  The quasicrystal(QC)s have unique lattice structure with the rotational
symmetry forbidden in the periodic crystals. The electric properties are far
from complete understanding. It has been unresolved whether the magnetic
long-range orders are realized in the QC. Here we report our theoretical
discovery of the ferromagnetic long-range order in the Tb-based QC. The
difficulty in past theoretical studies on the QC was lack of the microscopic
theory of the crystalline electric field (CEF), which is crucially important in
the rare-earth systems. By analyzing the CEF in the Tb-based QC, we clarify
that magnetic anisotropy plays a key role in realizing unique magnetic textures
in the Tb-based QC and approximant crystal (AC). By constructing the minimal
model, we show that various magnetic textures on the icosahedron at whose
vertices Tb atoms are located. We find that the hedgehog state is characterized
by the topological charge of one and the whirling-moment state is characterized
by unusually large topological charge of three. The hedgehog and
whirling-moment states are shown to be realized as antiferromagnetic orders
transcribed as the emergent monopole and antimonopole in the 1/1 AC. We find
that these states exhibit the topological Hall effect under applied magnetic
field accompanied by the topological as well as metamagnetic transition. Our
model and the determined phase diagram are expected to be relevant to the broad
range of the rare-earth based QCs and ACs with strong magnetic anisotropy,
which are useful not only to understand magnetism but also to explore novel
topological properties.
