Beyond skyrmions: Review and perspectives of alternative magnetic quasiparticles

Magnetic skyrmions have attracted enormous research interest since their discovery a decade ago. Especially the non-trivial real-space topology of these nano-whirls leads to fundamentally interesting and technologically relevant effects - the skyrmion Hall effect of the texture and the topological Hall effect of the electrons. Furthermore, it grants skyrmions in a ferromagnetic surrounding great stability even at small sizes, making skyrmions aspirants to become the carriers of information in the future. Still, the utilization of skyrmions in spintronic devices has not been achieved yet, among others, due to shortcomings in their current-driven motion. In this review, we present recent trends in the field of topological spin textures that go beyond skyrmions. The majority of these objects can be considered the combination of multiple skyrmions or the skyrmion analogues in different magnetic surroundings, as well as three-dimensional generalizations. We classify the alternative magnetic quasiparticles - some of them observed experimentally, others theoretical predictions - and present the most relevant and auspicious advantages of this emerging field.


I. INTRODUCTION
Over the last decades, information technology has become eminently relevant for our everyday lives. The recent conquest of modern IT applications, such as streaming services and cloud storages, has further intensified the demand for energy efficient data storage and manipulation. While current electronic solutions struggle to keep up with Moore's law 1 , new spintronic proposals have been suggested and may become relevant in the near future 2 .
One of the most auspicious and anticipated data storage devices is the racetrack memory. Originally proposed for utilizing domain walls as the carriers of information [3][4][5] , the bitsencoded by the presence or absence of the magnetic objectare written, deleted, moved and read in a narrow track. This quasi-one-dimensional setup is stackable enabling the possibility for an innately three-dimensional data storage with drastically increased bit densities. This non-volatile concept without mechanically moving parts surpasses current random access memories (RAM) and hard disk drives (HDD) in terms of a lower energy consumption and faster access times 4 .
Besides spintronics, topological matter is an aspiring research field which is why this review is concerned with noncollinear spin textures. The most prominent example is the magnetic skyrmion 6 . This whirl-like nano-object was first observed a decade ago 7 . Its topological protection gives it an enormous stability even at small sizes, which makes it a potential carrier of information in future data storage devices, such as the racetrack nanodevices [8][9][10] .
Besides great stability, the topological properties of skyrmions induce emergent electrodynamics, namely the topological Hall effect [11][12][13] and the skyrmion Hall effect [14][15][16] . While the first -an additional contribution to the Hall effect of electrons in the presence of a topologically non-trivial spin texture -may become favorable for detecting skyrmions 17,18 , the skyrmion Hall effect leads to a transverse deflection of skyrmions when they are driven by currents. This means that skyrmions are pushed towards the edge of the racetrack when a current is applied along the track, leading to pinning or even the loss of data. This is one of the reasons why no prototype of a skyrmion-based spintronic device exists today.
While there will be ongoing research for improving the applicability of magnetic skyrmions in spintronic devices, several alternative nano-objects have been predicted and observed during the last 6 years. Some of them promise even greater advantages compared to conventional skyrmions, which is why research in this direction will be drastically enhanced in the near future. In this review, we introduce and elaborate on these alternative magnetic quasiparticles. We establish a classification of the objects, explain methods to stabilize them, and compare their emergent electrodynamics to the case of conventional skyrmions.
We begin by elaborating on conventional magnetic skyrmions (Sec. II) to convey the differences to alternative magnetic quasiparticles later in the paper. We introduce how different types of skyrmions can be characterized topologically and geometrically (Sec. II A), explain various mechanisms for their stabilization (Sec. II B), and mediate the emergent electrodynamics (Sec. II C). Thereafter, we characterize and discuss the alternative magnetic quasiparticles (Sec. III). We distinguish three groups as visualized in Fig. 1: the fundamental excitations in ferromagnets (including different types of skyrmions; Fig. 1a), variations of these excitations (the combination of multiple excitations or excitations in other backgrounds than ferromagnets; Fig. 1b), and extensions (for example periodic arrays of magnetic objects or innately threedimensional spin textures; Fig. 1c). First, we address the objects that are closely related to skyrmions (Sec. III A), namely skyrmions with an arbitrary helicity 19,20 (Fig. 2b) and antiskyrmions 21 (Fig. 2a). Thereafter, we consider combinations of skyrmions (Sec. III B): bimerons [22][23][24] (Fig. 2d), biskyrmions 25,26 (Fig. 2e), skyrmioniums 27,28 (Fig. 2f), and antiferromagnetic skyrmions 29,30 (Fig. 2h). Finally, we discuss chiral bobbers 31,32 (Fig. 2j) and hopfions 33 (Fig. 2l) as non-trivial three-dimensional continuations of skyrmions in Sec. III C. Other magnetic quasiparticles (shown in Fig. 2) are briefly addressed and put in context while discussing the above objects. We conclude this review in Sec. IV. (c) Also, the fundamental and derived objects can be continued as periodic or non-periodic arrangements in two dimension or can be continued along the third dimensions trivially or non-trivially. The objects discussed more thoroughly in this review are typeset bold.

II. MAGNETIC SKYRMIONS
Skyrmions have originally been predicted in the 1960's in the context of particle physics 34,35 . The British nuclear physicist Tony Skyrme proposed a field-theoretical description of interacting pions and showed that the particle-like solutions are fermionic while pions themselves are bosonic. These solitons, described by a non-linear sigma model, are the threedimensional versions of what became known as skyrmions.
Today, skyrmions have been found in several fields of physics such as quantum Hall systems 36 , Bose-Einstein condensates 37 , liquid crystals 38 , particle physics 39 , string theory 40 and, as considered here, in magnetism 7 .
In this context, a skyrmion can be considered as a twodimensional object (Fig. 3a), that is continued trivially along the third dimension (Fig. 3c). Such skyrmion tubes or skyrmion strings have been observed for the first time in 2009 in MnSi by reciprocal-space measurements 7 and one year later using Lorentz transmission electron microscopy (LTEM) 41 .
The magnetic textures of these objects were in agreement with what had been predicted twenty years earlier 6 : Magnetic skyrmions in a ferromagnetic medium are characterized by a continuously changing magnetization density which is oriented oppositely in its center compared to the surrounding leading to a non-trivial real-space topology. These objects can occur as periodic lattices, like in the above publications, or as individual particles 41,42 .
The following discussion of conventional skyrmions is limited to their geometrical characterization, stabilizing mechanisms and emergent electrodynamics -all of which are a prerequisite for understanding the physics of the alternative magnetic quasiparticles. For a discussion of conventional skyrmions that goes beyond these points, we refer to one of the many review articles [43][44][45][46][47][48][49][50] .

