{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,13]],"date-time":"2026-11-13T13:26:58Z","timestamp":1794576418166,"version":"4.9.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"8111","license":[{"start":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T00:00:00Z","timestamp":1776816000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T00:00:00Z","timestamp":1776816000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Nature"],"published-print":{"date-parts":[[2026,4,23]]},"DOI":"10.1038\/s41586-026-10401-1","type":"journal-article","created":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T15:03:07Z","timestamp":1776870187000},"page":"869-873","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Symmetry classification of magnetic orders using oriented spin space groups"],"prefix":"10.1038","volume":"652","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7528-3995","authenticated-orcid":false,"given":"Yuntian","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4327-1209","authenticated-orcid":false,"given":"Xiaobing","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2523-0511","authenticated-orcid":false,"given":"Yutong","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jes\u00fas","family":"Etxebarria","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. Manuel","family":"Perez-Mato","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9843-2482","authenticated-orcid":false,"given":"Qihang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,22]]},"reference":[{"key":"10401_CR1","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1038\/nnano.2016.18","volume":"11","author":"T Jungwirth","year":"2016","unstructured":"Jungwirth, T., Marti, X., Wadley, P. & Wunderlich, J. Antiferromagnetic spintronics. Nat. Nanotechnol. 11, 231\u2013241 (2016).","journal-title":"Nat. Nanotechnol."},{"key":"10401_CR2","doi-asserted-by":"publisher","first-page":"015005","DOI":"10.1103\/RevModPhys.90.015005","volume":"90","author":"V Baltz","year":"2018","unstructured":"Baltz, V. et al. Antiferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018).","journal-title":"Rev. Mod. Phys."},{"key":"10401_CR3","doi-asserted-by":"publisher","first-page":"482","DOI":"10.1038\/s41578-022-00430-3","volume":"7","author":"L \u0160mejkal","year":"2022","unstructured":"\u0160mejkal, L., MacDonald, A. H., Sinova, J., Nakatsuji, S. & Jungwirth, T. Anomalous Hall antiferromagnets. Nat. Rev. Mater. 7, 482\u2013496 (2022).","journal-title":"Nat. Rev. Mater."},{"key":"10401_CR4","doi-asserted-by":"crossref","unstructured":"\u0160mejkal, L., Sinova, J. & Jungwirth, T. Emerging research landscape of altermagnetism. Phys. Rev. X 12, 040501 (2022).","DOI":"10.1103\/PhysRevX.12.040501"},{"key":"10401_CR5","doi-asserted-by":"crossref","unstructured":"Brinkman, W. F. & Elliott, R. J. Theory of spin-space groups. Proc. R. Soc. Lond. A Math. Phys. Sci. 294, 343\u2013358 (1966).","DOI":"10.1098\/rspa.1966.0211"},{"key":"10401_CR6","doi-asserted-by":"publisher","first-page":"538","DOI":"10.1016\/0031-8914(74)90157-8","volume":"76","author":"DB Litvin","year":"1974","unstructured":"Litvin, D. B. & Opechowski, W. Spin groups. Physica 76, 538\u2013554 (1974).","journal-title":"Physica"},{"key":"10401_CR7","volume":"12","author":"P Liu","year":"2022","unstructured":"Liu, P., Li, J., Han, J., Wan, X. & Liu, Q. Spin-group symmetry in magnetic materials with negligible spin-orbit coupling. Phys. Rev. X 12, 021016 (2022).","journal-title":"Phys. Rev. X"},{"key":"10401_CR8","volume":"14","author":"X Chen","year":"2024","unstructured":"Chen, X. et al. Enumeration and representation theory of spin space groups. Phys. Rev. X 14, 031038 (2024).","journal-title":"Phys. Rev. X"},{"key":"10401_CR9","first-page":"031037","volume":"14","author":"Z Xiao","year":"2024","unstructured":"Xiao, Z., Zhao, J., Li, Y., Shindou, R. & Song, Z.-D. Spin space groups: full classification and applications. Phys. Rev. X 14, 031037 (2024).","journal-title":"Phys. Rev. X"},{"key":"10401_CR10","first-page":"031039","volume":"14","author":"Y Jiang","year":"2024","unstructured":"Jiang, Y. et al. Enumeration of spin-space groups: towards a complete description of symmetries of magnetic orders. Phys. Rev. X 14, 031039 (2024).","journal-title":"Phys. Rev. X"},{"key":"10401_CR11","doi-asserted-by":"crossref","unstructured":"Corticelli, A., Moessner, R. & McClarty, P. A. Spin-space groups and magnon band topology. Phys. Rev. B 105, 064430 (2022).","DOI":"10.1103\/PhysRevB.105.064430"},{"key":"10401_CR12","unstructured":"Shubnikov, A. V. Symmetry and Antisymmetry of Finite Figures (Academy of Sciences of the Soviet Union, 1951)."