Altermagnetic Spin Splitting: Mechanisms & Insights

• Altermagnetic spin splitting is a momentum-dependent lift of spin degeneracy in compensated collinear magnets, enforced by symmetries like rotations and mirrors rather than inversion.
• Direct spectroscopic and transport studies in materials such as CrSb and Co1/4TaSe2 reveal splitting magnitudes up to 0.6 eV along specific momentum paths.
• Its anisotropic band structure and symmetry-enforced nodal degeneracies pave the way for novel device concepts and tunable spin transport without reliance on spin-orbit coupling.

Searching arXiv for papers on altermagnetic spin splitting to ground the article in the current literature. arXiv search query: "altermagnetic spin splitting" Altermagnetic spin splitting is the momentum-dependent, nonrelativistic lifting of spin degeneracy in a compensated collinear magnetic state whose opposite-spin sublattices are related by crystal rotations, mirrors, or nonsymmorphic operations rather than by inversion or translation. It combines zero net magnetization with exchange-scale band splitting and an alternating even-parity wave structure in reciprocal space, placing altermagnets conceptually between ferromagnets and conventional antiferromagnets. In this class of materials, the broken time-reversal symmetry of the ordered state does not produce a uniform ferromagnetic exchange shift of all bands; instead, it yields spin splitting that changes with momentum direction and is symmetry-enforced to vanish on selected nodal lines or planes. Direct spectroscopy in CrSb and Co$_{1/4}$TaSe$_2$, electrical probes in RuO$_2$, and extensions to magnons, photonic pseudospins, nanotubes, and multiferroic skyrmions have established altermagnetic spin splitting as a general symmetry-governed phenomenon rather than a conventional ferromagnetic or spin-orbit artifact (Reimers et al., 2023, Sprague et al., 18 Aug 2025, Chen et al., 2024, Qiu et al., 27 May 2026).

1. Concept and magnetic classification

Altermagnetic spin splitting arises in collinear magnets with compensated real-space order and zero net magnetization in the limit of vanishing spin-orbit coupling, yet with macroscopic time-reversal symmetry broken because the spin-up and spin-down sites are not connected by a translation or inversion. In ferromagnets, exchange splitting is tied to a net magnetic moment and is approximately uniform in momentum space. In conventional antiferromagnets, time reversal combined with a lattice translation or inversion often restores an effective degeneracy and suppresses intrinsic nonrelativistic spin splitting. In altermagnets, by contrast, the sublattices are connected by rotations, mirrors, glides, screws, or related operations, so the electronic structure becomes spin split at generic momenta despite compensation of the total moment (Naka et al., 2024, Reimers et al., 2023).

The resulting spin order is an even-parity wave order in momentum space. Depending on symmetry, the splitting can have $d$-, $g$-, or $i$-wave character. In layered Co$_{1/4}$TaSe$_2$, for example, the ARPES and DFT analysis identifies a $g$-wave pattern in which opposite-spin Fermi surfaces alternate in a sixfold-symmetric manner and remain degenerate on nodal symmetry planes (Sprague et al., 18 Aug 2025). In MnTe and hematite, the same terminology is used for $g$-wave spin polarization of the electronic or magnonic spectrum, with nodal planes or surfaces enforced by symmetry rather than by accidental band crossings (Belashchenko, 2024, Hoyer et al., 14 Mar 2025).

A central diagnostic is that, for generic $_2$0, the bands satisfy

$_2$1

while on symmetry-selected lines or planes they obey

$_2$2

This coexistence of generic splitting with nodal degeneracy is the hallmark of altermagnetic spin splitting and distinguishes it both from ferromagnetic exchange splitting and from conventional spin-orbit-driven Rashba or Dresselhaus effects (Sprague et al., 18 Aug 2025, Naka et al., 2024).

2. Symmetry structure and effective descriptions

The symmetry content of altermagnetic spin splitting is commonly encoded through momentum-dependent form factors. In a 2D $_2$3-wave altermagnet, a minimal two-sublattice model uses

$_2$4

with

$_2$5

This form factor changes sign under a $_2$6 rotation and vanishes on the nodal lines $_2$7, producing maximal splitting on antinodal directions and zero splitting on nodal directions (Sasioglu et al., 7 Jun 2026).