A. Topology and characterization
The topological character of a skyrmion can be comprehended by a stereographic projection: A two-dimensional skyrmion (Fig. 3a) can be constructed by rearranging the magnetic moments of a three-dimensional hedgehog (also called Bloch point; see Fig. 3b), where all moments on a sphere point along the radial direction. This sphere is opened at the bottom and flattened to a disk without changing the moments' orientations. The result is a topologically non-trivial magnetic object in two-dimensions.
In a continuous picture, where a skyrmion consists of a magnetization density m(r), this skyrmion cannot be transformed to a ferromagnetic state without discontinuous changes in the density. This is a manifestation of the non-trivial real-space topology quantified by the topological charge which is as an integral over the topological charge density The topological charge of a skyrmion can be determined more easily from its appearance due to the following transformation. One expresses the magnetization density in spherical coordinates with the azimuthal angle θ and the polar angle Φ and expresses the position vector in polar coordinates r = r(cos φ, sin φ). Exploiting the radial symmetry of the out-of-plane magnetization density θ = θ(r), the topological charge reads 43 The out-of-plane magnetization of a skyrmion is reversed comparing its center with its confinement. This is quantified by the first factor, the polarity The sign depends on the out-of-plane magnetization of the skyrmion host. Due to continuity of the magnetization density, the polar angle can only wrap around in multiples of 2π determining the second factor, the vorticity The two-dimensional integral has been simplified to a product of the polarity and the vorticity 51 allowing only for integer topological charges for different types of skyrmions. Besides their topological charges, different types of skyrmions can also differ by their in-plane magnetization. The polar angles of the position vector φ and the magnetization density Φ at each coordinate are related linearly, with an offset γ. This quantity is called helicity. With the polarity, vorticity and helicity we have established the three characterizing quantities for the different types of skyrmions, which can generally be expressed as Note, that the out-of-plane magnetization profile is simplified as a cosine function (radius r 0 ) and that the exact profile depends on the interaction parameters, the sample geometry, defects, and the presence of other quasiparticles.
As an example, the skyrmion in Fig. 3a has a positive polarity p = +1 and vorticity m = +1 leading to a topological charge of N Sk = +1. Since the in-plane component of the magnetization is always pointing along the radial direction, the helicity in this case is γ = 0. This type of skyrmion is called Néel skyrmion and is typically observed at interfaces 52 . On the contrary, the skyrmions in MnSi (e. g. from the initial observation 7 ) are called Bloch skyrmions. There, the in-plane components of the magnetization density are oriented perpendicularly with respect to the position vector. This toroidal configuration is characterized by a helicity of γ = ±π/2. In contrast to the polarity and the vorticity, the helicity is a continuous parameter allowing for skyrmions as intermediate states between Bloch and Néel skyrmions, as shown in Fig. 2b. Furthermore, the vorticity can in principle take any integer value constituting for example antiskyrmions for m = −1 (Fig. 2a) or higher-order (anti)skyrmions for |m| > 1 (Fig. 2c). Out of this manifold, Bloch 7 , Néel skyrmions 52 and skyrmions with an intermediate helicity 20 , as well as antiskyrmions 21 have been observed experimentally. Higher-order skyrmions 53,54 have been predicted.

B. Stabilizing mechanisms
Having discussed the different types of possible skyrmions mathematically, we will now address how these objects can be stabilized in ferromagnets. In such materials the magnetic moments {s i } are coupled via the exchange interaction Typically, the nearest-neighbor interaction is dominant. In a ferromagnet it favors a parallel alignment (J i,j = J > 0 for i, j nearest neighbors; J i,j = 0 otherwise) of the magnetic moments. In a continuous approximation, this term is expressed as In the continuous limit, a magnetic skyrmion would be stable due to the topological protection; it cannot be transformed into a uniform ferromagnet continuously, even though the ferromagnetic state would have the lower energy. However, since real skyrmions consist of magnetic moments and not a continuous density, this protection is not strict 55 in nature bringing forth the necessity of additional stabilizing interactions.
In theory, skyrmions have been stabilized by frustrated exchange interactions 19 , four-spin interactions, dipole-dipole interactions and Dzyaloshinskii-Moriya interactions 43 . In the following, we will focus on the latter two cases, since they are by far the most relevant mechanisms in nature.
Dzyaloshinskii-Moriya interaction. The Dzyaloshinskii-Moriya interaction 56,57 (DMI) is a chiral interaction responsible for the stability of most experimentally observed skyrmions. It is a correction due to spin-orbital coupling under a broken inversion symmetry and can be considered an antisymmetric exchange interaction, D ij = −D ji . The D ij are the Dzyaloshinskii-Moriya vectors, whose orientations account for the way the inversion symmetry is broken, satisfying the Moriya symmetry rules 57 .
For example, at an interface of a magnet and a heavy metal, like Co/Pt, the DMI vectors are oriented parallel to the interfacial plane, perpendicular to the bond of the Co atoms 58 . In a continuous approximation this is expressed as 59 Such an interaction favors a canting of magnetic moments. As a consequence, Néel-type skyrmions may be stabilized, as first observed at the interface of Fe and Ir (111) 52 . This type of DMI and the resulting skyrmions are well understood today.
In multistack systems of magnetic and heavy metal materials, the effective DMI strength can be tuned 60 changing the size and the stability of Néel skyrmions 61 . Different samples lead to different DMI vectors, since the inversion symmetry is broken in a different way. In Heusler materials, the layers are stacked in a way that the DMI vectors along the two bond directions have opposite signs favoring antiskyrmions energetically 21 . In B20 materials, such as MnSi, where the inversion symmetry is broken intrinsically, the DMI is expressed as leading to the stability of Bloch skyrmions 41 .
In principle, lower symmetric lattice structures or nanostructuring allow to generate further types of DMIs, leading to the stabilization of alternative topologically non-trivial excitations in magnets.
Dipole-dipole interaction. A second main mechanism for stabilizing skyrmions is the dipole-dipole interaction While the DMI typically stabilizes skyrmions smaller than a few hundred nanometers, the dipole-dipole interaction can stabilize objects of several micrometer in diameter 43 . Typically, these objects have an almost ferromagnetic center surrounded by a narrow domain wall. They are sometimes labeled 'bubble' but are topologically equivalent to skyrmions, given that they have the same N Sk . As a difference to the DMI, dipole-dipole interactions are achiral, meaning that they energetically favor Bloch skyrmions of both helicities γ = ±π/2 26,62 allowing even for their coexistence 63 .