},{"key":"10401_CR13","unstructured":"Zamorzaev, A. M. Generalization of Fedorov Groups. Candidate dissertation, Leningrad State Univ. (1953)."},{"key":"10401_CR14","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1103\/RevModPhys.40.359","volume":"40","author":"CJ Bradley","year":"1968","unstructured":"Bradley, C. J. & Davies, B. L. Magnetic groups and their corepresentations. Rev. Mod. Phys. 40, 359 (1968).","journal-title":"Rev. Mod. Phys."},{"key":"10401_CR15","doi-asserted-by":"publisher","first-page":"1078","DOI":"10.1063\/1.1714108","volume":"36","author":"WC Koehler","year":"1965","unstructured":"Koehler, W. C. Magnetic properties of rare-earth metals and alloys. J. Appl. Phys. 36, 1078\u20131087 (1965).","journal-title":"J. Appl. Phys."},{"key":"10401_CR16","doi-asserted-by":"publisher","first-page":"414","DOI":"10.1103\/PhysRev.151.414","volume":"151","author":"WC Koehler","year":"1966","unstructured":"Koehler, W. C., Cable, J. W., Wilkinson, M. K. & Wollan, E. O. Magnetic structures of holmium. I. The virgin state. Phys. Rev. 151, 414 (1966).","journal-title":"Phys. Rev."},{"key":"10401_CR17","first-page":"031042","volume":"12","author":"L \u0160mejkal","year":"2022","unstructured":"\u0160mejkal, L., Sinova, J. & Jungwirth, T. Beyond conventional ferromagnetism and antiferromagnetism: a phase with nonrelativistic spin and crystal rotation symmetry. Phys. Rev. X 12, 031042 (2022).","journal-title":"Phys. Rev. X"},{"key":"10401_CR18","doi-asserted-by":"publisher","first-page":"060501","DOI":"10.1063\/5.0198953","volume":"125","author":"SW Cheong","year":"2024","unstructured":"Cheong, S. W. & Huang, F. T. Kinetomagnetism of chirality and its applications. Appl. Phys. Lett. 125, 060501 (2024).","journal-title":"Appl. Phys. Lett."},{"key":"10401_CR19","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1038\/s41567-024-02750-3","volume":"21","author":"Q Liu","year":"2025","unstructured":"Liu, Q., Dai, X. & Bl\u00fcgel, S. Different facets of unconventional magnetism. Nat. Phys. 21, 329\u2013331 (2025).","journal-title":"Nat. Phys."},{"key":"10401_CR20","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1051\/anphys\/194812030137","volume":"3","author":"L N\u00e9el","year":"1948","unstructured":"N\u00e9el, L. Propri\u00e9t\u00e9s magn\u00e9tiques des ferrites; ferrimagn\u00e9tisme et antiferromagn\u00e9tisme. Ann. Phys. 3, 137\u2013198 (1948).","journal-title":"Ann. Phys."},{"key":"10401_CR21","doi-asserted-by":"publisher","first-page":"110501","DOI":"10.1063\/1.5144076","volume":"116","author":"J Finley","year":"2020","unstructured":"Finley, J. & Liu, L. Spintronics with compensated ferrimagnets. Appl. Phys. Lett. 116, 110501 (2020).","journal-title":"Appl. Phys. Lett."},{"key":"10401_CR22","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1038\/s41563-021-01139-4","volume":"21","author":"SK Kim","year":"2022","unstructured":"Kim, S. K. et al. Ferrimagnetic spintronics. Nat. Mater. 21, 24\u201334 (2022).","journal-title":"Nat. Mater."},{"key":"10401_CR23","unstructured":"Landau, L. D. & Lifshitz, E. M. in Electrodynamics of Continuous Media Ch. 28 (Pergamon Press, 1960)."},{"key":"10401_CR24","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1107\/S1600576716012863","volume":"49","author":"SV Gallego","year":"2016","unstructured":"Gallego, S. V. et al. MAGNDATA: towards a database of magnetic structures. I. The commensurate case. J. Appl. Crystallogr. 49, 1750\u20131776 (2016).","journal-title":"J. Appl. Crystallogr."},{"key":"10401_CR25","doi-asserted-by":"publisher","first-page":"1748","DOI":"10.1103\/PhysRevLett.43.1748","volume":"43","author":"SM Shapiro","year":"1979","unstructured":"Shapiro, S. M., Gurewitz, E., Parks, R. D. & Kupferberg, L. C. Multiple-q magnetic structure in CeAl2. Phys. Rev. Lett. 43, 1748 (1979).","journal-title":"Phys. Rev. Lett."},{"key":"10401_CR26","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-019-13968-8","volume":"11","author":"W Feng","year":"2020","unstructured":"Feng, W. et al. Topological magneto-optical effects and their quantization in noncoplanar antiferromagnets. Nat. Commun. 