Dynamical and static structure factors in hedgehog-antihedgehog order in an icosahedral 1/1 approximant crystal

Recent discoveries of magnetic long-range orders in the icosahedral quasicrystal and topological magnetic structures on the icosahedron (IC) as the hedgehog state and the antihedgehog state have attracted great interest. Here, we report our theoretical analysis of the dynamical as well as static structure of the hedgehog-antihedgehog order in the 1/1 approximant crystal (AC). By constructing the effective magnetic model for the rare-earth based AC, on the basis of the linear spin-wave theory, the excitation energy is shown to exhibit the reciprocal dispersion, as a consequence of preservation of the spatial inversion symmetry by the hedgehog-antihedgehog ordering. The static structure factor is shown to be expressed generally in the convolution form of the lattice structure factor and the magnetic structure factor on the IC(s) and the numerical calculation reveals the extinction rule. The dynamical structure factor shows that the high intensities appear in the low-energy branch along the $\Gamma$-X line and the R-$\Gamma$-M line in the reciprocal space.