C. Emergent electrodynamics
Magnetic skyrmions have successfully been generated and deleted in thin films (see reviews [43][44][45][46][47][48][49][50] for different methods). In order to constitute an operating data storage device, they also have to be driven and read. In this section we will discuss the topological Hall effect and the current-driven motion of skyrmions. This will be the foundation for our discussion of alternative magnetic quasiparticles later in this review and will reveal why skyrmions may not be the optimal candidates for such spintronic devices.
Topological Hall effect. The topological Hall effect of electrons is considered the hallmark of the skyrmion phase [11][12][13]17,18,[64][65][66][67][68][69] . To measure it, an electric field E is applied to the skyrmion host, and the current density j is detected. According to Ohm's law, E = ρj, the Hall effect of electrons is characterized by the transverse resistivity tensor element ρ xy . For a skyrmion crystal, it is commonly considered as three superimposed contributions 12 that can be isolated due to their distinct proportionalities: the ordinary Hall contribution 70 occurs due to an external magnetic field ρ HE xy ∝ B z , the anomalous Hall contribution 71 is due to spin-orbit coupling and, usually, a net magnetization ρ AHE xy ∝ M z , and the topological Hall contribution appears due to the presence of skyrmions or other topologically nontrivial spin textures.
The presence of a skyrmion leads to the emergence of an additional contribution to the Hall effect for the following reason: If an electron hops between two sites and reorients its spin, the original transfer integral t accumulates a complex phase factor 43 . This factor is the analogue of the Peierls phase, which characterizes the magnetic field in the ordinary Hall effect. In the skyrmionic case, it can also be related to an effective vector potential. In an adiabatic approximation, the corresponding field B em is called 'emergent field' and it is proportional to the topological charge density 43 This fictitious field can be used to easily relate the transverse deflection and the generation of a Hall effect with the presence of skyrmions (Fig. 4). As long as the electron spin and the texture are coupled strongly, the topological Hall effect is proportional to the number of skyrmions in the sample ρ THE xy ∝ B em,z ∝ N Sk . Skyrmion Hall effect. The non-trivial real-space topology of skyrmions also becomes apparent in the current-driven motion of the skyrmions themselves. One typically discusses two scenarios: the motion under spin-transfer torque (STT) 72 , where a spin-polarized current goes through the spin texture and reorients the magnetic moments of a skyrmion (Fig. 4), and the motion under spin-orbit torque (SOT) [73][74][75][76] , where a spin accumulation created by an electric current in the presence of spin-orbit interaction exerts a torque on the skyrmion texture. In both cases, the torque reorients the magnetic moments. This collective reorientation can be identified with a motion of the skyrmions. The SOT mechanism turns out to be more efficient 8 , since electron spins and magnetic moments can have a large misalignment leading to larger torques. Still, it has been predicted 8 and observed 15,16 that the skyrmions do not move parallel to the current, but experience a transverse deflection in this case. This phenomenon is called the skyrmion Hall effect and originates in the topological charge of the magnetic objects.
For the given reasons, in this review we focus on the SOT setup and will only occasionally mention the spin-transfer torque. Typically, one considers a bilayer of a ferromagnet, potentially hosting skyrmions, and a heavy metal. The applied current mainly flows in the heavy metal, where the spin Hall effect generates a pure spin current along the perpendicular direction with spins oriented perpendicular to both currents. In a racetrack, the perpendicularly injected spins are pointing along the narrow width of the track.
The reorientation of the magnetic moments can be modeled by the Landau-Lifshitz-Gilbert (LLG) equation 73,77,78 It is written here in the micromagnetic formulation, where the magnetization density has been discretized in small volumes with a normalized magnetization m i (the index has been dropped for simplicity). The first term on the right side describes the precession of each normalized magnetic moment around its space-and time-dependent effective magnetic field characterized by the free energy functional F accounting for all magnetic interactions. γ e = γ/µ 0 = 1.760 × 10 11 T −1 s −1 quantifies the gyromagnetic ratio of an electron. The second term is the Gilbert damping quantified by the dimensionless parameter α, leading to an alignment of the magnetic moment with its effective field after a certain propagation time without external perturbations. The last term is the torque τ . In the spin-orbit torque scenario it is given by the out-of-plane and in-plane torques. The out-of-plane torque term may change the shape of a skyrmion, but does not add a driving force to the effective equation of motion, which is why we consider only the in-plane torque term in the following. It reads 79 where d z is the thickness of the magnetic layer, M s is the saturation magnetization and θ SH is the spin-Hall angle describing the spin current θ SH j with spin orientation s. Numerically solving this equation for a skyrmion in a ferromagnet in the presence of an applied current leads to a motion of the skyrmion partially along the current direction (along the track), but also partially towards the track's edge. This transverse component is due to the topological charge of the skyrmion as can be found by considering the Thiele equa- This effective equation of motion can be derived from the LLG equation by assuming a perfectly rigid shape of the magnetic object and by considering only its center coordinate (with velocity v). The non-collinearity of the object is condensed in the gyroscopic vector G, the dissipative tensor D and the torque tensor I, which are calculated as follows (for a complete derivation see 84 ) The non-collinear object is then condensed to a single point. A circular Néel skyrmion for example is characterized by a topological charge of N Sk = p = ±1, a diagonal dissipative tensor with only D xx = D yy = 0 and an antisymmetric torque tensor with only I xy = −I yx = 0. For injected spins s ±y this yields a skyrmion Hall angle of For topological objects with more complicated I and D tensors (for example due to a broken rotational symmetry), this relation varies and can even allow for a vanishing skyrmion Hall effect, which can be seen as one main motivation for considering alternative magnetic quasiparticles.

III. ALTERNATIVE MAGNETIC QUASIPARTICLES
In this main section of the review, alternative magnetic quasiparticles are introduced and discussed concerning their perspective for spintronic applications, mainly for racetrack memories. In this regard, we will refer to the fundamentals established in the last section. We characterize the different types of objects related to skyrmions, address their stability and their emergent electrodynamics.
Fundamental excitations in ferromagnets. As presented in Fig. 1, we distinguish the fundamental excitations in ferromagnets from their variations and extensions. In panel a of Fig. 1 the fundamental excitations are shown. They can be separated into topologically trivial and non-trivial objects. The latter comprise merons and skyrmions of different varieties. As presented in the last section, topological excitations with a topological charge of ±1 have been found as Néel skyrmion 52 , Bloch skyrmion 7 , intermediate skyrmions 20 and antiskyrmion 21 (Fig. 2a). Furthermore, higher-order skyrmions 53 (Fig. 2c) have been predicted. Most attractive application-wise are the intermediate skyrmions and antiskyrmion, since they can be moved without the occurrence of a skyrmion Hall effect 85,86 , as will be presented in Sec. III A.
Variations of the topological excitations. In Sec. III B we discuss the combinations of skyrmions or merons to form new particles. Very promising are the combinations of two skyrmions with opposite topological charges: the antiferromagnetic skyrmion 29,30 and the skyrmionium 27,28 , which are predicted to move without a skyrmion Hall effect.
As shown in Fig. 1b, some of the composite objects can ambivalently be considered as skyrmionic excitation in different magnetic backgrounds. The bimeron [22][23][24] (Fig. 2d), for example, is on one side the combination of a meron and an antimeron, but also is a skyrmion in a ferromagnet which is magnetized in-plane. Likewise, the antiferromagnetic skyrmion 29,30 (Fig. 2h) is the combination of two ferromagnetic skyrmions with mutually reversed spins, but also is the fundamental topological excitation in a collinear antiferromagnet. The same holds for ferrimagnetic skyrmions 87,88 ( Fig. 2g) in ferrimagnets.
Extensions of the topological excitations. The fundamental objects as well as their variations can be extended in the sense that they can be arranged in a (pseudo-) twodimensional system, or that they can be extended along the third spacial dimension, see Fig. 1c.
The two-dimensional arrangements are typically rather trivial: Periodic crystals of skyrmions, antiskyrmions, bimerons, biskyrmions, antiferromagnetic skyrmions and other particles can form. However, such an arrangement can also be highly non-periodic like in a skyrmion glass 89,90 , or the arrangement can consist of multiple objects like for the meron-antimeron lattice, as observed recently 91 .

A. Different types of skyrmions
Skyrmions are characterized by the polarity p, vorticity m and helicity γ. All types of skyrmions behave similar under STT but differently under SOT. For example, the missing rotational symmetry of antiskyrmions brings about an anisotropic skyrmion Hall effect, which is highly relevant for racetrack applications.
The vorticity can also have integer values larger than 1. This characterizes higher-order skyrmions and antiskyrmions with |N Sk | > 1. These objects exhibit a topological Hall effect and a skyrmion Hall effect in the STT scenario just like skyrmions 93,94 but their rotational symmetry is broken, similar to antiskyrmions.
Furthermore, in this class of magnetic quasiparticles the fundamental excitations in in-plane magnetized samples are worth to be mentioned: merons (or vortices) which are closely related to skyrmions 95 . These objects are configurated like skyrmions near their centers but the magnetic moments at the edges of the particles do not point into the opposite out-ofplane direction compared to the centers but along in-plane directions, which is often related to the magnetic anisotropy. The azimuthal angle changes only by π/2 giving these objects a polarity and a topological charge of ±1/2. These objects become relevant for forming non-trivial textures like the bimeron [22][23][24] or the meron-antimeron crystal 91 but are less relevant themselves, since they are always situated in a coplanar but non-collinear in-plane magnet.
In this section, we focus on skyrmions with an intermediate helicity 19,20 and antiskyrmions 21 . Due to their non-trivial topological charge, both objects exhibit a topological Hall effect and a skyrmion Hall effect in the STT scenario. However, in the SOT scenario the possibility to move these objects parallel to an applied current exists 85,86 . In the following, we elaborate on this motion, the particles' stability and other nontrivial observations.

