Hopfion rings in a cubic chiral magnet
Abstract
Magnetic skyrmions and hopfions are topological solitons^{1}—welllocalized field configurations that have gained considerable attention over the past decade owing to their unique particlelike properties, which make them promising objects for spintronic applications. Skyrmions^{2,3} are twodimensional solitons resembling vortexlike string structures that can penetrate an entire sample. Hopfions^{4,5,6,7,8,9} are threedimensional solitons confined within a magnetic sample volume and can be considered as closed twisted skyrmion strings that take the shape of a ring in the simplest case. Despite extensive research on magnetic skyrmions, the direct observation of magnetic hopfions is challenging^{10} and has only been reported in a synthetic material^{11}. Here we present direct observations of hopfions in crystals. In our experiment, we use transmission electron microscopy to observe hopfions forming coupled states with skyrmion strings in B20type FeGe plates. We provide a protocol for nucleating such hopfion rings, which we verify using Lorentz imaging and electron holography. Our results are highly reproducible and in full agreement with micromagnetic simulations. We provide a unified skyrmion–hopfion homotopy classification and offer insight into the diversity of topological solitons in threedimensional chiral magnets.
Main
Topological magnetic solitons^{1} are localized magnetic textures that have properties similar to those of ordinary particles. In particular, they can mutually interact and move under the influence of external stimuli. Chiral magnetic skyrmions^{12,13,14,15} are notable examples of such objects. In thick threedimensional (3D) samples, the magnetization vector field of skyrmions usually forms vortexlike strings that can penetrate an entire sample from one surface to another. Such filamentary magnetic textures, which arise as a consequence of competition between Heisenberg exchange and the Dzyaloshinskii–Moriya interaction (DMI)^{16,17}, have been observed in noncentrosymmetric crystals using various experimental techniques^{2,3}.
Recent studies have shown that skyrmion strings in isotropic chiral magnets are not rigid textures, but can twist and bend^{18,19,20,21}. A prominent example of such elastic properties of skyrmion strings is the formation of skyrmion braids^{20}—ropelike structures composed of skyrmion strings that wind around each other. Here we report a fundamentally different phenomenon. We use stateoftheart transmission electron microscopy (TEM) and micromagnetic simulations to show that twisted skyrmion strings can be bent into rings in magnetic crystals, leading to the emergence of a different topology.
The concept of closed twisted skyrmion strings in field theories was first proposed by Ludwig Faddeev in 1975 (ref. ^{4}). Such structures are now commonly termed hopfions^{9}, after Heinz Hopf, who built the theory that underlies their homotopy classification^{22}. According to this theory, localized magnetization field configurations m(r) = M(r)/M_{s} can be classified according to the linkage of their fibres {m = m_{P}}, which, for any fixed points P on a unit sphere
, represent closed loops in
space. When any pair of such loops is linked once, as a pair of chain segments, the Hopf index of the field configuration is one, H = 1, and this configuration contains a hopfion.
Previous theoretical studies have predicted the existence of confined and isolated hopfions in chiral magnets^{8,23,24,25}. However, the mechanism for hopfion stabilization that we describe below is essentially different. In our experiment, hopfions appear as rings around skyrmion strings and remain stable exclusively owing to intrinsic interactions and not as a result of the sample shape. The confined geometry of the sample has an essential role only in the nucleation of these hopfions, and is not relevant for their stability. To distinguish between hopfions that appear around skyrmion strings from isolated hopfions and hopfions in a nanoscale multilayer disk^{11}, we refer to the former as hopfion rings.
Micromagnetic simulations of hopfion rings
Figure 1a–e illustrates the concept of hopfion rings and the process of their nucleation, according to the fieldswapping protocol used in our experimental setup below. In these micromagnetic simulations, we used a 0.5μmdiameter disk with a thickness of 180 nm, with the external magnetic field applied parallel to the central axis of the disk. The magnetic states shown in Fig. 1a–e were obtained by direct energy minimization of the micromagnetic functional for an isotropic chiral magnet, taking into account demagnetizing fields. We also took into account the presence of a thin damaged layer on the sample surface (Fig. 1a), which typically results from sample preparation by focused ion beam (FIB) milling. (See Methods for more details about micromagnetic calculations, the properties of the damaged layer and Lorentz image simulations.)
Figure 1a shows an initial state with three skyrmion strings, demonstrating a weak braiding effect and conical modulations along the sample thickness. In a confined geometry, such modulations exhibit an additional twist that resembles a vortexlike structure^{26,27} in any transverse section. Swapping the direction of the external magnetic field leads to a flip of the magnetization along the perimeter of the disk, as shown in Fig. 1b. To prevent the collapse of the skyrmions in a negative field, we apply a relatively weak field of −50 mT. Swapping the field back to the positive direction results in the appearance of edge modulations, representing a volume with magnetization pointing against the external field (Fig. 1c). On further increasing the field in the positive direction, the edge modulations contract towards the centre of the sample, forming a hopfion ring around three skyrmion strings (Fig. 1d). Figure 1d shows that each colour line (fibre) winds about the isosurface of the hopfion ring only once, which indicates that the Hopf index of the corresponding texture is nontrivial. The middle crosssection of the hopfion ring surrounding the three skyrmion strings shown in Fig. 1e resembles a skyrmion bag, as predicted theoretically for a twodimensional (2D) model of a chiral magnet^{28,29}. Our analysis shows that such textures have negative skyrmion topological charge, Q, in all transverse sections (see Methods). However, the rings shown in Fig. 1d represent hopfions, which is not possible in 2D materials^{30} and is different from skyrmion bags with a positive topological charge^{31}. It is therefore essential to distinguish between magnetic textures with hopfion rings and skyrmion bags.
