Classification of Magnetism and Altermagnetism in Quasicrystals

Abstract

Altermagnetism (AM), an unconventional magnetic phase characterized by zero net magnetism protected by symmetry(s) other than parity-time ($\mathcal{P}\mathcal{T}$) and a resulting spin-split band, has been studied exclusively in crystalline materials. Here, we extend the framework of AM to quasicrystals (QCs). We start from a comparison between the N\'{e}el state on the square lattice and that on a $D_4$-symmetric Thue-Morse QC, with both belonging to the same $d$-wave irreducible representation (IRRP) of the $D_4$ point group. Consequently, while the former is antiferromagnetism (AFM) protected by the combined $\mathcal{P}\mathcal{T}$ and translational symmetry, the lack of translational symmetry in the latter breaks the $\mathcal{P}\mathcal{T}$ symmetry, and the additional mirror or rotation symmetry protects AM. This example suggests that AM is more common in QCs than in crystals and can be easily explored through a point-group symmetry-based classification. Therefore, we classify magnetic phases in 2D $D_n$-symmetric QCs without spin-orbit coupling, by using IRRPs of $D_n$. Consequently, the identity IRRP represents ferromagnetism, the inversion-odd 1D IRRPs for twice-of-odd $n$ represent AFM, and all the remaining 1D IRRPs represent AM, protected by either mirror or rotation symmetry. We further take the Hubbard model to verify this result in various QCs with different symmetries. Our work highlights the QC as a natural platform where AM is common among magnetic phases.