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(Color online) Various spin textures result from Monte Carlo simulation. (a) helical. (b) helical + skyrmion. (c) skyrmion lattice and (d) ferromagnetic + skyrmion.
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The continuum limit of the tilted SU(2) spin model is shown to lead to a gauge Landau-Lifshitz equation which provides a unified description for various spin orders. For a definite gauge with zero field strength, we find a double periodic solution, where the conical spiral, in-plane spiral, helical, and ferromagnetic spin orders become special case...
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... strength of the DM interaction K/J = √ 2 tan (2π/9), we plot the phase diagrams in the plane of temperature versus the magnetic field based on our Monte Carlo simulations for both our model (11) and the conventional model. 14 The conventional model is our model without the anisotropic term and the typical spin orders involved are plotted in Fig. 5. Our results manifest that the landscape of those two phase diagrams are similar while the area ratio of the skyrmion lattice phase to the helical phase in our model is larger than that in the conventional model (see The helical phase and the ferromagnetic phase have no skyrmions and the skyrmion lattice phase has many skyrmions. In ...
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Citations
... Note that one is able to go beyond the standard thermodynamics by introducing a complex order parameter in the Ginzburg-Landau (GL) phenomenological theory for superconductors [25]. Meanwhile, various exotic spin orders [26,27], such as conical spiral, in-plane spiral, helical, as well as the ferromagnetic orders can be unified as different solutions of the Landau-Lifshitz equation with different nonabelian gauge potentials [28]. There the nonabelian gauge potential plays the role of connection that defines a nontrivial parallel transport resulting versatile scene frames. ...
The Ginzburg-Landau theory, which was introduced to phenomenologically describe the destruction of superconductivity by a magnetic field at the beginning, has brought up much more knowledge beyond the original one as a mean-field theory of thermodynamics states. There the complex order parameter plays an important role. Here we propose a macroscopic theory to describe the features of ferroelectrics by a two-component complex order parameter coupled to nonabelian gauge potentials that provide more freedom to reflect interplays between different measurables. Within this theoretical framework, some recently discovered empirical static and time-independent phenomena, such as vortex, anti-vortex, sprial orders can be obtained as solutions for different gauge potentials. It is expected to bring in a new angle of view with more elucidation than the traditional one that takes the polarization as order parameter.
... It is commonly used to study the classical dynamics of nanomagnetic systems such as thin films, nanoparticles, and spin chains [18][19][20][21]. It is capable of describing dynamical processes such as magnetic switching, spin waves, and skyrmions, between others, using both theoretical and numerical approaches [22][23][24][25][26][27][28][29]. Moreover, it has been shown that it can also describe quantum systems described by the open generalized Heisenberg model [30], showcasing an applicability for magnetic systems beyond classical magnets. ...
Magnetic materials and structures of macroscopic and microscopic size have been of interest during the past decades due to their application in data storage, flexible electronics, and medicine, among others. From the microscopic point of view, these systems are typically studied using the Landau-Lifshitz equation (LLE), while approaches such as the dumbbell model are used to study macroscopic magnetic structures. In this work we use both the LLE and the dumbbell model to study spin chains of various lengths under the effect of a time dependent-magnetic field, allowing us to compare qualitatively the results obtained by both approaches. This has allowed us to identify and describe in detail several frequency modes that appear, with additional modes arising as the length of the chain increases. Moreover, we find that the high-frequency modes tend to be absorbed by lower frequency ones as the amplitude of the field increases. The results obtained in this work are of interest not only to better understand the behavior of the macroscopic chain of spins but also to expand the tools available for qualitative studies of both macroscopic and microscopic versions of the studied system. This would allow to study of the qualitative behavior of microscopic systems (e.g. nanoparticles) using macroscopic arrays of magnets, and vice-versa.
... It is commonly used to study the classical dynamics of nanomagnetic systems such as thin films, nanoparticles, and spin chains [18][19][20][21]. It is capable of describe dynamic processes such as magnetic switching, spin waves, and skyrmions, between others, using both theoretical and numerical approaches [22][23][24][25][26][27][28][29]. Moreover, it has been shown that it can also describe quantum systems described by the open generalized Heisenberg model [30], showcasing an applicability for magnetic systems beyond classical magnets. ...
... It is remarkable that in the limiting case A ¼ κ 2 =2J; B ¼ 0, the Hamiltonian (1.1) can be written as the static version of the SUð2Þ gauged Oð3Þ NLσ model [19,20] H ¼ J 2 Z d 2 xð∂ k m þ A k × mÞ 2 ; k ¼ 1; 2; ð1:2Þ ...
... with a background gauge field A 1 ¼ ð−κ=J; 0; 0Þ; A 2 ¼ ð0; −κ=J; 0Þ. Though the DM term is usually introduced phenomenologically, a mathematical derivation of the Hamiltonian (1.2) with arbitrary A k has been developed recently [19]; i.e., it has been shown that the Hamiltonian can be derived mathematically in a continuum limit of the tilted (quantum) Heisenberg model ...
