4 b < 0 also leads to the thermodynamic stability of half- SLs (Fig. 4 (a)). The stabilization of such textures may be responsible for anomalous precursor effects in cubic helimagnets in zero field [7, 20, 21]. Chiral modulations in non-Heisenberg models . A gen- eralization of isotropic chiral magnets proposed in [7] replaces the usual Heisenberg-like exchange model by a non-linear sigma-model coupled to a modulus field with different stiffnesses. This yields a generalized gra- dient energy for a chiral isotropic system with a vec- tor order parameter, which is equivalent to the phe- nomenological theory in the director formalism [4, 7] $^{∑}$$_{i,j}$( ∂$_{i}$m$_{j}$ ) 2 → $^{∑}$$_{i,j}$( ∂$_{i}$m$_{j}$ ) 2 + (1 − η ) $^{∑}$$_{i,j}$( ∂$_{i}$m ) 2 = m 2 $^{∑}$$_{i,j}$( ∂$_{i}$n$_{j}$ ) 2 + η $^{∑}$$_{i}$( ∂$_{i}$m ) $^{2}$. Parameter η equals unity for a “Heisenberg” model, in chiral nematics η = 1 / 3 [4]. Confined chiral modulations are very sensitive to values of η < 1. The magnetic phase diagram calculated for η = 0 . 8 includes pockets with square half-SLs, hexagonal SLs with the magnetization in the center of the cells par- allel to the applied field, and helicoids with propagation transverse to the applied field (Fig. 4 b). The confinement effects on chiral Skyrmions strongly changes the picture of the formation and evolution of chi- ral modulated textures and shed new light on the prob- lem of precursor states observed as blue phases in chiral nematics [4] and in chiral magnets [7, 17–21]. The re- sults show that confinement / deconfinement transitions of localized string-like solitons can be realized in these condensed matter systems. They provide counterparts of formation mechanisms for extended microscopic mat- ter from topological solitons, as devised originally in the Skyrme model [22]. Acknowledgment. We thank K. v. Bergmann, G. Bihlmayer, S. Bl¨ ugel, H. Eschrig, S.V. Grigoriev, J.-H. Han, S. Heinze, R. M¨ ossner, C. Pappas, W. Selke, H. Wilhelm, and M. Zhitomirsky for discussions. Support by DFG project RO 2238/9-1 is gratefully acknowledged. The appendix includes technical comments on the mathematical structure of the interaction functional (1), on the mathematical relation between this model and those for different classes of noncentrosymmetric mag- netic systems and chiral liquid crystals. The equations for isolated and bound Skyrmions are provided along with a short information on the numerical methods to solve these problems. In noncentrosymmetric ferromagnets chiral asymmetry of exchange coupling related to the quantum-mechanical Cubic helimagnets. In other groups of chiral mag- nets the DM energy includes contributions with gradients along all three spatial directions. This stabilizes chiral modulations with propagation along the direction of an applied field as cone phases [13]. For the isotropic model Φ( m ) (1) the cone phase solution with the fixed magne- tization modulus and rotation of m around the applied magnetic field: ψ = z, cos( θ ) = h/m, m = | a − 1 / 2 | / 2, is the global energy minimum in the whole region where modulated states exist. For cubic helimagnets, the en- ergy density (1) has to be supplemented by anisotropic contributions, Φ$_{a}$ = b $^{∑}$$_{i}$( ∂m$_{i}$/∂x$_{i}$ ) 2 + k $^{∑}$$_{i}$m 4 $_{i}$, where b and k are reduced values of exchange and cubic anisotropies [13]. These anisotropic interactions impair the ideal harmonic twisting of the cone phase and lead to the thermodynamic stability of Skyrmion states, as shown in the equilibrium phase diagram Fig. 4 (a). The difference between the energy of the hexagonal Skyrmion lattice W$_{sk}$ and of the cone phase W$_{cone}$ calculated for the isotropic model, Δ W$_{min}$ = W$_{sk}$ − W$_{cone}$ , has min- ima along a curve h$_{x}$ ( a ) which reaches the critical point h$_{x}$ ( a$_{c}$ ) = 0 as Δ W$_{min}$ = 0 . 0784(0 . 25 − a )). Weak ex- change anisotropy of a cubic helimagnet, therefore, cre- ates a pocket around a$_{c}$ , where the hexagonal Skyrmion lattice becomes the global energy minimum in a field (Fig. 4 (a)). This case is realized in cubic helimag- nets with negative exchange anisotropy ( b < 0) as in MnSi [13]. This anisotropy effect provides a basic mech- anism, by which a Skyrmionic texture is stabilized in applied fields, as observed experimentally as so-called ”A-phase” in MnSi [17–19]. The exchange anisotropy fields [1, 16]. In the confinement pocket of the phase di- agram (III) the helicoids also exist only as bound states [16]. They have the lowest energy at lower fields, and the hexagonal SL corresponds to the global energy minimum at higher fields (Fig. 2 (c)). FIG. 4: (Color online) Magnetic phase diagrams of cubic heli- magnets with exchange anisotropy b = − 0 . 05 and the applied field along (111) (solid) and (001) (dashed) axes (a) contains regions with thermodynamically stable hexagonal SLs and half-SLs. Magnetic phase diagram of the modified isotropic model with η = 0 . 8 (b) has extended domains with thermo- dynamically stable helicoid, half-SLs and hexagonal SLs with the magnetization along the cell axes parallel to the magnetic field (hatched area). 0.21 0.23 0.25 0 2 4 a Half-Skyrmion Lattice Cone (a) Hexagonal Skyrmion Lattice h 10 2 a 0.18 0.22 0.26 Cone Saturated State Saturated State Helicoid (b) 0 4 8 Half- Skyrmion Lattice h 10 2 h$_{x}$ GLYPH<c103> GLYPH<c103> APPENDIX:DZYALOSHINSKII THEORY OF CHIRAL HELIMAGNETISM 1. Basic model.