Skyrmions with arbitrary helicity
One main problem upon utilizing magnetic skyrmions in racetrack devices is the skyrmion Hall effect. A possible solution is the utilization of skyrmions with a helicity γ different from that of Néel or Bloch skyrmions. In DMI systems this can in principle be achieved by considering a mix of interfacial and bulk DMI 85 or by considering interfacial DMI in materials where the dipole-dipole interaction (favoring Bloch skymions) is considerable 50,96 . Such objects have recently been observed using LTEM imaging 20 .
In order to understand the skyrmion Hall effect, we analyze the tensors in the Thiele equation. For a positive polarity, skyrmions with an arbitrary helicity are characterized by a topological charge of N Sk = +1 leading to a gyroscopic vector of G = −4π e z independent of the skyrmion's helicity. This independence holds also for the dissipative tensor which is diagonal with only D xx = D yy = 0. However, the torque tensor depends on the helicity γ, which implies that a SOT, characterized by spins s y that are injected from the perpendicular direction z, drives skyrmions with different helicities differently, resulting in different skyrmion Hall angles and trajectories. In general, the torque tensor of a skyrmion has the shape of a rotation matrix R z around the z axis For a particular helicity γ (Fig. 5a,b), the transverse motion of a skyrmion due to the gyroscopic force is compensated by a component of the driving force (Fig. 5g) so that the skyrmion moves along the applied electric current . Likewise, there exists a helicity for which the skyrmion moves perpendicular towards the edge. Due to the recent observation of these objects 20 , a verification of the theoretical predictions is highly anticipated.

Antiskyrmions
Antiskyrmions are characterized by a vorticity of m = −1, i. e., the in-plane magnetization rotates oppositely to the position vector. Unlike skyrmions (irrespective of their helicity), these particles are not rotationally symmetric. For this reason, the helicity has a different meaning for antiskyrmions. It is not a global offset between the polar angle of the position vector φ and the polar angle of the magnetization Φ, but distinguishes two axes along which the antiskyrmion has the profile of a Néel skyrmion with helicity 0 and π, respectively (dashed lines in Fig. 5f). The profile looks differently along other lines. For example, along the two bisectrices of these two axes the antiskyrmion has the same texture as a Bloch skyrmion with helicity ±γ/2, respectively.
Geometrically, an antiskyrmion can be constructed from a skyrmion by rotating all spins of the skyrmion by 180 • around a distinguished in-plane axis -say y. In this case, the x and z components of the magnetization change sign. The resulting antiskyrmion still has the same topological charge as the skyrmions since the polarity and the vorticity both change their signs. Furthermore, the rotation argument allows to also identify the DMI necessary to stabilize antiskyrmions: two of the DMI vectors need to change their sign 86,97 . This leads to the anisotropic DMI [Eq. (13)] presented in an earlier section. Note again, that the DMI vectors are determined by the crystal symmetry: While Néel skyrmions arise at interfaces, where heavy metal atoms are located directly below the bond of two magnetic atoms, for antiskyrmions a layered system is required with heavy metal atoms above and below two different bonds. This is the case in some Heusler materials, and indeed antiskyrmion crystals have been observed in Mn 1.4 Pt 0.9 Pd 0.1 Sn 21 , a Heusler material with D 2d symmetry that exhibits this particular type of DMI 86,98 . The antiskyrmions in this material have been shown to have long lifetimes at room temperature 99 . Since an antiskyrmion can be understood to consist of Bloch and Neel parts, the observed LTEM contrast (Fig. 7a) is unique. It exhibits two spots of high intensity and two spots of low intensity, corresponding to the different Bloch and Neel parts.
The change of the vorticity compared to skyrmions is highly relevant for the current-driven motion of antiskyrmions. The gyroscopic vector still points along z and the dissipative tensor D is still symmetric. However, the I tensor changes Like for the case of skyrmions with an arbitrary helicity, one can identify a particular helicity for which the skyrmion Hall effect is compensated for antiskyrmions. The important differ-ence is that the helicity of antiskyrmions corresponds merely to a rotation in real space. Consequently, applying the current along a certain direction leads to an antiskyrmion motion parallel to the current, as presented in Ref. 86. This allows to think of a racetrack, where the D 2d material is cut at an angle with respect to the high symmetry directions 100 . In a rotated coordinate system, where one axis points along the current direction, this leads to a rotation of the effective micromagnetic DMI vectors and generates a rotated antiskyrmion, i. e., an antiskyrmion with a certain γ. If it is cut under the correct angle, this racetrack allows for the desired straight-line motion of the bits without any transverse deflection (Fig. 5e,f). Another particularly interesting feature of antiskyrmion systems is that their anisotropic DMI is in conflict with the ubiquitous dipole-dipole interaction. While the first interaction allows only for antiskyrmions, the dipole-dipole interaction energetically favors Bloch skyrmions. The coexistence of both topologically distinct nano-objects has recently been observed by LTEM measurements 101,102 and confirmed by micromagnetic simulations 101 . Their distinct topological charges have later been confirmed by topological Hall measurements 103 . The DMI-stabilized antiskyrmions may show a square-shaped deformation due to the perturbative effect of the dipole-dipole interaction. The Bloch parts are increased and the Néel parts shrink in order to minimize the dipole-dipole energy 101 , as predicted in Ref. 104. The dipoledipole-mediated Bloch skyrmions, that come in two flavors (helicities γ = ±π/2), are elliptically deformed due to the DMI 101 . This finding allows to generalize the concept of the racetrack device using both, antiskyrmions and skyrmions, as the bits of information 100,101 (Fig. 8a). Such devices would be insensitive to diffusion or interactions between the nanoobjects, since non-periodic bit sequences are unproblematic in this case. The coexistence of skyrmions and antiskyrmions has also been predicted by frustrated exchange interactions 19 and for a very specific DMI tensor 98 . However, these two cases remain to be observed experimentally.

B. Combination of skyrmions or merons
In this section we present alternative magnetic quasiparticles that are formed as the combination of two or more skyrmions or merons. We begin with the bimeron [22][23][24] . It is the combination of two merons, which need to posses opposite vorticities in order to be geometrically compatible with each other and with the ferromagnetic background. Since also their polarities are mutually reversed, a bimeron is characterized by a topological charge of N Sk = ±1. It can therefore be considered a skyrmion in an in-plane magnetized material. Due to their missing rotational symmetry, bimerons are highly relevant for racetrack applications similar to antiskyrmions and intermediate skyrmion as presented in the last section.
Biskyrmions on the other hand, are formed by two partially overlapping skyrmions giving the new texture a topological charge of N Sk = ±2 25,26 . In order to be compatible, this time the two Bloch skyrmions need to have opposite helicities instead of vorticities.
When instead two skyrmions with opposite polarities are combined, the new object has a vanishing topological charge. This makes skyrmioniums 27,28 and antiferromagnetic skyrmions 29,30 the optimal candidates for spintronic applications. In the corresponding sections we discuss differences of these two objects, especially regarding the Hall effect of electrons.
Lastly, it is also worth mentioning that this list is nonexhaustive. For example, in synthetic antiferromagnets based on Co/Ru/Co films bimeronic topological defects (including both bi-vortices and bi-antivortices) living on domain walls were experimentally observed 105 ; also, a pair of helix edges is considered the fundamental topological excitation in a helical phase 106 .

