Non-Relativistic Spin Splitting Overview

• Non-relativistic spin splitting is the momentum-dependent energy difference between opposite-spin bands in compensated magnets, driven by exchange interactions and specific symmetry breakings.
• Effective Hamiltonians with tailored form factors reveal d-wave, g-wave, and twisted bilayer textures that quantify energy splitting and spin texture in momentum space.
• Experimental probes such as spin-ARPES, Kerr effect measurements, and linear magneto-birefringence validate NRSS and demonstrate its potential for engineering spin-polarized transport.

Searching arXiv for papers on non-relativistic spin splitting to ground the article in the supplied literature. Non-relativistic spin splitting (NRSS) is the momentum-dependent energy difference between opposite-spin bands that appears even when spin–orbit coupling is neglected. In compensated magnets, and especially in collinear antiferromagnets with zero net magnetization, NRSS arises from exchange interactions together with crystal and spin-space symmetries that fail to restore Kramers-like degeneracy. In its best-known form NRSS underlies altermagnetism, but recent work has also established compensated magnets that exhibit NRSS without belonging to the altermagnetic subclass, including systems with spin splitting already at the Brillouin-zone center (Bhowal et al., 23 Oct 2025, Yuan et al., 2024).

1. Definition and symmetry criteria

A standard definition writes the spin splitting as

$$\Delta \epsilon(\mathbf{k})=\epsilon(\mathbf{k},\uparrow)-\epsilon(\mathbf{k},\downarrow).$$

In compensated antiferromagnets, this quantity can be nonzero without any net magnetization and without SOC. The central symmetry statement is that if either $\Theta I$ or $UT$ is preserved, spins remain Kramers-degenerate at every $\mathbf{k}$; breaking both $\Theta I$ and $UT$ is necessary and sufficient for NRSS, placing the system in symmetry class SST-4 (Nathan et al., 13 Dec 2025). Here $\Theta$ is time reversal, $I$ inversion, $U$ a $180^\circ$ spin rotation within SU(2), and $\Theta I$0 a lattice translation that connects opposite-spin sublattices.

The same symmetry logic has been reformulated in spin-space-group language. In two-dimensional collinear antiferromagnets, spin degeneracy at each $\Theta I$1 is protected whenever the two AFM sublattices are related by any of $\Theta I$2, and NRSS requires that all these mappings be broken (Mavani et al., 12 Mar 2025). More generally, recent reviews distinguish three symmetry scenarios: inversion preserved and time reversal broken, giving a symmetric splitting even in $\Theta I$3; inversion broken and time reversal preserved, giving an antisymmetric splitting odd in $\Theta I$4; and simultaneous inversion and time-reversal breaking, allowing mixed even and odd contributions (Bhowal et al., 23 Oct 2025).

An additional criterion governs whether spin splitting can occur at $\Theta I$5. Yuan et al. showed that once both $\Theta I$6 and $\Theta I$7 are absent, a further condition is required for $\Theta I$8-point splitting: no proper or improper rotation $\Theta I$9 may remain in the little group at $UT$0 such that $UT$1 exchanges the two opposite-spin sublattices. If such an $UT$2 exists, the system can remain spin-degenerate at $UT$3 even while exhibiting NRSS away from $UT$4; if no such $UT$5 exists, a constant mass term can split the bands already at $UT$6 (Yuan et al., 2024).

2. Effective Hamiltonians and momentum-space textures

A minimal nonrelativistic description often takes the form

$UT$7

so that the two branches differ by

$UT$8

This form captures the exchange-driven splitting in collinear systems and makes explicit that the spin texture is encoded by a symmetry-determined form factor $UT$9 rather than by relativistic SOC (Sunko et al., 20 Nov 2025).

