Altermagnetism: a third magnetic class delimited by spin symmetry groups.

Modern research areas, ranging from topological condensed matter and dissipationless quantum transport to spintronics, are built on a magnetic symmetry group theory which entangles magnetism and relativistic spin-orbit coupling. Here we return to the non-relativistic foundations of collinear magnetism and delimit Type-I spin groups containing no spin-rotation symmetry element, Type-II with a spin-rotation symmetry, and Type-III with symmetry elements combining spin and crystal rotations. We show that they describe, respectively, ferromagnets with spin-polarized electron bands, antiferromagnets with spin-degenerate bands, and a third class with alternating spin-polarizations in locked momentum and crystal space. We refer to the third class as altermagnets and identify their topological invariants which have a form of an even-integer spin winding number in the momentum space. Our band structure analysis reveals a spin splitting by an electrostatic crystal field as a unique spectroscopy signature of altermagnetism. We find altermagnetic material candidates ranging from high N\'eel temperature insulators and metals to a parent cuprate of high-temperature superconductivity.