A complementary low-energy description for a $_2$8-type altermagnet writes

$_2$9

Here the altermagnetic term $_2$0 acts as a momentum-resolved Zeeman field that is collinear in spin space, anisotropic in momentum space, and nonrelativistic in origin. Unlike Rashba or Dresselhaus Hamiltonians, this term does not generate in-plane helical spin textures or spin-flip propagation; it preserves $_2$1 eigenstates while making the splitting strongly direction dependent (Liu et al., 7 Nov 2025).

In hexagonal CrSb, symmetry yields a $_2$2-wave structure with all spins lying in the basal plane. For in-plane momentum

$_2$3

the spin direction is

$_2$4

The spin texture therefore winds with twice the momentum angle and exhibits an effective threefold reciprocal-space pattern set by the magnetic space group, even though the crystal itself is hexagonal (Mineev, 16 Feb 2026).

These effective descriptions make explicit why the splitting vanishes on high-symmetry manifolds and appears on low-symmetry paths that connect them. In CrSb, the SX-ARPES experiment was designed precisely around such a path; in nanotubes derived from 2D altermagnets, dimensional projection converts the same nodal–antinodal structure into a chiral-angle-dependent 1D splitting. The general implication is that altermagnetic spin splitting is best understood as a symmetry-filtered exchange field in momentum space rather than as a uniform spin polarization (Reimers et al., 2023, Sasioglu et al., 7 Jun 2026).

3. Direct spectroscopic observation

The first direct spectroscopic benchmark in the literature is CrSb thin films. Spin-integrated soft X-ray ARPES on epitaxial CrSb(100) resolved a distinctive split band on the low-symmetry $_2$5–$_2$6 trajectory, which connects two symmetry-enforced degeneracy points. The observed splitting is approximately

$_2$7

and the relevant band lies just below the Fermi energy. High $_2$8 resolution was essential because the three-dimensional band structure would not reveal the splitting on the high-symmetry planes themselves. The measured dispersions match the altermagnetic calculation, preserve degeneracy at the endpoints, and are not reproduced by a nonmagnetic band structure; spin-orbit coupling introduces only minor changes, indicating that the effect is fundamentally exchange-driven (Reimers et al., 2023).

The structural origin in CrSb is also explicit. Ferromagnetic $_2$9 Cr planes are stacked antiferromagnetically along the easy $d$0-axis, but the two Cr sublattices are not related by inversion. Their local Sb coordination triangles are rotated by $d$1, which creates the alternating local crystal environments required for altermagnetic band splitting. ARPES selection rules further show that the visible branch has strong Cr $d$2-orbital character, especially $d$3, consistent with the polarization dependence of the photoemission signal (Reimers et al., 2023).

A layered realization was subsequently reported in Co$d$4TaSe$d$5. Magnetic susceptibility identifies type-A antiferromagnetic ordering with $d$6 K, while ARPES and DFT show a $d$7-wave Fermi-surface splitting that is maximal in the $d$8 plane. Measurements at 55 eV photon energy access this plane and reveal a clear two-feature structure in the momentum-distribution curves, whereas at 48 eV, corresponding roughly to $d$9, the splitting is strongly reduced or absent. The strongest separation occurs on the dog-bone pocket near $g$0, while bands remain essentially degenerate along $g$1. Above $g$2, the two-peak structure weakens, spectral weight redistributes, and the reconstructed low-temperature valence structure relaxes, tying the splitting directly to the ordered state (Sprague et al., 18 Aug 2025).

Beyond photoemission, symmetry-guided analysis of CrSb also connects the spin-split Fermi surfaces to quantum oscillations. The 2026 study of CrSb argues that the Fermi level is crossed by four spin-split bands—two tubular sheets along $g$3 and two closed pockets near $g$4—and that both oscillation frequencies and amplitudes should depend on magnetic field because the altermagnetic splitting itself is field dependent (Mineev, 16 Feb 2026). This broadens the experimental definition of direct observation from band imaging alone to a combined spectroscopic and Fermiology-based program.