The symmetry of $\mathbf{k}$0 defines the wave character of NRSS. For a planar $\mathbf{k}$1-wave altermagnet one may write

$\mathbf{k}$2

whereas a planar $\mathbf{k}$3-wave example is

$\mathbf{k}$4

The $\mathbf{k}$5-wave texture maps to a parity-even magnetic octupole, while the $\mathbf{k}$6-wave texture maps to a parity-even magnetic triakontadipole through the correspondence between momentum-space form factors and real-space magnetic multipoles (Sunko et al., 20 Nov 2025). In hematite below the Morin transition, first-principles multipole analysis identified ferroic ordering of rank-5 magnetic triakontadipoles as the lowest-order ferroically aligned magnetic multipoles in the absence of SOC, producing a $\mathbf{k}$7-wave NRSS (Verbeek et al., 2024).

Twisted bilayers add another texture class. In twisted bilayers of centrosymmetric antiferromagnets such as MnPSe$\mathbf{k}$8 and MnSe, symmetry analysis and first-principles calculations revealed “i-wave altermagnetism” with

$\mathbf{k}$9

together with a low-energy splitting

$\Theta I$0

This produces a six-lobed pattern in the Brillouin zone, distinct from both Rashba–Dresselhaus physics and the polynomial textures familiar in three-dimensional collinear altermagnets (Sheoran et al., 2023).

The effective Hamiltonian changes qualitatively in compensated magnets beyond altermagnets. Near $\Theta I$1, Yuan et al. derived

$\Theta I$2

with eigenvalues

$\Theta I$3

Here the constant term $\Theta I$4 splits the two spin bands even at $\Theta I$5; by contrast, in an altermagnet $\Theta I$6 by symmetry (Yuan et al., 2024). This sharpens the distinction between NRSS as a general phenomenon and altermagnetism as a particular symmetry subclass.

3. Material realizations

Rutile antiferromagnets provide a benchmark family. In ordered FeF$\Theta I$7 with G-type AFM and no SOC, density-functional calculations give an indirect band gap of approximately $\Theta I$8 eV and a robust splitting along $\Theta I$9 of approximately $UT$0 meV for bands near $UT$1, while $UT$2 and other high-symmetry lines remain spin-degenerate because of preserved $UT$3- and $UT$4-linked symmetries. In virtual-crystal FeOF the rutile nodal structure is retained, but the maximum $UT$5–$UT$6 splitting is enhanced to approximately $UT$7 meV. When local O/F short-range order is included, four nearly degenerate FeOF structures—$UT$8, $UT$9, $\Theta$0, and $\Theta$1—all retain large $\Theta$2 along $\Theta$3–$\Theta$4, ranging approximately from $\Theta$5 to $\Theta$6 meV, while only $\Theta$7 and $\Theta$8 exhibit $\Theta$9-point splitting (Nathan et al., 13 Dec 2025).

This FeOF result is notable because many studies of NRSS have focused on idealized, perfectly ordered crystals. In FeOF, local plane-to-plane anion correlations modify both the magnitude and the character of the splitting, and the effects are not fully captured in high-symmetry average structures. The same work further emphasizes heteroanionic compounds as a promising design space for NRSS antiferromagnets and notes that experimentally synthesized FeOF already exhibits a substantially higher Néel temperature, $I$0 K, than FeF$I$1, $I$2 K (Nathan et al., 13 Dec 2025).

A second class is provided by ternary magnetic nitrides and their ordered derivatives. In MnSiN$I$3, MnGeN$I$4, and MnSnN$I$5, DFT without SOC yields NRSS only away from high-symmetry zone-center directions, with $I$6 in the type-III magnetic space-group setting. By contrast, ordered Mn$I$7SiSnN$I$8 in space group $I$9 belongs to a type-I magnetic space group without SOC and shows a direct spin splitting at $U$0,

$U$1

while its band gap is reduced to $U$2 eV from $U$3 eV in MnSiN$U$4 (Yuan et al., 2024).

