Field and anisotropy driven transformations of spin spirals in cubic skyrmion hosts

We discuss distinctive features of spiral states in bulk chiral magnets such as MnSi and Cu$_2$OSeO$_3$ that stem from the effect of the cubic magneto-crystalline anisotropy. First of all, at both the helical-to-conical and the conical-to-ferromagnetic transitions, taking place at H$_{c1}$ and H$_{c2}$, respectively, the cubic anisotropy leads to reversible or irreversible jump-like reorientations of the spiral wavevectors. The subtle interplay between the easy and hard anisotropy axes gives rise to a phase transition between elliptically distorted conical states almost without any detectable change in the period. We show that the competition between on-site cubic and exchange anisotropy terms can also lead to oblique spiral states. Our work gives clear directions for further experimental studies to reveal theoretically predicted spiral states in cubic helimagnets beyond the aforementioned well-established states thus, can help to understand the magnetic phase diagram of these archetypal skyrmion hosts. In addition, we show that properties of isolated skyrmions such as inter-skyrmion attraction, orientation and/or nucleation are also rooted in the properties of host spirals states, in which skyrmions are stabilized.


I. INTRODUCTION
Following the pioneering theoretical works of Dzyaloshinskii 1,2 and the experimental findings in the 1980's, where the magnetic phase diagram of MnSi has been studied by different techniques and the results were summarized by Kadowaki 3 , chiral helimagnetism 4 experiences an increasing interest over the last years. This is mainly due to the the discovery of skyrmions 5-7 -two-dimensional localized whirls of the magnetization, which may be used in future information storage and data processing devices. Presently the field of skyrmionics encompasses activities to create real prototypes of skyrmion-based devices in nanosystems such as nanolayers 6,7 , nanowires 8 or nanodots 9,10 , exploiting the small size, topological protection and the ease with which skyrmions can be manipulated by electric currents. This swerve from bulk cubic helimagnets to nanosystems, witnessed over the last years, was stipulated by the reduced skyrmion stability in the former case, where skyrmions appear forming hexagonal lattices in a small pocket of the temperature-magnetic field phase diagram just below the transition temperature T C , the so-called A-phase.
Recent experimental findings, however, revealed a vast area of low-temperature skyrmion stability in the bulk insulating cubic helimagnet Cu 2 OSeO 3 , when the magnetic field is applied along the easy 001 crystallographic axis 11,13 . Remarkably, the theoretical explanation of this phenomenon is mainly based on the effect of single-ion cubic anisotropy on one-dimensional spiral states 14,15 . In fact, the ideal magnetization rotation in the conical phase, the main competitor of skyrmions, can be impaired by the easy and hard anisotropy axes for specific directions of the magnetic field. Through this mechanism, skyrmions, which are more resilient to anisotropyinduced deformations, due to their two-dimensional nature, gain stability over extended regions of the phase diagram. In general, it was shown that the cubic anisotropy has a direct impact on the spiral states and thus indirectly is at the origin of many remarkable phenomena for skyrmions.
The experimental findings provide hints that the cubic anisotropy in Cu 2 OSeO 3 initiates the nucleation of thermodynamically stable skyrmions via two different mechanisms 14 : either via rupture formation of metastable spiral states with wave vectors perpendicular to the field, or via torons 16 . The latter are localized particles consisting of two Bloch points at finite distance and a convexshaped skyrmion stretching between them. In Cu 2 OSeO 3 they can be generated by the inhomogeneous magnetic environment provided by the smooth transition between tilted spiral domains, which appear when the magnetic field is applied along the easy 001 crystallographic axis 11 . This tilted spiral phase constitutes a major deviation from the generic phase diagram of cubic chiral magnets 17,18 and is stabilized by a competition between on-site cubic and exchange anisotropies. This new phase may also give rise to new topological magnetic defects, such as isolated skyrmions, with interesting static and dynamic properties. Moreover, versatile skyrmion phases have been recently identified both at and above room temperature in a family of cubic chiral magnets: β-Mntype Co-Zn-Mn alloys with a different chiral space group from that of B20 compounds 12 .
