Recent discoveries of long-range magnetic order in Tsai-type hypermaterials have opened a new avenue for rationally designing magnetism in quasicrystals (QCs) and their approximant crystals (ACs). This review establishes a unified design framework based on three key principles: (1) electronic tuning via the valence-electron concentration (e/a), (2) control of spin anisotropy through crystal electric field effects, and (3) the use of structural degrees of freedom to relieve magnetic frustration. Together, these principles explain the emergence of ferromagnetic, antiferromagnetic, and spin-glass magnetic states in both ACs and QCs. Furthermore, by integrating experimental results, theoretical insights, and case studies, this review identifies key open problems —such as the microscopic mechanisms of magnetic ordering, universality classes in quasiperiodic systems, and the development of stable magnetic quasicrystals —and charts future directions for the field.
Labib et al. (Fri,) studied this question.