Altermagnetism with non-collinear spins

Altermagnetism is introduced as a category of magnetic states with ‘collinear’ antiferromagnetic spins and alternating variations of local structures around spins in such a way that the symmetry allows typical ferromagnetic behaviors. Altermagnets exhibiting ferromagnetic behaviors without any external perturbations (type-I) turn out to belong to the ferromagnetic point group. Other altermagnets (type-II and type-III) can have ferromagnetic behaviors only with external perturbations such as electric current or stress, 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 non-collinear spins and multiple local-structure variations.