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The Kramers’ degeneracy was born in the field of spectroscopy for systems with time-reversal symmetry. Under the additional condition of the inversion symmetry was applied also to the field of the solid-state physics for non-magnetic systems. Recently, it was shown that the extension of the Kramers’degeneracy to the antiferromagnetic systems has some limitations. Without spin-orbit coupling, some antiferromagnets does not present Kramers’degeneracy but a large non-relativistic spin-splitting due to the breaking of time-reversal symmetry. This antiferromagnetism without Kramers degeneracy was named altermagnetism. Altermagnetic compounds behave as conventional antiferromagnets in the real space and as ferromagnets in the k-space paving the way for new technological applications [1,2].

The presence of the altermagnetic phase strongly depends on the magnetic space group[3,4]. We investigate the altermagnetic properties of strongly-correlated transition metal oxides analyzing the Mott insulators Ca₂RuO₄ and YVO₃. In both cases, the orbital physics is extremely relevant in the t_2g subsector with the presence of an orbital-selective Mott physics in the first case and of a robust orbital-order in the second case [5]. I will briefly mention how the nonsymmorphic[6] symmetries and the dimensionality[7] affect the properties of the altermagnetic phase.

 Including the spin-orbit coupling, we study the effect of Dzyaloshinskii–Moriya interaction (DMI) in centrosymmetric and noncentrosymmetric altermagnets. Once time-reversal symmetry is broken in altermagnets, the DMI can produce weak ferromagnetism or weak ferrimagnetism from a purely relativistic effect[8]. The DMI that generated weak ferromagnetism in altermagnets has a staggered structure and the DMI can be enhanced by adapting to the staggered geometry the same strategies used to increase DMI in ferromagnetic multilayers[9]. The weak ferromagnetism from a purely relativistic effect is a property exclusively of the altermagnets that is not found in either ferromagnets or conventional antiferromagnets.[8]

1. L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501(2022)
2. C. Autieri, Nature 626, 482 (2024)
3. G. Cuono, R. M. Sattigeri, C. Autieri, T. Dietl, Phys. Rev. B 108, 075150 (2023)
4. M. J. Grzybowski, C. Autieri et al., Accepted in Nanoscale arXiv:2309.06422
5. G. Cuono, R. M. Sattigeri, J. Skolimowski, C. Autieri J. Magn. Magn. Mat. 586, 171163, (2023)
6. A. Fakhredine, R. Sattigeri, G. Cuono, C. Autieri, Phys.Rev. B 108, 115138 (2023)
7. R. M. Sattigeri, G. Cuono, C. Autieri, Nanoscale, 15, 16998 (2023)
8. C. Autieri et al. https://arxiv.org/abs/2312.07678 Submitted to PRB (2024)
9. A. Fakhredine, A. Wawro and C. Autieri J. Appl. Phys. 135, 035303 (2024)

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