Hyperspin Altermagnets.

The behavior of spin quantum in k-space is key to identifying altermagnets (AMs) as the third kind of fundamental collinear magnetism. By contrast, noncollinear magnets-though abundant in nature-lack well-defined spin quantum numbers, and the resulting spin textures are often highly complex, which limits their potential for next-generation spintronic applications. Here we propose hyperspin, which lives in a higher-dimensional space, to address these drawbacks. Through analyzing the commutation relations between spin and Hamiltonian for a class of noncollinear magnets, we reveal that it is a hyperspin, rather than the usual spin, that commutes with Hamiltonian. Unexpectedly, these noncollinear magnets should also show collinear spin-split bands in k-space like collinear AMs. We therefore classify such noncollinear magnets as hyperspin altermagnets (HAMs), as opposed to the usual collinear AMs. Our theory elucidates the fundamental physics of AMs and HAMs and provides a framework for exploring the wide range of noncollinear magnets that may possess other kinds of conserved quantities.