Relativistic and Nonrelativistic Spin-Splitting above and below the Fermi level in a g-wave Altermagnet

Nonrelativistic spin splitting (NRSS) challenges conventional wisdom about antiferromagnets by allowing spin-split electronic bands even in collinear orders with zero net magnetization. This sub-class of antiferromagnets, recently dubbed "altermagnets," enforces distinctive spin textures via spin-group symmetries in the crystal. However, direct experimental evidence for such symmetry-driven magnetism remains scarce, and distinguishing it from relativistic spin splitting presents additional challenges. Here, we combine first-principles calculations, symmetry analysis, and two spin-resolved spectroscopies—angle-resolved photoemission (spin-ARPES) and our newly developed spin- and angle-resolved electron reflection spectroscopy (spin-ARRES)—to achieve the first complete momentum-resolved mapping of relativistic (RSS) and nonrelativistic (NRSS) spin splitting in CoNb$_4$Se$_8$. By probing both the occupied (spin-ARPES) and unoccupied (spin-ARRES) electronic states in a single experiment, we uncover a series of momentum-dependent spin splitting phenomena each of which switch sign under sixfold rotations and persists far above and below the Fermi level. Crucially, the observed collapse of NRSS and the persistence of RSS near the Néel temperature distinguishes a genuine magnetic phase transition from local inversion symmetry breaking. Our work demonstrates, for the first time, the combined power of spin-ARPES and spin-ARRES in capturing the full spin texture across an extended energy range, positioning CoNb$_4$Se$_8$ as a prototype for exploring spin-group-based phenomena. These findings open new routes for engineering spin-based functionalities ranging from neuromorphic computing to unconventional superconductivity in layered antiferromagnets.