Here we undertake a systematic study of the effects imposed by cubic anisotropy on one-dimensional spiral states. First, we show that cubic anisotropy accounts for the reorientation processes of spiral states that occur not only at the low critical field H c1 , but also near the transition into the saturated state H c2 , an effect that might at a first sight seem counter-intuitive. Furthermore, we show that the existence of easy and hard anisotropy axes distorts elliptically the magnetization rotation and this effect may stabilise new spiral phases with no changes of the spiral orientation or period, but with detectable transitions between them. Our theoretical approach explains the behavior of the bulk helimagnets Cu 2 OSeO 3 , or MnSi, and provides a mechanism through which it is possible to generate thermodynamically stable skyrmionic spin textures over large fractions of their phase diagrams.

A. The general micromagnetic energy functional
Within the phenomenological theory introduced by Dzyaloshinskii 1 the magnetic energy density of a noncentrosymmetric ferromagnet with spatially dependent magnetization M can be written as +W a (m) (1) where A and D are coefficients of exchange and Dzyaloshinskii-Moriya interactions (DMI); H is an applied magnetic field; x i are the Cartesian components of the spatial variable. w D is composed of Lifshitz invariants L (k) i,j = M i ∂M j /∂x k − M j ∂M i /∂x k that are energy terms involving first derivatives of the magnetization with respect to the spatial coordinates. In the following, all calculations will be done for cubic helimagnets with w D = m · ∇ × m although the results may be applied for magnets with other symmetry classes 19 including different combinations of Lifshitz invariants. W 0 (M) includes only basic interactions essential to stabilize skyrmion and helical states and specifies their most general features attributed to all chiral ferromagnets. Next we discuss the physics governed by these dominant interactions. The effects of anisotropy W a (m), usually representing a smaller energy scale, will be discussed later.
B. One-dimensional spiral modulations within the isotropic theory In chiral magnets the Dzyaloshinskii-Moriya interactions w D play a crucial role in destabilizing the homo-  1. (color online) Schematic representations of the magnetization rotation in the conical phase in the presence of cubic anisotropy with kc > 0 (a) and kc < 0 (b) shown together with the magnetization traces in a space (c). Depending on the orientation of the cone propagation direction (z) with respect to the easy (green arrows) and hard (red arrows) anisotropy axes and the applied magnetic field (h||[001]) the energy density of the conical pahse can be increased or reduced. This phase undergoes a first-order phase transition into the ferromagnetic state at the critical field hc2 for kc > 0 and h||[001]. This is seen at the energy density difference between the conical and ferromagnetic phases, depicted in (d), and the magnetization curves shown in (e). The jumps in (e) occur at the field value where the energy density difference in (d) becomes zero, e.g., in the point B for kc = 0.1. For kc < 0 and h||[001] (f),(g) the mutual arrangement of easy and hard anisotropy axes, shown in (b) underlies a second-order phase transition.
geneous ferromagnetic arrangement and twisting it into a helix. At zero magnetic field the helices are singleharmonic modes forming the global minimum of the functional W 0 (M) 1 , M = M s [n 1 cos (q · r) + n 2 sin (q · r)]. n 1 , n 2 are the unit vectors in the plane of the magnetization rotation orthogonal to the wave vector q = 1/(2L D ) (n 1 ⊥ n 2 ; n 1 ⊥ q; n 2 ⊥ q). L D is proportional to the ratio of the counter-acting exchange and Dzyaloshinskii constants and introduces a fundamental length characterizing the periodicity of chiral modulations in chiral magnets, L D = A/D.
The helical modulations have a fixed rotation sense determined by the sign of Dzyaloshinskii-Moriya constant D and are continuously degenerate with respect to propagation directions of the helical modulations in space. An applied magnetic field lifts this degeneracy and stabilizes two types of one-dimensional modulations: cones and helicoids with propagation directions parallel and perpendicular to the applied magnetic field respectively.
