Altermagnetic Spin-Splitting in Compensated Magnets

• Altermagnetic spin-splitting is the nonrelativistic, momentum-dependent lifting of spin degeneracy in a compensated magnet, where anisotropic lattice distortions break translation symmetry.
• It exhibits d-wave, g-wave, or higher even-parity form factors that create extensive spin-split regions in the Brillouin zone interlaced with symmetry-protected nodal degeneracies.
• Its practical implications include spin-splitter currents, anisotropic magnetoresistance, and spin Hall-like effects arising from exchange-driven band anisotropy and ligand-mediated hybridization.

Altermagnetic spin-splitting is the nonrelativistic, momentum-dependent lifting of spin degeneracy in a collinear compensated magnet whose total magnetization vanishes, but whose spin-lattice symmetries do not enforce $E_\uparrow(\mathbf{k})=E_\downarrow(\mathbf{k})$ at generic $\mathbf{k}$. Its band structures combine large spin-split regions of the Brillouin zone with symmetry-protected nodal spin degeneracies, often in $d$-wave, $g$-wave, or related even-parity patterns, thereby placing altermagnets between ferromagnets and conventional antiferromagnets in both symmetry and transport phenomenology (Lee et al., 1 Dec 2025, Liang et al., 10 Jun 2026).

1. Symmetry class and distinction from ferromagnets and conventional antiferromagnets

In a conventional ferromagnet, one spin species is energetically favored throughout the crystal, the spin-resolved wavefunctions are uniformly distributed over equivalent atomic sites, and the system carries a nonzero net magnetization. In a conventional collinear antiferromagnet, spin-up and spin-down wavefunctions localize on opposite sublattices, but the local environments of those sublattices remain equivalent, so a combined symmetry such as $PT$, or translation plus time reversal, enforces spin degeneracy across the Brillouin zone. Altermagnets retain the collinear compensated character of antiferromagnets, yet display spin-split bands because the opposite-spin sublattices are related by rotations, mirrors, or nonsymmorphic operations rather than by a pure translation that restores degeneracy (Lee et al., 1 Dec 2025, Lee et al., 12 Feb 2026).

A convenient band-theoretic statement is that, in an altermagnet, $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$ for generic $\mathbf{k}$, while the opposite spin is found at a symmetry-related momentum $\mathbf{k}'$, $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$. This differs both from Stoner-like ferromagnetic splitting, where the sign of the splitting is globally uniform, and from relativistic Rashba- or Dresselhaus-type spin splitting, where $E_s(\mathbf{k}) = E_{-s}(-\mathbf{k})$ in a nonmagnetic inversion-broken crystal (Liang et al., 10 Jun 2026).

The characteristic momentum-space structure is not arbitrary. Altermagnetic band structures typically exhibit large anisotropic spin splitting in much of the Brillouin zone together with nodal spin degeneracies on symmetry-selected points, lines, or planes. Depending on the magnetic space group and orbital content, the form factor can be $\mathbf{k}$0-wave-like, $\mathbf{k}$1-wave-like, or of still higher even parity. In Co$\mathbf{k}$2NbSe$\mathbf{k}$3, for example, the low-energy splitting is explicitly described as $\mathbf{k}$4-wave, while in square-lattice minimal models and several oxide and fluoride examples the splitting has a $\mathbf{k}$5-wave-like pattern (Camerano et al., 20 May 2026, Lee et al., 1 Dec 2025).

A recurrent misconception is that any spin-split antiferromagnetic band structure must be spin-orbit-driven. The contemporary microscopic literature makes the opposite point: altermagnetic spin-splitting can already exist in the nonrelativistic limit, with spin-orbit coupling either absent or added later as a secondary perturbation that modifies optical and topological responses rather than generating the basic splitting itself (Lee et al., 1 Dec 2025, Liang et al., 10 Jun 2026).

2. Microscopic origin: alternating spin localization and broken translational equivalence

A concrete microscopic account identifies two ingredients as simultaneously necessary. The first is alternating spin-polarized wavefunction localization on magnetic sublattices, as in a collinear antiferromagnet: spin-up weight concentrates predominantly on one magnetic sublattice and spin-down weight on the other. The second is broken translational symmetry caused by distortions of the surrounding non-magnetic ion cages, which make those spin-hosting sublattices crystallographically inequivalent. Only their combination yields nonrelativistic spin splitting with zero net magnetization (Lee et al., 1 Dec 2025).

