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Quasicrystal (QC) possesses a unique lattice structure with rotationalsymmetry forbidden in conventional crystals. The electric property is far fromcomplete understanding and it has been a long-standing issue whether themagnetic long-range order is realized in the QC. The main difficulty was lackof microscopic theory to analyze the effect of the crystalline electric field(CEF) at the rare-earth atom in QCs. Here we show the full microscopic analysisof the CEF in Tb-based QCs. We find that magnetic anisotropy arising from theCEF plays a key role in realizing unique magnetic textures on the icosahedronwhose vertices Tb atoms are located at. Our analysis suggests that thelong-range order of the hedgehog characterized by the topological charge of oneis stabilized in the Tb-based QC. We also find that the whirling-moment stateis characterized by unusually large topological charge of three. The magnetictextures as well as the topological states are shown to be switched bycontrolling compositions of the non-rare-earth elements in the ternarycompounds. Our model is useful to understand the magnetism as well as thetopological property in the rare-earth-based QCs and approximant crystals.

Magnetism and topology in Tb-based icosahedral quasicrystal