No file available
To read the file of this research,
you can request a copy directly from the authors.
Lifetime, collapse and escape paths for hopfions in bulk magnets with competing exchange interactions
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
The lifetimes of magnetic hopfions on a discrete lattice with competing exchange interactions are calculated within the framework of the transition state theory for magnetic degrees of freedom. Three sets of discrete model parameters corresponding to the same continuous micromagnetic model are considered. Minimal energy paths for hopfion collapses were found on the multdimensional energy surface of the system. The activation energies of the collapse processes have been calculated. It turned out that the activation energy differs significantly for the three considered values of the parameters, which indicates the importance of lattice effects on this scale. Along with the collapse, the hopfion escape process through the sample boundary is studied. It is shown that this process does not require an activation energy. The lifetimes of hopfions are found and it is shown that they can exist only at temperatures of a few kelvins and practically cannot be generated due to thermal fluctuations.
ResearchGate has not been able to resolve any citations for this publication.
Hopfions are an intriguing class of string-like solitons, named according to a classical topological concept classifying three-dimensional direction fields. The search for hopfions in real physical systems has been ongoing for nearly half a century, starting with the seminal work of Faddeev. However, so far, realizations in bulk solids are missing. Here, we show that hopfions appear as emergent particles of the classical Heisenberg model with competing exchange interactions. This requires going beyond the model approach used in prior work and deriving a general micromagnetic energy functional directly from a spin-lattice Hamiltonian. We present a definite parameter space in which the existence of hopfions is possible. This opens a concrete vista to combine computational approaches such as density functional theory with material informatics to find magnetic crystals that can host hopfions. As proof of principle, we show how zero-field hopfions can be visualized by the means of off-axis electron holography in a transmission electron microscope.
Arising in many branches of physics, Hopf solitons are three-dimensional particle-like field distortions with nontrivial topology described by the Hopf map. Despite their recent discovery in colloids and liquid crystals, the requirement of applied fields or confinement for stability impedes their utility in technological applications. Here we demonstrate stable Hopf solitons in a liquid crystal material without these requirements as a result of enhanced stability by tuning anisotropy of parameters that describe energetic costs of different gradient components in the molecular alignment field. Nevertheless, electric fields allow for inter-transformation of Hopf solitons between different geometric embodiments, as well as for their three-dimensional hopping-like dynamics in response to electric pulses. Numerical modelling reproduces both the equilibrium structure and topology-preserving out-of-equilibrium evolution of the soliton during switching and motions. Our findings may enable myriads of solitonic condensed matter phases and active matter systems, as well as their technological applications. Hopf solitons are three-dimensional particle-like field distortions with nontrivial topology. Tai et al. show stable Hopf solitons in a liquid crystal material in the absence of an electric field or geometric confinement, their transformation and hopping-like dynamics in response to electric pulses.
The creation and annihilation of magnetic skyrmions are mediated by three-dimensional topological defects known as Bloch points. Investigation of such dynamical processes is important both for understanding the emergence of exotic topological spin textures, and for future engineering of skyrmions in technological applications. However, while the annihilation of skyrmions has been extensively investigated in two dimensions, in three dimensions the phase transitions are considerably more complex. We report field-dependent experimental measurements of metastable skyrmion lifetimes in an archetypal chiral magnet, revealing two distinct regimes. Comparison to supporting three-dimensional geodesic nudged elastic band simulations indicates that these correspond to skyrmion annihilation into either the helical and conical states, each exhibiting a different transition mechanism. The results highlight that the lowest energy magnetic configuration of the system plays a crucial role when considering the emergence and stability of topological spin structures via defect-mediated dynamics.
Among topological solitons, magnetic skyrmions are two-dimensional particle-like objects with a continuous winding of the magnetization, and magnetic Hopfions are three-dimensional objects that can be formed from a closed loop of twisted skyrmion strings. Theoretical models suggest that magnetic Hopfions can be stabilized in frustrated or chiral magnetic systems, and target skymions can be transformed into Hopfions by adapting their perpendicular magnetic anisotropy, but their experimental verification has been elusive so far. Here, we present an experimental study of magnetic Hopfions that are created in Ir/Co/Pt multilayers shaped into nanoscale disks, known to host target skyrmions. To characterize three-dimensional spin textures that distinguish Hopfions from target skyrmions magnetic images are recorded with surface-sensitive X-ray photoemission electron microscopy and bulk-sensitive soft X-ray transmission microscopy using element-specific X-ray magnetic circular dichroism effects as magnetic contrast. These results could stimulate further investigations of Hopfions and their potential application in three-dimensional spintronics devices.
