Identifying and designing altermagnetic crystals in real space

Altermagnetism is a compensated magnetic phase characterized by zero net magnetization and exchange-driven spin splitting. However, identifying altermagnets among collinear antiferromagnets usually requires full magnetic-space-group or spin-group analysis, which is not intuitive. Here we formulate a simple real-space criterion based on how the crystallographic operations of the host nonmagnetic structure permute the two opposite-spin sublattices. For simplicity, we focus on collinear compensated antiferromagnets whose magnetic primitive cell coincides with the host nonmagnetic crystallographic primitive cell. In this class, altermagnetic spin splitting is generally allowed unless an inversion-type operation exists that exchanges the two opposite-spin sublattices. First-principles calculations on representative noncentrosymmetric and centrosymmetric materials demonstrate this criterion. Similar rules can also be applied to low-dimensional crystals or quasicrystals. Our work reduces the identification of altermagnetism to a transparent real-space symmetry test and provides a practical route for discovering altermagnetic crystals.