3 tization modulus the field-driven transformation of the Skyrmion lattices evolves by distortions of the modulus profiles m ( ρ ) both in the hexagonal and square Skyrmion lattices while the equilibrium periods of the lattices do not change strongly with increasing applied field (Fig. 3). Despite the strong transformation of their internal structures the Skyrmion lattices preserve axisymmetric distribution of the magnetization near the centers of the Skyrmion lattice cells (Fig. 3). This remarkable property reflects the basic physical mechanism underlying the for- mation of Skyrmion lattices. The local energetic advan- tage of Skyrmion lattices over helicoids is due to a larger energy reduction in the “double-twisted” Skyrmion cell core compared to “single-twisted” helical states [4, 7]. This explains the unusual axial symmetry of the cell cores and their stability. An increasing magnetic field gradu- ally suppresses the antiparallel magnetization in the cell core reducing the energetic advantage of the “double- twist” and increases the overall energy of the condensed Skyrmion lattice. At critical field h$_{c}$ a first-order tran- sition occurs into the polarized paramagnetic state. In the interval h$_{c}$ < h < h$_{n}$ the hexagonal Skyrmion lattice exists as a metastable state. At the lability field h$_{n}$ the magnetization modulus in the cell center becomes zero (see magnetization profile for h = 0 . 042 in Fig. 3 (c)). At this field, the ”double-twist” region is suppressed, and the lattice loses its stability. At zero field with increasing temperature a the mag- netization modulus m in hexagonal and square lattices gradually decreases to zero at the ordering temperature. This is the instability-type nucleation transition into the paramagnetic phase : the order parameter m becomes zero at the transition point (as in the instability mode), however, the lattices retain their symmetry and the ar- rangement of axisymmetric Skyrmions up to the criti- cal point. In finite fields in the region (II) the attrac- tive Skyrmion-Skyrmion interaction means that multi- Skyrmions can always form bound states. Near a$_{c}$ such clusters of Skyrmions are more stable than the isolated Skyrmions. This is seen in Fig. 2 (a),(b). For tempera- tures above point B , a$_{B}$ < a < a$_{c}$ , the metastable SL as the densest infinite cluster is more stable than isolated Skyrmions. Thus, in region (II) of the phase diagram, there is clustering and, for higher temperatures, confine- ment of isolated Skyrmion excitations. The predicted Skyrmionic textures and the pre- cursor phenomena associated with the confinement of Skyrmions near the ordering transition are observable in magnetic materials with appropriate symmetry. In a large group of uniaxial noncentrosymmetric magnets the DM energy is described by gradients only along directions perpendicular to the axis (e.g. multiferroic BiFeO$_{3}$, space group R 3 c and antiferromagnetic Ba$_{2}$CuGe$_{2}$O$_{7}$, space group P ¯ 42$_{1}$ m [2, 15]). In such magnets one-dimensional modulations with the propagation vector in the basal plane ( helicoids ) exist in broad ranges of the magnetic Evolution of confined Skyrmions . In the vicinity of the ordering temperature the Skyrmion lattices exist within the confinement pocket (III). In this region Skyrmion states drastically differ from those in the main part of the phase diagram. Due to the ”softness” of the magne- magnetic field closely agrees with the behavior studied earlier for the low-temperature limit [5, 6] and the tran- sition mechanism at the high-field limit is of the nucle- ation type with isolated Skyrmion excitations appearing below the instability line h$_{0}$ . The temperature of the turning point p ( a$_{p}$ = − 0 . 75), a confinement tempera- ture T$_{c}$ determines the frontier between low- ( T < T$_{p}$ ) and high-temperature ( T$_{p}$ < a < T$_{c}$ ) chiral modulations. Usually Δ T = T$_{c}$ − T$_{p}$ ≪ T$_{c}$ and high-temperature mod- ulations are restricted by a narrow vicinity of the Neel temperature. According to calculations in [7] for MnSi high-temperature Skyrmions exist in temperature inter- val Δ T = T$_{c}$ − T$_{p}$ = 2 K below the Curie temperature T$_{c}$ . FIG. 3: (Color online) Numerically exact solutions for hexag- onal (a)–(c) and square (d)–(f) Skyrmion lattices within the confinement pocket. Contour plots of M$_{z}$ ( x, y ) and M ( x, y ) (a),(d) and their diagonal cross-sections (b),( e), (c),(f) are derived by minimization of Φ (1) with a = 0 . 23 and different values of the applied field. Solid (dashed) lines in panels (e), (f) indicate profiles for cells with (anti)parallel magnetization in the center.