  Altermagnetism is introduced as a category of magnetic states with vanishing
net magnetic moment, and consists of collinear alternating (i.e.,
antiferromagnetic-like) spins and alternating variations of local structures
around spins in such a way that the symmetry allows ferromagnetic behaviors
such as anomalous Hall effect and magneto-optical Kerr effects. Altermagnets
exhibiting ferromagnetic behaviors without any external perturbations (called
type-I) turn out to belong to the ferromagnetic point group, and represent a
form of weak ferromagnetic states. Other altermagnets (called type-II and
type-III) can have ferromagnetic behaviors only with external perturbations,
which conserve parity-time reversal (PT) symmetry. All types of altermagnets
themselves have broken PT symmetry. The concept of altermagnetism can be
extended to accommodate multiple spin and local-structure variations (thus,
include non-collinear spins), and this extended form of altermagnetism offers
an intriguing opportunity to leverage complementary advantages of both
ferromagnetism and antiferromagnetism, and thus holds great promise for, for
example, various spintronic applications.
