- Physical Review Journals
All JournalsPhysics Magazine

Physical Review X

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Unified Symmetry Classification of Magnetic Orders via Spin Space Groups: Prediction of Coplanar Even-Wave Phases

Ziyin Song1,2, Ziyue Qi1,2, Chen Fang1, Zhong Fang1,2, and Hongming Weng1,3,*

  • *Contact author: hmweng@iphy.ac.cn
PDF

Phys. Rev. X 16, 031038 – Published 13 August, 2026

DOI: https://doi.org/10.1103/zy7s-j86r

Abstract

Spin space groups (SSGs) impose fundamentally different constraints on magnetic configurations in real and reciprocal spaces. As a consequence, the correspondence between real-space and momentum-space spin arrangements is far richer than traditionally assumed. Building on the complete enumeration of SSGs, we develop a systematic, symmetry-based framework that classifies all possible spin arrangements allowed by these groups. This unified approach naturally incorporates conventional magnetic orders, altermagnetism, and coplanar odd-wave magnetism (including the previously proposed p-wave case) as distinct symmetry classes. Moreover, by determining the symmetry-allowed wave patterns of spin-polarized phases, our framework classifies coplanar odd-wave magnets beyond the p-wave case into higher-wave types (f, h, j, and l wave). Crucially, our classification predicts a variety of novel magnetic phases, highlighted by the discovery of the coplanar even-wave magnet(including d, g, and i wave): A state that is noncollinear in real space but hosts a collinear even-wave spin polarization in k space. Analysis of a minimal model reveals that this phase is characterized by nonquantized spin polarization and exhibits a novel mechanism for symmetry-enforced zero polarization on nondegenerate bands. Extending the framework from bulk crystals to layer SSGs appropriate for two-dimensional systems, we further predict layered counterparts and provide symmetry guidelines for designing bilayer coplanar odd-wave and even-wave magnets. We further validate this finding through first-principles calculations and propose CoCrO4 as a promising candidate for its experimental realization, thereby demonstrating the completeness and predictive power of the SSG-based classification of magnetic orders.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Classifying newly discovered magnetic materials has been limited by a piecemeal approach because a unified theoretical framework to systematically identify and predict unconventional spin patterns has been lacking. We resolved this limitation by using spin-space group symmetry to systematically categorize magnetic orders, allowing us to capture conventional magnets, altermagnets, and p-wave magnets under a single framework while predicting a new coplanar even-wave magnetic phase. In this proposed phase, spins form a coplanar antiferromagnetic pattern in real space, whereas the corresponding electronic bands display an even-wave spin polarization in momentum space. We demonstrated that this configuration enables the spin polarization of individual bands to vary continuously and even vanish entirely without the bands becoming degenerate. We identified cobalt chromate (CoCrO4) as a promising candidate material and showed how related phases can be engineered in layered structures. Our work establishes a practical, symmetry-guided taxonomy for discovering and designing unconventional magnetic materials with customized electronic and spintronic properties.

Article Text

Supplemental Material

References (64)

