Non-Relativistic Spin-Splitting (NRSS)

• NRSS is the lifting of spin degeneracy in electronic bands, primarily observed in compensated magnets without relying on spin–orbit coupling.
• It exhibits momentum-space textures with alternating spin signs and nodal structures, characterizing both altermagnetic and non-altermagnetic systems.
• The phenomenon enables large, exchange-driven spin splittings and offers tunable spintronic functionalities via strain, pressure, and electric polarization.

Searching arXiv for recent NRSS/altermagnetism papers to ground the article. Non-relativistic spin-splitting (NRSS) is the lifting of spin degeneracy in electronic bands with spin–orbit coupling disabled, most prominently in compensated magnetic systems such as collinear antiferromagnets. In its standard form, the splitting is written as $$\Delta E(\mathbf{k}) = E_{\uparrow}(\mathbf{k}) - E_{\downarrow}(\mathbf{k})$$, and a useful effective description is $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$, with $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$. Unlike Rashba or Dresselhaus splittings, NRSS does not rely on relativistic spin–orbit coupling and can persist in materials with zero net magnetization. Altermagnetism is the best-known subclass, but recent work has established that NRSS also occurs in compensated magnets that are not altermagnets, in compensated ferrimagnets, and in symmetry-engineered heterostructures and defect supercells (Bhowal et al., 23 Oct 2025, Dai et al., 12 Jun 2026).

1. Symmetry foundations

The central symmetry question is whether any antiunitary or composite symmetry maps $|\mathbf{k},\uparrow\rangle$ to $|\mathbf{k},\downarrow\rangle$ at the same momentum. If such a symmetry exists and squares to $-1$, Kramers-like spin degeneracy is enforced. In collinear antiferromagnets without SOC, the most important degeneracy-protecting operations are combined inversion–time reversal, written as $\Theta I$ or $PT$, and spin-reversal combined with a translation exchanging the magnetic sublattices, written as $UT$. NRSS becomes symmetry-allowed when these protections are absent (Nathan et al., 13 Dec 2025, Bhowal et al., 23 Oct 2025).

Spin-group formulations make this logic explicit. In the nonrelativistic limit, spin and lattice transform independently, so a general operation may be written as $[g_s \,\|\, g_l \,|\, \tau]$, with separate spin-space and real-space actions. This framework is essential because the same crystallographic operation can either preserve or fail to preserve spin degeneracy depending on whether it is accompanied by spin reversal, time reversal, or a translation connecting opposite-spin sublattices (Dai et al., 12 Jun 2026, Mavani et al., 12 Mar 2025).

A recurrent misconception is that NRSS is synonymous with altermagnetism. The broader classification is more general. The review literature distinguishes collinear, coplanar, and non-coplanar compensated magnets with NRSS, and separate work has shown that compensated antiferromagnets may exhibit $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$0 without belonging to the rotationally connected altermagnetic subclass (Bhowal et al., 23 Oct 2025, Yuan et al., 2024).

2. Momentum-space structures and subclasses

The momentum-space texture of NRSS is constrained by the symmetry that connects the opposite-spin sublattices. When a proper or improper rotation interconverts the sublattices, the splitting alternates in sign across the Brillouin zone and vanishes on symmetry-enforced nodal lines or planes. This is the standard altermagnetic situation, often described by $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$1-, $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$2-, or $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$3-wave textures. The review literature summarizes $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$4-wave patterns with two nodal planes, $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$5-wave patterns with four nodal planes, and $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$6-wave patterns with six nodal planes (Bhowal et al., 23 Oct 2025).

This distinction is visible in concrete models and materials. In two-dimensional polar bilayers, MnPSe$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$7 and strained MnPS$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$8 show altermagnetic $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$9-wave NRSS with nodal lines enforced by $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$0, whereas polar-stacked $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$1-CrTe$\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$2 shows non-altermagnetic NRSS because no crystal rotation swaps the opposite sublattices; in that case, spin splitting can occur even at the Brillouin-zone center in a compensated magnet (Mavani et al., 12 Mar 2025). A related extension was established for ordered Mn$\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$3SiSnN$\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$4, where the absence of any symmetry connecting the spin-structure motif pair allows a sizable nonrelativistic $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$5-point splitting, with $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$6 meV in the valence-edge region (Yuan et al., 2024).