Bimerons as in-plane skyrmions
The first object discussed in this category is the magnetic bimeron (Fig. 6a). It was first predicted in 2017 as a bimeron crystal 22 and shortly after also as an individual particle 23 .
As the name indicates, a bimeron consists of two meronsmore precisely of a meron and an antimeron with mutually reversed out-of-plane magnetizations (bottom panel in Fig. 6a), i. e. opposite polarities. This gives both subparticles the same topological charge of either +1/2 or −1/2 and the bimeron a topological charge of ±1 (middle panel in Fig. 6a). In this sense, it can also be considered a skyrmionic excitation in an in-plane magnet 23 . This becomes even more apparent from the following transformation: Starting from a conventional magnetic skyrmion, a rotation of all magnetic moments around a common in-plane axis by 90 • in magnetization space results in a bimeron texture. Such rotations leave the topological charge invariant. A bimeron can therefore ambivalently be understood as a meron-antimeron pair with opposite polarities or as a skyrmion in an in-plane magnet 23 .
Compared to the skyrmion, the bimeron is not characterized by an integer polarity, but the background magnetization can be rotated freely in the plane, mathematically speaking. This means that the class of bimerons has an additional continuous degree of freedom. A bimeron is characterized by two continuous angles instead of a discrete polarity and a continuous helicity. A possible characterization is that γ defines the angle of the connecting line between the two merons' centers with respect to the x axis and α defines the orientation of the net magnetization (parallel to the surrounding) with respect to the x axis, as indicated in Fig. 5e.
Up to now, only individual bimerons have been observed experimentally that were generated in a Py film via local vortex imprinting from a Co disk 24 (Fig. 7b). Furthermore, bubbles have been observed in in-plane magnets possibly pointing towards the existence of topologically non-trivial spin textures in in-plane magnetized samples 107 . Moreover, short skyrmion tubes were observed in MnSi whose cross-section along the tube direction has the profile of a bimeron 108 . However, the three-dimensional continuation of a bimeron would be a bimeron tube which is not realized there. The observed objects are more similar to skyrmions than to bimerons. The reason why only a handful of potential bimeron systems have been identified experimentally, may originate in the required DMI. One example generating one particular type of bimeron is given by 23 Since the DMI vectors are rotated around an in-plane axis compared to the interfacial DMI, the same must also apply to the texture. In this rather complicated setup, the DMI vectors along one direction are oriented in-plane, just like for interfacial and bulk DMI. However, along the other bond direction, the DMI vectors point out-of-plane, meaning that the inversion symmetry has to be broken in a particular way (an example is presented in Ref. 23). The required crystal structure lacks major symmetries, which is why most typical materials do not allow for a formation of bimerons. However, a similar DMI setup was recently calculated by first-principles calculations of Janus monolayers of the chromium trihalides Cr(I,Cl) 3 and Cr(I,Br) 3 for which bimerons have been stabilized in simulations 109 . Furthermore, it has been predicted that bimerons can be stabilized much more easily by frustrated exchange interactions. In contrast to the DMI, frustrated exchange interactions are achiral allowing to stabilize skyrmions, antiskyrmions, bimerons and even other objects likewise 19,22,23,110 . The only requirement for bimerons is that the external magnetic field has to be applied in the plane. Until now, topologically non-trivial spin textures with these properties stabilized by frustrated exchange interactions have never been observed. However, once this is the case, a rotation of the magnetic field will result in bimeron formation if the anisotropy is weak (or even better in the plane) and if the DMI is negligible. Without one of the two mechanisms bimerons can only exist as transition states that are unstable [111][112][113] . The observation of bimerons is highly desirable due to their special emergent electrodynamics. Similar to what has been presented for antiskyrmions and skyrmions with an arbitrary helicity (see last section), bimerons have a reduced symmetry compared to Bloch or Néel skyrmions. While the dissipative tensor and the gyroscopic vector are the same as for skyrmions and antiskyrmion, the global spin rotation manifests in the torque tensor, which is not antisymmetric anymore 23 . Following the same argumentation as for the antiskyrmion, there exists a specific current direction for which bimerons move parallel to the current (specific γ, α combinations), implying that specific bimerons can be used in racetracks as bits that move without skyrmion Hall effect 114 , as shown in Fig. 5c. In the STT scenario on the other hand, bimerons will always move under a skyrmion Hall effect, since the adiabatic torque transforms like the texture.
Furthermore, the bimeron is unique in terms of the topological Hall effect. Since a bimeron is an excitation in an in-plane magnet, the out-of-plane net magnetization vanishes and no anomalous Hall effect occurs. In a topological Hall measurement, the stabilizing field (typically along z for the skyrmion) is swept. For the bimeron this field has to point in-plane meaning that also no conventional Hall effect is measurable in the ρ xy tensor element. For this reason, bimerons allow for a pure measurement of the topological Hall effect upon variation of the in-plane field, establishing the optimal playground to investigate topological Hall physics 23 .
As a final remark, we would like to mention that sometimes the term bimeron is also used for an elongated skyrmion (or short helix segments), where both ends of the skyrmion have topological charges of ±1/2 and the center part is neutral [115][116][117][118] . This object is actually more similar to a skyrmion than to the bimeron in the sense of what has been discussed here.

Biskyrmions
The term biskyrmion is commonly used for two partially overlapping skyrmions (Fig. 6b) first observed in 2014 25 . In order to be geometrically compatible (the magnetization between the two skyrmions must be steady), both skyrmions need to posses a helicity difference of π, meaning that the in-plane magnetizations of both skyrmions are mutually reversed.
Following from our explanations about the possible stabilizing mechanisms of skyrmions in Sec. II B, biskyrmions can hardly be stable when the DMI is large. This chiral interaction would prefer one particular skyrmion helicity leading to a discontinuous magnetization when two skyrmions would partially overlap (in that case, even two attractive skyrmions would always remain separated by a ferromagnetic area, like in Ref. 119 ). On the contrary, the dipole-dipole interaction is achiral: it equally favors both types of Bloch skyrmions with helicities of γ = ±π/2, respectively 26,62 . When these two skyrmions are partially superimposed, the in-plane magnetizations between the skyrmions' centers point along the same direction, meaning that they are geometrically compatible. And indeed, it was found that already the short-range approximation of the dipole-dipole interaction leads to an attraction between two Bloch skyrmions with opposite helicities allowing to form magnetic biskyrmions 26 . Besides the stabilization of individual magnetic biskyrmions, biskyrmion lattices have been stabilized by this mechanism in micromagnetic simulations as well 26,120,121 .
Even before the theoretical prediction, biskyrmion lattices have been observed experimentally (Fig. 7c) in centrosymmetric materials 25,[122][123][124][125] like the layered manganite La 2−2x Sr 1+2x Mn 2 O 7 . In these materials the DMI is absent by symmetry, in agreement with the theoretically established stabilizing mechanism. Another possible origin for biskyrmion formation may be frustrated exchange interactions, as predicted in Ref. 126.
Regarding their emergent electrodynamics, biskyrmions behave similar to conventional skyrmions. Their topological charge of N Sk = ±2 (middle panel in Fig. 6b) leads to the emergence of a topological Hall effect and a skyrmion Hall effect in the STT scenario. The current-driven motion under SOT may turn out problematic, since the opposite helicities of the two Bloch skyrmions lead to sign reversed I tensors for the two subskyrmions. Consequently, they are driven along different directions eventually leading to the destruction of the biskyrmion. However, the lack of rotational symmetry -even in the m z component (bottom panel in Fig. 6b) -allows to utilize a rotation of biskyrmion for spintronic devices. As has been shown by micromagnetic simulations on a square lattice, the principle axis of a biskyrmion aligns with high-symmetry directions of the lattice (the second-nearest neighbor directions in this case), allowing in principle to switch between different configurations 26 .
As a closing remark, it is worth mentioning that the initial observations of biskyrmion crystals by LTEM imaging are under debate right now. In two recent publications 127,128 the authors showed that the unique Lorentz contrast can also arise due to a tilting effect. Tubes of topologically-trivial bubbles appear as skyrmion pairs with reversed in-plane magnetizations when viewed under an angle. Still, as explained, the stability of biskyrmions has recently been explained theoretically 26,120,121 , so the observed textures may indeed be biskyrmions. Alternative experimental techniques have to be utilized to dispel remaining doubts on the existence of magnetic biskyrmions.