Experimental observation of hopfion rings
Following the protocol suggested by the micromagnetic simulations, we demonstrated the nucleation of hopfion rings in a squareshaped FeGe sample of lateral size 1 μm × 1 μm and thickness 180 nm. (See Methods for details about sample preparation and the TEM experimental setup.) A representative example of a magnetic configuration with edge modulations after field swapping is shown in Fig. 1f. On increasing the external field, the bright contrast loop of the edge modulations contracts around two skyrmions. The final state (at 200 mT) corresponds to a hopfion ring around two skyrmion strings. The intermediate configurations at 180 mT and 190 mT are seldom observed in our experiments. In most cases, contraction of the edge modulations occurs abruptly on a timescale that is much shorter than the temporal resolution of the detector in our TEM.
As a result of the excitation of the magnetization induced by the abrupt contraction of the edge modulations and thermal fluctuations, the above protocol has a probabilistic character. The final state shown in Fig. 1f (at 200 mT) is an example of a successful contraction.
Notably, the sequential application of the above fieldswapping protocol can result in the nucleation of a second hopfion ring with high probability. Figure 1gshows an example of such a double hopfion ring around three skyrmion strings. Further images with multiple hopfion rings are provided in Extended Data Figs. 5and 6. These images illustrate the successful contraction of the second ring. However, because hopfion rings are metastable states, further application of the fieldswapping protocol can cause a transition to a lower energy state, resulting in the collapse of the hopfion rings.
The above protocol becomes more reliable at a higher sample temperature, indicating that thermal fluctuations have a key role in hopfionring nucleation. However, the range of applied magnetic fields over which the hopfion rings remain stable decreases gradually with increasing temperature. We found an optimal range of T = 180–200 K, over which the above protocol exhibits the greatest efficiency. Below 180 K, nucleation of hopfion rings is still possible, but typically requires more fieldswapping cycles. At a lower temperature, the edge modulations can move towards the edges of the sample and disappear. By contrast, at higher temperatures the edge modulations can contract towards the centre of the sample. Above 200 K, abrupt contraction of the edge modulations can lead to their collapse. The behaviour of the edge modulations at different temperatures can be explained by the presence of an energy barrier that prevents their contraction towards the centre of the domain. The probability of overcoming this energy barrier then becomes greater at high temperature.
The process of hopfionring nucleation is shown in Supplementary Videos 1–5. These videos were captured in situ at a temperature of T = 180 K. We performed several fieldswapping cycles in the first stage with a small amplitude of approximately ±50 mT. This step was designed to generate edge modulations that formed closed loops and propagated towards the centre of the sample. Once one or a few of these loops had been created, we gradually increased the applied magnetic field up to approximately 150 mT, which resulted in the formation of various hopfion rings.
The tilt angle of the external magnetic field to the plate normal is an essential parameter for hopfionring nucleation. In our experiment, we found that the tilt angle of the field should not exceed 5°. Otherwise, the edge modulations mainly form on one side of the sample, resulting in a strongly asymmetric configuration.
Diversity and infield evolution of hopfion rings
Figure 2 illustrates the evolution of diverse configurations of hopfion rings in increasing applied magnetic fields. It should be noted that the Lorentz TEM images shown in Fig. 2 were recorded with the sample at different temperatures, which can affect the stability range of the hopfion rings and skyrmions. As a result, the applied magnetic field is only indicated schematically above the figure. The complete series of images, showing the entire field of view and exact values of the external magnetic field and specimen temperature, is provided in Extended Data Figs. 1, 2 and 3. Other representative images of multiple skyrmions surrounded by a single hopfion ring are provided in Extended Data Fig. 4. Additional Lorentz TEM images illustrating the infield evolution of magnetic states with different topological charges and symmetries composed of skyrmions and hopfion rings are shown in Extended Data Figs. 5 and 6.
Figure 2 shows that the hopfion ring shrinks with increasing applied field. As a result, the distance between the skyrmions inside the ring decreases. By contrast, in ordinary skyrmion clusters (without a hopfion ring) the distance between the skyrmions increases with applied field^{32}. Figure 2 shows that the symmetry of the magnetic texture of skyrmions and hopfion rings also changes with increasing applied field. For instance, the hopfion ring shown in the bottom row has a triangular shape at low field. With increasing magnetic field, it adopts a pentagonal and then a circular shape. Such symmetry transitions are found to be reversible with respect to increasing and decreasing field.
With increasing field, the contrast of the hopfion rings in Lorentz TEM images becomes weaker, suggesting that the magnetic modulations become localized in a smaller volume. The distance between the hopfion ring and the skyrmions also decreases, with bright spots of skyrmion strings nearly touching the hopfion ring just before it collapses. After the collapse of the hopfion ring, the distance between the skyrmions increases abruptly. In most cases, bright spots of weak contrast, which are identified as chiral bobbers^{33} or dipole strings^{34}, are often observed. Their positions are marked by dashed circles in Fig. 2. Because chiral bobbers and dipole strings have nearly identical contrast in Lorentz TEM images^{35}, they cannot be distinguished reliably in these experiments. They disappear on increasing the applied magnetic field further. The appearance of such objects, which contain magnetic singularities, indicates that the collapse of a hopfion ring represents a topological transition.
