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We develop a theory of the magnetic field-induced formation of Skyrmioncrystal state in chiral magnets in two spatial dimensions, motivated by therecent discovery of the Skyrmionic phase of magnetization in thin film ofFe$_{0.5}$Co$_{0.5}$Si and in the A-phase of MnSi. Ginzburg-Landau functionalof the chiral magnet re-written in the CP$^1$ representation is shown to be aconvenient framework for the analysis of the Skyrmion states. Phase diagram ofthe model at zero temperature gives a sequence of ground states, helical spin$\rightarrow$ Skyrme crystal $\rightarrow$ ferromagnet, as the external field$B$ increases, in good accord with the thin-film experiment. In close analogywith Abrikosov's derivation of the vortex lattice solution in type-IIsuperconductor, the CP$^1$ mean-field equation is solved and shown to reproducethe Skyrmion crystal state.

Skyrmion Lattice in Two-Dimensional Chiral Magnet