Magnetic skyrmions are topologically protected spin textures that have nanoscale dimensions and can be manipulated by an electric current. These properties make the structures potential information carriers in data storage, processing and transmission devices. However, the development of functional all-electrical electronic devices based on skyrmions remains challenging. Here we show that the current-induced creation, motion, detection and deletion of skyrmions at room temperature can be used to mimic the potentiation and depression behaviours of biological synapses. In particular, the accumulation and dissipation of magnetic skyrmions in ferrimagnetic multilayers can be controlled with electrical pulses to represent the variations in the synaptic weights. Using chip-level simulations, we demonstrate that such artificial synapses based on magnetic skyrmions could be used for neuromorphic computing tasks such as pattern recognition. For a handwritten pattern dataset, our system achieves a recognition accuracy of ~89%, which is comparable to the accuracy achieved with software-based ideal training (~93%). The electrical current-induced creation, motion, detection and deletion of skyrmions in ferrimagnetic multilayers can be used to mimic the behaviour of biological synapses, providing devices that could be used for neuromorphic computing tasks such as pattern recognition.
Significance
While arising in theories in many branches of science, from particle physics to condensed matter and cosmology, stable three-dimensional topological solitons remained experimentally elusive until very recently. We now show that such solitons can be electrically and magnetically switched between states with the same or different Hopf indices. Richness and robustness of this switching promise technological applications in the new breeds of information displays and data storage devices, as well as may provide a test ground and new inspirations for the mathematical knot theory.
In triangular-lattice magnets, the coexistence of third-neighbor antiferromagnetic and nearest-neighbor ferromagnetic exchange interactions can induce rich magnetic phases including noncoplanar skyrmion crystals. Based on Monte Carlo simulation, we studied the dependence of magnetic phase transition on exchange interaction strength. Under the consideration of uniaxial anisotropy and magnetic field both perpendicular to the film plane, a large antiferromagnetic exchange interaction induces a high frustration. When the value of antiferromagnetic exchange interaction is one and a half times larger than the ferromagnetic one, a magnetic phase composed of canting spin stripes, never observed in the chiral magnets, forms. Interestingly, different canting spin stripes along three 120 degree propagation directions may coexist randomly in a magnetic phase, attesting that the canting spin stripes are three-fold degenerate states akin to helices and the multiple state of canting spin stripes is a circular configuration with zero skyrmion charge number. Moreover, skyrmions and antiskyrmions can be observed simultaneously in the configuration at the low temperature nearly close to 0 K, and their configuration and diameter properties are discussed. Finally, the mechanisms of skyrmion creation and annihilation are properly interpreted by comparing exchange and Zeeman energy terms.
Skyrmions are localized, topologically non-trivial spin structures which have raised high hopes for future spintronic applications. A key issue is skyrmion stability with respect to annihilation into the ferromagnetic state. Energy barriers for this collapse have been calculated taking only nearest neighbor exchange interactions into account. Here, we demonstrate that exchange interactions beyond nearest neighbors can be essential to describe stability of skyrmionic spin structures. We focus on the prototypical film system Pd/Fe/Ir(111) and demonstrate that an effective nearest-neighbor exchange or micromagnetic model can only account for equilibrium properties such as the skyrmion profile or the zero temperature phase diagram. However, energy barriers and critical fields of skyrmion collapse as well as skyrmion lifetimes are drastically underestimated since the energy of the transition state cannot be accurately described. Antiskyrmions are not even metastable. Our work shows that frustration of exchange interactions is a route towards enhanced skyrmion stability even in systems with a ferromagnetic ground state.
