Lifted Kramers spin degeneracy (LKSD) has been among the central topics of condensed-matter physics since the dawn of the band theory of solids1,2. It underpins established practical applications as well as current frontier research, ranging from magnetic-memory technology3-7 to topological quantum matter8-14. Traditionally, LKSD has been considered to originate from two possible internal symmetry-breaking mechanisms. The first refers to time-reversal symmetry breaking by magnetization of ferromagnets and tends to be strong because of the non-relativistic exchange origin15. The second applies to crystals with broken inversion symmetry and tends to be comparatively weaker, as it originates from the relativistic spin-orbit coupling (SOC)16-19. A recent theory work based on spin-symmetry classification has identified an unconventional magnetic phase, dubbed altermagnetic20,21, that allows for LKSD without net magnetization and inversion-symmetry breaking. Here we provide the confirmation using photoemission spectroscopy and ab initio calculations. We identify two distinct unconventional mechanisms of LKSD generated by the altermagnetic phase of centrosymmetric MnTe with vanishing net magnetization20-23. Our observation of the altermagnetic LKSD can have broad consequences in magnetism. It motivates exploration and exploitation of the unconventional nature of this magnetic phase in an extended family of materials, ranging from insulators and semiconductors to metals and superconductors20,21, that have been either identified recently or perceived for many decades as conventional antiferromagnets21,24,25.

Faculty of Mathematics and Physics Charles University Prague Czech Republic

Grupo de Investigación en Física Aplicada Departamento de Física Universidad del Norte Barranquilla Colombia

Institut de Physique École Polytechnique Fédérale de Lausanne Lausanne Switzerland

Institut für Physik Johannes Gutenberg Universität Mainz Mainz Germany

Institute of Physics Czech Academy of Sciences Prague Czech Republic

Institute of Semiconductor and Solid State Physics Johannes Kepler University of Linz Linz Austria

New Technologies Research Center University of West Bohemia Plzeň Czech Republic

Photon Science Division Paul Scherrer Institut Villigen Switzerland

Physik Institut Universität Zürich Zürich Switzerland

School of Physics and Astronomy University of Nottingham Nottingham United Kingdom

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Kramers HA. Théorie générale de la rotation paramagnétique dans les cristaux. Proc. Amsterdam Akad. 1930;33:959–972.

Wigner EP. Über die Operation der Zeitumkehr in der Quantenmechanik. Nachr. Ges. Wiss. Gottingen, Math. Phys. Kl. 1932;1932:546–559.

Chappert C, Fert A, Van Dau FN. The emergence of spin electronics in data storage. Nat. Mater. 2007;6:813–823. doi: 10.1038/nmat2024. PubMed DOI

Ralph DC, Stiles MD. Spin transfer torques. J. Magn. Magn. Mater. 2008;320:1190–1216. doi: 10.1016/j.jmmm.2007.12.019. DOI

Bader SD, Parkin S. Spintronics. Annu. Rev. Condens. Matter Phys. 2010;1:71–88. doi: 10.1146/annurev-conmatphys-070909-104123. DOI

Bhatti S, et al. Spintronics based random access memory: a review. Mater. Today. 2017;20:530–548. doi: 10.1016/j.mattod.2017.07.007. DOI

Manchon A, et al. Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems. Rev. Mod. Phys. 2019;91:035004. doi: 10.1103/RevModPhys.91.035004. DOI

Nagaosa N, Sinova J, Onoda S, MacDonald AH, Ong NP. Anomalous Hall effect. Rev. Mod. Phys. 2010;82:1539–1592. doi: 10.1103/RevModPhys.82.1539. DOI

Franz, M. & Molenkamp, L. (eds) Topological Insulators Vol. 6 (Elsevier, 2013).

Šmejkal L, Mokrousov Y, Yan B, MacDonald AH. Topological antiferromagnetic spintronics. Nat. Phys. 2018;14:242–251. doi: 10.1038/s41567-018-0064-5. DOI

Zang, J., Cros, V. & Hoffmann, A. (eds) Topology in Magnetism (Springer, 2018).

Tokura Y, Yasuda K, Tsukazaki A. Magnetic topological insulators. Nat. Rev. Phys. 2019;1:126–143. doi: 10.1038/s42254-018-0011-5. DOI

Vergniory MG, et al. A complete catalogue of high-quality topological materials. Nature. 2019;566:480–485. doi: 10.1038/s41586-019-0954-4. PubMed DOI

Šmejkal L, MacDonald AH, Sinova J, Nakatsuji S, Jungwirth T. Anomalous Hall antiferromagnets. Nat. Rev. Mater. 2022;7:482–496. doi: 10.1038/s41578-022-00430-3. DOI

Landau, L. D. & Lifshitz, E. M. Electrodynamics of Continuous Media 2nd edn (Pergamon Press, Oxford, 1984).

Winkler, R. Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, 2003).