The same taxonomy appears in bulk and moiré settings. Twisted bilayers of centrosymmetric MnPSe$\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$7 and MnSe develop $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$8-wave altermagnetic textures because twist breaks the symmetry that enforces monolayer spin degeneracy while retaining combined spin–lattice rotations that constrain the splitting pattern. Their full-Brillouin-zone maps show sixfold sign alternation and symmetry-enforced nodes (Sheoran et al., 2023). In hematite, the low-temperature collinear phase exhibits a $\Delta E(\mathbf{k}) = 2|\mathbf{g}(\mathbf{k})|$9-wave NRSS tied to rank-5 magnetic triakontadipoles, and in CoNb$|\mathbf{k},\uparrow\rangle$0Se$|\mathbf{k},\uparrow\rangle$1 the alternating sign of the measured spin polarization under successive $|\mathbf{k},\uparrow\rangle$2 rotations identifies a bulk $|\mathbf{k},\uparrow\rangle$3-wave altermagnetic phase (Verbeek et al., 2024, Dale et al., 2024).

A compact classification emerging from these studies is summarized below.

NRSS class	Symmetry hallmark	Representative systems
Rotational or mirror-connected altermagnetic NRSS	sign-alternating $|\mathbf{k},\uparrow\rangle$4 with nodal lines or planes; $|\mathbf{k},\uparrow\rangle$5 often protected	MnTe, MnPSe$|\mathbf{k},\uparrow\rangle$6, strained MnPS$|\mathbf{k},\uparrow\rangle$7, CoNb$|\mathbf{k},\uparrow\rangle$8Se$|\mathbf{k},\uparrow\rangle$9, twisted MnPSe$|\mathbf{k},\downarrow\rangle$0/MnSe
Non-altermagnetic compensated NRSS	no crystal rotation swaps opposite sublattices; $|\mathbf{k},\downarrow\rangle$1-point splitting allowed	$|\mathbf{k},\downarrow\rangle$2-CrTe$|\mathbf{k},\downarrow\rangle$3, Mn$|\mathbf{k},\downarrow\rangle$4SiSnN$|\mathbf{k},\downarrow\rangle$5, FeOF $|\mathbf{k},\downarrow\rangle$6/$|\mathbf{k},\downarrow\rangle$7 snapshots
Higher-multipole NRSS	$|\mathbf{k},\downarrow\rangle$8- or $|\mathbf{k},\downarrow\rangle$9-wave textures generated by ferroic magnetic octupoles or triakontadipoles	LaMnO$-1$0, $-1$1-Fe$-1$2O$-1$3

3. Microscopic mechanisms and material realizations

Several distinct microscopic routes produce NRSS. In rutile antiferromagnets, the operative picture is often formulated in terms of spin-structure motif pairs (SSMPs), meaning the local octahedral environments around the AFM sublattices. In FeF$-1$4, the differing motif geometries generate robust NRSS along $-1$5–M, while $-1$6 itself remains spin-degenerate because auxiliary operations such as $-1$7, $-1$8, $-1$9, and $\Theta I$0 still interconnect the SSMPs (Nathan et al., 13 Dec 2025). In heteroanionic FeOF, short-range O/F order modifies which of these operations survive. Four low-energy SRO snapshots—$\Theta I$1, $\Theta I$2, $\Theta I$3, and $\Theta I$4—all preserve strong non-SOC splitting along $\Theta I$5–M, but only $\Theta I$6 and $\Theta I$7 lose all SSMP-linking rotations and mirrors, thereby developing $\Theta I$8-point spin splitting absent in ordered FeF$\Theta I$9 and in the virtual-crystal model (Nathan et al., 13 Dec 2025).