Figure 3 shows exotic states with negative and positive topological charges obtained using the above protocol in a 180nmthick sample. Magnetic textures with such contrast in our experiments are observed less frequently than those depicted in Fig. 2 (see also Extended Data Fig. 5). The figure shows that a hopfion ring surrounding skyrmion strings can have different sizes, which seem to be limited only by the geometry of the sample. The latter observation agrees with theoretical predictions for 2D skyrmion bags^{28}, which also have no limitation in their size. The theoretical Lorentz TEM images shown in Fig. 3 are in excellent agreement with the experimental images.
Figure 4 and Extended Data Figs. 7 and 8 show corresponding magnetic textures obtained by direct energy minimization of the micromagnetic functional. In particular, Fig. 4 shows the 3D magnetic textures of the first four states in Fig. 3a–d, and Extended Data Fig. 8 shows magnetic textures of the other four states in Fig. 3eh. By starting from different initial states, we found slightly different stable configurations for most of the configurations shown in Fig. 3. Figure 4shows the magnetic textures with the lowest energies.
The hopfion rings shown in Fig. 4a,b,d are located in the middle plane of the sample. However, this is not always the case. In Fig. 4c, the hopfion ring is shifted slightly towards one of the surfaces and is also stretched slightly along the z axis. A similar configuration is shown in Extended Data Fig. 8a. In these cases, the projected magnetization of the hopfion ring onto the xy plane overlaps with the magnetization of the interior ring, leading to a visible contrast difference in Lorentz TEM images of the corresponding magnetic textures. For example, the outer rings shown in Fig. 4c,e have higher contrast when compared to the other images. The contrast in the theoretical images shows the same features. Additional comparisons between experimental and theoretical Lorentz TEM images and electron optical phase images are provided in Extended Data Figs. 9 and 10.
Skyrmion–hopfion topological charge
It is essential to find a complete topological classification of the observed magnetic textures. Our homotopygroup analysis (see Methods) reveals that these textures are classified according to their skyrmion–hopfion topological charge, which can be represented by an ordered pair of integers, (Q, H). Notably, the Hopf charge H here depends not only on the intrinsic structure of the hopfion rings but also on their linking to skyrmion strings. For instance, for the configuration shown in Fig. 4d (right), if there were no skyrmion string at the centre, the Hopf charge would be −1. However, linkage to the skyrmion string results in an actual Hopf charge of +1. Consequently, when such a ring detaches continuously from the string, there must be a profound transformation of its internal structure, leading to its conversion into an antihopfion (see Supplementary Video 6).
Despite the presence of hopfion rings in all of the magnetic textures shown in Fig. 4, it is worth emphasizing that we do not specify the Hopf charge for the cases depicted in Fig. 4b,c. Because the inner rings are near the open boundaries of the plate, the compactification condition is not satisfied (see Methods), and a homotopy classification based on the Hopf charge is unsuitable in this case. Another type of texture that does not satisfy the compactification condition is fractional hopfions^{24,36}.
Discussion
In our experiments, the sizes of the hopfion rings are much smaller than the inplane sample sizes, and hopfion rings can move in at least two spatial dimensions. In thicker samples, we speculate that hopfion rings can also move and interact with each other in the third spatial dimension—along the skyrmion strings. Figure 4e–h and Extended Data Fig. 8e–h illustrate a wide diversity of stable solutions for hopfion rings in bulk systems (see Methods). On the basis of the general principles of classical field theory, it is understood that the Hopf charge of skyrmion strings can be affected by longitudinal twists of skyrmions, as well as by skyrmion braiding^{37,38,39}. However, owing to the chiral nature of the DMI present in the system studied here, stable states with skyrmion twists of multiples of 2π are not observed. Nevertheless, we speculate that such states might be possible in systems that have frustrated exchange interactions^{9,40}. On the other hand, the phenomenon of skyrmion braiding has already been demonstrated in chiral magnets^{20}. Extended Data Fig. 8h shows an example of a skyrmion braid with two hopfion rings and H = 12. This example can be compared with Fig. 4h, which shows straight skyrmion strings surrounded by two hopfion rings and H = 10.
A notable property of the hopfion solutions is the presence of a zero mode arising from the intrinsic symmetry of the Hamiltonian (see Methods). The rotational–translational motion of the hopfion ring along the string preserves energy (see Supplementary Videos 7 and 8), which suggests high mobility of the hopfion rings.
Our findings open broad perspectives for the study of the dynamical^{41,42} and transport properties^{43,44} of hopfion rings, and will have practical applications in spintronics, neuromorphic computing and other technologies.
Methods
Micromagnetic simulations
The micromagnetic approach was followed in this work. The total energy of the system includes the exchange energy, the DMI energy, the Zeeman energy and the energy of the demagnetizing fields^{45}:
(1)
where m(r) = M(r)/M_{s} is a unit vector field that defines the direction of the magnetization, M_{s} = ∣M(r)∣ is the saturation magnetization and μ_{0} is the vacuum permeability (μ_{0} ≈ 1.257 μN A^{−2}). The constants
and
are the exchange stiffness and the isotropic bulk DMI, respectively. The ratio between
and
defines the equilibrium period of the conical phase,
. The magnetic field in equation (1), B = B_{ext} + ∇ × A_{d}, is the sum of the external magnetic field and the demagnetizing field, where A_{d}(r) is the component of the magnetic vector potential induced by the magnetization. For the calculations in the bulk system, we set the external magnetic field
, where
is the conical phase saturation field in the absence of demagnetizing fields^{20}. We used the following material parameters for FeGe^{20,33}:
= 4.75 pJ m^{−1},
= 0.853 mJ m^{−2} and M_{s} = 384 kA m^{−1}. For the 0.5μmdiameter and 180nmthick disk sample depicted in Fig. 1a–e, the calculations were performed on a mesh with 256 × 256 × 64 cuboids. Calculations for the 1 μm × 1 μm × 180 nm sample were performed on a mesh with 400 × 400 × 72 cuboids. For the bulk magnet, we exclude dipole–dipole interactions and consider a domain of size 5L_{D} × 5L_{D} × 10L_{D} under periodic boundary conditions on a mesh with 256 × 256 × 512 cuboids.