  1. C. J. Bradley and B. L. Davies, Magnetic groups and their corepresentations, Rev. Mod. Phys. 40, 359 (1968).
  2. Ron Lifshitz, Magnetic point groups and space groups, in Encyclopedia of Condensed Matter Physics (Second Edition), edited by Tapash Chakraborty (Academic Press, Oxford, 2024), pp. 1–10.
  3. Daniel B. Litvin, Magnetic Group Tables: 1-, 2- and 3-Dimensional Magnetic Subperiodic Groups and Magnetic Space Groups (International Union of Crystallography, Chester, 2013).
  4. D. B. Litvin and W. Opechowski, Spin groups, Physica 76, 538 (1974).
  5. W. F. Brinkman and R. J. Elliott, Theory of spin-space groups, Proc. R. Soc. A 294, 343 (1966).
  6. Pengfei Liu, Jiayu Li, Jingzhi Han, Xiangang Wan, and Qihang Liu, Spin-group symmetry in magnetic materials with negligible spin-orbit coupling, Phys. Rev. X 12, 021016 (2022).
  7. Zheng Liu, Mengjie Wei, Wenzhi Peng, Dazhi Hou, Yang Gao, and Qian Niu, Multipolar anisotropy in anomalous Hall effect from spin-group symmetry breaking, Phys. Rev. X 15, 031006 (2025).
  8. Yuntian Liu, Xiaobing Chen, Yutong Yu, Jesús Etxebarria, J. Manuel Perez-Mato, and Qihang Liu, Symmetry classification of magnetic orders using oriented spin space groups, Nature (London) 652, 869 (2026).
  9. Jian Yang, Zheng-Xin Liu, and Chen Fang, Symmetry invariants and classes of quasiparticles in magnetically ordered systems having weak spin-orbit coupling, Nat. Commun. 15, 10203 (2024).
  10. Peng-Jie Guo, Yi-Wen Wei, Kai Liu, Zheng-Xin Liu, and Zhong-Yi Lu, Eightfold degenerate fermions in two dimensions, Phys. Rev. Lett. 127, 176401 (2021).
  11. Pengfei Liu, Ao Zhang, Jingzhi Han, and Qihang Liu, Chiral Dirac-like fermion in spin-orbit-free antiferromagnetic semimetals, Innovation 3, 100343 (2022).
  12. A. Corticelli, R. Moessner, and P. A. McClarty, Spin-space groups and magnon band topology, Phys. Rev. B 105, 064430 (2022).
  13. Xiaobing Chen, Yuntian Liu, Pengfei Liu, Yutong Yu, Jun Ren, Jiayu Li, Ao Zhang, and Qihang Liu, Unconventional magnons in collinear magnets dictated by spin space groups, Nature (London) 640, 349 (2025).
  14. Sheng Zhang, Ziyin Song, Zhong Fang, Hongming Weng, and Zhijun Wang, IRSSG: An open-source software package for spin space groups, Comput. Phys. Commun. 326, 110190 (2026).
  15. Ziyin Song, A. Z. Yang, Yi Jiang, Zhong Fang, Jian Yang, Chen Fang, Hongming Weng, and Zheng-Xin Liu, Constructions and applications of irreducible representations of spin-space groups, Phys. Rev. B 111, 134407 (2025).
  16. Libor Šmejkal, Jairo Sinova, and Tomas Jungwirth, Beyond conventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022).
  17. Libor Šmejkal, Jairo Sinova, and Tomas Jungwirth, Emerging research landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022).
  18. Igor Mazin (The PRX Editors), Editorial: Altermagnetism—A new punch line of fundamental magnetism, Phys. Rev. X 12, 040002 (2022).
  19. Yusuke Noda, Kaoru Ohno, and Shinichiro Nakamura, Momentum-dependent band spin splitting in semiconducting MnO2: A density functional calculation, Phys. Chem. Chem. Phys. 18, 13294 (2016).
  20. Libor Šmejkal, Rafael González-Hernández, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets, Sci. Adv. 6, eaaz8809 (2020).
  21. Kyo-Hoon Ahn, Atsushi Hariki, Kwan-Woo Lee, and Jan Kuneš, Antiferromagnetism in RuO2 as d-wave Pomeranchuk instability, Phys. Rev. B 99, 184432 (2019).
  22. Lin-Ding Yuan, Zhi Wang, Jun-Wei Luo, Emmanuel I. Rashba, and Alex Zunger, Giant momentum-dependent spin splitting in centrosymmetric low-z antiferromagnets, Phys. Rev. B 102, 014422 (2020).
  23. Ling Bai, Wanxiang Feng, Siyuan Liu, Libor Šmejkal, Yuriy Mokrousov, and Yugui Yao, Altermagnetism: Exploring new frontiers in magnetism and spintronics, Adv. Funct. Mater. 34, 2409327 (2024).