11, 118 (2020).","journal-title":"Nat. Commun."},{"key":"10401_CR27","doi-asserted-by":"publisher","first-page":"961","DOI":"10.1038\/s41567-023-02017-3","volume":"19","author":"H Takagi","year":"2023","unstructured":"Takagi, H. et al. Spontaneous topological Hall effect induced by non-coplanar antiferromagnetic order in intercalated van der Waals materials. Nat. Phys. 19, 961\u2013968 (2023).","journal-title":"Nat. Phys."},{"key":"10401_CR28","doi-asserted-by":"crossref","unstructured":"Litvin, D. B. Spin translation groups and neutron diffraction analysis. Acta Crystallogr. A 29, 651\u2013660 (1973).","DOI":"10.1107\/S0567739473001658"},{"key":"10401_CR29","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1146\/annurev-matsci-070214-021008","volume":"45","author":"JM Perez-Mato","year":"2015","unstructured":"Perez-Mato, J. M. et al. Symmetry-based computational tools for magnetic crystallography. Annu. Rev. Mater. Res. 45, 217\u2013248 (2015).","journal-title":"Annu. Rev. Mater. Res."},{"key":"10401_CR30","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1016\/0022-3697(58)90076-3","volume":"4","author":"I Dzyaloshinsky","year":"1958","unstructured":"Dzyaloshinsky, I. A thermodynamic theory of \u201cweak\u201d ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4, 241\u2013255 (1958).","journal-title":"J. Phys. Chem. Solids"},{"key":"10401_CR31","doi-asserted-by":"publisher","first-page":"017205","DOI":"10.1103\/PhysRevLett.112.017205","volume":"112","author":"H Chen","year":"2014","unstructured":"Chen, H., Niu, Q. & MacDonald, A. H. Anomalous Hall effect arising from noncollinear antiferromagnetism. Phys. Rev. Lett. 112, 017205 (2014).","journal-title":"Phys. Rev. Lett."},{"key":"10401_CR32","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1038\/nature15723","volume":"527","author":"S Nakatsuji","year":"2015","unstructured":"Nakatsuji, S., Kiyohara, N. & Higo, T. Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature. Nature 527, 212\u2013215 (2015).","journal-title":"Nature"},{"key":"10401_CR33","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.aaz8809","volume":"6","author":"L \u0160mejkal","year":"2020","unstructured":"\u0160mejkal, L., Gonz\u00e1lez-Hern\u00e1ndez, R., Jungwirth, T. & Sinova, J. Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets. Sci. Adv. 6, eaaz8809 (2020).","journal-title":"Sci. Adv."},{"key":"10401_CR34","doi-asserted-by":"publisher","first-page":"036702","DOI":"10.1103\/PhysRevLett.130.036702","volume":"130","author":"RD Gonzalez Betancourt","year":"2023","unstructured":"Gonzalez Betancourt, R. D. et al. Spontaneous anomalous Hall effect arising from an unconventional compensated magnetic phase in a semiconductor. Phys. Rev. Lett. 130, 036702 (2023).","journal-title":"Phys. Rev. Lett."},{"key":"10401_CR35","doi-asserted-by":"publisher","first-page":"144426","DOI":"10.1103\/PhysRevB.92.144426","volume":"92","author":"W Feng","year":"2015","unstructured":"Feng, W., Guo, G. Y., Zhou, J., Yao, Y. & Niu, Q. Large magneto-optical Kerr effect in noncollinear antiferromagnets Mn3X (X = Rh, Ir, Pt). Phys. Rev. B 92, 144426 (2015).","journal-title":"Phys. Rev. B"},{"key":"10401_CR36","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1038\/s41566-017-0086-z","volume":"12","author":"T Higo","year":"2018","unstructured":"Higo, T. et al. Large magneto-optical Kerr effect and imaging of magnetic octupole domains in an antiferromagnetic metal. Nat. Photon.\u00a012, 73\u201378 (2018).","journal-title":"Nat. Photon."},{"key":"10401_CR37","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1103\/PhysRev.120.91","volume":"120","author":"T Moriya","year":"1960","unstructured":"Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91 (1960).","journal-title":"Phys. Rev."},{"key":"10401_CR38","doi-asserted-by":"publisher","first-page":"A1334","DOI":"10.1103\/PhysRev.133.A1334","volume":"133","author":"GF Herrmann","year":"1964","unstructured":"Herrmann, G. F. Magnetic resonances and susceptibility in orthoferrites. Phys. Rev. 133, A1334 (1964).","journal-title":"Phys. Rev."},{"key":"10401_CR39","doi-asserted-by":"publisher","first-page":"850","DOI":"10.1063\/1.1713506","volume":"35","author":"PL Richards","year":"1964","unstructured":"Richards, P. L. Antiferromagnetic resonance in CoF2, NiF2, and MnCo3. J Appl. Phys. 35, 850\u2013851 (1964).","journal-title":"J Appl. Phys."