Altermagnetism Classification

Abstract: Altermagnets are defined as magnetic states with fully compensated spin angular momenta (spins) and broken PT (P: parity, T: time reversal, PT: parity times time reversal) symmetry. We classify three kinds of altermagnets: M-type: broken T symmetry and nonzero net magnetic moments, S-type: broken T symmetry and zero net magnetic moments, and A-type: unbroken T and broken P symmetries. Furthermore, strong altermagnets have spin split bands through exchange coupling in the non-relativistic limit, i.e., for zero spin-orbit coupling (SOC), and weak altermagnets has spin split bands only with non-zero SOC. These strong vs. weak altermagnets can be identified from the total number of symmetric orthogonal spin rotation operations. These classifications of altermagnets will be an essential guidance for the future research on altermagnetism.