Skyrmioniums
Up to now, we have discussed fundamental and composite magnetic objects with a finite topological charge. If however two skyrmions with opposite topological charges are combined to a new particle, the topological charge vanishes. If both objects would have the same polarity, this would require the combination of objects with opposite vorticities, e. g. a skyrmion and an antiskyrmion. Although these objects have been found to coexist in D 2d materials 101 , a combination of both has not yet been observed.
If however both objects have the same vorticities but opposite polarities, they can even be stabilized by bulk or interfacial DMI -the interaction that is typically dominating in skyrmion hosts. In fact, it has been found that under certain conditions an increase of the DMI constant leads to the formation of a skyrmion inside of another skyrmion with mutually reversed spins 27 (Fig. 6c). This object is called skyrmionium or 2π-skyrmion 130,134-142 since the azimuthal angle rotates by 2π (instead of π) when going from the object's center to the confinement. The outer skyrmion has the shape of a ring and is characterized by the opposite topological charge compared to the inner skyrmion (middle panel in Fig. 6c).
Skyrmioniums have recently been observed in a thin ferromagnetic film on top of a topological insulator 137 and have been created by laser pulses 136 . Furthermore, their generation has been predicted by the perpendicular injection of spin currents 27,139,143 or via alternating the out-of-plane orientation of an external magnetic field 138 .
The concept of the 2π-skyrmion can be extended to kπskyrmions, theoretically stable for certain parameters when the DMI is increased even further. Experimentally, such objects have been seen in nanodisks 130 . There, the texture is mainly stabilized by the confinement and the value of k is determined by the disk radius. Such objects are also called target skyrmions due to the peculiar profile of the magnetization's z component (Fig. 6c). In these disks the value of k is often not integer since the magnetic moments are not pointing perfectly out-of-plane at the confinement (Fig. 7d). Another texture related to kπ-skyrmions are so called skyrmion bags 144,145 where multiple skyrmions (instead of just one) are positioned next to each other inside of a skyrmion ring.
Due to the vanishing topological charge and the observation in experiments, magnetic skyrmioniums seem to be highly relevant for spintronic applications in the future. Their emergent electrodynamics will be discussed in the following. The trivial real-space topology on a global level leads to the absence of a skyrmion Hall effect 27 . For this reason, Néel-type skyrmioniums move without skyrmion Hall effect even in the SOT scenario. Still, the finite topological charges of the two subskyrmions become apparent: the skyrmionium deforms upon motion along the track. The inner skyrmion pushes towards the racetrack edge but is confined by the outer ring skyrmion that pushes transversally along the other direction. Both trans- verse motions compensate each other on a global level but still this effect constitutes a barrier for the maximum driving current 27,143 . When the current is too large, the transverse forces become too strong and the skyrmionium unzips 27 . Consequently, both skyrmionium parts annihilate. Summarizing, the absence of the skyrmion Hall effect solves an important problem: a skyrmionium moves parallel to an applied current directly in the middle of a racetrack avoiding pinning at the edges. Still, the skyrmion Hall effect of the subsystems is problematic as the current density cannot be increased much higher than that of a conventional skyrmion system. For the topological Hall effect one has to distinguish between global and local arguments as well. On a global level (measuring the Hall voltage of the charge accumulation on a mesoscopic length scale), the Hall effect is absent due to the vanishing topological charge. If, however, the detection is performed locally on the length scale of the skyrmionium size, one can find signatures of both subsystems. In Ref. 143 it has been shown that the topological Hall effect of skyrmioniums in a racetrack exhibits a unique triple-peak feature originating from the alternating topological charges of the ring-and center-skymions. This effect can therefore be used to detect skyrmioniums electrically, in a similar way as presented for skyrmions 17,18 .