The stability of magnetic skyrmions against thermal fluctuations and external perturbations is investigated within the framework of harmonic transition state theory for magnetic degrees of freedom. The influence of confined geometry and atomic scale non-magnetic defects on the skyrmion lifetime is estimated. It is shown that a skyrmion on a track has lower activation energy for annihilation and higher energy for nucleation if the size of the skyrmion is comparable with the width of the track. Two mechanisms of skyrmion annihilation are considered: inside the track and escape through the boundary. For both mechanisms, the dependence of activation energy on the track width is calculated. Non-magnetic defects are found to localize skyrmions in their neighborhood and strongly decrease the activation energy for creation and annihilation. This is in agreement with experimental measurements that have found nucleation of skyrmions in presence of spin-polarized current preferably occurring near structural defects.
Topological solitons are knots in continuous physical fields classified by nonzero Hopf index values. Despite arising in theories that span many branches of physics, from elementary particles to condensed matter and cosmology, they remain experimentally elusive and poorly understood. We introduce a method of experimental and numerical analysis of such localized structures in liquid crystals that, similar to the mathematical Hopf maps, relates all points of the medium’s order parameter space to their closed-loop preimages within the three-dimensional solitons. We uncover a surprisingly large diversity of naturally occurring and laser-generated topologically nontrivial solitons with differently knotted nematic fields, which previously have not been realized in theories and experiments alike. We discuss the implications of the liquid crystal’s nonpolar nature on the knot soliton topology and how the medium’s chirality, confinement, and elastic anisotropy help to overcome the constraints of the Hobart-Derrick theorem, yielding static three-dimensional solitons without or with additional defects. Our findings will establish chiral nematics as a model system for experimental exploration of topological solitons and may impinge on understanding of such nonsingular field configurations in other branches of physics, as well as may lead to technological applications.
Conjugate gradient methods for energy minimization in micromagnetics are compared. The comparison of analytic results with numerical simulation shows that standard conjugate gradient method may fail to produce correct results. A method that restricts the step length in the line search is introduced, in order to avoid this problem. When the step length in the line search is controlled, conjugate gradient techniques are a fast and reliable way to compute the hysteresis properties of permanent magnets. The method is applied to investigate demagnetizing effects in NdFe12 based permanent magnets. The reduction of the coercive field by demagnetizing effects is μ0ΔH = 1.4 T at 450 K.
We present a new type of a thermodynamically stable magnetic state at
interfaces and surfaces of chiral magnets. The state is a soliton solution of
micromagnetic equations localized in all three dimensions near a boundary and
contains a singularity, but nevertheless has a finite energy. Both features
combine to a quasi-particle state for which we expect unusual transport and
dynamical properties. It exhibits high thermal stability and thereby can be
considered as promising object for fundamental research and practical
applications in spintronic devices. We provide arguments that such a state can
be found in different B20-type alloys e.g. Mn$_{1-x}$Fe$_x$Ge,
Mn$_{1-x}$Fe$_x$Si, Fe$_{1-x}$Co$_x$Si.
An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based density-functional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surface, and dissociative adsorption of CH4 on an Ir(111) surface. (C) 2000 American Institute of Physics. [S0021-9606(00)70546-0].
A modification of the nudged elastic band method for finding minimum energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring constants are used to increase the density of images near the top of the energy barrier to get an improved estimate of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir(111) and H-2 on Si(100) using plane wave based density functional theory are presented. (C) 2000 American Institute of Physics. [S0021-9606(00)71246-3].
Three-dimensional (3D) magnetic textures attract much attention from researchers due to their fascinating structures and dynamic behaviors. The magnetic hopfion is a prominent example of a 3D magnetic texture. Here, we numerically study the mutual conversion between a Néel-type hopfion and a Néel-type toron under an external magnetic field. We also investigate the excitation modes of hopfions and torons in a film with strong perpendicular magnetic anisotropy. It is found that the Néel-type hopfion could be a stable state in the absence of the external magnetic field, and its diameter varies with the out-of-plane magnetic field. The Néel-type hopfion may transform into a Néel-type toron at an out-of-plane magnetic field of about 20 mT, where the cross-section structure is a Néel-type skyrmion. The hopfion and toron show different excitation modes in the presence of an in-plane microwave magnetic field. Our results provide a method to realize the conversion between a Néel-type hopfion and a Néel-type toron, which also opens a path to distinguish between a Néel-type hopfion and a Néel-type toron based on their excitation modes.