},{"key":"10401_CR40","doi-asserted-by":"publisher","first-page":"094438","DOI":"10.1103\/PhysRevB.109.094438","volume":"109","author":"H Watanabe","year":"2024","unstructured":"Watanabe, H., Shinohara, K., Nomoto, T., Togo, A. & Arita, R. Symmetry analysis with spin crystallographic groups: disentangling effects free of spin-orbit coupling in emergent electromagnetism. Phys. Rev. B 109, 094438 (2024).","journal-title":"Phys. Rev. B"},{"key":"10401_CR41","unstructured":"Hellenes, A. B. et al. P-wave magnets. Preprint at https:\/\/arxiv.org\/abs\/2309.01607 (2023)."},{"key":"10401_CR42","doi-asserted-by":"publisher","first-page":"206401","DOI":"10.1103\/PhysRevLett.122.206401","volume":"122","author":"D Zhang","year":"2019","unstructured":"Zhang, D. et al. Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect. Phys. Rev. Lett. 122, 206401 (2019).","journal-title":"Phys. Rev. Lett."},{"key":"10401_CR43","doi-asserted-by":"publisher","first-page":"064202","DOI":"10.1103\/PhysRevMaterials.3.064202","volume":"3","author":"JQ Yan","year":"2019","unstructured":"Yan, J. Q. et al. Crystal growth and magnetic structure of MnBi2Te4. Phys. Rev. Mater. 3, 064202 (2019).","journal-title":"Phys. Rev. Mater."},{"key":"10401_CR44","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-021-21154-y","volume":"12","author":"SXM Riberolles","year":"2021","unstructured":"Riberolles, S. X. M. et al. Magnetic crystalline-symmetry-protected axion electrodynamics and field-tunable unpinned Dirac cones in EuIn2As2. Nat. Commun. 12, 999 (2021).","journal-title":"Nat. Commun."},{"key":"10401_CR45","first-page":"031006","volume":"15","author":"Z Liu","year":"2025","unstructured":"Liu, Z. et al. Multipolar anisotropy in anomalous Hall effect from spin-group symmetry breaking. Phys. Rev. X 15, 031006 (2025).","journal-title":"Phys. Rev. X"},{"key":"10401_CR46","doi-asserted-by":"publisher","first-page":"11169","DOI":"10.1103\/PhysRevB.54.11169","volume":"54","author":"G Kresse","year":"1996","unstructured":"Kresse, G. & Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).","journal-title":"Phys. Rev. B"},{"key":"10401_CR47","doi-asserted-by":"publisher","first-page":"1758","DOI":"10.1103\/PhysRevB.59.1758","volume":"59","author":"G Kresse","year":"1999","unstructured":"Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758 (1999).","journal-title":"Phys. Rev. B"},{"key":"10401_CR48","doi-asserted-by":"publisher","first-page":"3865","DOI":"10.1103\/PhysRevLett.77.3865","volume":"77","author":"JP Perdew","year":"1996","unstructured":"Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).","journal-title":"Phys. Rev. Lett."},{"key":"10401_CR49","doi-asserted-by":"publisher","first-page":"1396","DOI":"10.1103\/PhysRevLett.78.1396","volume":"78","author":"JP Perdew","year":"1997","unstructured":"Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple: erratum. Phys. Rev. Lett. 78, 1396 (1997).","journal-title":"Phys. Rev. Lett."},{"key":"10401_CR50","doi-asserted-by":"publisher","first-page":"5188","DOI":"10.1103\/PhysRevB.13.5188","volume":"13","author":"HJ Monkhorst","year":"1976","unstructured":"Monkhorst, H. J. & Pack, J. D. Special points for Brillouin-zone integrations. Phys. Rev. B 13, 5188 (1976).","journal-title":"Phys. Rev. B"}],"container-title":["Nature"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41586-026-10401-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41586-026-10401-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41586-026-10401-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T15:03:07Z","timestamp":1779375787000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41586-026-10401-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,22]]},"references-count":50,"journal-issue":{"issue":"8111","published-print":{"date-parts":[[2026,4,23]]}},"alternative-id":["10401"],"URL":"https:\/\/doi.org\/10.1038\/s41586-026-10401-1","relation":{},"ISSN":["0028-0836","1476-4687"],"issn-type":[{"value":"0028-0836","type":"print"},{"value":"1476-4687","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,22]]},"assertion":[{"value":"28 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 March 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}