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DM energy w$_{D}$ consists of Lifshitz invariants [1] 

For noncentrosymmetric uniaxial ferromagnets DM func- tionals w$_{D}$ are listed in [5]. In particular, for isotropic and cubic helimagnets w$_{D}$ ( m ) = L ( y ) xz + L ( x ) zy + L ( z ) xy = m · rot m [13] (for details see appendix). Functional (1) includes only the basic ( isotropic ) interactions essential to stabilize chiral modulations. This depends on three internal variables (components of the magnetization vec- tor m ) and two control parameters, the reduced magnetic field amplutude h and the ”effective” temperature a ( T ). In chiral magnetism models (1) play a fundamental role and are similar to the Frank energy in liquid crystals and Landau-Ginzburg functional in superconductivity. We consider 2D chiral modulations homogeneous along the applied field h || z and modulated in the plane perpendic- ular h . 

Isolated and embedded Skyrmions . The equations min- imizing functional (1) include solutions for axisymmet- ric localized states ( isolated Skyrmions ), ψ = ψ ( φ ), and θ ( ρ ), m ( ρ ) where we use spherical coordinates for the magnetization ( m, θ, ψ ) and cylindrical coordinates for the spatial variables ( ρ, φ, z ). The solutions ψ ( φ ) for all noncentrosymmetric classes have been derived in [5]. For cubic helimagnets and uniaxial systems with n 22 sym- metry, ψ = φ − π/ 2 (Fig. 1 a). Profiles θ ( ρ ) and m ( ρ ) of isolated Skyrmions are derived from numerical solutions of the Euler equations, as given in Ref. 7 and common for all noncentrosymmetric classes with Lifshitz invariants, with the boundary conditions θ (0) = π , dm/dρ (0) = 0, θ ( ∞ ) = 0, m ( ∞ ) = m$_{0}$ ( m$_{0}$ is the magnetization in the saturated state). For extended textures of two- dimensional models, the functional has been investigated by numerical energy minimization using finite-difference discretization on rectangular grids with adjustable grid spacings [7]. 

Isolated Skyrmions (Fig. 1 (a)) exist only below a crit- ical line h$_{0}$ and condense into a hexagonal lattice (Fig. 2) 

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below a field h$_{c}$ , which marks the transition between the Skyrmion lattice (SL) and the homogeneous paramag- netic state (see Fig. 2 (a), (b)). Near the ordering tem- perature a square half-Skyrmion lattice (Fig. 1 (c)) has lower energy than the hexagonal lattice. Half-SLs con- sist of cells with up and down magnetization in the cen- ter and in-plane magnetization along the cell boundaries. Such cells have a topological charge 1 / 2. In the hexag- onal SLs the magnetization at the boundaries (center) is (anti)parallel to the applied field. The cells bear unit topological charges. 

Confinement . By solving the linearized Euler equa- tions for the asymptotics of isolated Skyrmions Δ m = ( m − m$_{0}$ ) , θ ∝ exp( − κρ ) ( ρ ≫ 1) one finds three distinct regions in the magnetic phase diagram with different character of Skyrmion-Skyrmion interactions (Fig. 2 (a)): repulsive interactions in a broad temperature range (area (I)) are changed to attractive interaction at higher tem- peratures (area (II)). Finally in area (III) near the order- ing temperature a$_{c}$ = 0 . 25 strictly confined Skyrmions exist. These regions are separated by the line 

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with turning points p ( − 0 . 75 , √ 2 / 4), q (0 . 06 , 0 . 032 √ 5), and u (-0.5, 0) (dashed line in Fig. 2 (a)). In the major part of the phase diagram, the evolution of SLs under a 