Following the arguments presented in a previous study^{35}, a thin surface layer of the isotropic chiral magnet crystal is damaged during FIB milling and can be effectively approximated by material parameters that are identical to those of the bulk crystal, but with the DMI coupling constant set to zero. In the previous report^{35}, the thickness of the FIBdamaged layer of an FeGe nanocylinder was estimated to be 6 ± 1 nm. According to another report^{21}, the thickness of the damaged layer of an FeGe needlelike sample is around 10 nm. Here, we assume an intermediate thickness for the damaged layer of 7.5 nm (corresponding to three surface cuboids).
It should be noted that the presence or absence of a damaged layer in our simulations has almost no effect on the stability of the solutions shown in Fig. 4. The contrast in theoretical Lorentz TEM images in Fig. 3 also does not change significantly when the presence of a damaged layer is ignored. However, the presence of a damaged surface layer has a crucial role in hopfionring nucleation. In the simulations, through the application of a magnetic field in the negative and positive directions with respect to the z axis, we only succeeded in observing hopfionring nucleation, as shown in Fig. 1a–d, in the presence of a damaged surface layer.
Statically stable solutions of the Hamiltonian (equation (1)) were found by using the numerical energy minimization method described previously^{20} using the Excalibur code^{46}. The solutions were doublechecked using the publicly available software Mumax^{47}. In the Supplementary information, we also provide three Mumax scripts, which can be used to reproduce the results of our micromagnetic simulations. Script I allows the hopfionring nucleation depicted in Fig. 1a–d to be reproduced. Because the states depicted in Fig. 1a,d are two states with different energies that are stabilized in identical conditions, the transition between them requires additional energy pumping. The energy balance between these states depends on the applied field. In the experimental setup, this infield transition is enhanced by thermal fluctuation and, as a result, has a probabilistic character. To make the nucleation of the hopfion ring deterministic (reproducible), in the micromagnetic simulations we use an abrupt switch of the magnetic field (with a step of around 100 mT) to overcome the barrier between the metastable states. Script II, with minor modifications of the initial states discussed in the next section, can be used to reproduce the states shown in Fig. 4. For a description of Script III, see the following section.
Initial state for hopfion rings in micromagnetic simulations
On the basis of experimental observations and theoretical analysis, we noticed that the presence of the conical phase around different localized states results in an additional contribution to the electron optical phase shift that changes around the perimeter of the sample. To obtain the magnetic textures in nanoscale samples, we used initial configurations corresponding to a superposition of cylindrical domains, with their magnetization pointing up and down, embedded in a conical phase and with an additional phase modulation resembling a vortex in the xy plane of the form
(2)
where k = 2π/L_{D} is the wave number. In another study^{27}, similar vortexcone configurations were discussed in the context of screw dislocations in bulk chiral magnets. Here, a magnetic configuration approximated by equation (2) appears owing to an interplay between shortrange interactions (Heisenberg exchange and DMI) and a longrange demagnetizing field. This effect has previously been observed in samples of confined geometry^{26,32,33}.
Representative examples of two initial states are illustrated in Extended Data Fig. 9a. Stable magnetic states obtained from these initial states after energy minimization are shown in Extended Data Fig. 9b,c. The state with a compact hopfion ring not only has lower energy, but also provides contrast in theoretical Lorentz TEM images that accurately match experimental images (Extended Data Fig. 9e,f). The results shown in Extended Data Fig. 9 can be reproduced by using Mumax Script II.
For simulations of the bulk, skyrmion strings with hopfion rings were embedded into the uniform conical phase:
(3)
For hopfion rings, we used the following toroidal ansatz:
(4)
(5)
where R_{1} and R_{2} are the minor and major radii of a torus, respectively,
and
. We also refer the reader to the Mumax Script III for initial state implementation. By default, Script III can reproduce a complex configuration in the bulk system, as shown in Extended Data Fig. 8h. With minor modifications, it can also be used to replicate all other states.
Simulations of electron optical phaseshift and Lorentz TEM images
By using the phase object approximation and assuming that the electron beam is antiparallel to the z axis, the wave function of an electron beam can be written as follows^{48}:
(6)
where φ(x, y) is the magnetic contribution to the phase shift^{49}
(7)
e is an elementary (positive) charge (around 1.6 × 10^{−19} C) and h is Planck’s constant (approximately 6.63 × 10^{−34} m^{2} kg s^{−1}). Because our approach for the solution of the micromagnetic problem recovers the magnetic vector potential A_{d}, simulation of the electron optical phase shift is straightforward.