Armitage NP, Mele EJ, Vishwanath A. Weyl and Dirac semimetals in three-dimensional solids. Rev. Mod. Phys. 2018;90:015001. doi: 10.1103/RevModPhys.90.015001. DOI

Krempaský J, et al. Disentangling bulk and surface Rashba effects in ferroelectric α-GeTe. Phys. Rev. B. 2016;94:205111. doi: 10.1103/PhysRevB.94.205111. DOI

Sante DD, Barone P, Bertacco R, Picozzi S. Electric control of the giant Rashba effect in bulk GeTe. Adv. Mater. 2013;25:509–513. doi: 10.1002/adma.201203199. PubMed DOI

Šmejkal L, Sinova J, Jungwirth T. Beyond conventional ferromagnetism and antiferromagnetism: a phase with nonrelativistic spin and crystal rotation symmetry. Phys. Rev. 2022;12:031042. doi: 10.1103/PhysRevX.12.031042. DOI

Šmejkal L, Sinova J, Jungwirth T. Emerging research landscape of altermagnetism. Phys. Rev. 2022;12:040501. doi: 10.1103/PhysRevX.12.040501. DOI

Gonzalez Betancourt RD, et al. Spontaneous anomalous Hall effect arising from an unconventional compensated magnetic phase in a semiconductor. Phys. Rev. Lett. 2023;130:036702. doi: 10.1103/PhysRevLett.130.036702. PubMed DOI

Mazin II. Altermagnetism in MnTe: origin, predicted manifestations, and routes to detwinning. Phys. Rev. B. 2023;107:L100418. doi: 10.1103/PhysRevB.107.L100418. DOI

Néel L. Magnetism and local molecular field. Science. 1971;174:985–992. doi: 10.1126/science.174.4013.985. PubMed DOI

Kunitomi N, Hamaguchi Y, Anzai S. Neutron diffraction study on manganese telluride. J. Phys. 1964;25:568–574. doi: 10.1051/jphys:01964002505056800. DOI

Šmejkal L, Železný J, Sinova J, Jungwirth T. Electric control of Dirac quasiparticles by spin-orbit torque in an antiferromagnet. Phys. Rev. Lett. 2017;118:106402. doi: 10.1103/PhysRevLett.118.106402. PubMed DOI

Naka M, et al. Spin current generation in organic antiferromagnets. Nat. Commun. 2019;10:4305. doi: 10.1038/s41467-019-12229-y. PubMed DOI PMC

González-Hernández R, et al. Efficient electrical spin splitter based on nonrelativistic collinear antiferromagnetism. Phys. Rev. Lett. 2021;126:127701. doi: 10.1103/PhysRevLett.126.127701. PubMed DOI

Naka M, Motome Y, Seo H. Perovskite as a spin current generator. Phys. Rev. B. 2021;103:125114. doi: 10.1103/PhysRevB.103.125114. DOI

Ma H-Y, et al. Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current. Nat. Commun. 2021;12:2846. doi: 10.1038/s41467-021-23127-7. PubMed DOI PMC

Šmejkal L, Hellenes AB, González-Hernández R, Sinova J, Jungwirth T. Giant and tunneling magnetoresistance in unconventional collinear antiferromagnets with nonrelativistic spin-momentum coupling. Phys. Rev. X. 2022;12:011028.

Šmejkal L, González-Hernández R, Jungwirth T, Sinova J. Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets. Sci. Adv. 2020;6:eaaz8809. doi: 10.1126/sciadv.aaz8809. PubMed DOI PMC

Samanta K, et al. Crystal Hall and crystal magneto-optical effect in thin films of SrRuO3. J. Appl. Phys. 2020;127:213904. doi: 10.1063/5.0005017. DOI

Naka M, et al. Anomalous Hall effect in κ-type organic antiferromagnets. Phys. Rev. B. 2020;102:075112. doi: 10.1103/PhysRevB.102.075112. DOI

Hayami S, Kusunose H. Essential role of the anisotropic magnetic dipole in the anomalous Hall effect. Phys. Rev. B. 2021;103:L180407. doi: 10.1103/PhysRevB.103.L180407. DOI

Mazin II, Koepernik K, Johannes MD, González-Hernández R, Šmejkal L. Prediction of unconventional magnetism in doped FeSb2. Proc. Natl Acad. Sci. 2021;118:e2108924118. doi: 10.1073/pnas.2108924118. PubMed DOI PMC

Naka M, Motome Y, Seo H. Anomalous Hall effect in antiferromagnetic perovskites. Phys. Rev. B. 2022;106:195149. doi: 10.1103/PhysRevB.106.195149. DOI

Feng Z, et al. An anomalous Hall effect in altermagnetic ruthenium dioxide. Nat. Electron. 2022;5:735–743. doi: 10.1038/s41928-022-00866-z. DOI

Bose A, et al. Tilted spin current generated by an antiferromagnet. Nat. Electron. 2022;5:263–264. doi: 10.1038/s41928-022-00758-2. DOI

Bai H, et al. Observation of spin splitting torque in a collinear antiferromagnet RuO2. Phys. Rev. Lett. 2022;128:197202. doi: 10.1103/PhysRevLett.128.197202. PubMed DOI