  24. T. Jungwirth, J. Sinova, P. Wadley, D. Kriegner, H. Reichlová, F. Krizek, H. Ohno, and L. Šmejkal, Altermagnetic spintronics, Nat. Phys. (2026).
  25. Anna Birk Hellenes, Tomáš Jungwirth, Rodrigo Jaeschke-Ubiergo, Atasi Chakraborty, Jairo Sinova, and Libor Šmejkal, P-wave magnets, arXiv:2309.01607.
  26. Bjørnulf Brekke, Pavlo Sukhachov, Hans Gløckner Giil, Arne Brataas, and Jacob Linder, Minimal models and transport properties of unconventional p-wave magnets, Phys. Rev. Lett. 133, 236703 (2024).
  27. Rinsuke Yamada et al., A metallic p-wave magnet with commensurate spin helix, Nature (London) 646, 837 (2025).
  28. Atasi Chakraborty, Anna Birk Hellenes, Rodrigo Jaeschke-Ubiergo, Tomás Jungwirth, Libor Šmejkal, and Jairo Sinova, Highly efficient non-relativistic Edelstein effect in nodal p-wave magnets, Nat. Commun. 16, 7270 (2025).
  29. Zi-Ting Sun, Xilin Feng, Ying-Ming Xie, Benjamin T. Zhou, Jin-Xin Hu, and K. T. Law, Pseudo-Ising superconductivity induced by p-wave magnetism, Phys. Rev. B 112, 214504 (2025).
  30. Qihang Liu, Xi Dai, and Stefan Blügel, Different facets of unconventional magnetism, Nat. Phys. 21, 329 (2025).
  31. Jian-Keng Yuan, Zhiming Pan, and Congjun Wu, Unconventional magnetism in spin-orbit coupled systems, Phys. Rev. B 113, 014426 (2026).
  32. Congjun Wu and Shou-Cheng Zhang, Dynamic generation of spin-orbit coupling, Phys. Rev. Lett. 93, 036403 (2004).
  33. Congjun Wu, Kai Sun, Eduardo Fradkin, and Shou-Cheng Zhang, Fermi liquid instabilities in the spin channel, Phys. Rev. B 75, 115103 (2007).
  34. Xun-Jiang Luo, Jin-Xin Hu, and K. T. Law, Spin symmetry criteria for odd-parity magnets, arXiv:2510.05512.
  35. Xiaobing Chen, Weizhao Chen, and Qihang Liu, The rise of unconventional magnetism, arXiv:2603.27505.
  36. Mengli Hu, Mikel I. Iraola, Paul McClarty, Jeroen van den Brink, and Maia G. Vergniory, Non-collinear altermagnetic phases in the Mott insulator NiS2, arXiv:2603.01329.
  37. Zhenyu Xiao, Jianzhou Zhao, Yanqi Li, Ryuichi Shindou, and Zhi-Da Song, Spin space groups: Full classification and applications, Phys. Rev. X 14, 031037 (2024).
  38. Yi Jiang, Ziyin Song, Tiannian Zhu, Zhong Fang, Hongming Weng, Zheng-Xin Liu, Jian Yang, and Chen Fang, Enumeration of spin-space groups: Toward a complete description of symmetries of magnetic orders, Phys. Rev. X 14, 031039 (2024).
  39. Xiaobing Chen, Jun Ren, Yanzhou Zhu, Yutong Yu, Ao Zhang, Pengfei Liu, Jiayu Li, Yuntian Liu, Caiheng Li, and Qihang Liu, Enumeration and representation theory of spin space groups, Phys. Rev. X 14, 031038 (2024).
  40. Gemma de la Flor, Bernd Souvignier, Gotzon Madariaga, and Mois I. Aroyo, Layer groups: Brillouin-zone and crystallographic databases on the Bilbao crystallographic server, Acta Crystallogr. Sect. A 77, 559 (2021).
  41. See Supplemental Material at http://link.aps.org/supplemental/10.1103/zy7s-j86r for details of minimal models, the mapping to altermagnetic phases, candidate materials, response-tensor analysis, and first-principles calculations, which includes Refs. [42–48].
  42. Yichi Zhang, Hua Bai, Jiankun Dai, Lei Han, Chong Chen, Shixuan Liang, Yanzhang Cao, Yingying Zhang, Qian Wang, Wenxuan Zhu, Feng Pan, and Cheng Song, Electrical manipulation of spin splitting torque in altermagnetic RuO2, Nat. Commun. 16, 5646 (2025).
  43. Yibo Fan, Qian Wang, and Wei Wang, Robust magnetic-field-free perpendicular magnetization switching by manipulating spin polarization direction in RuO2/[Pt/Co/Pt] heterojunctions, ACS Nano 18, 26350 (2024).
  44. Y. Zhang, Y. Sun, H. Yang, J. Železný, Stuart P. P. Parkin, Claudia Felser, and Binghai Yan, Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds mn3x (x=Ge, Sn, Ga, Ir, Rh, and Pt), Phys. Rev. B 95, 075128 (2017).
  45. T. Farajollahpour, R. Ganesh, and K. V. Samokhin, Berry curvature-induced transport signature for altermagnetic order, npj Quantum Mater. 10, 77 (2025).