Two-dimensional and moiré platforms substantially enlarge the material space. Twisted bilayers of MnPSe$U$5 and MnSe show linear spin-splitting coefficients $U$6 in the range $U$7–$U$8 meV·Å for a $U$9 twist and can reach up to $180^\circ$0 meV·Å at other commensurate angles (Sheoran et al., 2023). A broader survey of twisted bilayer altermagnets including CoCl$180^\circ$1, AX$180^\circ$2 ($180^\circ$3 Mn, V; $180^\circ$4 Cl, Br, I), NiF$180^\circ$5, NiBr$180^\circ$6, FeS, CoS, MnTe$180^\circ$7, MnSe$180^\circ$8, and RuSe reported fitted $180^\circ$9 values from a few $\Theta I$00 meV·Å up to $\Theta I$01 meV·Å, with local values at $\Theta I$02 exceeding $\Theta I$03 meV·Å for some heavy-halide cases (Pathak et al., 23 Feb 2026).

Heterostructures introduce another distinction between global splitting and nodal altermagnetism. In MnPS$\Theta I$04/TMDC heterostructures, stacking S1 shows global spin splitting with $\Theta I$05 at generic $\Theta I$06, whereas stacking S2 exhibits altermagnetic-like band crossings with exact degeneracies on nodal lines. For MnPS$\Theta I$07/MoS$\Theta I$08, the calculated no-SOC splitting along $\Theta I$09 in S1 grows from $\Theta I$10 meV to $\Theta I$11 meV as $\Theta I$12 increases from $\Theta I$13 to $\Theta I$14, while S2 retains zero splitting on the nodal cut and finite splitting away from it (Wrzos et al., 27 Nov 2025). This suggests that interface registry and twist can toggle between qualitatively different NRSS regimes within a single materials platform.

4. Control mechanisms and design principles

A major theme in recent work is that NRSS can be controlled by structural degrees of freedom rather than by SOC engineering. In oxide perovskites with Pbnm symmetry, Bandyopadhyay et al. traced NRSS to the combination of octahedral or antipolar distortions, an antiferromagnetic dipole pattern, and the resulting ferroically ordered magnetic multipoles. For LaMnO$\Theta I$15, the lowest nonzero contribution in A-type AFM is an octupolar term

$\Theta I$16

and the paper proposed three engineering strategies: modifying the A-site cation size, strain engineering, and electric-field control in superlattice structures (Bandyopadhyay et al., 21 Mar 2025).

A related universal framework expresses the spin splitting as

$\Theta I$17

with the active magnetic multipoles $\Theta I$18 controlled by atomic distortions $\Theta I$19. In this picture, a dominant distortion can be driven by electric field, strain, or chemical radius mismatch, giving

$\Theta I$20

Concrete case studies include centrosymmetric MnF$\Theta I$21 under strain, where DFT gives $\Theta I$22 meV and $\Theta I$23 meV; ferroelectric BaCuF$\Theta I$24, where polarization reversal flips the sign of the NRSS and the equilibrium splitting is set by approximately $\Theta I$25 meV in the DFT band structure; and LaMnO$\Theta I$26/R!MnO$\Theta I$27 superlattices, where $\Theta I$28 along $\Theta I$29 grows from approximately $\Theta I$30 meV in La/Eu to approximately $\Theta I$31 meV in La/Gd (Ray et al., 24 Sep 2025).

Ferroelectricity provides a symmetry-selective switching route. A general theory for collinear antiferromagnets defines switching operators that simultaneously invert the electric polarization $\Theta I$32 and the NRSS $\Theta I$33, with the key operation

$\Theta I$34

up to stabilizer-group elements. In a two-band description,

$\Theta I$35

and symmetry requires $\Theta I$36 to be an odd functional of $\Theta I$37 to first order. The case studies span a quasi-one-dimensional zigzag graphene nanoribbon with $\Theta I$38 meV at the valence-band maximum, bilayer Nb$\Theta I$39I$\Theta I$40 with a $\Theta I$41 meV NRSS at $\Theta I$42, and three-dimensional MnSe$\Theta I$43 with $\Theta I$44 meV along $\Theta I$45–$\Theta I$46 (Dai et al., 12 Jun 2026). In two-dimensional polar AFM bilayers, the same principle distinguishes polarization-even and polarization-odd NRSS. Type II and Type III stackings satisfy

$\Theta I$47

so the full spin texture reverses under ferroelectric switching, whereas Type I does not (Mavani et al., 12 Mar 2025).

Hydrostatic pressure is another control parameter. In MnTe, neutron diffraction with in situ pressure showed that applying pressure significantly increases the Néel temperature but decreases the ordered magnetic moment. The fitted relation is

$\Theta I$48

and DFT indicates that pressure increases band hopping and exchange while reducing the local Mn moment through enhanced orbital hybridization and electron delocalization. The same analysis states that the maximum valence-band NRSS increases by approximately $\Theta I$49 per GPa for compressions up to approximately $\Theta I$50 (Carlisle et al., 13 May 2025). In FeSb$\Theta I$51, by contrast, pressure progressively reduces the exchange-like splitting: the band-24 splitting along $\Theta I$52–$\Theta I$53 decreases from approximately $\Theta I$54 eV at $\Theta I$55 GPa to approximately $\Theta I$56 eV at $\Theta I$57 GPa, while the band-26 splitting along $\Theta I$58–$\Theta I$59 falls from approximately $\Theta I$60 eV to approximately $\Theta I$61 eV (Bhandari et al., 29 Jul 2025).

Defect and interface engineering can also induce NRSS where it is otherwise forbidden. Twin boundaries in compensated magnets provide a structural route in which a boundary rotation or mirror, combined with a ferromagnetic domain wall, enforces an $\Theta I$62- or $\Theta I$63-type sublattice exchange rather than a $\Theta I$64- or $\Theta I$65-type relation. In BiCoO$\Theta I$66 with $\Theta I$67 ferroelastic twin boundaries, DFT yields spin splitting up to approximately $\Theta I$68 eV and a minimal fitting form

$\Theta I$69

In CoO$\Theta I$70 with $\Theta I$71, $\Theta I$72, and $\Theta I$73 twin boundaries, the splitting reaches approximately $\Theta I$74 eV and follows

$\Theta I$75

A corresponding tight-binding theory gives the scaling

$\Theta I$76

where $\Theta I$77 is the twin-boundary density (Eggestad et al., 18 Nov 2025).

5. Experimental probes

Direct spectroscopic evidence for NRSS has recently been obtained in the intercalated transition-metal dichalcogenide CoNb$\Theta I$78Se$\Theta I$79, identified as a $\Theta I$80-wave altermagnet. Spin-ARPES on the occupied states resolved two electron pockets at $\Theta I$81 with a momentum splitting $\Theta I$82 Å$\Theta I$83 and energy splittings of $\Theta I$84 meV and $\Theta I$85 meV at symmetry-related momenta. Spin- and angle-resolved electron reflection spectroscopy (spin-ARRES) then extended the observation to unoccupied states up to approximately $\Theta I$86 eV above $\Theta I$87, showing six lobes of spin-polarized reflectivity whose sign alternates every $\Theta I$88. The temperature dependence showed suppression of NRSS at the Néel temperature $\Theta I$89 K, while a residual splitting of approximately $\Theta I$90–$\Theta I$91 meV remained at $\Theta I$92 K (Dale et al., 2024).

Magneto-optical Kerr effect calculations provide a complementary probe in FeOF. With SOC included only in the optical response, the Kerr angle is written as

$\Theta I$93

For FeF$\Theta I$94, FeOF in the virtual-crystal approximation, and the $\Theta I$95 and $\Theta I$96 short-range-ordered FeOF structures, the polar Kerr signal is zero by symmetry. For the $\Theta I$97 and $\Theta I$98 structures, finite $\Theta I$99 produces nonzero Kerr rotations over a broad optical window from $UT$00 to $UT$01 eV, with distinct spectral line shapes that could be used to locally identify $UT$02-split domains in FeOF (Nathan et al., 13 Dec 2025).