Antiferromagnetic skyrmions
The dynamic properties and the stability of single antiferromagnetic skyrmions have first been predicted in antiferromagnets 29,146 and synthetic antiferromagnetic bilayers 147 . Shortly after, they have been extended to two-sublattice antiferromagnetic skyrmion crystals 148 . Like skyrmioniums, antiferromagnetic skyrmions can be understood as the combination of two skyrmions with mutually reversed spins. Therefore, they are characterized by a vanishing topological charge. However, in the present case, the subskyrmions are not spatially separated but intertwined. For this reason, the magnetization density vanishes locally and the Néel order parameter, the main order parameter for antiferromagnets, can be considered instead of the magnetization. Calculating the topological charge with this parameter gives ±1. Thus, the antiferromagnetic skyrmions are still skyrmions from the viewpoint of topology, but exhibit different dynamics compared to ferromagnetic skyrmions. These antiferromagnetic dynamics can also be described by the Thiele equation but by using the Néel order parameter 149 .
Due to the compensated topological charge for the magnetization, an antiferromagnetic skyrmion moves without skyrmion Hall effect 29,147 , similar to what has been discussed for the skyrmionium. However, the two subsystems are coupled much stronger and do not allow for a deformation due to a pairwise opposing transverse motion, as was the case for the skyrmionium. Typical for antiferromagnetic spin textures, the antiferromagnetic skyrmions can be propelled much faster by currents compared to conventional skyrmions. Velocities on the scale of kilometers per second have been simulated 29,147,150 . This makes antiferromagnetic skyrmions the ideal carriers of information for data storage devices. Additionally, it was theoretically shown that the diffusion constant 29 of antiferromagnetic skyrmions is high in systems with small damping unlike for their ferromagnetic counterparts, thus showing a potential in driving them with temperature gradients. Furthermore, they do not exhibit stray fields, which potentially allows for a denser stacking of quasi onedimensional racetracks upon building the three-dimensional storage device.
In terms of the required DMI, the stabilization of antiferromagnetic skyrmions is not problematic 29,151 . Two skyrmions with mutually reversed spins (i. e. two skyrmions with opposite polarity and a helicity difference of π) are energetically favored by the same type of DMI. Furthermore, both subskyrmions need to be coupled antiferromagnetically quite strongly, to accomplish the antiparallel alignment of the corresponding magnetic moments. And indeed, very recently, the bilayer-type antiferromagnetic skyrmions have been observed in synthetic antiferromagnets 30,131 (Fig. 7e) at room temperature. In Ref. 30 the small stray fields resulting from the bilayer setup have been detected using magnetic force microscopy (MFM). In Ref. 131 the authors explain a method to prepare synthetic antiferromagnets with a tunable net moment. While they can achieve a complete compensated system, they deliberately prepared also a system with a small net moment to be able to perform magneto-optical Kerr effect (MOKE) measurements.
For antiferromagnetic skyrmion-based logic 152 or racetrack 29 applications, a controlled generation process is needed. Conventional skyrmions have for example been generated by directed, deterministic approaches such as spin torques or magnetic fields (see reviews [43][44][45][46][47][48][49][50] ). These approaches are, however, difficult to utilize for antiferromagnetic skyrmions, since all vectorial quantities would have to act either only on one of the two subskyrmions, so that the other generates automatically due to the strong antiferromagnetic coupling, or they would have to act on both subskyrmions with an opposite sign. Both is hardly feasible, since magnetic fields for example cannot change their sign on the length scale of the lattice constant. Stochastic processes, like the generation of nano-objects at defects or from the confinement (as shown for conventional skyrmions 153,154 ), appear to be more advantageous in this regard. The stabilization of antiferromagnetic skyrmions becomes even more challenging when an antiferromagnetic skyrmion crystal shall be stabilized. By analogy with skyrmion crystals, a stabilizing magnetic field is inevitable. It has to be oriented along ±z for the two subsystems, respectively. This problem may be circumvented by growing a potential antiferromagnetic skyrmion host on top of a collinear antiferromagnet with the same crystal structure at the interface. By doing so, a staggered magnetic field is mimicked by the exchange interaction at the interface 148 .
For antiferromagnetic skyrmions in a single layer (not the case of a synthetic antiferromagnet bilayer) the detection is another problem. Both, the magnetization and topological charge density of the magnetization, are compensated globally and locally (middle and bottom panels in Fig. 6d). These antiferromagnetic skyrmions would therefore appear invisible for the real-space techniques, such as MFM or LTEM. Furthermore, anomalous and topological Hall signatures do not exist. Luckily, a different hallmark has been predicted: the topological spin Hall effect 148,155,156 . The resulting signal is the analogue of the conventional spin Hall effect, but originates in the non-collinearity of the spin texture. The topological spin Hall effect can most easily be comprehended if one assumes two electronically uncoupled subskyrmions (the results hold also for the coupled case): Due to the opposite spin alignment, the emergent fields of the two subskyrmions are oriented oppositely. This leads to a transverse deflection of the electrons in opposite directions. The two species of electrons align with their respective texture and can therefore be considered as 'spin up' and 'spin down' states, again due to the opposite spin alignment.
Summarizing, one can have a positive feeling about the utilization of antiferromagnetic skyrmions in spintronic devices in the future. Synthetic antiferromagnetic skyrmions have been observed and, just recently, the current-driven motion of synthetic antiferromagnetic skyrmions has been realized 131 . Furthermore, single layer antiferromagnetic skyrmions have been predicted. The topological spin Hall effect may play an essential role for observing these objects despite completely compensated magnetizations, stray fields and topological charge densities of the magnetization. Moreover, even in antiferromagnetic insulators the skyrmions were predicted and can potentially be moved by an electrically created anisotropy gradient 157 or by thermal gradients.
Signatures of the favorable emergent electrodynamics of antiferromagnetic skyrmions have also been seen for a similar object: the ferrimagnetic skyrmion 87,88 (Fig. 2g). It consists of two coupled subskyrmions with mutually reversed spins, similar to the antiferromagnetic skyrmion. However, the magnetic moments have different magnitudes on the two sublattices leading to an uncompensated magnetization, which allowed for the detection of ferrimagnetic skyrmions in GdFeCo films by X-ray imaging 87 (Fig. 7f). When these objects are driven by spin currents, there exists a critical temperature at which the skyrmion Hall effect is absent 88 . At this temperature, the angular momentum is compensated, even though the magnetization is not, due to different gyromagnetic ratios for the magnetic moments in the different sublattices 88 . Experimentally, this complete compensation of the skyrmion Hall effect still lacks observation, but a reduced skyrmion Hall angle of θ Sk = 20 • has been observed at room temperature 87 . Similar experiments will certainly be performed for antiferromagnetic skyrmions in the future. Moreover, it was recently experimentally observed that in ferrimagnetic insulators near the compensation temperature domain walls driven by SOT can move at speeds reaching 6 km/s 158 , thus hinting that the same speeds can be achieved as well by ferrimagnetic skyrmions in insulators in the future.
As a closing remark, we would like to mention that the idea to combine skyrmions on different sublattices has been generalized in several works. Three skyrmion crystals can for example be intertwined as stabilized by Monte Carlo simulations 159,160 . However, these objects do not exhibit the advantages of antiferromagnetic skyrmions because the topological charge for the magnetization is finite in this case.

C. Three-dimensional objects
In the last main section we present three-dimensional solitons and extensions of skyrmions. We begin with the trivial case: two-dimensional skyrmions, antiskyrmions, bimerons and other objects are extended as tubes along the third dimension. Still, these tubes can show interesting interactions and can even have a varying helicity along the tube 164 . We focus here on two more fundamental variations of such tubes.
First, we discuss chiral bobbers 31,32 (Fig. 2j). These are skyrmion tubes that end in a Bloch point (the hedgehog in Fig. 3b, also shown in an experimental measurement in Fig. 7g). Interestingly, the chiral bobbers can appear in the same sample as skyrmion tubes. Both objects can be well distinguished experimentally, allowing to think of an improved racetrack storage device with both objects as the bits of information 32 (Fig. 8b). On the other hand, skyrmion tubes can also bend and form closed loops called hopfions 33 (Fig. 2l). These objects are unique from a fundamental point of view since they are characterized by a second topological invariant, the hopf number.

Chiral bobbers
As explained, the trivial continuation of skyrmions along the third dimension are skyrmion tubes. In MnSi for example, they penetrate the whole sample. However, materials are not always perfect, so it can happen that two skyrmion tubes merge to one [165][166][167] . This decrease of topological charge along a tube's cross section is caused by a Bloch point (Fig. 3b) with a singularity in its center. Similarly, a single skyrmion tube can end in or begin from a Bloch point [165][166][167][168] . In this case, the resulting magnetic object is called chiral bobber 31,32,169 (Fig. 2j). It has been predicted 31 and observed experimentally by LTEM measurements 32 in B20 materials. Fig. 7g shows a schematic representation of a pair of two chiral bobbers, as well as an experimental measurement of a Bloch point in Ni 80 Fe 20 .
One unique characteristics of a chiral bobber is that the location of the Bloch point, with respect to the sample's surface, is fixed 31 . It depends only on the ratio of exchange to DMI and is for example independent of the sample thickness 31 . Like a fishing bobber, the chiral bobber is 'swimming' always close to the confinement of the sample. For this reason, the chiral bobber must be seen as a distinct magnetic soliton despite its phenomenology as a discontinued skyrmion tube. Furthermore, this means that the volume which is occupied by a chiral bobber is fixed with respect to thickness variations, and so is the soliton's energy. For a skyrmion tube on the other hand, the occupied volume and the energy scale linearly with the thickness. For this reason, skyrmions in these materials become unstable above a critical sample thickness which is where chiral bobbers are energetically preferred 31 . It has been experimentally confirmed that chiral bobbers do not exist below a critical thickness 32 .
In order to stabilize chiral bobbers, a specific experimental procedure had to be applied in Ref. 32: The magnetic field was rotated approximately 10 • out of the normal direction and then the sample was magnetized and demagnetized a few times. After the nucleation of the chiral bobbers and skyrmion tubes, the field was rotated back to the conventional out-of-plane direction. A possible nucleation mechanism is the formation of chiral bobbers from edge dislocations of the spin spiral phase, similar to the formation of skyrmion tubes. Skyrmion tubes and chiral bobbers have been distinguished by the LTEM phase shift which is proportional to the occupied volume of the magnetic object, i. e., chiral bobbers exhibit a weaker intensity than skyrmion tubes 32 .
Even though a chiral bobber is not topologically protected (mathematically speaking the texture can be pushed out of the magnet via the confinement), it possesses a similar energy barrier compared to the skyrmion tube 31 . This allows for a coexistence of skyrmion tubes and chiral bobbers allowing to think of an alternative racetrack storage device where the '1' and '0' bits are encoded by the presence of a skyrmion tube or a chiral bobber, respectively (Fig. 8b). This solves the problem that the bits do not have to be located at precise positions. Instead, information can be encoded also as an irregular sequence 32 .
The two different objects can be read via their net magnetization or via their Hall signal. As has been shown numerically in Ref. 170, chiral bobbers exhibit an increasing Hall conductivity upon increasing the sample thickness, while this signal remains constant for skyrmion tubes.
The lack of topological protection may become problematic for the current-driven motion of chiral bobbers. As micromagnetic simulations of the STT scenario revealed, chiral bobbers do not only move along the sample but also the Bloch point propagates towards the edge along the tube direction until the chiral bobber has disappeared 167 . The SOT scenario remains to be investigated.
As a final remark, besides their presence in chiral bobbers, Bloch points can form three-dimensional Bloch anti-Bloch crystals with their counterparts [171][172][173] (Fig. 2k). These crystals can be seen as the three-dimensional analogue of a skyrmion lattice. Recently, it has even been shown that in MnSi 1−x Ge x a topological transition between the skyrmion lattice and the this Bloch anti-Bloch lattice occurs 174 . (c) Magnetic hopfions as innately three-dimensional solitons can also be used as carriers of information in racetrack devices. Since they are anisotropic, in principle, they can be tilted 162,163 and potentially be switched to encode data.

