Ligand-mediated Origin of Altermagnetic Spin-Splitting

The gist The splitting is captured by a short-range tight-binding model, establishing its local origin <ref:2605.21284#pg2> This work proposes a computational framework to uncover the chemical origin of altermagnetism through a real-space decomposition of hopping processes <ref:2605.21284#pg4>.

Altermagnetism and its Significance

Altermagnetism is an interesting class of collinear magnets that combines zero net magnetization with momentum-dependent spin-split electronic bands Unlike conventional ferromagnets, they retain spin-polarized band structures through the interplay of crystal symmetry and magnetic order. This unique feature makes them promising for spintronics, enabling spin functionality without stray fields or magnetic interference, with potential applications in spin-current generation, anomalous transport, and ultrafast memory and logic devices [12–16]. The growing interest has stimulated an extensive search for realistic altermagnetic materials, combining first-principles predictions with experimental characterization to identify robust platforms for spin-split electronic structure [17–20].

Computational Framework

The paper uses first-principles calculations and Wannier Hamiltonian engineering to uncover the microscopic bonding contributions of altermagnetic spin splitting in the g-wave altermagnet Co1/4NbSe2. The approach involves constructing a Wannier-based tight-binding Hamiltonian from the DFT electronic structure and systematically controlling the real-space hopping processes in the system (see Methods for details). To guide parameter selection, researchers first analyzed the spatial range of electronic processes by examining hopping amplitudes as a function of interatomic distance (Fig. 2a), along with their corresponding histogram (Fig. 2b). They then progressively truncated the Wannier Hamiltonian in real space using cutoff distances rc1 and rc2, indicated in Fig. 2a–b in orange and green, respectively.

Identifying the Origin of Splitting

The analysis showed that retaining only the dominant short-range hoppings within rc1 = 3.85 ˚A (orange and green bonds in Fig. 2c) is already sufficient to reproduce the main band dispersions, along with the characteristic altermagnetic spin splitting of the full system (Fig. 2d). This indicates that the essential mechanisms are predominantly local and do not rely on longer-range hopping processes. Conversely, further reducing the cutoff to rc2 = 3.102 ˚A, which eliminates the Nb–Nb hopping network (orange bonds in Fig. 2c), substantially modifies the spectrum, driving the system toward an insulating state with only a weak residual spin splitting. This shows that capturing the overall band structure of Co1/4NbSe2 requires the inclusion of Nb–Nb hoppings (orange bonds in Fig. 2c), which sustain the dominant altermagnetic response.

Hopping-Resolved Analysis

The researchers decomposed the truncated short-range Hamiltonian as H(rc1) = H0 + XXYHXY, where H0 is the Hamiltonian matrix containing minimal hoppings terms needed to reproduce NbSe2 bands close to the Fermi level: hoppings between Nb-Se, and Nb-Nb. Starting from the NbSe2 host sector H0, they selectively activated individual couplings to determine their role in generating altermagnetic band splitting. The inclusion of HCoSe completes the full Hamiltonian H(rc1) and fully recovers the g-wave altermagnetic spin splitting (see Fig. 3d). This term directly introduces bond anisotropy tied to the Co sublattice, leading to a pronounced reconstruction of the electronic structure.

Ligand-Mediated Mechanism

The key finding is that the splitting is not rooted in direct magnetic-ion hopping, but is instead primarily generated through ligand-mediated Co–Se–Nb pathways. These processes transfer the symmetry of the magnetic sublattice to itinerant metallic states, establishing a direct real-space mechanism for altermagnetic band splitting in metallic systems. The analysis demonstrated that although HNbCo does not itself contain anisotropic spin-dependent hoppings between antiferromagnetically ordered Co sites typically required to generate g-wave altermagnetism, it nevertheless induces a finite spin splitting through its indirect coupling to the anisotropic Se network. Ultimately, the dominant mechanism arises from HCoSe, which transfers the symmetry-breaking anisotropy of the Co sublattice to the itinerant Nb bands. The continuous tuning of Co–Se hybridization strength via a scaling dimensionless parameter λ demonstrates that altermagnetic spin splitting is continuously transmitted to the itinerant sector through ligand-mediated hybridization. This smooth evolution provides direct microscopic evidence of its origin.