4. Transport responses and device concepts

Transport manifestations of altermagnetic spin splitting are distinguished by their dependence on the Néel vector and by the absence, in the idealized mechanism, of any requirement for spin-orbit coupling. In RuO$g$5, the proposed spin-splitting effect generates a spin current

$g$6

with spin polarization determined by the Néel vector rather than by conventional spin Hall geometry. In $g$7-RuO$g$8/Co bilayers, the corresponding spin-splitting magnetoresistance (SSMR) has an anomalous angular dependence,

$g$9

with $i$0 for current along $i$1 at 50 K and 3 T, rather than the $i$2 minimum expected for ordinary SMR. The reported $i$3 ratio is about $i$4 at 50 K, and its pronounced temperature and thickness dependence is used to infer a Néel vector along $i$5 and long-range magnetic order in thin films (Chen et al., 2024).

A related RuO$i$6 study demonstrates electrical manipulation of the spin splitting torque (SST). In that work, ASSE generates a time-reversal-odd spin current whose polarization is parallel to the Néel vector; current-induced switching of the Néel vector is achieved by spin-orbit torque in Pt/RuO$i$7 heterostructures and verified by Hall transport and XMLD. RuO$i$8(100) shows Néel-vector-dependent SST, whereas RuO$i$9(110) does not provide the same tunable channel. In the switching experiments, the reported critical current densities are about $_{1/4}$0 for Pt/RuO$_{1/4}$1(100) and $_{1/4}$2 for Pt/RuO$_{1/4}$3(110) (Zhang et al., 2024).

Thermal and nonlocal responses extend the same physics beyond dc magnetoresistance. A four-terminal nonequilibrium Green’s-function study predicts a spin splitting Nernst effect in which a longitudinal temperature gradient drives opposite-spin electrons to opposite transverse edges, generating a pure spin current without spin-orbit coupling or net magnetism. A notable symmetry result is

$_{1/4}$4

which is opposite to the sign structure of the conventional spin Nernst effect (Yi et al., 4 Sep 2025). In diffusive hybrid structures, drift–diffusion theory predicts both the forward spin-splitter effect and its inverse, including nonlocal spin-valve signals whose zero-field amplitude is proportional to $_{1/4}$5, with $_{1/4}$6 the ferromagnetic detector polarization and $_{1/4}$7 the Néel vector (Sigales et al., 8 Feb 2026).

Real-space transport signatures have also been proposed. In a multi-terminal $_{1/4}$8-type altermagnet, the momentum-dependent splitting induces a real-space spin precession that modulates the Hall voltage. The oscillation period,

$_{1/4}$9

provides a direct measure of the spin-splitting strength $_2$0, and the effect remains visible under dephasing $_2$1 and crystalline warping (Liu et al., 7 Nov 2025). Under nonuniform driving, semiclassical and lattice Keldysh calculations predict electric and spin current vortices in altermagnets even in the Ohmic regime; the paper emphasizes that no corresponding swirls appear in ferromagnets because uniform exchange splitting lacks the necessary anisotropic $_2$2 structure (Herasymchuk et al., 10 Jul 2025).

Strain can act as a direct transport control knob. In $_2$3-wave altermagnetic MnTe, elastic shear strain induces a finite spin splitting effect even though the nonrelativistic unstrained crystal has vanishing spin conductivity tensor. The calculated spin splitting gauge factor exceeds 20 near the valence-band maximum, and the paper stresses that proper inclusion of Rashba-Dresselhaus spin-orbit coupling is essential for quantitatively correct transport modeling (Belashchenko, 2024).