In the fluoride prototype MnF$\mathbf{k}$6, the relevant non-magnetic cages are distorted Mn–F octahedra. The local cage on one magnetic site may be elongated along one diagonal, while the cage on the opposite-spin site is elongated along the perpendicular diagonal. Because Mn–F hybridization depends on these local distortions, a state whose weight resides mainly on one sublattice acquires a different energy from its opposite-spin partner on the other sublattice, even though the total number of up and down spins remains equal (Lee et al., 1 Dec 2025).

The same logic has been reformulated in explicitly bonding-based language for Co$\mathbf{k}$7NbSe$\mathbf{k}$8. There, first-principles and Wannier analyses show that the dominant contribution to the $\mathbf{k}$9-wave spin splitting does not come from direct magnetic-ion hopping alone, but from short-range ligand-mediated hybridization. Co local moments imprint anisotropy onto Se $d$0 orbitals, and those ligand states transfer the anisotropy to itinerant Nb-derived bands. This establishes a local real-space mechanism that complements symmetry classification: the decisive channel is ligand-assisted coupling rather than a purely abstract allowed term (Camerano et al., 20 May 2026).

A further extension is that the required anisotropy need not be crystallographically pre-imposed. In a two-orbital square-lattice model, staggered antiferromagnetism coexisting with staggered orbital order generates anisotropic opposite-spin sublattices related by rotation, not translation or inversion, and thereby produces robust altermagnetic spin-splitting spontaneously. This suggests that altermagnetism can emerge as an interaction-driven instability even when the nonmagnetic lattice itself has no sublattice anisotropy (Leeb et al., 2023).

3. Minimal Hamiltonians, nodal structure, and momentum-space form factors

A widely used microscopic model for a $d$1-wave altermagnet on a square lattice employs two magnetic sublattices and one effective orbital per site. In the basis $d$2, the Hamiltonian is written as

$d$3

with

$d$4

$d$5

$d$6

Here $d$7 is an atomic exchange-driven spin splitting term that changes sign between the two sublattices, while $d$8 is an anisotropic third-neighbor hopping generated by the distorted non-magnetic cages (Lee et al., 1 Dec 2025).

The corresponding eigenvalues are

$d$9

From this form, the spin splitting is governed by the product $g$0, and it vanishes unless both $g$1 and $g$2. The model therefore makes the microscopic criterion explicit: alternating exchange fields alone are insufficient, and cage-induced hopping anisotropy alone is insufficient; both must coexist (Lee et al., 1 Dec 2025).

The same Hamiltonian explains the nodal structure. Spin degeneracy occurs at the zone center and zone boundary and along $g$3 and $g$4, where symmetry maps a spin-up state to a spin-down state of the same energy. Away from those lines, $g$5 becomes finite, producing a four-lobed, $g$6-wave-like pattern with large splitting near the diagonal sectors of the Brillouin zone. A useful estimate is

$g$7

which makes transparent why large exchange and large anisotropic hopping cooperate (Lee et al., 1 Dec 2025).

Other symmetry settings lead to different nodal manifolds without changing the central principle. In the hybrid manganese chloride $g$8, spin-space-group analysis yields

$g$9

so the planes $PT$0 and $PT$1 are spin-degenerate, while finite nonrelativistic splitting appears elsewhere in the Brillouin zone. In nanotubes rolled from a two-dimensional $PT$2-wave altermagnet, dimensional projection turns the parent momentum-space form factor into a one-dimensional chiral-angle dependence,

$PT$3

so nodal and antinodal tube orientations inherit vanishing and maximal splitting, respectively (Liang et al., 10 Jun 2026, Sasioglu et al., 7 Jun 2026).

4. Material realizations and direct experimental evidence

The model picture has been validated by first-principles work on MnF$PT$4, where the top valence bands derived from Mn $PT$5 orbitals display clear nonrelativistic spin splitting with zero net magnetization. The calculated splitting is direction dependent, and wavefunction analysis shows that the spin-up and spin-down Bloch states elongate along different crystallographic directions in accordance with the alternating orientation of the Mn–F cages. The $PT$6-derived bands exhibit larger splitting than the $PT$7-derived bands because the relevant Mn–F bond elongation makes the associated hopping more dispersive (Lee et al., 1 Dec 2025).