A second route is multipolar. In Pbnm oxide perovskites, antiferroic charge multipoles created by octahedral rotations, Jahn–Teller distortions, and antipolar A-site displacements combine with AFM dipolar order to generate ferroic magnetic octupoles. In LaMnO$PT$0, A-type, C-type, and G-type AFM orders couple to different quadrupolar patterns and thereby select different $PT$1-space harmonics for NRSS: $PT$2, $PT$3, and $PT$4, respectively (Bandyopadhyay et al., 21 Mar 2025). Hematite generalizes this logic to rank-5 order: below the Morin transition, ferroically ordered magnetic triakontadipoles arise from simultaneous antiferroic ordering of charge hexadecapoles and magnetic dipoles, and these triakontadipoles generate the observed $PT$5-wave NRSS (Verbeek et al., 2024).

Ligand asymmetry and sublattice-selective crystal fields provide another mechanism. In GdAlSi, even-parity magnetic octupoles $PT$6 and the toroidal quadrupole $PT$7 exist already without SOC and produce a $PT$8-wave altermagnetic texture in which the spin-up and spin-down Fermi pockets are rotated by $PT$9 with respect to one another (Nag et al., 2023). In CoNb$UT$0Se$UT$1, the Symmetry-Constrained Adaptive Basis makes the crystal-field swapping between the two Co sublattices explicit under the $UT$2 screw and glide operations, and the nonrelativistic exchange field then converts the resulting sublattice polarization into an alternating $UT$3-wave spin splitting (Dale et al., 2024).

Recent work also shows that interfaces and boundaries can create NRSS even when the bulk crystal forbids it. MnPS$UT$4/TMDC heterostructures display two distinct nonrelativistic regimes depending on stacking: S2 hosts altermagnetic-like band crossings, whereas S1 exhibits global spin splitting characteristic of symmetry-breaking NRSS (Wrzos et al., 27 Nov 2025). Twin boundaries in BiCoO$UT$5 and CoO$UT$6, when combined with ferromagnetic domain walls, generate twin-boundary-induced NRSS with nodal surfaces dictated by the boundary supercell symmetry rather than by the bulk space group (Eggestad et al., 18 Nov 2025).

4. External control and design principles

One of the most active directions in NRSS research is external control. Electric polarization provides a particularly direct route. In polar antiferromagnetic bilayers, opposite polarization states may or may not reverse the sign of the nonrelativistic spin splitting depending on the symmetry operator that connects them. In MnPSe$UT$7 bilayers, AB and BA are related by $UT$8, so the NRSS sign does not switch with $UT$9. In strained MnPS$[g_s \,\|\, g_l \,|\, \tau]$0 and in $[g_s \,\|\, g_l \,|\, \tau]$1-CrTe$[g_s \,\|\, g_l \,|\, \tau]$2, AB and BA are related by $[g_s \,\|\, g_l \,|\, \tau]$3, so $[g_s \,\|\, g_l \,|\, \tau]$4 and the spin texture reverses under polarization switching (Mavani et al., 12 Mar 2025). A later group-theoretical generalization established minimal switching sets for one-, two-, and three-dimensional collinear antiferromagnets, including altermagnets and compensated ferrimagnets, and identified $[g_s \,\|\, g_l \,|\, \tau]$5 as the universal minimal switcher in three dimensions (Dai et al., 12 Jun 2026).

Strain and pressure are equally effective because NRSS is tied to crystal-field anisotropy and exchange pathways. In MnTe, hydrostatic pressure increases $[g_s \,\|\, g_l \,|\, \tau]$6 from about $[g_s \,\|\, g_l \,|\, \tau]$7 K at ambient pressure to about $[g_s \,\|\, g_l \,|\, \tau]$8 K at $[g_s \,\|\, g_l \,|\, \tau]$9 GPa while reducing the ordered moment, and DFT shows that compression modifies the nonrelativistic splitting in a band-selective way: the first valence band splitting increases, whereas the first conduction band splitting decreases (Carlisle et al., 13 May 2025). In FeSb$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$00, pressure shifts the symmetry-enforced spin-up/spin-down nodes at $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$01 and A below the Fermi level and suppresses the NRSS of band 24 along $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$02–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$03–M from about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$04 meV to about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$05 meV and of band 26 along A–Z–A′ from about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$06 meV to about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$07 meV by $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$08 GPa (Bhandari et al., 29 Jul 2025). In oxide perovskites, compressive strain enhances octahedral rotations and the associated ferroic magnetic octupoles, thereby increasing the splitting along the symmetry-allowed directions (Bandyopadhyay et al., 21 Mar 2025).

