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Ligand-mediated origin of altermagnetic spin splitting

Luigi Camerano1,2,*, Federico Bisti1, and Gianni Profeta1,2

  • 1Department of Physical and Chemical Sciences, University of L'Aquila, Via Vetoio, 67100 L'Aquila, Italy
  • 2CNR-SPIN L'Aquila, Via Vetoio, 67100 L'Aquila, Italy

  • *Contact author: luigi_camerano@outlook.it
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Phys. Rev. B 114, 264408 – Published 5 November, 2026

DOI: https://doi.org/10.1103/s83n-w474

Abstract

Altermagnets host spin-split electronic bands despite zero net magnetization, opening new routes for spintronics beyond conventional ferromagnets. Going beyond symmetry-based classifications, which specify allowed terms but not their hierarchy, here we use first-principles calculations and Wannier Hamiltonian engineering to uncover the microscopic bonding contributions of altermagnetic spin splitting in the g-wave altermagnet Co1/4NbSe2. We show that the splitting is captured by a short-range tight-binding model, establishing its local origin. By selectively controlling hopping channels, we demonstrate that the dominant contribution arises not from direct magnetic-ion hopping, but from ligand-mediated hybridization that transfers anisotropy to itinerant states. This identifies ligand-assisted coupling as a dominant mechanism driving altermagnetic spin splitting in intercalated metals, providing a microscopic bridge between minimal models and material-specific electronic structure calculations in altermagnets.

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References (70)