In the Fresnel mode of Lorentz TEM, neglecting aberrations other than defocus, aperture functions and sources of incoherence and blurring, the wave function at the detector plane can be written in the form
(8)
where the kernel is given by the expression
(9)
the relativistic electron wavelength is
(10)
Δz is the defocus of the imaging lens, c is the speed of light (approximately 2.99 × 10^{8} m s^{−1}), U is the microscope accelerating voltage and m_{e} is the electron rest mass (around 9.11 × 10^{−31} kg). The image intensity is then calculated using the expression
(11)
For more details about the calculation of Lorentz TEM images, see ref. ^{20}.
Homotopygroup analysis
Skyrmion topological charge
For magnetic textures localized in the plane area
, such that at the boundary of this area ∂Ω the magnetization field m(∂Ω) = m_{0}, the classifying group is the second homotopy group of the space
at the base point m_{0}, and there is an isomorphism to the group of integers (Abelian group with respect to addition):
(12)
This implies that any continuous magnetic texture satisfying the above criteria of localization can be attributed to an integer number, which is commonly referred to as the skyrmion topological charge (or skyrmion topological index), and can be calculated as follows:
(13)
where
(14)
is the vector of curvature^{50,51}, which is also known (up to a prefactor) as the gyrovector or vorticity^{10,52,53,54}, and r_{1}, r_{2} and r_{3} are local righthanded Cartesian coordinates.
The unit field m can be parameterized on the
sphere using polar and azimuthal angles Θ and Φ, respectively, in the form
. The corresponding topological invariant, up to the sign, is the degree of mapping of the skyrmion localization area onto the sphere^{55}, which can be calculated using the top part of equation (13), assuming that r_{1} and r_{2} lie in the skyrmion plane. It should be noted that the sign of the integral in the top part of equation (13) depends on the choice of the orientation of the coordinate system. For example, in Fig. 4a the sign of Q depends on whether the r_{3} axis is parallel or antiparallel to the z axis and equals − 11 or 11, respectively. The condition in the bottom part of equation (13) removes this ambiguity. A justification for this statement, based on the theory of fundamental invariants, can be found in a previous study^{56}. The local coordinate system (r_{1}, r_{2}, r_{3}) for calculating the topological charge Q of a particular skyrmion is chosen according to the condition in the bottom part of equation (13).
For skyrmions that have different m_{0} in the global coordinate system, equation (12) is not globally applicable because the base points^{57} are different. However, isomorphisms to the group of integers can always be completed through continuous individual transformations of vector fields to match the vectors m_{0} to one base point.
Here, we use the same convention for the sign of the topological charge as previous reports^{20,28,29,58}, such that an elementary Blochtype or Neeltype skyrmion has Q = −1.
Hopfion topological charge
For a magnetic texture localized within the 3D domain
, with a fixed magnetization m(∂Ω) = m_{0} at the boundary ∂Ω of the domain, the classifying group corresponds to the third homotopy group of the space
at the base point m_{0}:
(15)
The corresponding topological charge, which is known as the Hopf invariant, can be calculated using Whitehead’s formula^{51,59}:
(16)
Skyrmion–hopfion topological charge
To analyse the continuous texture localized on a segment of a skyrmion string, we use the compactification approach and other methods of algebraic topology^{60}. First, we note that, owing to the invariance of Q along a skyrmion string, the lower and upper crosssections bounding the skyrmion string segment are related by a trivial transformation. This implies that the dimension along the skyrmion string can be compactified to a circle
. Second, we note that the conical phase and the phase with uniform magnetization m_{0} are equivalent to each other up to a trivial transformation. By exploiting this observation, one can compactify the remaining two dimensions. The magnetic texture localized on a segment of a skyrmion string can be treated as if it is confined within a solid torus
. The noncollinearities of m are then localized inside Ω, while everywhere on its surface
the magnetization m(∂Ω) = m_{0} is fixed.
Thereby, the homotopy classification arises from a continuous map from a onepoint compactified solid torus to the spin space:
(17)
By using the homeomorphism of the quotient spaces
and
, as well as the homotopy equivalence between
and
, we find a homotopy equivalent map
(18)
Taking the base point, m_{0}, as a point common to the wedge sum, we immediately find the homotopy group
(19)
where π_{2} and π_{3} correspond to equations (12) and (15), respectively, and the components of the topological charge are subject to equations (13) and (16), respectively. The topological index for the textures depicted in Fig. 4 and Extended Data Fig. 8 then represents the ordered pair of integers (Q, H).
Calculation of topological charges
To compactify the textures obtained in micromagnetic simulations, we used the nested box approach. This method involves fixing and placing the box containing the studied texture at the centre of a slightly larger computational box. The computational box has periodic boundary conditions along the z direction, and the remaining boundaries are fixed
. To ensure continuity of the vector field m in the transition regions between the nested boxes, we minimized the Dirichlet energy, ∫dr∣∇ m∣^{2}. Next, to calculate Fand the topological charge Q, we used a previously proposed lattice approach^{61}. The vector potential of the divergencefree field F was obtained by evaluating the integral:
(20)
The Hopf index H was then determined by numerically integrating equation (16).
For additional verification, we also computed the index H by calculating the linking number for curves in real space that corresponded to two different points on the spin sphere^{62}.