5. Microscopic origin and materials design

Early symmetry classifications identify which spin-splitting harmonics are allowed, but they do not by themselves determine microscopic hierarchy. Subsequent work has therefore focused on real-space mechanisms. One route is purely interaction driven. A two-orbital square-lattice model shows that altermagnetism can form spontaneously through the coexistence of staggered antiferromagnetism and staggered orbital order, even when the lattice itself does not preimprint inequivalent magnetic sublattices. In that Hartree–Fock phase, the $_2$4 orbital rotation between sublattices generates the required symmetry structure, producing robust $_2$5-wave altermagnetic band splitting and a large spin-splitter conductivity. The calculated spin-splitter angle can reach about $_2$6 (Leeb et al., 2023).

A second route is ligand mediated. In Co$_2$7NbSe$_2$8, Wannier-Hamiltonian engineering shows that the splitting has a short-range local origin and is dominated not by direct magnetic-ion hopping but by Co–Se–Nb hybridization. Truncation analyses demonstrate that Nb–Nb hopping establishes the metallic host, Se–Se hopping restores three-dimensional host connectivity, Nb–Co coupling contributes only indirectly, and Co–Se hopping is the decisive channel that transfers magnetic anisotropy from the Co sublattice to Nb-derived itinerant states at the Fermi level. Scaling the Co–Se hybridization continuously tunes the splitting, providing direct microscopic evidence for ligand-assisted altermagnetic coupling (Camerano et al., 20 May 2026).

Perovskites illustrate a distinct structural mechanism. In distorted $_2$9 compounds, the GdFeO$g$0-type octahedral rotation generates sublattice-dependent anisotropic hopping between transition-metal $g$1 orbitals. When combined with collinear $g$2 antiferromagnetic order, especially C-type order, this produces nonrelativistic spin-split bands without requiring spin-orbit coupling. The same review emphasizes that the anomalous Hall effect in these materials is a separate, SOC-dependent phenomenon that additionally requires next-nearest-neighbor hopping, whereas the basic spin splitting and spin-current generation do not (Naka et al., 2024).

Hybrid molecular magnets add chemically programmable symmetry control. In $g$3, first-principles calculations and spin-space-group analysis predict compensated altermagnetic order with sign rules governed by chirality, polarity, and magnetic domain. Reversing both chirality and polarity, or reversing the magnetic domain alone, inverts the spin splitting throughout the Brillouin zone, whereas reversing chirality alone or polarity alone changes the sign only in symmetry-selected regions. With spin-orbit coupling, the Kerr rotation follows a different rule: reversing chirality or magnetic order flips the Kerr sign, but reversing only the polar variant does not (Liang et al., 10 Jun 2026). This separation between exchange-dominated spin splitting and SOC-enabled readout is an important design principle.

6. Reduced dimensionality, bosonic analogues, and topological textures

Dimensional reduction can convert the momentum anisotropy of a 2D altermagnet into a geometry-controlled 1D splitting. Rolling a 2D $g$4-wave altermagnet into a nanotube projects the parent form factor onto the tube axis, producing a lowest-subband splitting

$g$5

The splitting is maximal for antinodal orientations $g$6 and $g$7, vanishes for the nodal orientation $g$8, and reverses sign between the two antinodal axes. Density-functional calculations for nanotubes derived from checkerboard V$g$9O reproduce this nodal–antinodal selection rule, and the same mechanism persists in V$g$0OSe$g$1, V$g$2OSeTe, and Fe$g$3SSe (Sasioglu et al., 7 Jun 2026).

Photonic systems provide a bosonic counterpart. The first experimental orbital altermagnetic photonic crystal realizes momentum-dependent pseudospin splitting in a staggered gyromagnetic lattice of YIG resonators, protected by antiunitary $g$4 symmetry. Its local basis

$g$5

combines a pseudospin doublet with a local orbital $g$6 doublet, and the crucial term in the tight-binding model has a $g$7-wave form factor,

$g$8

Measured band structures and iso-frequency contours on a $g$9 array show alternating pseudospin polarization, while near 14.2 GHz the experiment demonstrates pseudospin-selective transmission and filtering under circularly polarized excitation (Qiu et al., 27 May 2026).