For helicoids, analytical solutions for the polar an- [010] [001] [100] [010] [001] [100] [010] [001] [100] [010] [   . In a sufficiently high magnetic field H H the helicoid infinitely expands and transforms into a system of isolated non-interacting 2π-domain walls (kinks) separating domains with the magnetization along the applied field 1,20 . The dimensionless value of this critical field is The azimuthal angle ψ, on the contrary, is fixed by the different forms of the Lifshitz invariants. In particular ψ = π/2 for Bloch helicoids in cubic helimagnets and ψ = 0 for Neel helicoids or cycloids in polar skyrmion hosts with C nv symmetry such as the lacunar spinels GaV 4 S 8 21,22 , GaV4Se8 23 , GaMo4S8 24 or VOSe2O5 25 . In this latter class of materials, the polar axial symmetry gives rise to Lifshitz invariants different from that of the cubic helimagnets (Eq. (1)) and the modulation vectors are restricted to the plane perpendicular to the highsymmetry axis.
The conical state combines the properties of the ferromagnetic and the helical states as a compromise between the Zeeman and DM energies. The conical spirals retain their single-harmonic character with ψ = z/2L (1) the conical phase is the global minimum over the whole range of applied fields where modulated states exist (0 < h < h d ), whereas helicoids and skyrmion lattices can exist only as metastable states.

III. SPIRAL STATES IN HELIMAGNETS WITH CUBIC ANISOTROPY
In the following we discusse the influence of the cubic anisotropy on the one-dimensional modulated states. W a (m) in Eq. (1) can be written as: where K c is the coefficient of the cubic magnetic anisotropy and the vector field of the magnetization is given by m chiral magnets. Thus, it is essential to recognize that these weaker interactions play a crucial role in determining the stability limits of the different modulated states on the corresponding phase diagrams.
In the case of single-ion cubic anisotropy, the energy of the conical phase can be either increased or decreased depending on the mutual arrangement of the easy anisotropy axes and the propagation direction of the cones. A good starting point is a simple analysis of the homogeneous state in a system with the easy cubic anisotropy axis parallel to the applied magnetic field as described in Ref. 26 and shown in Fig. 2. In this case, the manifold of energy extrema provides a hint on the distortions of the conical state. The rotation of the magnetization becomes in tune with a complex landscape of the energy term associated, with the cubic anisotropy ( Fig. 1). The direction of the wave vector q itself is also specified by the competing Zeeman and cubic anisotropy energies. These basic concepts are clearly manifested, in particular, in the critical fields at the standard phase diagrams of bulk cubic helimagnets such as MnSi and Cu 2 OSeO 3 : the critical fields H c1 , where oblique spirals align along the field, and H c2 , where the conical phase undergoes a phase transition to the ferromagnetic state. A.
The conical-to-ferromagnetic phase transition at the critical field Hc2.
In this section we present solutions for spiral states, which we obtained by exact numerical minimization of the energy functional, given by Eq. (1), including the cubic anisotropy K c (2). For this purpose, we consider the dimensionless anisotropy constant, k c = K c A/D 2 .
a. First-or second-order phase transitions between the conical and the ferromagnetic state.
In the present section, we consider the solutions for the conical state in the field applied along [001] or [111] for both signs of k c . As it will become evident later, firstorder phase transitions occur when the magnetic field is applied along the easy anisotropy axes. In this case, on its way to saturation, the magnetization is forced to embrace hard anisotropy axes, as shown in Fig. 1 (a) and (c). This is an energetically unfavourable configuration, that forces a discontinuous transition of the magnetisation towards the magnetic field. On the contrary, if the magnetic field points towards a hard anisotropy axis, the minima of the cubic anisotropy gradually align along the field and underlie the continuous second-order phase transition between the conical and the ferromagnetic state, as illustrated by Fig. 1 (b) and (c).
In the following, we carry out a comprehensive analysis of the magnetization rotation in the conical phase by considering the surface energy plots shown in first row of Fig. 2, and the three-dimensional polar plots of the cubic anisotropy energy depicted in the second row of Figs. 2. The different topologies of these surface plots underly the particular types of the phase transitions.
