Ground-State Phase Diagram of the Effective Model with Antiferromagnetic Nearest-Neighbor and Ferromagnetic Next-Nearest-Neighbor Interactions in 1/1 Periodic Approximants to Icosahedral Quasicrystals

By performing complete enumeration of all configurations of magnetic moments, we have determined the ground-state phase diagram of the effective model with antiferromagnetic nearest-neighbor interactions $J_1$ and ferromagnetic next-nearest-neighbor interactions $J_2$ on the 1/1 periodic approximant to icosahedral quasicrystal. Depending on the ratio of $J_2\mathrm{/}{J}_1$ and the magnetic anisotropy arising from the crystalline electric field, various non-collinear ferrimagnetic orders and antiferromagnetic orders are shown to be stabilized, whose magnetic space groups are identified as $R\bar{3}$, $C{2}^{\mathrm{\prime }}\mathrm{/}{m}^{\mathrm{\prime }}$, $P{m}^{\mathrm{\prime }}{m}^{\mathrm{\prime }}2$ and $I_{\mathrm{P}}{m}^{\mathrm{\prime }}{\bar{3}}^{\mathrm{\prime }}$. We have revealed the topological as well as magnetic property of each ground state with degeneracy, which is useful to understand experimental observations of magnetic structures and domains in the rare-earth-based 1/1 approximants.