A direct spectroscopic demonstration has been reported for the intercalated transition-metal dichalcogenide Co$PT$8TaSe$PT$9. Magnetic susceptibility establishes type-A antiferromagnetic order with $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$0 K, while ARPES and DFT show a Fermi surface in which spin degeneracy is preserved on symmetry-enforced nodal manifolds but lifted elsewhere, particularly along $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$1. Temperature-dependent ARPES further reveals band shifts and closing of energy gaps above $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$2, consistent with suppression of the altermagnetic state (Sprague et al., 18 Aug 2025).

Hybrid materials enlarge the same phenomenology. In $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$3, first-principles calculations predict a compensated altermagnetic state with an insulating gap of about $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$4 eV and a maximum valence-band splitting near $E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$5 meV along C–$E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$6–D. The sign and momentum pattern of the splitting are controlled by a symmetry-related manifold of chirality, polarity, and magnetic domain states, so the same altermagnetic texture can be globally inverted either by reversing the magnetic domain or by reversing chirality and polarity together (Liang et al., 10 Jun 2026).

Real-space Hamiltonian engineering has also clarified why layered metallic compounds can host robust splitting without a net moment. In Co$E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$7NbSe$E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$8, a short-range Wannier Hamiltonian reproduces the spin-split DFT bands, and selectively removing hopping channels shows that ligand-mediated Co–Se–Nb processes are essential, whereas direct Nb–Co hopping is subdominant. This establishes that altermagnetic spin-splitting can be both local in real space and strongly visible in itinerant bands (Camerano et al., 20 May 2026).

5. Transport manifestations: spin-splitter currents, torques, and magnetoresistance

The most characteristic transport consequence is the spin-splitter effect: an electric field in a suitable crystallographic direction generates a transverse spin current even without spin-orbit coupling. In RuO$E(s,\mathbf{k}) \neq E(-s,\mathbf{k})$9, this nonrelativistic spin-momentum locking gives rise to a time-reversal-odd spin Hall response, termed the altermagnetic spin-splitting effect (ASSE). First-principles calculations show that realistic bulk RuO$\mathbf{k}$0 and (001)/(101) films on TiO$\mathbf{k}$1 are likely nonmagnetic at small $\mathbf{k}$2, whereas strained (100) and (110) films exhibit strain-induced altermagnetic spin splitting and strong ASSE even at $\mathbf{k}$3 (Lee et al., 12 Feb 2026).

For strained (100) RuO$\mathbf{k}$4, the dominant time-reversal-odd component reaches $\mathbf{k}$5 at $\mathbf{k}$6, corresponding to a spin Hall angle of about $\mathbf{k}$7. In strained (110) RuO$\mathbf{k}$8, large longitudinal time-reversal-odd spin currents appear, such as $\mathbf{k}$9 and $\mathbf{k}'$0. These results reconcile earlier contradictory reports by tying strong ASSE to epitaxial strain and orientation rather than to an intrinsic bulk magnetic ground state (Lee et al., 12 Feb 2026).

A diffusive transport theory reformulates the same physics in terms of coupled charge and spin chemical potentials. In the collinear regime,

$\mathbf{k}'$1

where the tensor $\mathbf{k}'$2 encodes nonrelativistic altermagnetic spin splitting. The first relation describes the inverse spin-splitter effect, and the second describes charge-to-spin conversion. In a nonlocal geometry, the resulting voltage depends on the relative orientation of the ferromagnetic detector polarization and the altermagnetic Néel vector, reproducing the main phenomenology of altermagnet-based spin valves and Hanle precession (Sigales et al., 8 Feb 2026).

Several experimentally accessible responses derive from the same band splitting. A four-terminal NEGF study on a $\mathbf{k}'$3-type square-lattice model visualizes the real-space spin accumulation generated by the spin-splitter effect, shows that edges induce oscillatory accumulation patterns, and proves that the effect vanishes at half filling because of a combined particle-hole and spin-reversal symmetry. Moderate impurity scattering of the order of the hopping amplitude does not destroy the effect (Hallberg et al., 24 Mar 2025). A related transport signature is spin-splitting magnetoresistance in RuO$\mathbf{k}'$4/Co bilayers, where the angular phase shift of the magnetoresistance tracks the Néel-vector orientation and reveals a sizable spin-splitting contribution distinct from conventional spin Hall magnetoresistance (Chen et al., 2024). In Pt/RuO$\mathbf{k}'$5/Py structures, the same nonrelativistic splitting generates spin splitting torque with spin polarization parallel to the Néel vector, and electrical switching of the RuO$\mathbf{k}'$6 Néel vector correspondingly changes the torque symmetry (Zhang et al., 2024).