For k c > 0 and h = 0 ( Fig. 2 (a)) the equilibrium states of the homogeneous state correspond to the easy axes of cubic anisotropy oriented along the 001 crystallographic directions. Maxima of the cubic anisotropy energy correspond to the hard 111 directions. Let us consider the helical mono-domain state where the mag- [010] [001] [111] [111] [111] h=0.20  netization for h = 0 rotates in the plane (001). While rotating, the magnetization leaves one energy minimum corresponding to 001 directions and, rotating through the saddle point between hard axes 111 , gets into another energy minimum along a 001 direction. The trace of the magnetization in the conical phase is shown by the yellow line in Fig. 2 . When applying a magnetic field along the [001] axis, the magnetization in the conical phase is forced to rotate in the vicinity of hard 111 axes, as shown in Fig. 2 (b) for h = 0.2. This underlies the subsequent first-order phase into the ferromagnetic state, which becomes evident, by e.g. the magnetization jumps in the magnetization curves shown in Fig. 1 (e). In Fig. 1 (d) we plot the difference between the energy densities of the conical spiral and the ferromagnetic states. In this plot the point B for k c = 0.1 stands for the first-order phase transition that takes place when the energy difference between the two phases vanishes. Logically the same situation should also occur for h||[111] and k c < 0, where three hard anisotropy axes 001 impose distortions to the conical phase and underlie the magnetization jump into the ferromagnetic state.
For k c < 0 and h = 0 (Fig. 2 (c), (d)), the equilibrium states of the cubic anisotropy energy correspond to the 111 -type crystallographic directions. Thus, the maxima of the cubic anisotropy are the hard 001 -type directions, and if a magnetic field is applied along a 001 axis, the angle between the easy directions and the field is 70.6 • . In this case, the magnetization in the conical phase rotates so that it sweeps the easy directions 111 ( Fig. 1 (b)). Even at zero field the magnetic structure is conical, thus, it has a non-zero uniform magnetization (see magnetization curves in Fig. 1 (g)). However, this low-field conical phase is in reality a metastable solution overshadowed by the oblique spiral states with q-vectors along the easy 111 -type directions. In a finite magnetic field the global minima of the cubic anisotropy gradually approach the field direction and thus display a secondorder phase transition to the ferromagnetic state ( Fig.  2 (d)). The same principle is valid also for h||[111] and k c > 0, where the angle between the field direction and the easy 001 axes is 54.7 • , whereas for the hard 111 axes the angle is 70.6 • . In this case the topology of the corresponding polar plots would allow the gradual alignment of the magnetization along the hard 111 direction with increasing magnetic field.
b. Conical domains in the ferromagnetic state.
As a consequence of the first-order phase transition that occurs for either k c > 0 and h||  in the field polarised state adopting the configuration illustrated in Fig. 3. Such domains exhibit both types of the rotational sense at their boundaries. At the left side of the depicted conical droplet ( Fig. 3 (a)-(c)) the magnetization in the conical phase adjusts to the homogeneous state with the "correct" twist supported by DMI. This leads to energy gain as shown by the negative energy density difference in the xy plane, i.e., for a cross-section perpendicular to the magnetic field h and the conical qvector, shown in Fig. 3 (d)). At the right side of the conical droplet, the magnetization acquires the "wrong" in-plane twist, which results in an energetic "penalty", as shown in Fig. 3 (d).
A similar behaviour is also found along the z direction, which is parallel to h and q. As illustrated in Fig. 3 (e), (f) the energy density of the conical droplet exhibits alternating negative and positive contributions corresponding to "correct" and "wrong" senses of the magnetization rotation at the boundary to the field polarised state. When the magnetic field is increased from the conical to the ferromagnetic phase, the conical droplet may either continuously expand for h < h B , as seen in Fig. 1 (d) or gradually shrink for h > h B , a feature that underlies the kinetic character of such first-order phase transition. Remarkably, when the magnetic field is decreased from the field polarized to the conical state, the energy density shown in Fig. 3 (f) leads to an "in-phase" merge of neighboring conical droplets that initially may be formed with some phase shift with respect to each other. Then, the region with the positive energy density at the boundary of one conical droplet is annihilated by the corresponding part with the negative energy of another droplet, a feature that underlies in-phase "zipping" and "unzipping" of conical droplets.

B. The helical-to-conical phase transition at the critical field Hc1
In this section we present qualitative results on the spiral reorientation processes that occur for k c < 0 and for different directions of the applied magnetic field (Fig.  2 (e)). Numerically exact solutions for the spiral states are obtained by direct minimization of the energy functional, given by Eq. (1), including K c in Eq. (2), and are discussed elsewhere 27 . Since k c > 0 favours easy axes of 111 type, the ground state at zero field consists of domains of spiral states with the corresponding directions of the q-vectors. Fig. 2 (d) shows a schematic layout of the q-vectors in stable and metastable spiral states with respect to the field and the crystallographic directions.