  46. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  47. John P. Perdew, Kieron Burke, and Matthias Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  48. Hendrik J. Monkhorst and James D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  49. Ran He, Dan Wang, Nannan Luo, Jiang Zeng, Ke-Qiu Chen, and Li-Ming Tang, Nonrelativistic spin-momentum coupling in antiferromagnetic twisted bilayers, Phys. Rev. Lett. 130, 046401 (2023).
  50. Yunxi Qi, Jun Zhao, and Hui Zeng, Spin-layer coupling in two-dimensional altermagnetic bilayers with tunable spin and valley splitting properties, Phys. Rev. B 110, 014442 (2024).
  51. Jianke Tian, Jia Li, Hengbo Liu, Yan Li, Ze Liu, Linyang Li, Jun Li, Guodong Liu, and Junjie Shi, Spin-layer coupling in an altermagnetic multilayer: A design principle for spintronics, Phys. Rev. B 111, 035437 (2025).
  52. Samuel V. Gallego, J. Manuel Perez-Mato, Luis Elcoro, Emre S. Tasci, Robert M. Hanson, Koichi Momma, Mois I. Aroyo, and Gotzon Madariaga, MAGNDATA: Towards a database of magnetic structures. I. The commensurate case, J. Appl. Crystallogr. 49, 1750 (2016).
  53. Marc Pernet, G. Quezel, Jean Coing-Boyat, and Félix Bertaut, Structures magnétiques des chromates de cobalt et de nickel, Bull. Minéral. 92, 264 (1969).
  54. S. Kume, F. Kanamaru, Y. Shibasaki, M. Koizumi, K. Yasunami, and T. Fukuda, Generation of high pressure oxygen and synthesis of CoCrO4 with CrVO4 type structure, Rev. Sci. Instrum. 42, 1856 (1971).
  55. Luis Elcoro, Jesus Etxebarria, J. Manuel Perez-Mato, and Emre S. Tasci, Automatic calculation of symmetry-adapted tensors under spin-group symmetry: STENSOR, a new tool of the Bilbao crystallographic server, J. Appl. Crystallogr. 59, 640 (2026).
  56. H. A. Jahn, Note on the Bhagavantam-Suryanarayana method of enumerating the physical constants of crystals, Acta Crystallogr. 2, 30 (1949).
  57. S. V. Gallego, J. Etxebarria, L. Elcoro, E. S. Tasci, and J. M. Perez-Mato, Automatic calculation of symmetry-adapted tensors in magnetic and non-magnetic materials: A new tool of the Bilbao crystallographic server, Acta Crystallogr. Sect. A 75, 438 (2019).
  58. Jesus Etxebarria, J. Manuel Perez-Mato, Emre S. Tasci, and Luis Elcoro, Crystal tensor properties of magnetic materials with and without spin–orbit coupling. Application of spin point groups as approximate symmetries, Acta Crystallogr. Sect. A 81, 317 (2025).
  59. Rui-Chun Xiao, Yuanjun Jin, Zhi-Fan Zhang, Zi-Hao Feng, Ding-Fu Shao, and Mingliang Tian, tensorsymmetry: A package to get symmetry-adapted tensors disentangling spin-orbit coupling effect and establishing analytical relationship with magnetic order, Comput. Phys. Commun. 318, 109872 (2026).
  60. Hikaru Watanabe, Kohei Shinohara, Takuya Nomoto, Atsushi Togo, and Ryotaro Arita, Symmetry analysis with spin crystallographic groups: Disentangling effects free of spin-orbit coupling in emergent electromagnetism, Phys. Rev. B 109, 094438 (2024).
  61. Rafael González-Hernández, Libor Šmejkal, Karel Výborný, Yuta Yahagi, Jairo Sinova, Tomá š Jungwirth, and Jakub Železný, Efficient electrical spin splitter based on nonrelativistic collinear antiferromagnetism, Phys. Rev. Lett. 126, 127701 (2021).
  62. Jakub Železný, Yang Zhang, Claudia Felser, and Binghai Yan, Spin-polarized current in noncollinear antiferromagnets, Phys. Rev. Lett. 119, 187204 (2017).
  63. Mengli Hu, Oleg Janson, Claudia Felser, Paul McClarty, Jeroen van den Brink, and Maia G. Vergniory, Spin Hall and Edelstein effects in chiral non-collinear altermagnets, Nat. Commun. 16, 8529 (2025).
  64. Ziyin Song, SSGClassify, GitHub repository, https://github.com/zine-phy/SSGClassify (2026).

Outline

Information

Phys. Rev. X 16, 031038– Published 13 August, 2026
  • Received 3 February 2026
  • Revised 13 May 2026
  • Accepted 14 July 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation