Non-Relativistic Odd-Parity Magnetism

• Non-relativistic odd-parity magnetism is defined by Bloch-state spin polarization that is antisymmetric under momentum inversion, yielding compensated spin splitting without relying on relativistic spin-orbit coupling.
• It encompasses various magnetic orders—collinear, coplanar, and noncoplanar—employing symmetry analyses such as effective antiunitary protection and spin-group classifications to differentiate types.
• The framework integrates orbital, exchange, and loop-current mechanisms, revealing practical implications for topological band structures, transport responses, and emergent nonrelativistic spin textures.

Non-relativistic odd-parity magnetism denotes a class of unconventional magnetic states in which the Bloch-state spin polarization is antisymmetric under momentum inversion, $\bm S(\bm k)=-\bm S(-\bm k)$, even though the system is symmetry-compensated and the spin splitting does not originate from relativistic spin-orbit coupling. Recent work has generalized the concept from noncollinear magnets to collinear, coplanar, and noncoplanar orders, to odd-parity altermagnetic and orbital constructions, and to topological, Floquet, and magnonic settings, establishing odd-parity magnetism as a symmetry-governed non-relativistic band phenomenon rather than a single microscopic mechanism (Yu et al., 3 Jan 2025, Luo et al., 7 Oct 2025).

1. Classification and defining characteristics

Within the current symmetry taxonomy of unconventional magnetism, odd-parity magnets form one of the pure-parity classes, alongside even-parity magnets identified with altermagnetism. In the odd-parity class, all nonzero spin-texture components are odd under $\bm k\to-\bm k$, whereas even-parity magnets satisfy $s_\alpha(\bm k)=s_\alpha(-\bm k)$. A broader representation-theoretic framework further identifies hybrid-parity magnets, where different Cartesian spin components have different parities, and unconstrained-parity magnets, where no symmetry relates $\bm k$ and $-\bm k$ and the parity of the spin texture is ill-defined (Luo et al., 20 May 2026).

Spin-group analyses subdivide odd-parity magnets into three canonical types. Type-I odd-parity magnets are collinear and retain only one nonzero spin component, for example $s_i(\bm k)=-s_i(-\bm k)$ with $s_{j,l}(\bm k)=0$. Type-II odd-parity magnets are coplanar and retain two odd components, $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$ with $s_l(\bm k)=0$. Type-III odd-parity magnets are noncoplanar and satisfy $s_x(\bm k),s_y(\bm k),s_z(\bm k)$ all odd in momentum (Luo et al., 7 Oct 2025).

This parity structure distinguishes odd-parity magnetism from both ordinary ferromagnetism and even-parity altermagnetism. In the symmetry classification of unconventional magnetism, odd-parity textures automatically imply compensation because opposite momenta carry opposite spin polarization, whereas even- and hybrid-parity classes require additional symmetries to ensure zero net magnetization. A recurring theme in the literature is therefore that odd-parity magnetism is best understood as “magnetism in momentum space,” with the real-space magnetic texture serving as the symmetry source of a translation-invariant, non-relativistic spin splitting (Luo et al., 20 May 2026).

2. Spin-group symmetry and effective antiunitary protection

The general spin-space-group constraint on the spin texture is written as

$\bm k\to-\bm k$0

for a symmetry element $\bm k\to-\bm k$1, where $\bm k\to-\bm k$2, $\bm k\to-\bm k$3 is a spin rotation, and $\bm k\to-\bm k$4 acts in real space. In this formulation, odd-parity spin splitting follows from specific combinations of unitary and antiunitary operations that relate $\bm k\to-\bm k$5 and $\bm k\to-\bm k$6, and the full spin texture transforms as a representation of an effective momentum-space point group $\bm k\to-\bm k$7 (Luo et al., 7 Oct 2025).

One important symmetry route is the presence of an effective inversion-like or time-reversal-like operation that maps $\bm k\to-\bm k$8 to $\bm k\to-\bm k$9 while reversing all nonzero spin components. In the broader classification language, the operation $s_\alpha(\bm k)=s_\alpha(-\bm k)$0 enforces

$s_\alpha(\bm k)=s_\alpha(-\bm k)$1

and therefore pure odd parity for every nonzero component. This is the symmetry archetype behind fully odd spin textures, whereas other operations generate even or hybrid parity by reversing only selected components (Luo et al., 20 May 2026).

Concrete realizations often use effective antiunitary symmetries rather than microscopic time reversal. In bilayer odd-parity altermagnetism generated from orbital order, the key symmetry is

$s_\alpha(\bm k)=s_\alpha(-\bm k)$2

which yields

$s_\alpha(\bm k)=s_\alpha(-\bm k)$3

while generally allowing $s_\alpha(\bm k)=s_\alpha(-\bm k)$4. The symmetry is built by stacking two noncentrosymmetric ferromagnetic monolayers in an interlayer antiferromagnetic configuration and applying an in-plane layer-flip operation so that the top layer becomes the time-reversal copy of the bottom one (Zhuang et al., 25 Aug 2025).

A closely related antiunitary structure appears in spinless $s_\alpha(\bm k)=s_\alpha(-\bm k)$5-wave orbital magnetism. There the combined translation-time-reversal symmetry $s_\alpha(\bm k)=s_\alpha(-\bm k)$6 enforces $s_\alpha(\bm k)=s_\alpha(-\bm k)$7, while $s_\alpha(\bm k)=s_\alpha(-\bm k)$8 implies $s_\alpha(\bm k)=s_\alpha(-\bm k)$9. Together they produce

$\bm k$0

which is the operational definition of $\bm k$1-wave orbital magnetism in that model (Li et al., 20 Apr 2026).

Helimagnets realize the same logic through generalized Bloch symmetry. For a coplanar single-$\bm k$2 spiral, translation by a primitive lattice vector combined with a compensating spin rotation allows the perpendicular spin component to satisfy $\bm k$3. This establishes that, in the non-relativistic limit, spiral magnetic order generically enforces an antisymmetric spin texture, subject to caveats when a lattice translation induces a $\bm k$4 spin rotation and preserves a two-fold spin degeneracy (Larsen et al., 9 Apr 2026).

3. Microscopic mechanisms and lattice constructions

A major line of work derives odd-parity spin splitting directly from coplanar antiferromagnetic exchange in nonsymmorphic crystals. In a systematic study of 421 conventional period-doubling antiferromagnetic systems, microscopic models were constructed for 119 cases, with three competing symmetry-allowed translation-invariant orders: odd-parity spin splitting, nematic order, and scalar odd-parity order. The induced odd-parity spin order is controlled by the vector $\bm k$5, and the spin-splitting energy scale is described as generically large. The same framework identifies 325 irreducible representations for $\bm k$6-wave spin splittings, 84 for $\bm k$7-wave, and 12 for $\bm k$8-wave splittings, and it connects scalar odd-parity order to a non-zero Berry-curvature dipole without SOC (Yu et al., 3 Jan 2025).

A distinct mechanism replaces SOC by orbital angular momentum. In odd-parity altermagnetism originated from orbital orders, hopping between orbitals with magnetic-quantum-number difference $\bm k$9 carries a geometrical phase

$-\bm k$0

so the spin splitting is generated by orbital order rather than by $-\bm k$1. On a noncentrosymmetric square lattice with dimerization and $-\bm k$2 orbitals, this produces $-\bm k$3-wave altermagnets; on a hexagonal $-\bm k$4-symmetric lattice, the same mechanism yields $-\bm k$5-wave altermagnets. The lattice symmetry determines the allowed angular form: $-\bm k$6 gives $-\bm k$7-wave, while $-\bm k$8 gives $-\bm k$9-wave odd-parity altermagnetism (Zhuang et al., 25 Aug 2025).

Sublattice and loop currents provide a further non-relativistic route. In the Haldane-Hubbard model and its generalizations to 2D and 3D bipartite lattices, sublattice currents break $s_i(\bm k)=-s_i(-\bm k)$0, generate momentum-dependent sublattice imbalance, and convert compensated collinear magnetic order into odd-parity altermagnetism. In the spinless loop-current construction of $s_i(\bm k)=-s_i(-\bm k)$1-wave orbital magnetism, electrons circulate around plaquettes, the net macroscopic magnetization cancels, and the Bloch states acquire a momentum-dependent orbital texture measured by the modern-theory expression for $s_i(\bm k)=-s_i(-\bm k)$2 (Lin, 12 Mar 2025, Li et al., 20 Apr 2026).