Hopfions
A different way to transform a skyrmion tube into an innately three-dimensional object is to form a torus. These objects are known as hopfions (as in Fig. 2l) and have been introduced back in 1931 175 . They are topologically characterized by the Hopf invariant 176,177 where B em is the emergent field and A is the corresponding vector potential fulfilling the condition ∇ × A = B em . In the simplest case, the cross-section texture is a bimeron with a topological charge of ±1 and the magnetization rotates once going around a circle of fixed radius around the hopfions center (as in Fig. 2l). This yields the Hopf number of ±1 as a product of the cross-section topological charge and the number of magnetization windings around the torus. From a mathematical point of view, several types of hopfions can be constructed. For example, the cross section can have higher topological charges or the magnetization can rotate more than once going around a circle. Also, the Hopf number is increased by more complicated configurations, e. g., by linking multiple hopfions 178 .
Hopfions have first been predicted in 1975 in a Skyrme-Faddeev model 179 . Over the following years, hopfions have been found in hydrodynamics 180 , electrodynamics 181 and other fields of physics, and recently they have been stabilized in micromagnetic simulations 33,178,[182][183][184] . The objects can be stabilized in chiral nanodisks due to the confinement 33 and have even been stabilized without magnetic fields 178 .
Their emergent electrodynamics are promising due to a globally compensated emergent field that is however finite locally (largest inside of the tube that forms the torus). This toroidal emergent field leads to the deflection of current electrons perpendicular to the hopfion plane: One half of the object leads to a positive Hall resistance while the other half gives a negative signal. While both signals cancel on a global level, a local measurement of a hopfion in a potential racetrack device yields a unique signature as simulated in Ref. 162 . Furthermore, the topological Hall signal depends on the orientation of the hopfion bringing about the possibility to switch electrical signals (Fig. 8c): the Hall voltage always emerges perpendicularly to the Hopfion plane. Therefore, it is expected to occur in a pure manner since it is characterized by a different Hall resistance tensor element than the anomalous and ordinary Hall effects 162 .
Likewise, when driven by torques, the hopfions move along the current direction and do not experience a skyrmion Hall effect 185 . The locally finite field leads however to a tilting of the hopfion plane, as was first predicted in Ref. 162 and later confirmed by micromagnetic simulations 163 . On the one hand, if this effect is kept small, hopfions are promising for racetrack storage applications. On the other hand, the tilting mechanism may even be utilized for other spintronic applications because it allows to switch between different hopfion configurations.

IV. PERSPECTIVES AND CONCLUSION
Summarizing this review, we have discussed the stability and the emergent electrodynamics of skyrmions and related alternative magnetic quasiparticles. We have classified the manifold of particles (Fig. 1) in fundamental excitations (topologically trivial and non-trivial), variations of these excitations and continuations. Out of all fundamental excitations, skyrmions with an arbitrary helicity 19,20 and antiskyrmions 21 have been discussed as technologically relevant objects. The variations comprise topological excitations in a different magnetic background and the combination of multiple subparticles. Here we have discussed in detail bimerons [22][23][24] , biskyrmions 25,26 , skyrmioniums 27,28 , and antiferromagnetic skyrmions 29,30 . 'Continuation' means that all of these objects can be arranged in two dimensions (periodically and non-periodically) and can be continued along the third dimension. For the innately three-dimensional objects, we have laid focus on chiral bobbers 31,32 and hopfions 33 .
Out of the presented objects (Fig. 2) antiferromagnetic skyrmion are often considered the optimal bits for spintronic applications. Their compensated magnetic texture allows to drive them by currents at enormously high velocities of up to several kilometers per second and the absence of the skyrmion Hall effect eradicates the problem of pinning of the bits at the edges. Furthermore, their local compensation of magnetization renders stray fields small allowing for a dense stacking of racetracks in a three-dimensional buildup.
Moreover, antiskyrmions seem to be highly promising. They are easy to detect and most experimental techniques working for the conventional skyrmions can be carried over. These objects can move parallel to an applied current allowing to consider antiskyrmions as the carriers of information in racetrack devices. One further problem that occurs whenever the '0' and '1' bits of information are constituted by the presence or absence of a magnetic quasiparticle is that the positions of the bits cannot be precisely fixed. On the long time scale the information may become falsified for example due to the repulsive interaction between the objects. A solution may be delivered by using two distinct objects to constitute the bits because then irregular sequences are not problematic. A promising example is using skyrmions and antiskyrmions in systems with D 2d symmetry 97,98,100-102 (Fig. 8a). Also it is conceivable to use a three-dimensional approach with chiral bobbers and skyrmion tubes as bits of information 32 (Fig. 8b).
Furthermore, it seems worthwhile to look into other applications of topologically non-trivial quasiparticles. Conventional skyrmions have been considered for utility in logic de-vices 112 , transistors 186 , magnetic tunel junctions [187][188][189] , nanooscillators 190 , as microwave devices 191 or magnonic devices 192 , neuromorphic applications 193,194 , as well as reservoir computing 195,196 , stochastic computing 197 and quantum computing 198,199 etc. Future will tell if the alternative quasiparticles are favorable also in these regards.
As a closing remark, we would like to mention that the presented characterization scheme is not complete. Multiple skyrmions can in principle be combined to form new objects; e. g., a quadskyrmion instead of a biskyrmion. Furthermore, one can combine multiple objects and change the magnetic background; antiferromagnetic versions of skyrmioniums 200,201 , skyrmion bags 145 and bimerons [202][203][204] have already been predicted. Since antiferromagnetic skyrmions have been observed just recently, the desire arises to find also antiferromagnetic versions of other nano-objects.