Multipole Moments Under Square Vortex and Skyrmion Crystals

Non-coplanar spin textures such as magnetic vortices and skyrmions manifest themselves in unusual physical phenomena owing to their topologically nontrivial properties. Here, we investigate emergent multipole moments under vortex and skyrmion crystals in the centrosymmetric tetragonal system. Depending on the vorticity and helicity of the vortex or skyrmion, various multipole moments, including magnetic toroidal and electric toroidal multipoles, are induced on the atomic scale. In particular, the vortex and skyrmion spin textures consisting of multiple spin density waves give rise to density waves in terms of other multipole moments. Our results reveal a close relationship between non-coplanar multiple-Q spin textures and multipole moments.