Net and Compensated Altermagnetism from Staggered Orbital Order: Layer-Dependent Spin Splitting in Sr n + 1 Cr n O 3 n + 1

Altermagnets are collinear antiferromagnets with non-relativistic spin splitting of the electronic states. We identify the Ruddlesden-Popper chromates Sr_{n+1}Cr_{n}O_{3n+1}, including the perovskite SrCrO_{3}, as candidate materials in which altermagnetism can emerge from spontaneous orbital ordering rather than crystal symmetry. First-principles calculations reveal a layer-dependent spin splitting whose net effect depends on the relative alignment of spin and orbital order: when spin and orbital orders align in adjacent layers, the system exhibits overall spin splitting and thus altermagnetism. When either the spin or orbital order is reversed between adjacent layers, the sign of the spin splitting reverses as well. As a result, the splitting remains finite within each layer but cancels overall. We refer to this compensated state, consisting of two altermagnetic sublattices with opposite spin splitting, as antialtermagnetism. In the Ruddlesden-Popper chromates, odd-n members can host both altermagnetic and antialtermagnetic states, whereas even-n compounds and the perovskite limit are strictly antialtermagnetic.