Minimal models for altermagnetism

Altermagnets feature vanishing net magnetization, like antiferromagnets, but exhibit time-reversal symmetry breaking and momentum-dependent spin-split band structures. Motivated by the fact that all proposed altermagnets have paramagnetic states with multiple magnetic ions in the unit cell, we develop a class of realistic minimal models for altermagnetism through a comparative analysis of the magnetic atom Wyckoff site symmetry and the space group symmetry. Specifically, we develop electronic models for all centrosymmetric space groups with magnetic atoms occupying inversion symmetric Wyckoff positions with multiplicity two. These forty models include monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic materials and describe d-wave, g-wave, and i-wave altermagnetism. We further define and examine an altermagnetic susceptibility and mean field instabilities within a Hubbard model to reveal that these models have altermagnetic ground states. We shed insight on why most altermagnets form in nonsymmorphic space groups. We also provide the symmetry-required form of the spin-orbit coupling and show it yields a Berry curvature that is linear in this coupling for all forty models. We apply our models to representative cases of RuO2, MnF2, FeSb2, kappa-Cl, CrSb, and MnTe.