... Following the procedure to obtain a gauged NLσ model from a spin system, as discussed in Ref. [19], we consider a generalization of the SU(3) Heisenberg model defined by the Hamiltonian ...
We study two dimensional soliton solutions in the CP2 nonlinear sigma model with a Dzyaloshinskii-Moriya type interaction. First, we derive such a model as a continuous limit of the SU(3) tilted ferromagnetic Heisenberg model on a square lattice. Then, introducing an additional potential term to the derived Hamiltonian, we obtain exact soliton solutions for particular sets of parameters of the model. The vacuum of the exact solution can be interpreted as a spin nematic state. For a wider range of coupling constants, we construct numerical solutions, which possess the same type of asymptotic decay as the exact analytical solution, both decaying into a spin nematic state.
... It is remarkable that in the limiting case A = κ 2 /2J, B = 0, the Hamiltonian (1.1) can be written as the static version of the SU(2) gauged O(3) NLσ-model [19,20] ...
... with a background gauge field A 1 = (−κ/J, 0, 0), A 2 = (0, −κ/J, 0). Though the DM term is usually introduced phenomenologically, a mathematical derivation of the Hamiltonian (1.2) with arbitrary A k has been developed recently [19], i.e.; it has been shown that the Hamiltonian can be derived mathematically in a continuum limit of the tilted (quantum) Heisenberg model ...
... Following the procedure to obtain a gauged NLσ-model from a spin system, as discussed in Ref. [19], we consider a generalization of the SU(3) Heisenberg model defined by the Hamiltonian ...
We study two dimensional soliton solutions in the $CP^2$ nonlinear $\sigma$-model with a Dzyaloshinskii-Moriya type interaction. First, we derive such a model as a continuous limit of the $SU(3)$ tilted ferromagnetic Heisenberg model on a square lattice. Then, introducing an additional potential term to the derived Hamiltonian, we obtain exact soliton solutions for particular sets of parameters of the model. The vacuum of the exact solution can be interpreted as a spin nematic state. For a wider range of coupling constants, we construct numerical solutions, which possess the same type of asymptotic decay as the exact analytical solution, both decaying into a spin nematic state.
... In three-dimensional bulk magnets, skyrmion lattices are stabilized by thermal fluctuations above the helical state [4]. Interestingly, in the thin-film, two-dimensional limit, the skyrmion lattice is stable over a wide range of the phase diagram [10][11][12]. ...
... Such anisotropy, if Ising-like, stabilizes Ising-type vertices. A perpendicular magnetic field will similarly stabilize the Ising vertices, and in particular favor the skyrmion lattice, consistent with the well-known fact that external fields can stabilize skyrmion lattice phases [1,2,[10][11][12]. ...
We study untwisted heterobilayers of ferromagnetic and antiferromagnetic van der Waals materials, with in particular a Dzyaloshinskii-Moriya interaction in the ferromagnetic layer. A continuum low energy field theory is utilized to study such systems. We develop a phase diagram as a function of the strength of inter-layer exchange and Dzyaloshinskii-Moriya interactions, combining perturbative and strong coupling analyses with numerical simulations using Landau-Lifshitz-Gilbert equations. Various moir\'e-periodic commensurate phases are found, and the commensurate-incommensurate transition is discussed. Among the commensurate phases, we observe an interesting skyrmion lattice phase wherein each moir\'e unit cell hosts one skyrmion.
... Controlling of magnetic skyrmions is an essential issue regarding their potential application in devices. Nowadays, various schemes have been demonstrated such as, the use of electric currents, [14][15][16][17][18][19] temperature gradient, [20][21][22] microwaves, 23,24) or strain. 25) In particular, manipulating magnetic skyrmions by an electric field has been achieved through the voltage-controlled magnetic anisotropy in ultrathin metal films 26,27) , or the magnetoelectric coupling in multiferroic insulator. ...
We demonstrate that an electric field could activate the three low-energy eigenmodes of the Néel-type skyrmion lattice via the electrically induced Dzyaloshinskii–Moriya interaction. In particular, we predict that the relative intensity of the clockwise rotation mode against the counter-clockwise rotation mode is significantly enhanced for the electrical activation in comparison with the magnetic activation. We also discover that the electrically and magnetically active modes obey unique selection rules. These findings promise a fresh pathway towards energy-efficient electrical manipulation of skyrmion excitations for future skyrmion-based magnonics.
