Non-relativistic spin effects in symmetry-broken antiferromagnets shaped by auxiliary symmetry

While traditional relativistic (Rashba) spin splitting in energy bands requires (a) breaking inversion symmetry, (b) spin-orbit coupling (SOC) in (c) nonmagnets, recent discovery of non-relativistic spin splitting (NRSS) [Yuan, Wang, Luo, Rashba, and Zunger, Phys. Rev. B 102, 014422 (2020)] is enabled by (a’) breaking space-time reversal and translation-spin-rotation symmetries, (b’) in the absence of SOC in (c’) antiferromagnets (AFM). Here we find additional, auxiliary symmetries including spin-interconverting and polar symmetries, which can shape the spin splitting and polarization, yet all in AFM subgroups [PRX 15, 031076 (2025)]. These include: (i) α-type such as BiCrO₃ and Mn₂ScSbO₆ with spin splitting at the Brillouin Zone center yet small magnetization correlated with the band gap; The magnitude of NRSS in α-type AFM is closely related to the asymmetry of opposite-spin density of states. (ii) β-type (altermagnets, having rotation spin-interconverting symmetries), and (iii) polar γ-type (having exclusively reflection spin-interconverting symmetries) such as Mn₄Nb₂O₉ and FeScO₃. We distinguish α-type AFM from compensated ferrimagnets: The former has (i) zero magnetization at zero temperature and almost zero magnetization with perturbed level of occupation and (ii) same species of opposite-spin sublattices, whereas the latter has (i’) zero magnetization at zero temperature but large magnetization at finite temperatures or (ii’) different species of opposite-spin sublattices.