  1. L. Šmejkal, R. González-Hernández, T. Jungwirth, and J. Sinova, Crystal time-reversal symmetry breaking and spontaneous Hall effect in collinear antiferromagnets, Sci. Adv. 6, eaaz8809 (2020).
  2. L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conventional ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022).
  3. L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging research landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022).
  4. P. G. Radaelli, Tensorial approach to altermagnetism, Phys. Rev. B 110, 214428 (2024).
  5. P. Liu, J. Li, J. Han, X. Wan, and Q. Liu, Spin-group symmetry in magnetic materials with negligible spin-orbit coupling, Phys. Rev. X 12, 021016 (2022).
  6. C. Song, H. Bai, Z. Zhou, L. Han, H. Reichlova, J. H. Dil, J. Liu, X. Chen, and F. Pan, Altermagnets as a new class of functional materials, Nat. Rev. Mater. 10, 473 (2025).
  7. H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Multifunctional antiferromagnetic materials with giant piezomagnetism and noncollinear spin current, Nat. Commun. 12, 2846 (2021).
  8. M. Hu, X. Cheng, Z. Huang, and J. Liu, Catalog of c-paired spin-momentum locking in antiferromagnetic systems, Phys. Rev. X 15, 021083 (2025).
  9. X. Chen, J. Ren, Y. Zhu, Y. Yu, A. Zhang, P. Liu, J. Li, Y. Liu, C. Li, and Q. Liu, Enumeration and representation theory of spin space groups, Phys. Rev. X 14, 031038 (2024).
  10. Y. Jiang, Z. Song, T. Zhu, Z. Fang, H. Weng, Z.-X. Liu, J. Yang, and C. Fang, Enumeration of spin-space groups: Toward a complete description of symmetries of magnetic orders, Phys. Rev. X 14, 031039 (2024).
  11. S.-W. Cheong and F.-T. Huang, Altermagnetism with non-collinear spins, npj Quantum Mater. 9, 13 (2024).
  12. R. González-Hernández, L. Šmejkal, K. Výborný, Y. Yahagi, J. Sinova, T. Jungwirth, and J. Železný, Efficient electrical spin splitter based on nonrelativistic collinear antiferromagnetism, Phys. Rev. Lett. 126, 127701 (2021).
  13. D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, and E. Y. Tsymbal, Spin-neutral currents for spintronics, Nat. Commun. 12, 7061 (2021).
  14. Y. Guo, X. Zhang, Z. Huang, J. Chen, Z. Luo, J. Zhang, J. Li, Z. Zhang, J. Zhao, X. Han, and H. Wu, Quantum materials for spintronic applications, npj Spintron. 2, 36 (2024).
  15. Z. Guo, X. Wang, W. Wang, G. Zhang, X. Zhou, and Z. Cheng, Spin‐polarized antiferromagnets for spintronics, Adv. Mater. 37, 2505779 (2025).
  16. R. Zarzuela, R. Jaeschke-Ubiergo, O. Gomonay, L. Šmejkal, and J. Sinova, Transport theory and spin-transfer physics in d-wave altermagnets, Phys. Rev. B 111, 064422 (2025).
  17. J. Krempaský, L. Šmejkal, S. W. D'Souza, M. Hajlaoui, G. Springholz, K. Uhlířová, F. Alarab, P. C. Constantinou, V. Strocov, D. Usanov, W. R. Pudelko, R. González-Hernández, A. Birk Hellenes, Z. Jansa, H. Reichlová, Z. Šobáň, R. D. Gonzalez Betancourt, P. Wadley, J. Sinova, D. Kriegner, et al., Altermagnetic lifting of Kramers spin degeneracy, Nature (London) 626, 517 (2024).
  18. T. Osumi, S. Souma, T. Aoyama, K. Yamauchi, A. Honma, K. Nakayama, T. Takahashi, K. Ohgushi, and T. Sato, Observation of a giant band splitting in altermagnetic MnTe, Phys. Rev. B 109, 115102 (2024).
  19. Y.-P. Zhu, X. Chen, X.-R. Liu, Y. Liu, P. Liu, H. Zha, G. Qu, C. Hong, J. Li, Z. Jiang, X.-M. Ma, Y.-J. Hao, M.-Y. Zhu, W. Liu, M. Zeng, S. Jayaram, M. Lenger, J. Ding, S. Mo, K. Tanaka, et al., Observation of plaid-like spin splitting in a noncoplanar antiferromagnet, Nature (London) 626, 523 (2024).