Several later proposals explicitly remove the common restriction to noncollinear spin textures. Floquet engineering under circularly polarized, elliptically polarized, or bicircular light can induce odd-parity spin splitting in ordinary 2D collinear antiferromagnets; depending on crystal symmetry and polarization state, the induced texture can be $s_i(\bm k)=-s_i(-\bm k)$3-wave or $s_i(\bm k)=-s_i(-\bm k)$4-wave, and its sign can be reversed by changing light chirality (Huang et al., 28 Jul 2025). In collinear altermagnets, a phase-locked two-color linearly polarized drive or a translationally invariant $s_i(\bm k)=-s_i(-\bm k)$5-odd loop-current order introduces a $s_i(\bm k)=-s_i(-\bm k)$6 perturbation that converts native even-parity splitting into a tunable mixed-parity spin texture (Yu, 4 May 2026). A different engineered route uses an sAFM/metal/sAFM van der Waals trilayer, where a half-translation stacking forces $s_i(\bm k)=-s_i(-\bm k)$7, exposes a dominant biquadratic interaction, and drives a filling-controlled transition from a collinear $s_i(\bm k)=-s_i(-\bm k)$8-symmetric phase to an orthogonal $s_i(\bm k)=-s_i(-\bm k)$9-wave phase with $s_{j,l}(\bm k)=0$0 (Kim et al., 11 Feb 2026).

4. Band topology, emergent SOC analogues, and transport responses

Odd-parity magnetism is tightly coupled to topological band structures. In bilayer odd-parity altermagnets generated from orbital orders, the decoupled layers can each enter a Chern-insulating regime with opposite Chern numbers, so the total charge Chern number vanishes while the spin Chern number is nonzero. The resulting state is a quantum spin Hall insulator with helical edge states and quantized spin Hall conductance,

$s_{j,l}(\bm k)=0$1

with $s_{j,l}(\bm k)=0$2 in the examples shown (Zhuang et al., 25 Aug 2025).

Odd-parity magnets can also be intrinsically $s_{j,l}(\bm k)=0$3-topological. Spin-group symmetry criteria identify odd-parity magnetic phases with an effective time-reversal symmetry $s_{j,l}(\bm k)=0$4, allowing a nontrivial $s_{j,l}(\bm k)=0$5 classification. In the model realization on a bilayer breathing kagome lattice, Wilson-loop calculations for the lowest two bands and nanowire spectra with helical edge states demonstrate this topological possibility explicitly (Luo et al., 7 Oct 2025).

A further development is the realization of nonrelativistic SOC analogues in odd-parity coplanar magnets. By symmetry-guided stacking of two monolayer odd-parity altermagnets, bilayer coplanar magnets can realize Rashba-, Weyl-, and Dresselhaus-type spin textures without relativistic SOC. For the Rashba case, the projected low-energy Hamiltonian is

$s_{j,l}(\bm k)=0$6

and the same platform supports a spin Edelstein effect and fully gapped chiral topological superconducting phases with Chern number $s_{j,l}(\bm k)=0$7 when proximitized by an $s_{j,l}(\bm k)=0$8-wave superconductor and subjected to a Zeeman field (Liu et al., 22 Jun 2026).

Response functions sharply distinguish odd-parity from even-parity magnetism. In the symmetry classification of unconventional magnetism, odd-parity textures are the natural setting for the Edelstein effect, while even-parity altermagnetic textures are associated with spin-current responses; hybrid-parity magnets can support both simultaneously (Luo et al., 20 May 2026). Microscopic calculations on CeNiAsO show a substantial non-relativistic Edelstein response $s_{j,l}(\bm k)=0$9 that changes sign with chemical potential (Yu et al., 3 Jan 2025). In the vdW heterostructure proposal, the Edelstein kernel $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$0 vanishes in the collinear $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$1-symmetric phase and becomes large in the orthogonal $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$2-wave phase, remaining robust even for sizable Rashba coupling $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$3 (Kim et al., 11 Feb 2026). In Fe-based superconductors with coplanar magnetic order, the non-relativistic odd-parity state exhibits a finite out-of-plane Berry curvature and nonlinear anomalous Hall effect, while the Edelstein response vanishes without SOC and acquires finite in-plane components only after SOC tilts the spins into the $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$4-plane (Dsouza et al., 29 Aug 2025).

