Exact non-circular symmetric N -skyrmions in helical magnets without inversion symmetry

Topological objects of intricate structures have been found in a wide range of systems, like cosmology to liquid crystal1, DNA chains2, superfluid 3He (Ref. 3), quantum Hall magnets4,5, Bose-Einstein condensates6 etc. The topological spin texture in helical magnets, observed in recently7,8, makes a new entry into this fascinating phenomenon. The unusual magnetic behaviour of helimagnet MnSi, noticed in recent years9−12, prompted the suspicion that the magnetic states arising in such crystals are of topological nature13−15. Experiments7,8 based on the topological Hall effect confirmed such topologically nontrivial states as the skyrmions16, located on a plane17,18 perpendicular to the applied magnetic field. However, the available models close to MnSi, investigating the formation of skyrmion states4,5,6,13,14, are based mostly on approximate or numerical methods. We present here a theoretical model for chiral magnets with competing exchange and Dzyaloshinskii-Moriya type interaction, which leads to an exact skyrmionic solution with integer topological charge N . Such topologically stable spin states with analytic solution on a two-dimensional plane19 show helical structures of partial order, without inversion and circular symmetry. These exact N-skyrmions, though represent higher excited states, correspond to the lowest energy stable configuration in each topological sector and are likely to appear in MnSi under suitable experimental conditions. The present exact topological solitons, with explicit noncircular symmetry could be applicable also to other fields, where skyrmions are observed, especially in natural systems with less symmetries. Topological properties can be revealed through a mapping from a continuum space to a differentiable manifold. Therefore for describing topological objects one has to shift from the lattice to a continuum picture. In a magnetic model with the spin configuration varying slowly over the lattice spacing, one can approximate using a long wavelength description, the d dimensional lattice to a continuum space R. At the same time the associated spin Sj would go to a vector field n (x), a = 1, 2, 3 of unit length |n|2 = 1, at the classical limit, after a proper renormalisation20, which might induce topological invariants of different nature, at different dimensions d, under suitable boundary conditions 20−23. We intend to use this construction on a two-dimensional xy-plane perpendicular to the applied magnetic field, to simulate the result on MnSi found through the topological quantum Hall experiment. The physically motivated boundary condition demands that at large distances: |x| = ρ → ∞, the spin field n(x) should go to its vacuum solution, orienting itself to a fixed vector n∞ = δa3, along the applied field. This condition in turn introduces a nontrivial topology by identifying the infinities of the coordinate space to a single point, which compactifies the vector space R2 to a sphere S2 x and defines thus a