Derivation of zero mode
The zero mode is obtained by analysing the symmetries of the Hamiltonian presented in the supplementary material of a previous report^{32}. Without dipole–dipole interactions, the energy density of the bulk system in equation (1) is invariant under the following transformations from
to m(r) and vice versa:
(21)
where
(22)
and z_{0} is an arbitrary parameter, which, in the most general case, defines the screwlike motion of an entire magnetic texture about the z axis with pitch 2π/k = L_{D}. Of note, there are at least two cases in which the transformation (equations (21) and (22)) does not affect the magnetic texture, meaning that
holds for any value of z_{0}. The first case is rather trivial and corresponds to the conical phase with the wave vector aligned parallel to the z axis (see equation (3)). The second case is particularly intriguing, and involves the skyrmion string in the conical phase, in which the primary axis of the string aligns with the rotation axis^{32}. Applying the transformation in equations (21) and (22) to the skyrmion string with a hopfion ring describes the screw motion of the hopfion ring around the string representing a zero mode. The parameter z_{0}, in this case, denotes the displacement of the hopfion ring along the string. Supplementary Videos 7 and 8 provide a visualization of such a screw motion of two different hopfion rings depicted in Fig. 4e,f. Evidence for such zero mode in other 3D solitons can be found in previous reports^{8,63}.
Specimen preparation
FeGe TEM specimens were prepared from a single crystal of B20type FeGe using a FIB workstation and a liftout method^{20}.
Magnetic imaging in the transmission electron microscope
The Fresnel defocus mode of Lorentz imaging and offaxis electron holography were performed in an FEI Titan 60300 TEM operated at 300 kV. For both techniques, the microscope was operated in Lorentz mode with the sample at first in magneticfieldfree conditions. The conventional microscope objective lens was then used to apply outofplane magnetic fields to the sample of between −0.15 and + 1.5 T. A liquidnitrogencooled specimen holder (Gatan model 636) was used to vary the sample temperature between 95 and 380 K. Fresnel defocus Lorentz images and offaxis electron holograms were recorded using a 4k × 4k Gatan K2 IS direct electron counting detector. Lorentz images were recorded at a defocus distance of 400 μm, unless otherwise specified. Multiple offaxis electron holograms, each with a 4 s exposure time, were recorded to improve the signaltonoise ratio and analysed using a standard fast Fourier transform algorithm in Holoworks software (Gatan). The magnetic induction map shown in Fig. 1 was obtained from the gradient of an experimental magnetic phase image.
Supplementary Videos 1–5 show in situ Lorentz TEM images captured at a defocus distance of approximately 400 μm and at a sample temperature of 180 K. Each video begins with several cycles of field swapping, in which the applied magnetic field alternates between positive and negative directions perpendicular to the plate. The field amplitude is limited to 50 mT or less. As the magnetic field in the transmission electron microscope is provided by the objective lens, this alternating field leads to a visible rotation of the image on the screen. Counterclockwise rotation indicates an increase in the field towards the viewer and vice versa. These fieldswapping cycles induce edge modulations that propagate towards the centre of the sample. The primary objective of this cycle is to generate edge modulations that propagate out of the free edges. After a few fieldswapping cycles, closed loops near the centre of the square sample form. To enhance visibility, the playback speed of all of the videos has been doubled. Once at least one closed loop has been formed away from the sample edges, the magnetic field is increased to approximately 150 mT, resulting in the nucleation of a hopfion ring. Supplementary Video 1 illustrates the nucleation of a double hopfion ring.
Supplementary Video 4 reveals instabilities of the hopfion ring. After hopfionring nucleation, the magnetic field was initially decreased below a threshold value of approximately 50 mT, causing the hopfion ring to lose its shape and elongate over the sample. Subsequently, the field was increased again, leading to the reformation of a compact hopfion ring surrounding six skyrmion strings. Finally, the field was increased further above 190 mT, leading to the collapse of the hopfion ring. The concluding frame of Supplementary Video 4 depicts a cluster of six skyrmions without a hopfion ring.
Data availability
Source data for TEM images are available at https://doi.org/10.5281/zenodo.8281078. Source data for micromagnetic simulations are provided with the paper.
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Acknowledgements
We thank H. Du for help with specimen preparation, and J. V. Vas and Q. Lan for their help with the video capturing of hopfionring nucleation. This project has received funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme (grant 856538; project ‘3D MAGiC’). F.Z. acknowledges financial support from the Fundamental Research Funds for the Central Universities and the National Natural Science Fund for Excellent Young Scientists Fund Program (Overseas); N.S.K. and S.B. acknowledge financial support from the Deutsche Forschungsgemeinschaft through SPP 2137 ‘Skyrmionics’ (grants KI 2078/11 and BL 444/162, respectively); F.N.R. acknowledges support from the Swedish Research Council; and R.E.D.B. is grateful for financial support from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Programme (grant 823717; project ‘ESTEEM3’) and the Deutsche Forschungsgemeinschaft (project ID 405553726; TRR 270).
Funding
Open access funding provided by Forschungszentrum Jülich GmbH.
Author information
Authors and Affiliations

SpinX Institute, Electron Microscopy Center, School of Physics and Optoelectronics, State Key Laboratory of Luminescent Materials and Devices, GuangdongHong KongMacao Joint Laboratory of Optoelectronic and Magnetic Functional Materials, South China University of Technology, Guangzhou, China
Fengshan Zheng

Ernst RuskaCentre for Microscopy and Spectroscopy with Electrons, Forschungszentrum Jülich, Jülich, Germany
Fengshan Zheng, Wen Shi & Rafal E. DuninBorkowski

Peter Grünberg Institute, Forschungszentrum Jülich, Jülich, Germany
Fengshan Zheng, Nikolai S. Kiselev, Wen Shi, Stefan Blügel & Rafal E. DuninBorkowski

Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, Jülich, Germany
Nikolai S. Kiselev & Stefan Blügel

Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
Filipp N. Rybakov

Institute of Microstructure and Properties of Advanced Materials, Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing, China
Luyan Yang
Contributions
F.Z. and N.S.K. conceived the project and designed the experiments. F.Z., L.Y. and W.S. performed the TEM experiments and data analysis. N.S.K. and F.N.R. developed the theory and performed numerical simulations. F.N.R. performed homotopygroup analysis. N.S.K., F.Z. and F.N.R. prepared the manuscript. All authors discussed the results and contributed to the final manuscript.