The extrinsic sector has also been worked out semiclassically. In FeSb$\mathbf{k}'$7, impurity-induced side-jump and skew-scattering contributions can dominate the transverse spin-splitter response, and the resulting extrinsic spin conductivity is time-reversal even rather than time-reversal odd. The total spin Hall angle can reach values around $\mathbf{k}'$8, indicating that asymmetric impurity scattering is not merely a correction but potentially the leading channel for spin-splitter currents in dirty metallic altermagnets (Sarkar et al., 26 Feb 2026).

6. Thermal, optical, bosonic, and design extensions

Because the splitting is exchange-driven and symmetry-controlled, altermagnetic responses extend beyond charge transport. Under a longitudinal temperature gradient, a four-terminal altermagnet can exhibit a spin splitting Nernst effect, in which opposite spins are deflected transversely without spin-orbit coupling or net magnetization. In the model proposed for this effect, the transverse group velocity acquires a term proportional to the longitudinal wave vector, so the Nernst coefficient is controlled by Fermi energy, temperature, sample size, and the angular factor $\mathbf{k}'$9, with the notable symmetry relation $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$0 rather than the antisymmetric tensor structure of the conventional spin Nernst effect (Yi et al., 4 Sep 2025).

Magnonic and optical channels inherit the same underlying symmetry. In metallic altermagnets, electron–magnon scattering does not split the equilibrium magnon dispersion, but a charge current can generate a transverse magnon spin current whose polarization is along the Néel vector, a magnonic analogue of the electronic spin-splitter effect. This current exhibits both chemical-potential dependence and strong temperature dependence and can reach magnitudes of order $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$1 of the electronic spin conductivity near resonances (Sourounis et al., 2024). In the chiral hybrid manganese chloride, adding spin-orbit coupling produces a magneto-optical Kerr response whose sign flips under chirality reversal or magnetic-domain reversal, while polar-variant reversal alone leaves the Kerr sign unchanged (Liang et al., 10 Jun 2026).

Low-dimensional and bosonic realizations sharpen the geometric aspect of the phenomenon. Rolling a two-dimensional $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$2-wave altermagnet into a nanotube projects the parent momentum-space texture onto a one-dimensional spectrum whose splitting follows $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$3, vanishing for nodal tube orientations and reaching extrema for antinodal orientations (Sasioglu et al., 7 Jun 2026). In a photonic crystal, an antiunitary $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$4 symmetry combined with local $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$5-orbital $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$6 modes produces momentum-dependent pseudospin splitting with a $E(s,\mathbf{k}) = E(-s,\mathbf{k}')$7-wave form factor and alternating pseudospin polarization, enabling pseudospin filtering and pseudospin splitting of electromagnetic waves (Qiu et al., 27 May 2026).

Several design principles recur across these disparate platforms. Strong intra-atomic exchange or local magnetic polarization must coexist with a mechanism that transfers anisotropy to itinerant states, whether through distorted ligand cages, ligand-mediated hybridization, spontaneous orbital order, or dimensional projection. This suggests that ligand chemistry, point-group engineering, epitaxial strain, and domain control are as central as magnetic exchange itself (Lee et al., 1 Dec 2025, Camerano et al., 20 May 2026). The most persistent open questions concern spin-orbit coupling, electronic correlations, lattice dynamics, and the definition of the phase itself. Minimal nonrelativistic models neglect SOC and dynamical correlations by construction, yet both can reshape topological and magneto-optical observables. Domain walls may involve concomitant lattice dislocations because reversing the sign of the exchange field is equivalent to reversing the sign of the anisotropic hopping in the simplest models. The RuO$E(s,\mathbf{k}) = E(-s,\mathbf{k}')$8 controversy further shows that strain, thickness, and extrinsic disorder can determine whether an experimentally relevant sample is genuinely altermagnetic or effectively nonmagnetic (Lee et al., 1 Dec 2025, Jia et al., 24 Jun 2026).

Altermagnetic spin-splitting is therefore best understood not as a single band anomaly but as a symmetry-governed exchange phenomenon that links real-space anisotropy, sublattice-selective wavefunction localization, and momentum-dependent spin polarization. Its importance lies precisely in this conjunction: it offers ferromagnet-like spin-polarized bands and rich transport functionality while retaining antiferromagnetic compensation and vanishing net magnetization.