Short-range order and disorder are no longer regarded as merely destructive. In FeOF, cluster-expansion plus Monte Carlo analysis identified four nearly degenerate SRO motifs within $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$09 meV per formula unit, and DFT shows that all four retain large NRSS along $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$10–M with $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$11–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$12 meV near the Fermi level (Nathan et al., 13 Dec 2025). Large supercells relax to $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$13 beyond about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$14, corresponding to $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$15–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$16 nm domains, so realistic samples are expected to contain heterogeneous nanoscale regions whose electronic structure is set by local anion correlations rather than by a high-symmetry average (Nathan et al., 13 Dec 2025).

Twist, electric field, and boundary engineering extend control to low dimensions. Twisting bilayers of centrosymmetric antiferromagnets generates $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$17-wave NRSS with linear coefficients up to $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$18–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$19 meVÅ and up to about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$20 meVÅ near $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$21 twist (Sheoran et al., 2023). In a broader survey of twisted bilayer altermagnets, extracted $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$22 coefficients span $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$23–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$24 eVÅ in several cases, out-of-plane electric fields produce Zeeman-type splittings up to about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$25–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$26 meV at $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$27 MV/cm, and anisotropic strain can drive reversible $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$28 wave transitions that activate finite spin conductivity (Pathak et al., 23 Feb 2026). A conceptually different control route is Floquet driving: circular or elliptical light generates a valley-odd mass term $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$29, which lifts spin degeneracy in antiferromagnets even when static NRSS is absent, and bath engineering then allows steady-state pure spin currents and net spin accumulation without SOC (Li et al., 30 Jul 2025).

5. Experimental signatures and diagnostics

Direct observation of NRSS requires probes that distinguish nonrelativistic spin splitting from relativistic or ferromagnetic effects. Spin-resolved ARPES has become the principal occupied-state probe. In CoNb$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$30Se$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$31, spin-ARPES directly resolves the alternating $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$32-wave spin texture below the Fermi level, while spin- and angle-resolved electron reflection spectroscopy (spin-ARRES) extends the measurement to unoccupied states and shows the same sign alternation every $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$33 around $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$34 at about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$35 eV above $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$36 (Dale et al., 2024). Temperature-dependent spin-ARPES further shows suppression of the NRSS at the Néel temperature $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$37 K, providing direct evidence of an altermagnetic phase transition, while residual splitting above $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$38 suggests coexistence of altermagnetic fluctuations and SOC effects (Dale et al., 2024).

Conventional ARPES also reveals the interplay between NRSS and topology. In GdAlSi, angle-resolved photoemission confirms Fermi arcs on the (001) surface, complementing the calculated SOC-free altermagnetic splitting of about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$39 meV and establishing the coexistence of NRSS with a Weyl semimetal state (Nag et al., 2023). For insulating or weakly conducting materials, the diagnostic space is broader. The review literature points to magnetic Compton scattering, where the magnetic Compton profile $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$40 integrates the spin-polarized momentum density and can detect NRSS-like momentum-space structure even when ARPES is impractical (Bhowal et al., 23 Oct 2025).

Optical probes are especially sensitive to symmetry. In FeOF, polar magneto-optical Kerr calculations predict null Kerr spectra for FeF$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$41, VCA-FeOF, $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$42, and $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$43, but finite Kerr rotations over a broad spectral range for the $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$44-split $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$45 and $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$46 short-range-order motifs. Because the distinct dielectric tensors produce distinguishable Kerr spectra, MOKE becomes an experimental fingerprint of SRO-driven electronic structure (Nathan et al., 13 Dec 2025). The same work notes that extreme anti-reflection-enhanced MOKE microscopy can resolve nm-scale magnetic domains and is therefore suited to probing the predicted $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$47–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$48 nm SRO domains in FeOF (Nathan et al., 13 Dec 2025).