We theoretically study orientational structures in chiral magnetics and cholesteric liquid crystal (CLC) nanosystems confined in the slab geometry. Our analysis is based on the model that, in addition to the exchange and the Dzyaloshinskii-Moriya interactions, takes into account the bulk and surface anisotropies. In CLC films, these anisotropies describe the energy of interaction with external magnetic/electric field and the anchoring energy assuming that magnetic/electric anisotropy is negative and the boundary conditions are homeotropic. We have computed the phase diagram and found that the ground state of the film is represented by various delocalized structures depending on the bulk and surface anisotropy parameters, κ^{b} and κ^{s}. These include the z helix and the z cone states, the oblique, and the x helicoids. The minimum energy paths connecting the ground state and metastable helicoids and the energy barriers separating these states are evaluated. We have shown that there is a variety of localized topological structures such as the skyrmion tube, the toron, and the bobber that can be embedded in different ground states including the z cone (conical phase) and tilted fingerprint states. We have also found the structure called the leech that can be viewed as an intermediate state between the toron and the skyrmion tube.
Resonant spin dynamics of topological spin textures are correlated with their topological nature, which can be employed to understand this nature. In this study, we present resonant spin dynamics of three-dimensional topological spin texture, i.e., Neel and Bloch hopfions. Using micromagnetic simulations, we stabilize Bloch and Neel hopfions with bulk and interfacial Dzyaloshinskii-Moriya interaction (DMI), respectively. We identify the ground state spin configuration of both hopfions, effects of anisotropies, geometric confinements, and demagnetizing fields. To confirm topological nature, Hopf number is calculated for each spin texture. Then, we calculate the resonance frequencies and spin-wave modes of spin precessions under multiple magnetic fields. Unique resonance frequencies and specific magnetic field dependence can help to guide experimental studies to identify the three-dimensional topological spin texture of hopfions in functioning chiral magnets when imaging is not possible.
Topological protection of chiral magnetic structures is investigated by taking a two-dimensional magnetic skyrmion as an example. The skyrmion lifetime is calculated based on harmonic transition state theory for a discrete lattice model using various values of the ratio of the lattice constant and the skyrmion size. Parameters of the system corresponding to exchange, anisotropy and Dzyaloshinskii-Moriya interaction are chosen in such a way as to keep the energy and size of the skyrmion unchanged for small values of the lattice constant, using scaling relations derived from continuous micromagnetic description. The number of magnetic moments included in the calculations reaches more than a million. The results indicate that in the limit of infinitesimal lattice constant, the energy barrier for skyrmion collapse approaches the Belavin-Polyakov lower bound of the energy of a topological soliton in the σ-model, the entropy contribution to the pre-exponential factor in the Arrhenius rate expression for collapse approaches a constant and the skyrmion lifetime can, for large enough number of spins, correspond to thermally stable skyrmion at room temperature even without magnetic dipole–dipole interaction.
A new method for the numerical computation of the lifetimes of magnetic states within harmonic transition state theory (HTST) has been developed. In the simplest case, the system is described by a Heisenberg-like Hamiltonian with short- range interaction. Calculations are performed in Cartesian coordinates. Constraints on the values of magnetic moments are taken into account using Lagrange multipliers. The pre-exponential factor in the Arrhenius law in HTST is written in terms of the determinants of the Hessian of energy at the minima and saddle points on the multidimensional energy surface. An algorithm for calculating these determinants without searching for eigenvalues of the Hessian but using recursive relations is proposed. The method allows calculating determinants for systems containing millions of magnetic moments. This makes it possible to calculate the pre-exponential factor and estimate the lifetimes of micron-scale topological structures with atomic resolution, which until now has been impossible using standard approaches. The accuracy of the method is demonstrated by calculating 2D and 3D skyrmionic structures.
Three-dimensional topological solitons attract a great deal of interest in fields ranging from particle physics to cosmology, but remain experimentally elusive in solid-state magnets. Here we numerically predict magnetic heliknotons, an embodiment of such nonzero-Hopf-index solitons localized in all spatial dimensions while embedded in a helical or conical background of chiral magnets. We describe conditions under which heliknotons emerge as metastable or ground-state localized nonsingular structures with fascinating knots of magnetization field in widely studied materials. We demonstrate magnetic control of three-dimensional spatial positions of such solitons, as well as show how they interact to form moleculelike clusters and possibly even crystalline phases comprising three-dimensional lattices of such solitons with both orientational and positional order. Finally, we discuss both fundamental importance and potential technological utility of magnetic heliknotons.