  20. O. J. Amin, A. Dal Din, E. Golias, Y. Niu, A. Zakharov, S. C. Fromage, C. J. B. Fields, S. L. Heywood, R. B. Cousins, F. Maccherozzi, J. Krempaský, J. H. Dil, D. Kriegner, B. Kiraly, R. P. Campion, A. W. Rushforth, K. W. Edmonds, S. S. Dhesi, L. Šmejkal, T. Jungwirth, et al., Nanoscale imaging and control of altermagnetism in MnTe, Nature (London) 636, 348 (2024).
  21. B. Jiang, M. Hu, J. Bai, Z. Song, C. Mu, G. Qu, W. Li, W. Zhu, H. Pi, Z. Wei, Y.-J. Sun, Y. Huang, X. Zheng, Y. Peng, L. He, S. Li, J. Luo, Z. Li, G. Chen, H. Li, et al., A metallic room-temperature d-wave altermagnet, Nat. Phys. 21, 754 (2025).
  22. F. Zhang, X. Cheng, Z. Yin, C. Liu, L. Deng, Y. Qiao, Z. Shi, S. Zhang, J. Lin, Z. Liu, M. Ye, Y. Huang, X. Meng, C. Zhang, T. Okuda, K. Shimada, S. Cui, Y. Zhao, G.-H. Cao, S. Qiao, et al., Crystal-symmetry-paired spin–valley locking in a layered room-temperature metallic altermagnet candidate, Nat. Phys. 21, 760 (2025).
  23. A. P. Sakhya, M. I. Mondal, M. Sprague, R. B. Regmi, A. K. Kumay, H. Sheokand, I. I. Mazin, N. J. Ghimire, and M. Neupane, Electronic structure of a layered altermagnetic compound CoNb4Se8, arXiv:2503.16670.
  24. N. Dale, O. A. Ashour, M. Vila, R. B. Regmi, J. Fox, C. W. Johnson, E. S. Barnard, A. Fedorov, A. Stibor, N. J. Ghimire, and S. M. Griffin, Relativistic and nonrelativistic spin splitting above and below the Fermi level in a g-wave altermagnet, arXiv:2411.18761.
  25. A. De Vita, C. Bigi, D. Romanin, M. D. Watson, V. Polewczyk, M. Zonno, F. Bertran, M. B. Petersen, F. Motti, G. Vinai, M. Tuniz, F. Cilento, M. Cuoco, B. M. Andersen, A. Kreisel, L. J. D'Onofrio, O. J. Clark, M. T. Edmonds, C. Candelora, M. Xu, et al., Robust spin splitting and fermiology in the layered altermagnet CoNb4Se8, Adv. Mater. e75367 (2026).
  26. X. Yang, Z. Q. Wang, and H. H. Fu, Anisotropic spin transport enhanced by zero‐dimensional weyl nodes in altermagnetic weyl semimetals, Adv. Funct. Mater. 36, 2502516 (2025).
  27. W. Lu, S. Feng, Y. Wang, D. Chen, Z. Lin, X. Liang, S. Liu, W. Feng, K. Yamagami, J. Liu, C. Felser, Q. Wu, and J. Ma, Signature of topological surface bands in altermagnetic Weyl semimetal CrSb, Nano Lett. 25, 7343 (2025).
  28. M. Leiviskä, R. Firouzmandi, K.-H. Ahn, P. Kubaščik, Z. Soban, S. P. Bommanaboyena, C. Müller, D. Kriegner, S. Sailler, D. Reustlen, M. Lammel, K. K. Bestha, M. Hývl, L. Šmejkal, J. Železný, A. U. B. Wolter, M. Scheufele, J. Fischer, M. Opel, S. Geprägs, et al., Spin Hall magnetoresistance at the altermagnetic insulator/Pt interface, Phys. Rev. Mater. 9, 084403 (2025).
  29. Z. Cui, Z. Zhu, X. Duan, B. Hao, X. Chen, J. Zhang, and T. Zhou, Symmetry‐driven unconventional magnetoelectric coupling in perovskite altermagnets: From bulk to the two‐dimensional limit, Adv. Sci. e00004 (2026).
  30. T. Zhang, L. Yuan, J. M. Rondinelli, H. A. Fertig, and S. Zhang, Tunable hidden altermagnetic spin splitting in layered ruddlesden–popper oxides, Nano Lett. 26, 2778 (2026).
  31. S. Bandyopadhyay, S. Picozzi, and S. Bhowal, Designing nonrelativistic spin splitting in oxide perovskites, Phys. Rev. B 112, 064405 (2025).
  32. L. Camerano, A. O. Fumega, J. L. Lado, A. Stroppa, and G. Profeta, Multiferroic nematic d-wave altermagnetism driven by orbital-order on the honeycomb lattice, npj 2D Mater. Appl. 9, 75 (2025).
  33. C. Yu and J. L. Lado, Intrinsic i-wave altermagnetism in 2D graphene superlattices, Adv. Sci. e78167 (2026).