Karube S, et al. Observation of spin-splitter torque in collinear antiferromagnetic RuO2. Phys. Rev. Lett. 2022;129:137201. doi: 10.1103/PhysRevLett.129.137201. PubMed DOI

Strocov VN, et al. Soft-X-ray ARPES facility at the ADRESS beamline of the SLS: concepts, technical realisation and scientific applications. J. Synchrotron Radiat. 2014;21:32–44. doi: 10.1107/S1600577513019085. PubMed DOI

Kriegner D, et al. Multiple-stable anisotropic magnetoresistance memory in antiferromagnetic MnTe. Nat. Commun. 2016;7:11623. doi: 10.1038/ncomms11623. PubMed DOI PMC

Ebert H, Ködderitzsch D, Minár J. Calculating condensed matter properties using the KKR-Green’s function method—recent developments and applications. Rep. Prog. Phys. 2011;74:096501. doi: 10.1088/0034-4885/74/9/096501. DOI

Braun J, Minár J, Ebert H. Correlation, temperature and disorder: recent developments in the one-step description of angle-resolved photoemission. Phys. Rep. 2018;740:1–34. doi: 10.1016/j.physrep.2018.02.007. DOI

Zhang P, et al. A precise method for visualizing dispersive features in image plots. Rev. Sci. Instrum. 2011;82:043712. doi: 10.1063/1.3585113. PubMed DOI

Kriegner D, et al. Magnetic anisotropy in antiferromagnetic hexagonal MnTe. Phys. Rev. B. 2017;96:214418. doi: 10.1103/PhysRevB.96.214418. DOI

Ishizaka K, et al. Giant Rashba-type spin splitting in bulk BiTeI. Nat. Mater. 2011;10:521–526. doi: 10.1038/nmat3051. PubMed DOI

Braun J, et al. Exploring the XPS limit in soft and hard x-ray angle-resolved photoemission using a temperature-dependent one-step theory. Phys. Rev. B. 2013;88:205409. doi: 10.1103/PhysRevB.88.205409. DOI

Hoesch M, et al. Spin-polarized Fermi surface mapping. J. Electron. Spectrosc. Relat. Phenom. 2002;124:263–279. doi: 10.1016/S0368-2048(02)00058-0. DOI

Kriegner D, Wintersberger E, Stangl J. Xrayutilities: a versatile tool for reciprocal space conversion of scattering data recorded with linear and area detectors. J. Appl. Crystallogr. 2013;46:1162–1170. doi: 10.1107/S0021889813017214. PubMed DOI PMC

Damascelli A, Hussain Z, Shen Z-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 2003;75:473–541. doi: 10.1103/RevModPhys.75.473. DOI

Strocov V. Photoemission response of 2D electron states. J. Electron. Spectrosc. Relat. Phenom. 2018;229:100–107. doi: 10.1016/j.elspec.2018.09.001. DOI

Powell CJ, Jablonski A. Surface sensitivity of Auger-electron spectroscopy and X-ray photoelectron spectroscopy. J. Surf. Anal. 2011;17:170–176. doi: 10.1384/jsa.17.170. DOI

Strocov VN, et al. Three-dimensional electron realm in VSe2 by soft-x-ray photoelectron spectroscopy: origin of charge-density waves. Phys. Rev. Lett. 2012;109:086401. doi: 10.1103/PhysRevLett.109.086401. PubMed DOI

Weber F, et al. Three-dimensional Fermi surface of 2H–NbSe2: implications for the mechanism of charge density waves. Phys. Rev. B. 2018;97:235122. doi: 10.1103/PhysRevB.97.235122. DOI

Schröter N, et al. Observation and control of maximal Chern numbers in a chiral topological semimetal. Science. 2020;369:179–183. doi: 10.1126/science.aaz3480. PubMed DOI

Strocov VN, et al. High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies. J. Synchrotron Radiat. 2010;17:631–643. doi: 10.1107/S0909049510019862. PubMed DOI PMC

Dil JH. Spin and angle resolved photoemission on non-magnetic low-dimensional systems. J. Phys. Condens. Matter. 2009;21:403001. doi: 10.1088/0953-8984/21/40/403001. PubMed DOI

Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B. 1996;54:11169–11186. doi: 10.1103/PhysRevB.54.11169. PubMed DOI

Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996;78:1396–1396. doi: 10.1103/PhysRevLett.78.1396. PubMed DOI

Minár J. Correlation effects in transition metals and their alloys studied using the fully self-consistent KKR-based LSDA + DMFT scheme. J. Phys. Condens. Matter. 2011;23:253201. doi: 10.1088/0953-8984/23/25/253201. DOI

Lloyd P. Wave propagation through an assembly of spheres: II. The density of single-particle eigenstates. Proc. Phys. Soc. 1967;90:207. doi: 10.1088/0370-1328/90/1/323. DOI

Lloyd P, Smith P. Multiple scattering theory in condensed materials. Adv. Phys. 1972;21:69–142. doi: 10.1080/00018737200101268. DOI

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Anisotropic magnetoresistance in altermagnetic MnTe

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Direct observation of altermagnetic band splitting in CrSb thin films