The current development to employ magnetic skyrmions in novel spintronic device designs has led to a demand for room-temperature-stable skyrmions of ever smaller size. We present extensive studies on skyrmion stability in atomistic magnetic systems in two- and three-dimensional geometries. We show that for materials described by the same micromagnetic parameters, the variation of the atomistic exchange between different neighbors, the stacking order, and the number of layers of the atomic lattice can significantly influence the rate of the thermally activated decay of a skyrmion. These factors alone are important considerations, but we show that their combination can open up novel avenues of materials design in the search for sub-10 nm skyrmions, as their lifetime can be extended by several orders of magnitude.
Efficient algorithms for the calculation of minimum energy paths of magnetic transitions are implemented within the geodesic nudged elastic band (GNEB) approach. While an objective function is not available for GNEB and a traditional line search can, therefore, not be performed, the use of limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) and conjugate gradient algorithms in conjunction with orthogonal spin optimization (OSO) approach is shown to greatly outperform the previously used velocity projection and dissipative Landau-Lifschitz dynamics optimization methods. The implementation makes use of energy weighted springs for the distribution of the discretization points along the path and this is found to improve performance significantly. The various methods are applied to several test problems using a Heisenberg-type Hamiltonian, extended in some cases to include Dzyaloshinskii-Moriya and exchange interactions beyond nearest neighbors. Minimum energy paths are found for magnetization reversals in a nano-island, collapse of skyrmions in two-dimensional layers and annihilation of a chiral bobber near the surface of a three-dimensional magnet. The LBFGS-OSO method is found to outperform the dynamics based approaches by up to a factor of 8 in some cases.
Magnetic skyrmions are whirls in the magnetization whose topological winding number promises stability against thermal fluctuations and defects. They can only decay via singular spin configurations. We analyze the corresponding energy barriers of skyrmions in a magnetic monolayer for two distinct stabilization mechanisms, i.e., Dzyaloshinskii-Moriya interaction (DMI) and competing interactions. Based on our numerically calculated collapse paths on an atomic lattice, we derive analytic expressions for the saddle-point textures and energy barriers of large skyrmions. The sign of the spin stiffness and the sign of fourth-order derivative terms in the classical field theory determines the nature of the saddle point and thus the height of the energy barrier. In the most common case for DMI-stabilized skyrmions, positive stiffness and negative fourth-order term, the saddle-point energy approaches a universal upper limit described by an effective continuum theory. For skyrmions stabilized by frustrating interactions, the stiffness is negative and the energy barrier arises mainly from the core of a singular vortex configuration.
Topological magnetic textures have attracted considerable interest since they exhibit new properties and might be useful in information technology. Magnetic hopfions are three-dimensional (3D) spatial variations in the magnetization with a nontrivial Hopf index. We find that, in ferromagnetic materials, two types of hopfions, Bloch-type and Néel-type hopfions, can be excited as metastable states in the presence of bulk and interfacial Dzyaloshinskii-Moriya interactions, respectively. We further investigate how hopfions can be driven by currents via spin-transfer torques (STTs) and spin-Hall torques (SHTs). Distinct from 2D ferromagnetic skyrmions, hopfions have a vanishing gyrovector. Consequently, there are no undesirable Hall effects. Néel-type hopfions move along the current direction via both STT and SHT, while Bloch-type hopfions move either transverse to the current direction via SHT or parallel to the current direction via STT. Our findings open the door to utilizing hopfions as information carriers.
Two-dimensional topological solitons, commonly called Skyrmions, are extensively studied in solid-state magnetic nanostructures and promise many spintronics applications. However, three-dimensional topological solitons dubbed hopfions have not been demonstrated as stable spatially localized structures in solid-state magnetic materials. Here we model the existence of such static solitons with different Hopf index values in noncentrosymmetric solid magnetic nanostructures with a perpendicular interfacial magnetic anisotropy. We show how this surface anisotropy, along with the Dzyaloshinskii-Moriya interactions and the geometry of nanostructures, stabilize hopfions. We demonstrate knots in emergent field lines and computer simulate Lorentz transmission electron microscopy images of such solitonic configurations to guide their experimental discovery in magnetic solids.
