 2026 N1 - https://doi.org/10.11648/j.ajmp.20261502.11 DO - 10.11648/j.ajmp.20261502.11 T2 - American Journal of Modern Physics JF - American Journal of Modern Physics JO - American Journal of Modern Physics SP - 13 EP - 23 PB - Science Publishing Group SN - 2326-8891 UR - https://doi.org/10.11648/j.ajmp.20261502.11 AB - In the present work, we report on a theoretical investigation of the magnetoelectric parameters of the helical Heisenberg multiferroic spin chain model. The spin-wave approximation in the bosonization process is used to evaluate the energy spectrum of the quadratic form of the model as that of regular quantum gases. The quantized form of the model is conveniently treated using the canonical ensemble in terms of the free energy developed by the use of Landau theory. Following the Fermi-Dirac statistics of quantum gases, the joint effect of Dzyaloshinskii -Mriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions on the magnetoelectric properties of the helical multiferroic spin structure controlled by a static electric field in the y-direction and a magnetic field in the z-direction is quantified. The magnetization, susceptibility, and electric polarization are used as measurable parameters. The work convincingly establishes that the juxtaposition of KSEA interactions to DM interactions is worthy since, on the one hand, it amplifies the magnetic property of the system, and on the other hand makes it possible to control the phase transition dynamics induced in such materials by the symmetric inversion due to the DM interactions. At low temperature, the measurable parameters are closely related to ferroelectricity and ferromagnetism in multiferroic materials, where all effects favor the symmetrization mechanism and hence magnetoelectric properties. These properties are critical in spintronics and information storage control. VL - 15 IS - 2 ER -
Department of Physics and Engineering Sciences, University of N’Djamena, N’Djamena, Chad
Department of Physics, University of Yaounde 1, Yaounde, Cameroon
Faculty of Engineering and Technology, University of Buea, Buea, Cameroon
Department of Physics, University of Dschang, Dschang, Cameroon
Department of Physics, University of Yaounde 1, Yaounde, Cameroon
Institute of Molecules and Materials of Le Mans, Le Mans University, Le Mans, France
Figure 1. Magnetic field-dependent magnetizationmeasured under the joint effect of DM and KSEA interaction, with in (M1a) and in (M1b). The other parameters are and .
Figure 2. Electric field-dependent magnetizationmeasured under the joint effect of DM and KSEA interaction, with in (M2a) and in (M2b). The other parameters are and .
Figure 3. Magnetic field-dependent magnetic susceptibility measured under the joint effect of DM and KSEA interaction, with in (S1a) and in (S1b). The other parameters are and .
Figure 4. Electric field-dependent magnetic susceptibility measured under the joint effect of DM and KSEA interaction, with in (S2a) and in (S2b). The other parameters are: and .
Figure 5. Magnetic field-dependent electric polarization measured under the joint effect of DM and KSEA interaction, with in (P1a) and in (P1b). The other parameters are and .
Figure 6. Electric field-dependent electric polarization measured under the joint effect of DM and KSEA interaction, with in (P2a) and in (P2b). The other parameters areand .
Figure 7. Magnetic field -dependent electric susceptibility measured under joint effect of DM and KSEA interaction, with in (Se1a) and in (Se1b). The other parameters are: and .
Figure 8. Electric field -dependent electric susceptibility measured under joint effect of DM and KSEA interaction, with in (Se2a) and in (Se2b). The other parameters are: and .
Figure 9. Magnetic susceptibility-dependent electric susceptibility phase diagram.
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