  34. Z. Zhu, X. Duan, J. Zhang, B. Hao, I. Žutić, and T. Zhou, Two-dimensional ferroelectric altermagnets: From model to material realization, Nano Lett. 25, 9456 (2025).
  35. J. Sødequist and T. Olsen, Two-dimensional altermagnets from high throughput computational screening: Symmetry requirements, chiral magnons, and spin-orbit effects, Appl. Phys. Lett. 124, 182409 (2024).
  36. Y. Liu, J. Yu, and C.-C. Liu, Twisted magnetic van der Waals bilayers: An ideal platform for altermagnetism, Phys. Rev. Lett. 133, 206702 (2024).
  37. M. Roig, A. Kreisel, Y. Yu, B. M. Andersen, and D. F. Agterberg, Minimal models for altermagnetism, Phys. Rev. B 110, 144412 (2024).
  38. R. González-Hernández and B. Uribe, Model Hamiltonian for altermagnetic topological insulators, Phys. Rev. B 112, 184101 (2025).
  39. M. Dürrnagel, H. Hohmann, A. Maity, J. Seufert, M. Klett, L. Klebl, and R. Thomale, Altermagnetic phase transition in a Lieb metal, Phys. Rev. Lett. 135, 036502 (2025).
  40. M. Dürrnagel, L. Klebl, T. Müller, R. Thomale, and M. Klett, Extended s-wave altermagnets arXiv:2508.20163.
  41. J. Issing, J. Seufert, M. Klett, S. Mazumdar, Y. Iqbal, R. Thomale, and A. Maity, Engineering altermagnetic orders on the square-kagome lattice through sublattice interference, Phys. Rev. B 114, 115128 (2026).
  42. V. Leeb, A. Mook, L. Šmejkal, and J. Knolle, Spontaneous formation of altermagnetism from orbital ordering, Phys. Rev. Lett. 132, 236701 (2024).
  43. P. Das, V. Leeb, J. Knolle, and M. Knap, Realizing altermagnetism in Fermi-Hubbard models with ultracold atoms, Phys. Rev. Lett. 132, 263402 (2024).
  44. V. P. Kravchuk, K. V. Yershov, J. I. Facio, Y. Guo, O. Janson, O. Gomonay, J. Sinova, and J. van den Brink, Chiral magnetic excitations and domain textures of g-wave altermagnets, Phys. Rev. B 112, 144421 (2025).
  45. N. Jana, A. Chakraborty, A. Mukherjee, and A. Agarwal, Layered topological antiferromagnetic metal at room temperature: YbMn2Ge2, Phys. Rev. B 112, 075127 (2025).
  46. N. Jana, A. Chakraborty, A. Mukherjee, and A. Agarwal, Correlation-driven orbital order realizes 2D metallic altermagnetism, arXiv:2603.25426.
  47. S. Bhowal and N. A. Spaldin, Ferroically ordered magnetic octupoles in d-wave altermagnets, Phys. Rev. X 14, 011019 (2024).
  48. S. Han, D. Jo, I. Baek, S. Cheon, P. M. Oppeneer, and H.-W. Lee, Harnessing magnetic octupole Hall effect to induce torque in altermagnets, Phys. Rev. Lett. 135, 076705 (2025).
  49. R. Jaeschke-Ubiergo, V.-K. Bharadwaj, W. Campos, R. Zarzuela, N. Biniskos, R. M. Fernandes, T. Jungwirth, J. Sinova, and L. Šmejkal, Atomic altermagnetism, arXiv:2503.10797.
  50. D. Šabani, C. Bacaksiz, and M. V. Milošević, Beyond orbitally resolved magnetic exchange in Crl3 and Nil2, Phys. Rev. Lett. 135, 036704 (2025).
  51. L. Camerano, D. Mastrippolito, D. Pierucci, J. Dai, M. Tallarida, L. Ottaviano, G. Profeta, and F. Bisti, Darkness in interlayer and charge density wave states of 2H-TaS2, Phys. Rev. B 111, L121112 (2025).
  52. W. Li, J. T. Reichanadter, S. Wu, J. S. Oh, R. Basak, S. C. Haley, S. Wang, J. E. C. Mata, E. Vescovo, D. Lu, M. Hashimoto, C. Klewe, S. Sarker, J. L. McChesney, A. Frañó, J. G. Analytis, R. J. Birgeneau, J. B. Neaton, and Y. He, Electronic origin of delicate antiferromagnetism in FexNbS2, Phys. Rev. Lett. 136, 216503 (2026).