A magnetic Hopfion is a three-dimensional topological soliton that consists of a closed loop of a twisted magnetic Skyrmion string. The results of numerical simulations are presented that demonstrate the existence of a stable Hopfion in a nanocylinder of a chiral magnet and an explicit analytic expression is shown to provide a reasonable approximation to the numerically computed Hopfion. A mechanism is suggested to create the Hopfion from a target Skyrmion by introducing an interfacial perpendicular magnetic anisotropy.
Magnetic skyrmions are small swirling topological defects in the magnetization texture. Their stabilization and dynamics depend strongly on their topological properties. In most cases, they are induced by chiral interactions between atomic spins in non-centrosymmetric magnetic compounds or in thin films with broken inversion symmetry. Skyrmions can be extremely small, with diameters in the nanometre range, and behave as particles that can be moved, created and annihilated, which makes them suitable for 'abacus'-type applications in information storage and logic technologies. Until recently, skyrmions had been observed only at low temperature and, in most cases, under large applied magnetic fields. An intense research effort has led to the identification of thin-film and multilayer structures in which skyrmions are now stable at room temperature and can be manipulated by electrical currents. The development of skyrmion-based topological spintronics holds promise for applications in the mid-term furure, even though many challenges, such as the achievement of writing, processing and reading functionalities at room temperature and in all-electrical manipulation schemes, still lie ahead. © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
The mechanism and activation energy for the annihilation of a magnetic skyrmion is studied by finding the minimum energy path for the transition in a system described by a Heisenberg-type Hamiltonian extended to include dipole-dipole, Dzyaloshinskii-Moriya, and anisotropy interactions so as to represent a Co monolayer on a Pt(111) surface. The annihilation mechanism involves isotropic shrinking of the skyrmion and slow increase of the energy until the transition state is reached after which the energy drops abruptly as the ferromagnetic final state forms. The maximum energy along the minimum energy path, which gives an estimate of the activation energy within the harmonic approximation of transition state theory, is found to be in excellent agreement with direct Langevin dynamics simulations at relatively high temperature carried out by Rohart et al. [Phys. Rev. B 93, 214412 (2016)]. The dipole-dipole interaction, the computationally most demanding term in the Hamiltonian, is found to be important but its effect on the stability of the skyrmion and shape of the transition path can be mimicked accurately by reducing the anisotropy constant in the Hamiltonian.
Magnetic skyrmions are chiral quasiparticles that show promise for the transportation and storage of information. On a fundamental level, skyrmions are model systems for topologically protected spin textures and can be considered as the counterpart of topologically protected electronic states, emphasizing the role of topology in the classification of complex states of condensed matter. Recent impressive demonstrations of the control of individual nanometre-scale skyrmions — including their creation, detection, manipulation and deletion — have raised expectations for their use in future spintronic devices, including magnetic memories and logic gates. From a materials perspective, it is remarkable that skyrmions can be stabilized in ultrathin transition metal films, such as iron — one of the most abundant elements on earth — if in contact with materials that exhibit high spin–orbit coupling. At present, research in this field is focused on the development of transition-metal-based magnetic multilayer structures that support skyrmionic states at room temperature and allow for the precise control of skyrmions by spin-polarized currents and external fields.
The Hopf fibration is an example of a texture: a topologically stable, smooth, global configuration of a field. Here we demonstrate the controlled sculpting of the Hopf fibration in nematic liquid crystals through the control of point defects. We demonstrate how these are related to torons by use of a topological visualization technique derived from the Pontryagin-Thom construction.
Topological properties and dynamics of magnetic skyrmions
- N Nagaosa
- Y Tokura
N. Nagaosa and Y. Tokura, Topological properties and
dynamics of magnetic skyrmions, Nature Nanotechnology 8, 899 (2013).
Field-driven dynamics of magnetic hopfions
- D Raftrey
- P Fischer
D. Raftrey and P. Fischer, Field-driven dynamics of
magnetic hopfions, Phys. Rev. Lett. 127, 257201 (2021).
An expression of hopf's invariant as an integral
- J H C Whitehead
J. H. C. Whitehead, An expression of hopf's invariant as
an integral, Proc. Nat. Acad. Sci. U.S.A. 33, 117 (1947).