For H [001] (first panel in Fig. 2 (e)), all four equivalent spiral states with q 111 are at an angle of 35.2 • with respect to the field. As also discussed in Ref. 27, with increasing magnetic field, these wave vectors first rotate gradually and then jump by a first-order phase transition towards the field direction.
For H [111] (second panel in Fig. 2 (e)), only the spiral that is co-aligned with the magnetic field is stable, whereas the spirals along the other equivalent wave vectors q 111 become metastable. With decreasing magnetic field only one spiral (or conical) domain with the q-vector along the field persists and at zero field the other spiral domains are not recovered.
For H [110] (third panel in Fig. 2 (e)), there are two sets of 111 directions composing smaller and larger angles with respect to the field. Thus the field-driven transition into the conical phase is a two-step process: the energetically unfavorable spiral states with the angle 90 • reorient first, whereas the stable spirals with the angle 27.35 • gradually rotate and eventually flip along the field. With decreasing magnetic field the transitions at both critical fields should occur in a reproducible manner, possibly with some hysteresis.   For H|| [110], independently on the sign of the cubic anisotropy constant k c , the q-vector of the conical state is surrounded by two hard and two easy anisotropy axes 111 and 100 . This rosette of anisotropy axes leads to the characteristic elliptical distortion of the conical state, which moreover changes when the magnetization rotation occurs in the exterior or interior of the rosette. In particular for k c < 0 and low field values, the trajectory of the rotating magnetization is elongated in-plane along the hard 100 axes (Fig. 4 (a), (b)). With increasing magnetic field the magnetization approaches the rosette, penetrates the rosette's interior and changes its elliptical distortion being now elongated along the easy 111 axes (Fig. 4 (c), (d)).
For k c > 0, the same process takes place, but with inverse switch of the elliptical distortion (Fig. 5). The ratios of the ellipse major and minor axes for both signs of the anisotropy constant are plotted as a function of the magnetic field for different values of k c in Fig. 4 (e) and Fig. 5 (e). From these curves as well as from the magnetization curves (Fig. 4 (f)), it becomes clear that the phase transition between these two conical phases is more abrupt for the negative values of k c . This might be explained by the different angles the easy and hard anisotropy axes compose with respect to the field -35.3 • for 111 and 45 • for 100 . Indeed, when k c < 0, the rotating magnetization tries to avoid two hard axes [100] and [010] and this results in the jump of the magnetization from the rosette's exterior to its interior. For k c > 0 [100] and [010] axes are easy anisotropy directions. Thus, although the sense of ellipticity changes, no jump of the magnetization is observed. An important point is that the phase transition between the two different conical states leaves the periodicity of the spiral state almost unaffected. In fact, the period changes just by a few percent for low values of the cubic anisotropy (less than 4% in Fig. 4 (g) for k c = −0.2), and this change is hardly detectable by e.g. neutron scattering experiments. The field dependence of dm h /dh (Figs. 4 inset of (f) and Fig. 5 (f)), on the contrary, provides a clear hint for this transition as it shows a λ-shaped spike only for k c < 0.

D.
Spiral flip near the saturation field Hc2.
In the following we will show that spiral reorientation processes akin to those at H c1 may intervene also near the saturation field H c2 . Jumps of the spiral q-vectors are less expected although they may be realized for relatively moderate values of the cubic anisotropy constant (Fig. 6). Indeed, the conical phase co-aligned with the field is considered to be the most favorable state that benefits from the Zeeman interaction. In the following, we show theoretically that a tilted spiral state may originate from the interplay between Zeeman and cubic anisotropy interactions, which is generic to chiral magnets, and thus may be important for explaining the experimental data of specific cubic helimagnets. The spiral jumps are also sensitive to the direction of the applied magnetic field and the value of k c , since relatively large values of k c are needed to implement particular reorientations.