  53. L. Camerano, E. A. Martínez, V. Porée, L. Martella, D. Mastrippolito, D. Pierucci, F. D'Orazio, P. M. Sheverdyaeva, P. Moras, E. Della Valle, T. Yu, M. Hoesch, C. M. Polley, T. Balasubramanian, A. Nicolaou, L. Ottaviano, V. N. Strocov, G. Profeta, and F. Bisti, Emergent 3D fermiology and magnetism in an intercalated van der Waals system, Adv. Sci. 13, e76533 (2026).
  54. R. B. Regmi, H. Bhandari, B. Thapa, Y. Hao, N. Sharma, J. McKenzie, X. Chen, A. Nayak, M. El Gazzah, B. G. Márkus, L. Forró, X. Liu, H. Cao, J. F. Mitchell, I. I. Mazin, and N. J. Ghimire, Altermagnetism in the layered intercalated transition metal dichalcogenide CoNb4Se8, Nat. Commun. 16, 4399 (2025).
  55. J. N. Graham, T. J. Hicken, R. B. Regmi, M. Janoschek, I. Mazin, H. Luetkens, N. J. Ghimire, and Z. Guguchia, Local probe evidence supporting altermagnetism in Co1/4NbSe2, Phys. Rev. Res. 8, 023361 (2026).
  56. M. Sprague, M. I. Mondal, A. P. Sakhya, R. B. Regmi, S. Sadhukhan, A. K. Kumay, H. Sheokand, I. I. Mazin, N. J. Ghimire, and M. Neupane, Observation of altermagnetic spin splitting in an intercalated transition metal dichalcogenide Nat. Commun. 17, 9935 (2025).
  57. P. Park, W. Cho, C. Kim, Y. An, Y.-G. Kang, M. Avdeev, R. Sibille, K. Iida, R. Kajimoto, K. H. Lee, W. Ju, E.-J. Cho, H.-J. Noh, M. J. Han, S.-S. Zhang, C. D. Batista, and J.-G. Park, Tetrahedral triple-Q magnetic ordering and large spontaneous Hall conductivity in the metallic triangular antiferromagnet Co1/3TaS2, Nat. Commun. 14, 8346 (2023).
  58. H. Takagi, R. Takagi, S. Minami, T. Nomoto, K. Ohishi, M.-T. Suzuki, Y. Yanagi, M. Hirayama, N. D. Khanh, K. Karube, H. Saito, D. Hashizume, R. Kiyanagi, Y. Tokura, R. Arita, T. Nakajima, and S. Seki, Spontaneous topological Hall effect induced by non-coplanar antiferromagnetic order in intercalated van der Waals materials, Nat. Phys. 19, 961 (2023).
  59. See Supplemental Material at http://link.aps.org/supplemental/10.1103/s83n-w474 for discussion of bands without Nb ions and role of Nb-Co hopping.
  60. F. Weber, R. Hott, R. Heid, L. L. Lev, M. Caputo, T. Schmitt, and V. N. Strocov, Three-dimensional Fermi surface of 2h−NbSe2: Implications for the mechanism of charge density waves, Phys. Rev. B 97, 235122 (2018).
  61. E. Day-Roberts, H. Wu, O. Erten, and A. S. Botana, An altermagnetic materials library in intercalated transition-metal dichalcogenides, arXiv:2601.02481.
  62. L. Camerano, L. Martella, L. Battaglia, F. Giannessi, F. Camilli, L. Lozzi, P. M. Sheverdyaeva, P. Moras, L. Ottaviano, G. Profeta, and F. Bisti, Plasmon engineering in intercalated 2H-TaS2, Nano Lett. 26, 9083 (2026).
  63. G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993).
  64. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  65. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  66. N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
  67. I. Souza, N. Marzari, and D. Vanderbilt, Maximally localized Wannier functions for entangled energy bands, Phys. Rev. B 65, 035109 (2001).
  68. N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
  69. G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, et al., Wannier90 as a community code: New features and applications, J. Phys.: Condens. Matter 32, 165902 (2020).
  70. T. Cole, S. Coh, and D. Vanderbilt, Python tight binding (PythTB) (Version 2.0.0) [Computer software] Zenodo, 2025, https://doi.org/10.5281/zenodo.17595433.

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Phys. Rev. B 114, 264408– Published 5 November, 2026
  • Received 22 May 2026
  • Revised 19 October 2026
  • Accepted 20 October 2026
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