Spin splitting without symmetry: a nearly compensated ferrimagnet and the origin of altermagnetism
The spin splitting of collinear altermagnets is usually attributed to the crystal symmetry relating the opposite-spin sublattices --- a rotation or mirror in place of the translation or inversion of a conventional antiferromagnet. We show that this symmetry organizes the splitting but does not produce it; its origin is the anisotropic arrangement of the magnetic orbitals and their ligands, which also fixes the symmetry itself. Two results establish this. A triclinic ($P1$) \ce{Mn} oxide, a fully compensated ferrimagnet whose two inequivalent sublattices are related by no symmetry, is spin split throughout the Brillouin zone; repositioning the same ligands, atom for atom, to restore an inversion centre relating the two metal sites collapses the splitting to a residual two orders of magnitude smaller, so the splitting follows from the arrangement rather than the symmetry. Symmetry, we show formally, can only ever reverse or forbid a splitting that already exists: it cannot create one between sublattices an arrangement has left electronically identical, nor where the underlying orbitals carry no anisotropy to arrange in the first place. Organizing the analysis through an operation-resolved symmetry hierarchy and reading the crystal classes as a single axis, the enforced nodal planes fall from two to one to zero across the orthorhombic, monoclinic, and triclinic classes while the arrangement-generated splitting persists to the base. The altermagnet and the fully compensated ferrimagnet are thus the two ends of one axis --- the same splitting, organized by symmetry at one end and unorganized at the other --- and the search for compensated spin-split magnets is a search over orbital arrangements rather than symmetry labels.