Corresponding authors
Correspondence to Fengshan Zheng, Nikolai S. Kiselev or Filipp N. Rybakov.
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Competing interests
The authors declare no competing interests.
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Extended data figures and tables
Extended Data Fig. 1 Field evolution of the magnetic state with double hopfion rings and three skyrmions.
This is an extended version of Fig. 2 (top row), showing overfocus Lorentz TEM images recorded at a specimen temperature of 95 K. In a, the skyrmion string is attached to the outer ring. In b, the skyrmion moves to the left edge of the sample but becomes more visible in c with increasing field. In d, the skyrmion moves to the top lefttop corner. Between e and f, the texture rotates slightly. In g and h, one can observe compression of the rings. The outer hopfion ring collapses at 301 mT to either a chiral bobber or a dipole string, as marked by a dashed circle in i. In j and k, the single ring continues to contract with increasing field. The entire system collapses at 343 mT to three bobbers or dipole strings (see l).
Extended Data Fig. 2 Field evolution of the magnetic state with a hopfion ring.
a, An extended version of Fig. 2 (second row), showing overfocus Lorentz TEM images of a hopfion ring around four skyrmions recorded at a specimen temperature of 180 K. The outer ring collapses at 358 mT to a chiral bobber or a dipole string, as marked by a dashed circle. b, An extended version of Fig. 2(third row), showing overfocus Lorentz TEM images recorded at a specimen temperature of 180 K. At higher applied magnetic fields (399, 444 and 450 mT), skyrmions inside the hopfion ring have fivefold symmetry. The hopfion ring collapses at 467 mT, and the nucleation of a chiral bobber or a dipole string is not observed in this case.
Extended Data Fig. 3 Field evolution of the magnetic state with a hopfion ring and six skyrmions.
This is an extended version of Fig. 2 (bottom row), showing overfocus Lorentz TEM images recorded at a specimen temperature of 200 K. In a, two skyrmions are attached to the hopfion ring. In b, skyrmions move to the lefttop corner of the sample. In c, the skyrmion cluster surrounded by the hopfion ring takes a pentagon shape. With increasing field, in d, e, f, g, and h, the contrast of the hopfion ring gradually decreases. In i, j, and k, the skyrmions return to the sample center and interact with the hopfion ring. l, The hopfion ring collapses at 316 mT, and the nucleation of a chiral bobber or a dipole string is not observed in this case.
Extended Data Fig. 4 Lorentz TEM images of magnetic textures of topological charge −1 to −12 with a single hopfion ring.
Overfocus Lorentz TEM images are shown. The number of skyrmions inside the hopfion ring, N_{Sk}, is labelled in the upper right corner of each image. The applied magnetic field is indicated above each image. The images were recorded at a specimen temperature of between ~180 and ~200 K.
Extended Data Fig. 5 Examples of exotic magnetic states with hopfion rings and skyrmion strings.
Overfocus and underfocus Lorentz TEM images recorded at a specimen temperature of between ~180 and ~200 K are shown. The magnitude of the applied magnetic field is marked on the left of each image. The images in a, b, d, e, g, and i show the magnetic states depicted in Fig. 4 but for the entire field of view of the sample. c and f are the magnetic states with a single skyrmion string and multiple hopfion rings. The magnetic state in h is similar to that in c but with an extra skyrmion inbetween internal and external hopfion rings. The intermediate state in j is similar to that shown in Fig. 1f at 190 mT.
Extended Data Fig. 6 Field evolution of magnetic states with multiple hopfion rings.
The images are arranged in rows. a, Magnetic textures with three hopfion rings. The outermost hopfion ring collapses at 131 mT. The two inner hopfion rings collapse at 139 mT. b, Double hopfion rings coupled to one skyrmion string. The double rings show nonhomogeneous contrast at lower applied fields owing to interaction with the specimen edge. With increasing applied field, the ring detaches from the specimen edge and the two hofion rings show homogeneous contrast. The two hopfion rings collapse at 227 mT. c, The magnetic state of a positive topological charge surrounded by a single hopfion ring. With increasing field, the state evolves into a skyrmion surrounded by two hopfion rings at 213 mT and collapses to a single skyrmion at 256 mT. The applied magnetic field is labelled above each image. The Lorentz TEM images were recorded overfocus at a defocus distance of 400 μm at a specimen temperature of 95 K.
Extended Data Fig. 7 Micromagnetic simulations of exotic magnetic states with hopfion rings.