Conclusion

The work identifies ligand-assisted hybridization as the key channel through which altermagnetic spin splitting emerges in the metallic state. These results identify a clear hierarchy of microscopic processes underlying altermagnetic spin splitting, where HCoSe is dominant and transfers anisotropy to Nb bands. The study provides a bridge between model Hamiltonian approaches and realistic materials realizations of altermagnetism. It suggests a design principle for altermagnets based on controlling ligand physics rather than direct magnetic pathways.

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Ligand-mediated Origin of Altermagnetic Spin-Splitting Luigi Camerano,1, 2, ∗ Federico Bisti,1 and Gianni Profeta1, 2 Department 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 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 the key mechanism of altermagnetic spin splitting and provides a microscopic bridge between minimal models and symmetry guided first-principles material searches, enabling real-space design of altermagnetic functionality. Altermagnetism has garnered intense interest as a new class of collinear magnets that combine zero net magnetization with momentum-dependent spin-split electronic bands [1–8]. Unlike conventional ferromagnets, they retain spin-polarized band structures through the interplay of crystal symmetry and magnetic order [5, 9–11]. This unique feature makes them promising for spintronics, enabling spin functionality without stray fields or magnetic interference, with potential applications in spin-current generation, anomalous transport, and ultrafast memory and logic devices [12–16]. This growing interest has stimulated an extensive search for realistic altermagnetic materials, combining first-principles predictions with experimental characterization to identify robust platforms for spin-split electronic structure [17–20]. Candidate systems now span metals [21–25], semimetals [26, 27], oxides [28–31], and layered compounds [32–36]. In parallel, minimal Hamiltonian approaches serve as an important theoretical framework, providing a symmetry-based conceptual foundation for understanding altermagnetism. In particular, tight-binding (TB) models on square, honeycomb, Lieb and related lattices [32, 37–44] have shown how compensated magnetic sublattices, orbital anisotropy, and crystal symmetry can generate momentum-dependent spin splitting in the absence of both net magnetization and relativistic spin-orbit coupling. Together, these studies have opened the way to a classification of altermagnetic states in terms of ferroically ordered magnetic octupoles [45, 46] and symmetry-harmonic decompositions of the spin density [47], while identifying characteristic transport responses and providing a unified framework for momentum-dependent spin textures. However, most existing models rely on hopping parameters chosen to satisfy symmetry constraints to capture the existence of altermagnetism, without identifying its microscopic origin. In particular, the specific realspace electronic hopping processes, orbital pathways, and chemical bonds determining the altermagnetic spin splitting in real materials remain largely overlooked. In this paper, we propose a computational framework to uncover the chemical origin of altermagnetism through a real-space decomposition of hopping processes, enabled by first-principles Wannier Hamiltonian engineering. As a representative case, we focus on an intercalated transition metal dichalcogenide (TMD), a system in which chemical composition and symmetry, and consequently the altermagnetic splitting, can be effectively tuned via straightforward processes such as intercalation. Intercalated layered TMDs with transition metals atoms like Fe, Ni, Co, Mn provide a particularly clean platform to address this problem. Indeed, in the TMD family, strong arXiv:2605.21284v1 [cond-mat.mtrl-sci] 20 May 2026

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FIG. 1. a) Atom-resolved projected band structure of Co1/4NbSe2 along high-symmetry lines for kz!= 0 from DFT calculations.

1. Real-space design of altermagnetic functionality: The improved AI system can design new metallic altermagnets by controlling ligand physics rather than direct magnetic pathways, enabling a real-space framework for designing metallic altermagnetic functionality in layered quantum materials.

2. Mechanism disentanglement: The system can perform a disentanglement of distinct hopping pathways, including direct magnetic–magnetic, ligand-assisted, and host-mediated contributions to determine the specific origin of altermagnetic spin splitting in any given material structure.

3. Tunable spin splitting control: The AI can quantify the mechanism by continuously tuning the Co–Se hybridization strength via a scaling dimensionless parameter lambda to observe how a momentumdependent spin polarization develops in both Co- and Se-derived bands and is subsequently transferred to Nb-derived states.