2 a aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa aaaaaaaaaaaaaa (a) ( ) c I III p A B u h$_{c}$ h$_{0}$ h$_{n}$ h$_{n}$ h$_{n}$ h q h Skyrmion Lattice Hexagonal 0.4 0.2 0.0 -1.0 -0.5 0.0 0.25 0.23 0.25 a 5 0 h 10 2 GLYPH<c103> GLYPH<c97> Helicoid Hexagonal SL Square SL GLYPH<c98> 0 0.1 aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa (b) B h$_{c}$ h$_{0}$ 0.1 0.2 a h$_{c}$ II Isolated Skyrmions m/m 0 1.0 0.5 0.0 5 10 15 0 0 (a) (b) (c) GLYPH<c112> GLYPH<c114> GLYPH<c113> FIG. 2: (Color online) Field h vs. temperature a phase diagram. (a) Areas with repulsive (I) (shaded), attractive (II) Skyrmions (hatched), and strictly confined pocket (III) (cross-hatched) are separated by h ⋆ (dashed line), Eq. (3). Isolated Skyrmions collapse at critical line h$_{0}$ . Above this line no static Skyrmions exist. Below line h$_{c}$ Skyrmions con- dense into a hexagonal lattice In region (II), the hexagonal Skyrmion lattice (SL) exists as metastable state up to the nucleation field h$_{n}$ . For temperatures between points A and B , h$_{n}$ < h$_{0}$ , for larger temperatures a$_{B}$ < a < a$_{c}$ = 0 . 25 the isolated Skyrmions disappear at lower fields than the dense SL, h$_{0}$ > h$_{n}$ . For clarity, line h$_{n}$ is only schematically given in panel (a), numerically exact data are shown in panel (b). De- tail of the phase diagram (c) near the ordering temperature shows the existence regions for different modulated states. Lines for first order transitions: α square half-SL ↔ hexago- nal SL, β helicoid ↔ hexagonal SL. Line γ marks the continu- ous transition from the conical equilibrium phase in isotropic systems to the paramagnetic phase. FIG. 1: (Color online) (a) Equilibrium solutions of local- ized Skyrmions, shown as cross section in the inset, display strong localization of θ ( ρ ) (blue) and weak variation of the magnetization modulus m ( ρ ) (red) in a broad range of tem- perature and field ( a, h ). Exemplary profiles for a = − 0 . 25, h = 0 . 25. (b) At lower fields, isolated Skyrmions condense into a dense-packed hexagonal lattice. (c) Near the ordering temperature a$_{c}$ = 0 . 25 square lattice solutions with staggered half-Skyrmion cells arise. 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 . 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 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) 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 h ⋆ = $^{√}$2 ± P ( a )( a + 1 ± P ( a ) / 2) , P ( a ) = √ 3 + 4 a (3) L ( k ) ij = m$_{i}$ ( ∂m$_{j}$/∂x$_{k}$ ) − m$_{j}$ ( ∂m$_{i}$/∂x$_{k}$ ) . (2)