Magnetic order in Tsai-type icosahedral quasicrystals and periodic approximants.

Quasicrystals (QCs), a class of aperiodic crystals with long-range order and non-crystallographic rotational symmetry, provide a unique platform for exploring emergent magnetism beyond conventional periodic solids. Although theoretical studies have long suggested that magnetic order is compatible with quasiperiodicity, long-range magnetic order in real QCs has only recently been realized through design principles established from their periodic approximants. This review presents three complementary and hierarchical principles for understanding and controlling magnetism in Tsai-type QCs and periodic approximants: (i) tuning the valence electron concentration (e/a) to control the sign and strength of Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions, (ii) exploiting crystal electric field (CEF) anisotropy to select spin orientation and stabilize noncoplanar magnetic textures, and (iii) controlling structural degrees of freedom to manipulate frustration. Together, these principles provide a phenomenological framework for understanding and organizing magnetic phase selection in quasiperiodic materials. They also provide a unified perspective on the contrasting emergence of long-range magnetic order and spin-glass-like freezing in Tsai-type quasicrystals. Critical behaviour near magnetic transitions, emerging opportunities and outstanding challenges are also discussed, highlighting future directions for discovering novel magnetic states in QCs.