... Therefore, several experimental techniques like neutron scattering [8], Lorentz transmission microscopy [9], magnetic transmission x-ray microscopy [15], magneto optical Kerr effect (MOKE) microscopy [11] or Spin polarized scanning tunneling microscopy [13] have been used to investigate magnetic Skyrmions. In addition to the experimental investigations combined analytical and numerical calculations [16,17] as well as Monte Carlo [7,[18][19][20] and spin dynamics simulations [1,14,[21][22][23][24] have been performed to get a deeper understanding of the topic and more informations about statics, thermodynamics and the dynamics of magnetic Skyrmions. Especially spin dynamics simulations can be used to describe the dynamics of magnetic Skyrmions. ...
... The phase diagram is similar to phase diagrams calculated with classical Monte Carlo calculations in [7,19]. However, there is one significant difference. ...
... However, there is one significant difference. In [7,19] the Skyrmion phase is divided into a Skyrmion lattice (SkX) and a Skyrmion gas phase marked as SkX + FM. To see the transition between the Skyrmion lattice and Skyrmion gas phase thermal fluctuations are necessary. ...
A self-consistent mean field theory is introduced and used to investigate the thermodynamics and spin dynamics of an S = 1 quantum spin system with a magnetic Skyrmion. The temperature dependence of the Skyrmion profile as well as the phase diagram are calculated. In addition, the spin dynamics of a magnetic Skyrmion is described by solving the time dependent Schrödinger equation with additional damping term. The Skyrmion annihilation process driven by an electric field is used to compare the trajectories of the quantum mechanical simulation with a semi-classical description for the spin expectation values using a differential equation similar to the classical Landau-Lifshitz-Gilbert equation.
... Recent rapid advances on Skyrmion research [10][11][12][13] has spurred interests in magnon transport through the Skyrmionic spin texture background. [14][15][16][17][18][19][20][21][22] Since Skyrmion is but a quantized form of spin chirality which acts as a source of emergent magnetic flux on the magnon particles, magnon transport in the Skyrmion matter is an excellent example of chirality-driven spin transport phenomena which is gaining attention as a new frontier in magnonics. [23][24][25] The transport manifestation of the spin chirality can be pictured as follows. ...
... This kind of emergent magnetism can be manifested as the formation of topologically non-trivial bands of magnons [26][27][28][29][30][31][32][33][34] in the momentum space, or as localized emergent magnetic fields in real space. [14][15][16][17][18][19][20][21][22] We formulate the rest of the review by starting with a theory of thermally driven spin transport in purely spin systems and cascading down to the magnon picture by making mean-field approximations. Contacts with many existing magnon-based theories of thermal Hall transport can be established this way. ...
... All three spin response functions S ab , E ab , N ab derived as Eqs. (7), (14), and (15) apply equally well to disordered spin systems as well as in their ordered phases. ...
Experimental and theoretical aspects of Hall-type transport of spins in magnetic insulators are reviewed. A general formalism for linear response theory of thermal Hall transport in the spin model is developed, which is general enough to be applicable to both the magnon and the paramagnetic, spin-liquid regimes. The expression of the energy current operator in the spin language is shown to be closely related to the spin chirality operator. Recent experiments on magnon-mediated thermal Hall transport in the two-dimensional kagome, and three-dimensional pyrochlore ferromagnetic insulators are reviewed in light of the multi-band magnon theory of Hall transport, and compared to the more mysterious thermal Hall transport found in the putative quantum spin ice material. As realizations of spin-chirality driven magnon transport in the real space, we review the general theory of emergent gauge fields governing the magnon dynamics in the textured magnet, and discuss its application to the magnon-Skyrmion scattering problem. Topological magnon Hall effect driven by the Skyrmion texture is discussed.
... The knowledge about the energy landscape of skyrmionic systems is of huge importance for tailoring these nontrivial magnetic structures according to the requirements of data storage devices. While there are numerous MC studies 30,[34][35][36] dealing with the phase diagram, here we report on MC simulations regarding the kinetic properties of Sk and FM states in ultrathin magnetic films as a function of temperature and magnetic field. In our investigations, we find that the energy landscape of a skyrmionic system is very different from that of magnetic atomic clusters and nanoislands exhibiting two-state behavior with degenerate energy minima, which are symmetric with respect to the energy barrier 37,38 . ...
The switching between topologically distinct skyrmionic and ferromagnetic states has been proposed as a bit operation for information storage. While long lifetimes of the bits are required for data storage devices, the lifetimes of skyrmions have not been addressed so far. Here we show by means of atomistic Monte Carlo simulations that the field-dependent mean lifetimes of the skyrmionic and ferromagnetic states have a high asymmetry with respect to the critical magnetic field, at which these lifetimes are identical. According to our calculations, the main reason for the enhanced stability of skyrmions is a different field dependence of skyrmionic and ferromagnetic activation energies and a lower attempt frequency of skyrmions rather than the height of energy barriers. We use this knowledge to propose a procedure for the determination of effective material parameters and the quantification of the Monte Carlo timescale from the comparison of theoretical and experimental data.