Linear magneto-birefringence has been proposed as a symmetry-guided bulk probe of altermagnetic order. The key idea is a direct mapping between even-parity NRSS textures in momentum space and ferroically ordered magnetic multipoles in real space. In this framework, $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$49-wave octupoles produce field-linear changes in the symmetric dielectric tensor, whereas $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$50-wave triakontadipoles require strain to activate linear-optical magneto-birefringence. The resulting selection rules distinguish diagonal and off-diagonal birefringent channels and provide a route to domain imaging (Sunko et al., 20 Nov 2025). This suggests a broader experimental principle: NRSS is most cleanly identified when the measured response is odd under magnetic-domain reversal but persists with SOC disabled in theory.

6. Functional implications and unresolved issues

NRSS is attractive for spintronics because it allows large spin-polarized responses without net magnetization. Exchange-driven splittings can reach the $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$51–$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$52 eV range, larger than typical Rashba-like energy scales, and can generate spin currents, anisotropic transport, and Hall-like responses in compensated magnets (Bhowal et al., 23 Oct 2025). In MnTe, pressure simultaneously raises $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$53 and tunes the NRSS, suggesting a route toward band-selective hole versus electron control in altermagnetic transport (Carlisle et al., 13 May 2025). In FeOF, the predicted $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$54-split configurations resemble ferromagnets in some responses and may enable uncompensated spin currents because there is no cancellation between opposite momentum quadrants (Nathan et al., 13 Dec 2025). In twisted bilayer altermagnets, strain-induced $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$55 transitions activate finite transverse spin conductivity and enhance the spin-splitter angle up to $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$56 (Pathak et al., 23 Feb 2026).

The coexistence of NRSS with other quantum phenomena further broadens its relevance. GdAlSi combines a non-centrosymmetric collinear AFM, Weyl points, and nonrelativistic spin splitting in a single material, motivating device concepts such as the proposed spin twister valve and spin junction transistor (Nag et al., 2023). Floquet-driven antiferromagnets add a dynamical route: the induced NRSS can support pure spin currents in the SOC-free case and, with asymmetric leads, net spin accumulation without relying on conventional Edelstein physics (Li et al., 30 Jul 2025). General ferroelectric-control frameworks extend this to nonvolatile switching of $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$57 and $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$58 across one-, two-, and three-dimensional platforms (Dai et al., 12 Jun 2026).

Several open issues remain. One is conceptual: altermagnetism is a strict symmetry subclass, whereas NRSS is the broader phenomenon. This distinction matters because $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$59-split compensated magnets such as Mn$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$60SiSnN$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$61, $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$62-CrTe$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$63, and some FeOF SRO motifs fall outside the usual altermagnetic definition while remaining fully nonrelativistic spin-split systems (Yuan et al., 2024, Mavani et al., 12 Mar 2025). Another is experimental disentanglement: residual above-$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$64 spin splitting in CoNb$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$65Se$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$66 shows that SOC and altermagnetic fluctuations can coexist, so temperature dependence, domain control, and SOC-off calculations remain essential for interpretation (Dale et al., 2024). A further practical issue is materials stability and operating temperature. Here the outlook is mixed but promising: FeOF is already synthesized and has $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$67 K, markedly higher than FeF$$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$68, while pressure raises MnTe to about $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$69 K at $$H(\mathbf{k}) = \varepsilon(\mathbf{k})\mathbb{I} + \mathbf{g}(\mathbf{k})\cdot\boldsymbol{\sigma}$$70 GPa (Nathan et al., 13 Dec 2025, Carlisle et al., 13 May 2025).

Taken together, the recent literature presents NRSS as a symmetry-governed, exchange-driven phenomenon spanning bulk oxides, rutiles, perovskites, topological semimetals, van der Waals bilayers, twisted moirés, heterostructures, and defect-engineered supercells. The field has moved from idealized symmetry classification to experimentally resolved band structures and to explicit control through polarization, strain, pressure, twist, disorder, and light. This suggests that the main organizing principle is no longer the search for a single “altermagnetic” motif, but the broader design of symmetry environments in which compensated magnetism can produce large, tunable, and diagnostically distinct non-relativistic spin splitting.