Linear magneto-birefringence (LMB) has been proposed as a direct optical probe of altermagnetic order. In this framework the field-linear part of the dielectric tensor is

$UT$03

so the LMB susceptibility tensor is

$UT$04

Diagonal LMB isolates octupolar components such as $UT$05 through $UT$06, while off-diagonal LMB isolates components such as $UT$07 through $UT$08. For $UT$09-wave multipoles the formalism extends to strain-linear LMB,

$UT$10

which provides domain imaging and a benchmark for theoretical models of the NRSS texture (Sunko et al., 20 Nov 2025).

These probes address a long-standing difficulty of the field: NRSS often appears most clearly along non-high-symmetry momentum directions or in nanoscale domains. The FeOF study explicitly notes short-range-ordered domains of $UT$11–$UT$12 nm and argues that extreme anti-reflection MOKE microscopy can spatially resolve approximately $UT$13 nm domains, making SRO-driven NRSS experimentally accessible (Nathan et al., 13 Dec 2025).

6. Functional consequences and conceptual scope

A common misconception is that NRSS and altermagnetism are interchangeable terms. The current literature does not support that identification. Altermagnets form one subgroup, typically with symmetry-protected nodes and $UT$14, but compensated magnets with type-I magnetic space groups can exhibit NRSS with a constant $UT$15-point splitting, and polar bilayer constructions can realize both altermagnetic and non-altermagnetic NRSS (Yuan et al., 2024, Mavani et al., 12 Mar 2025). MnPS$UT$16/TMDC heterostructures offer an explicit interface example: S2 hosts altermagnetic-like band crossings, while S1 shows global spin splitting characteristic of symmetry-breaking NRSS (Wrzos et al., 27 Nov 2025).

The functional implications follow from the large exchange scale of the splitting. In the “beyond altermagnet” Mn$UT$17SiSnN$UT$18, the direct $UT$19-point splitting $UT$20 eV leads to the estimate

$UT$21

which at room temperature gives $UT$22, i.e. nearly fully-polarized currents. The same paper argues that removing the previous NRSS constraint on $UT$23 may facilitate spin-current generation without cancellation arising from alternating spin polarizations (Yuan et al., 2024).

In moiré systems, NRSS becomes a transport knob. Twisted bilayer altermagnets subject to diagonal strain $UT$24 acquire finite transverse spin conductivity and a “spin-splitter angle”

$UT$25

with calculated values reaching approximately $UT$26 in tb-CoCl$UT$27 and up to approximately $UT$28 in tb-MnTe$UT$29 for a $UT$30 strain (Pathak et al., 23 Feb 2026). In MnPS$UT$31/TMDC heterostructures, adding SOC in the TMDC layer yields a valley splitting

$UT$32

so rotating the MnPS$UT$33 Néel axis tunes the valley splitting continuously from $UT$34 down to $UT$35 (Wrzos et al., 27 Nov 2025).

The broader scope of NRSS now includes compensated antiferromagnets with collinear, coplanar, and non-coplanar spin arrangements (Bhowal et al., 23 Oct 2025). A further conceptual extension appears in triplet superconductors, where the triplet order parameter induces a wave-vector-dependent spin texture of Bogoliubov quasiparticles and an electric-field-driven Edelstein effect even though the quasiparticle spectrum remains twofold spin-degenerate (Li et al., 15 May 2026). This suggests that the central NRSS idea—nonrelativistic intertwining of exchange or order-parameter structure with momentum-resolved spin character—has become a unifying language across several branches of condensed-matter theory, while its strict band-splitting form remains most sharply realized in compensated magnetic materials.