Collinear and Coplanar Altermagnetism in a Penrose Quasicrystal

We generalize the previous group-theory based classification of collinear magnetism in the quasicrystal (QC) to general magnetic states. Taking the Penrose lattice as an example, we find that all magnetic states belonging to 2D irreducible representations (IRRPs) of the $D_5$ point group, including collinear states and coplanar ones, correspond to altermagnetism (AM). Unlike conventional collinear AM wherein zero magnetism is protected by, e.g. rotation or mirror reflection, which globally flip the spin in the magnetic pattern, the zero magnetism in these 2D-IRRP states is instead protected by a special group-theory-orthogonality theorem that basis functions belonging to different IRRPs are always mutually orthogonal. Taking the single-orbital Penrose-Hubbard as an example, our linear-response theory based calculations show that the largest portion of the parameter space is occupied by these 2D-IRRP states, and our further Ginzburg-Landau theory based analysis reveals that the polarization directions of the two basis magnetic states in the 2D IRRPs can be either parallel or perpendicular, leading to collinear or coplanar magnetic states. The special group-theory-orthogonality compensation mechanism and the absence of the PT symmetry finally leads to AM states, which are to be verified by the spin split in the spin-dependent angle-resolved-photoemission-spectrum and the spin conductance. These intriguing collinear and coplanar AM can also emerge in other QCs.