Staggered Dzyaloshinskii-Moriya inducing weak ferromagnetism in centrosymmetric altermagnets and weak ferrimagnetism in noncentrosymmetric altermagnets

The Dzyaloshinskii-Moriya interaction (DMI) has explained successfully the weak ferromagnetism in {\it some} centrosymmetric antiferromagnets. However, in the last years, it was generally claimed that the DMI is not effective in such systems. We reconciled these views by separating the conventional antiferromagnets from altermagnets. Altermagnets are collinear magnets having zero magnetization preserved by crystal symmetries in the non-relativistic limit. The spin-up and spin-down sublattices can be connected either by a proper rotation or by a combination of a rotation and a mirror or an inversion symmetry. Consequently, the system shows even-parity wave spin order in the k-space lifting the Kramer's degeneracy in the non-relativistic band structure leading to unconventional magnetism. The DMI can create weak ferromagnetism or weak ferrimagnetism in centrosymmetric and in noncentrosymmetric altermagnets while it is not effective in conventional antiferromagnets. Once the spin-orbit coupling is included in an altermagnetic system (where the time-reversal symmetry is broken), the components of spin moments of the two sublattices along the N\'eel vector are antiparallel but the other two spin components orthogonal to the N\'eel vector can be null, parallel or antiparallel. In cases where we have different bands showing parallel and antiparallel spin components at the same time, the magnetic order results in weak ferrimagnetism. If we restrain to high-symmetry directions for the N\'eel vector, we find weak ferrimagnetism only in the noncentrosymmetric compound MnSe among the examined cases.