As an illustrative example, we consider the case with H||[111] and easy 111 axes of the cubic anisotropy (Fig.  6). The q-vector undergoes a first-order transition from the conical state ( Fig. 6 (a),(b)) into a tilted spiral state with almost constant tilt angle for a range of magnetic field values (Fig. 6 (c),(d)), with both the tilt angle and the field range dependent on k c . The sketches in Fig. 6 (c)-(d) are instructive to deduce the nature of this tilted spiral state. Indeed, in an increasing magnetic field, the magnetization in the conical state sweeps only two 111 anisotropy axes ( Fig. 6 (a), (b)). In tilted state, however, by sacrificing the Zeeman energy, the magnetization may try to embrace all four 111 axes (Fig. 6 (c), (d)). This leads to the maximum tilt angle 70.6 • , reached for the higher k c values (Fig. 6 (e)). This phase transition is accompanied by characteristic jumps in the magnetization curve. An interesting point is that such a reorientation phase transition requires a threshold value of k c . For k c = −0.1 not even a hint of the metastable tilted spiral becomes apparent in the angle-dependent energy density (like that in Fig. 6 (f)).
Conceptually the same reorientation process may occur also for H|| [110].

IV. OBLIQUE SPIRAL STATES DUE TO THE COMPETING ANISOTROPY INTERACTIONS
Tilted spiral states also may originate from the interplay between competing anisotropic spin interactions, which is generic to chiral magnets. In fact, such an interplay gives a rather good control over oblique states with smooth variation of the tilt angle. As an instructive example, we consider spiral reorientation processes due to the competition between cubic anisotropy k c with the easy axes 100 and exchange anisotropy k ea < 0, i.e. with easy 111 axes. This exchange anisotropy is defined as w ea = k ea ∂mi ∂xi 2 .
For the present setup of easy anisotropy axes, the tilted spiral state is found only for h|| [001]. Above a critical field value, the conical spiral ( Fig. 1 (a), Fig. 2 (b)) begins to tilt towards one of the four body diagonals, the 111 directions, as shown in Fig. 7 (a). Such a tilted spiral state appears when |k ea | exceeds a critical value, which is slightly lower than 0.05 for k c = 0.1. Fig. 7 (b) shows the dependence of the spiral energy on the tilt angle φ with q varying in the (110) plane, for h = 0.2 and h = 0.25. As h increases, φ grows, reaches its maximal value and then decreases back to zero, corresponding to a return into the conical spiral state (Fig. 7 (c)). The maximal tilt angle, φ max , which depends on the ratio of the competing fourth-order and exchange anisotropies, is plotted in Fig. 7 (d). As the exchange anisotropy increases above the critical value, |k ea | > 0.14, the state with q 111 is stabilised even at zero magnetic field.
This effect can be understood as follows: the competing anisotropies give rise to new minima of the conical spiral energy in the q-space. If only the fourth-order anisotropy k c is taken into account, the global energy minimum for H 001 is obtained for q H. However, the additional anisotropy term in Eq. (2) with an appropriate sign of k ea shifts these minima away from [001] towards the four directions 111 and thus underlies an instability of the conical state ( Fig. 7 (a)).

V. DISCUSSION AND CONCLUSIONS
To conclude, we have considered the magnetic fielddriven reorientation transitions of spiral states that occur at the critical fields H c1 and H c2 . We have shown that the theoretical model (1) predicts richer phase diagrams that are much more complex than assumed so far. Novel phases are stabilised, such as the an oblique spiral state that has recently been reported for the bulk helimagnet Cu 2 OSeO 3 11,13 . Moreover, phase transitions related to the ellipticity switch of spiral states may occur like the one that has been theoretically predicted for H||[110] but has not been yet observed experimentally. Our theoretical findings can thus serve as a guideline for planning future experiments, that would systematically explore the role of impact of specific geometries of easy and hard anisotropy axes, on the ground states of chiral helimagnets. In the following we discuss the impact these exotic spiral states may have on the stabilisation and properties of chiral skyrmions.

