Pinning Effects of Exchange and Magnetocrystalline Anisotropies on Skyrmion Lattice

Reorientation of skyrmion crystal (SkX) with respect to crystallographic axes is believed to be insensitive to anisotropies of fourth order in spin-orbit coupling, for which sixth order terms are considered for explanation. Here, we show that this is wrong due to an oversimplified assumption that SkX possesses hexagonal symmetry. When the deformation of SkX is taken into account, fourth order anisotropies such as exchange anisotropy and magnetocrystalline anisotropy have pinning \textcolor{red}{(in this work, the word ‘pinning’ refers to the reorientation effects of intrinsic anisotropy terms)} effects on SkX. In particular, we reproduce some experiments of MnSi and Fe$_{1-x}$Co$_{x}$Si by considering the effect of fourth order magnetocrystalline anisotropy alone. We reproduce the $30^\circ$ rotation of SkX in Cu$_2$OSeO$_3$ by considering the combined effects of the exchange and magnetocrystalline anisotropies. And we use the exchange anisotropy to explain the reorientation of SkX in VOSe$_2$O$_5$.