Topological textures can also host switchable altermagnetic spin splitting. In BiFeO$_2$00, the skyrmion phase is predicted to exhibit a nonvanishing altermagnetic splitting analogous to the collinear state, with magnitude $_2$01 eV in the electronic structure. The sign of the splitting reverses with ferroelectric polarization, and room-temperature control is demonstrated through electric-field switching at $_2$02 kV/cm, with circular photogalvanic effect readout of the helicity-dependent spin photocurrent. The reported skyrmion radius is $_2$03 nm, and the switching is reversible over multiple field cycles (Wang et al., 11 Jan 2026).

The concept extends further to collective excitations. In hematite, a four-sublattice spin-wave theory predicts a $_2$04-wave altermagnetic magnon splitting of approximately 2 meV within a total magnon bandwidth of about 100 meV. Capturing the splitting in a Heisenberg description requires exchange interactions extending at least to the 13th neighbor, because the 13th shell is the first with the necessary symmetry-inequivalent bond types. Spin-orbit-induced anisotropy and DMI do not obscure the key altermagnetic features, and the work predicts that inelastic neutron scattering should resolve the splitting. The same study identifies a third-order nonlinear magnon spin splitter effect, implying transverse heat-to-spin conversion without an external magnetic field (Hoyer et al., 14 Mar 2025).

7. Debates, misconceptions, and current directions

A persistent misconception is that altermagnetic spin splitting is simply a disguised spin-orbit effect or a weak-ferromagnetic artifact. The direct ARPES study of CrSb argues the opposite: the observed splitting is exchange driven, appears on low-symmetry paths rather than as a rigid global spin shift, and is only weakly modified by spin-orbit coupling (Reimers et al., 2023). The RuO$_2$05 transport literature makes the same distinction at the response level, separating the nonrelativistic SSE or ASSE channel from conventional relativistic spin Hall backgrounds (Chen et al., 2024).

A second misconception is that any compensated antiferromagnet should show the effect. The perovskite overview explicitly states that if $_2$06 symmetry is preserved, spin splitting is prohibited even when time reversal is broken (Naka et al., 2024). Likewise, Co$_2$07TaSe$_2$08 shows that degeneracy survives on nodal planes and selected directions, so the phenomenon depends not merely on compensation but on the detailed crystal–magnetic symmetry relation between sublattices (Sprague et al., 18 Aug 2025).

The most visible controversy concerns RuO$_2$09. One line of work interprets SSMR in $_2$10-RuO$_2$11/Co as evidence for a sizable altermagnetic component and long-range order in thin films (Chen et al., 2024). A later systematic first-principles study argues that realistic small $_2$12 likely leaves bulk RuO$_2$13 and $_2$14 films nonmagnetic in the absence of extrinsic effects, while $_2$15 and $_2$16 films can acquire strain-induced altermagnetic spin splitting even at $_2$17 (Lee et al., 12 Feb 2026). That study attributes inconsistent experimental reports to overestimated $_2$18, strain relaxation, thickness dependence, and defects or interfaces that induce extrinsic magnetism. Taken together, these results place orientation, strain state, and interfacial conditions at the center of the RuO$_2$19 debate.

Direct observation also remains nontrivial because the splitting is symmetry selective and may vanish on commonly measured high-symmetry cuts. This has motivated proposals for indirect but geometry-sensitive probes, including current vortices under nonuniform fields, inverse spin-splitter voltages in hybrid strips, and Hall-voltage oscillations from altermagnetic spin precession (Herasymchuk et al., 10 Jul 2025, Sigales et al., 8 Feb 2026, Liu et al., 7 Nov 2025). This suggests that future progress will rely on combined spectroscopic, transport, and symmetry-resolved analysis rather than on any single experimental signature.

In its mature form, the subject is no longer limited to identifying candidate materials. Current work treats altermagnetic spin splitting as a unifying framework for exchange-driven momentum-space anisotropy across electrons, magnons, pseudophotons, and topological textures; for chemically programmable sign control through chirality, polarity, and ligand pathways; and for device functions based on Néel-vector-controlled spin transport without ferromagnetic stray fields.