A. Spiral states and the internal structure of isolated skyrmions
Remarkably, the internal properties of spiral states are directly imprinted on the functionalities of chiral skyrmions. Indeed, isolated skyrmions acquire a magnetization distribution according to a surrounding "parental" or host state. In a state that is homogeneously magnetized along the field, axially-symmetric skyrmion particles may form that underlie a repulsive inter-skyrmion potential 28 . On the other hand, when surrounded by a conical phase with the wave vector along the skyrmion axes, skyrmions develop a threedimensional structure: the central core region nearly preserves the axial symmetry, whereas the domain-wall region, that connects the core with the embedding conical state acquires a crescent-like shape. This asymmetric profile of the cross-section forms a screw-like modulation along the skyrmion axis, matching the rotating magnetization of the conical phase at all cross-sections 29 . To reduce the energy excess necessarily related to such a rotation, isolated skyrmions tend to form compact clusters with the unimpeded magnetization rotation in their interior.
Experimentally, clusters of skyrmions have been observed in thin (70 nm) single-crystal samples of Cu 2 OSeO 3 using transmission electron microscopy 30 . In a bulk Cu 2 OSeO 3 for H||[001], skyrmion clusters surrounded by the conical state exist up to the field H c2 of the first-order phase transition between the cone and the FM state ( Fig. 2 (a), (b)). With the onset of the homogeneous state, the inter-skyrmion attraction turns into repulsion and the constituent skyrmions disperse.
If the number of isolated skyrmions is sufficiently high, they may form a SkL with an equilibrium period that exists above the H c2 line, as observed experimentally 13,14 . However, if the number of skyrmions is low, they keep moving away from each other and the skyrmion signal in the experiment should fade away with time. Thus, the following questions related to the skyrmion structure and skyrmion-skyrmion interaction arise: (i) What would be the impact of the elliptically distorted conical states for H||[110] (Fig. 4, 5) or the oblique spiral states (Fig. 7)? Does the skyrmion attraction become anisotropic in these cases? (ii) How does the structure of skyrmion clusters distort during the spiral flips in Fig. 6? Do skyrmions avert this sort of spiral jumps or are they instead annihilated via Bloch points? B. Spiral states as a background to accommodate differently oriented skyrmions Recently it was found that skyrmion tubes within the conical phase may orient either along or perpendicular to a wave vector 35,36 . The first type of skyrmions perfectly blends into the homogeneously saturated state after the cone closes in a strong magnetic field either by the first-order phase transition ( Fig. 2 (b)) or by the second-order one (Fig. 2 (d)). The second type of skyrmions perfectly blends into the spiral state and represents two merons with equally distributed topological charge Q = 1/2. Then, a crossover between the two types of isolated skyrmions occurs for an intermediate value of an applied magnetic field. Thus, the found oblique spiral states (Fig. 6, 7) may underlie new skyrmion states oblique with respect to the field and their controllable switch by the field.

C. Spiral states and the nucleation of skyrmions
Skyrmion nucleation may occur via different mechanisms specified by the stable and metastable spiral states 14,15 . In particular, the low-temperature stable SkL in the bulk-insulating cubic helimagnet Cu2OSeO3 for H||[001] may appear via rupture formation of spiral states with wave vectors perpendicular to the field 14 , along [100] and [010]. In this rather conventional mechanism also observed in the first papers on skyrmion visualization 6,7 , the pairs of merons behave as if they were free particles. The properties of such skyrmions and merons within the spiral background were recently investigated in Refs. 32 and 33. The essential difference in this case is that the spiral states are metastable solutions, but give rise to a stable SkL.
The experimental findings in Cu 2 OSeO 3 14 also hint that skyrmions may appear from the conical and the homogeneous states near the critical field H cr2 . This may occur via formation of magnetic torons of finite length accompanied by their elongation due to the cubic or ex-change anisotropy, which leads to the negative energy density of skyrmion cores and their mutual attraction. Such a homogeneous nucleation mechanism via toronsspatially localized three-dimensional particles composed of a skyrmion filament of finite length cupped by two Bloch points terminating its prolongation -was recently discussed in Refs. 16 and 34. In this sense, all sorts of the magnetization inhomogeneities provided, e.g., by the domain walls between spiral domains near the flip transition ( Fig. 6) or by the superstructure formed from the rotating oblique spiral (Fig. 7), represent a favorable medium for the nucleation and stabbilisation of skyrmions.