Each column represents a single hopfion ring obtained by energy minimization, assuming open boundary conditions. The calculations were performed for the applied magnetic fields indicated above each column. The first three rows show the magnetization vector field in the top, middle and bottom planes of the square sample, whose size (1 μm × 1 μm × 180 nm) is identical to that used in the experiment. An alternative representation of the magnetization vector fields of the magnetic states by means of isosurfaces is provided in Fig. 4 in the main text (for the magnetic states shown in the first four columns) and Extended Data Fig. 8 (for the magnetic states shown in columns 5 to 8). The two bottom rows show theoretical overfocus and underfocus Lorentz TEM images calculated for a defocus distance of 400 μm. The dashed squares indicate the field of view that was used in Fig. 3 in the main text.
Extended Data Fig. 8 Micromagnetic simulations of magnetic textures with hopfion rings in a confined sample and in the bulk.
a–d, As in Fig. 4a–d, but for the other experimentally observed magnetic states shown in Fig. 3e–h with Q = −1, 0, +1 and +3. e–h, Different configurations of coupled skyrmion strings and hopfion rings in the bulk. The skyrmion–hopfion index given by the ordered pair of integers (Q, H) is specified for each state. The hopfion rings in e are interlinked. In f, two hopfion rings (top and bottom) are located on individual strings, and one hopfion ring (in the middle) surrounds two strings. In g, two hopfion rings share one skyrmion string and have one extra individual string. The braiding of the skyrmion strings in h increases the hopfion index compared to the state shown in Fig. 4h. Note that, for the states shown in a–d, the Hopf charge is uncertain (see the main text for details).
Extended Data Fig. 9 Micromagnetic simulations for hopfion rings, with modulations penetrating the whole thickness, and a comparison of the contrast with experimental measurements.
a, Three initial states. b,c, Corresponding magnetic textures after energy minimization. In b, a crosscut of the simulated state is shown; in c, the same magnetic texture is visualized using isosurfaces m_{z} = 0. The calculations for all three cases were conducted for an external magnetic field of 243 mT, with a very small tilt angle not exceeding 2°. For details, see Mumax Script II in the Supplementary information. d, Corresponding crosscuts for the median plane and theoretical phase images calculated for each of the three states. e, Corresponding theoretical Lorentz TEM images, assuming a defocus distance of 400 μm, as in the experimental setup. f, Experimental Lorentz TEM images recorded in an external magnetic field of 243 mT and at a defocus distance of 400 μm. The experimental images match the theoretical images well for States 1a and 1b with a compact hopfion ring. In State 1a, the vortex is centred on the hopfion ring, whereas in State 1b this is not the case. Note the bumps on the hopfion ring in State 1b visible in the experimental and theoretical Lorentz images. For State 2, which exhibits magnetic modulations extending along the thickness, the theoretical Lorentz TEM images show two distinct features: a strongly contrasting bright ring in the overfocus image and an additional bright ring in the underfocus image. No highintensity bright ring in the overfocus image or second bright ring in the underfocus image are observed in the experimental images.
Extended Data Fig. 10 Quantitative comparison of experimental and theoretical phaseshift images.
a, Experimental electron optical phaseshift image recorded using offaxis electron holography at T = 180 K in an external magnetic field of 200 mT. b, Theoretical phaseshift image calculated for the equilibrium configuration in the micromagnetic model, as shown in d. The grey levels used to display the magnitude of the phase shift, φ, are identical for the experimental and theoretical images. c, Line profiles of the phase shift extracted along the paths marked in a and b. The experimental and theoretical profiles indicate that the skyrmion inside the hopfion ring induces an electron optical phase shift of approximately 3.5 radians, whereas the hopfion ring induces an electron optical phase shift of approximately 1.0 radians. d, Magnetic texture obtained from micromagnetic simulations for an external magnetic field of 200 mT. The magnetization field is represented by the isosurface m_{z} = 0.
Supplementary information
Supplementary Data
This zipped file contains Supplementary Scripts 1–3. Supplementary Script 1: This Mumax script allows to reproduce the results of micromagnetic simulations presented in Fig. 1. Supplementary Script 2: This Mumax script allows to reproduce the results of micromagnetic simulations presented in Extended Data Fig. 9. Supplementary Script 3: This Mumax script allows to reproduce the results of micromagnetic simulations presented in Extended Data Fig. 8h.
Supplementary Video 1
This video shows the in situ nucleation of a double hopfion ring, similar to that illustrated in Fig. 1g, but for four skyrmion strings.
Supplementary Video 2
This video shows the in situ nucleation of a hopfion ring on two skyrmion strings, similar to that illustrated in Fig. 1f.
Supplementary Video 3
This video shows the in situ nucleation of an exotic configuration featuring a hopfion ring surrounding four skyrmion strings and a skyrmionium.
Supplementary Video 4
This video shows the in situ nucleation of a hopfion ring that surrounds six skyrmion strings. As the external magnetic field is decreased below the elliptic instability threshold for skyrmions, the magnetic texture loses its compact form. However, as the field is increased further, the contrast reemerges, revealing the presence of the hopfion ring around six skyrmion strings.
Supplementary Video 5
This video shows the in situ nucleation of another exotic configuration similar to that illustrated in Extended Data Fig. 6i.
Supplementary Video 6
This video visualizes the homotopy transformation between two states with (Q, H) = (1, 1). The first state corresponds to an unlinked skyrmion string and hopfion, while the second state corresponds to a hopfion ring linked to a skyrmion string. Note the colour change on the isosurface of the hopfion ring throughout the transformation process.
Supplementary Video 7
This video visualizes the zero mode exhibited by the elementary hopfion ring depicted in Fig. 4e.
Supplementary Video 8
This video visualizes the zero mode exhibited by the sophisticated shaped hopfion ring depicted in Fig. 4f.
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