5. Materials platforms and experimental access

The clearest direct experimental demonstration to date is the spin-spiral type-II multiferroic NiI$s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$5. In this material the spiral order is chiral, induces ferroelectric polarization through the spin-current mechanism, and supports an odd-parity spin polarization satisfying

$s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$6

Non-collinear DFT calculations show that the spin splitting is already strong without SOC, the dominant spin component is perpendicular to the spiral plane, and switching chirality reverses the spin polarization. Experimentally, pyroelectric current changes sign under opposite poling fields, the polarization is about $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$7, and the current peak appears near $s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$8 K. Thin-flake measurements of zero-bias photocurrent and, especially, the circular photogalvanic effect show that the CPGE signal reverses when the chirality is switched by electric-field cooling and is nearly absent along the ferroelectric polarization direction, where it is about two orders of magnitude smaller (Song et al., 29 Apr 2025).

Helimagnetic materials provide a complementary route in which the odd-parity texture is reconstructed from the primitive crystallographic unit cell by the generalized Bloch theorem. MnI$s_{i,j}(\bm k)=-s_{i,j}(-\bm k)$9, MnTe$s_l(\bm k)=0$0, and NiI$s_l(\bm k)=0$1 were analyzed in this way. In MnI$s_l(\bm k)=0$2, DFT predicts $s_l(\bm k)=0$3 spiral order, and the spin splitting is maximized near the physical ground-state ordering vector; in MnTe$s_l(\bm k)=0$4, odd-parity spin polarization survives at the Fermi level in a helimagnetic metal; in NiI$s_l(\bm k)=0$5, nearly degenerate $s_l(\bm k)=0$6 and $s_l(\bm k)=0$7 states both show a clear $s_l(\bm k)=0$8-wave odd-parity structure with nodal lines orthogonal to $s_l(\bm k)=0$9. The same work emphasizes that the magnitude of spin splitting correlates strongly with odd-orbital, especially $s_x(\bm k),s_y(\bm k),s_z(\bm k)$0-type, character (Larsen et al., 9 Apr 2026).

Fe-based superconductors provide a multiorbital realization tied to nonsymmorphic symmetry. In FeSe, a coplanar magnetic state at $s_x(\bm k),s_y(\bm k),s_z(\bm k)$1 exhibits the predicted $s_x(\bm k),s_y(\bm k),s_z(\bm k)$2-wave splitting proportional to $s_x(\bm k),s_y(\bm k),s_z(\bm k)$3, and the DFT spin splitting reaches about $s_x(\bm k),s_y(\bm k),s_z(\bm k)$4 eV despite the highly nodal form factor (Yu et al., 3 Jan 2025). A related study of coplanar Fe-based magnetic order in $s_x(\bm k),s_y(\bm k),s_z(\bm k)$5 finds that the splitting magnitude depends sensitively on the out-of-plane hopping parameter $s_x(\bm k),s_y(\bm k),s_z(\bm k)$6 and on the Fermi energies of the hole and electron pockets; FeSe is predicted to show larger splitting than LaFeAsO, for which DFT finds only a few meV for realistic moments (Dsouza et al., 29 Aug 2025).

The search space is already substantial. A symmetry-and-microscopy study of exchange-driven odd-parity magnetism identifies 67 materials in the Magndata database to which its theory applies (Yu et al., 3 Jan 2025). A spin-group classification based directly on odd-parity symmetry criteria identifies 48 candidate materials, spanning type-I, type-II, and type-III odd-parity magnets (Luo et al., 7 Oct 2025). Related symmetry work on $s_x(\bm k),s_y(\bm k),s_z(\bm k)$7- and $s_x(\bm k),s_y(\bm k),s_z(\bm k)$8-symmetric Ising spin order identifies 16 candidate materials whose parity-breaking perturbations can induce non-relativistic odd-parity spin splittings (Yu et al., 12 Mar 2026). Additional proposed platforms include vdW $s_x(\bm k),s_y(\bm k),s_z(\bm k)$9 heterostructures (Kim et al., 11 Feb 2026), rare-earth tetraborides such as TbB$\bm k\to-\bm k$00 with component-resolved mixed parity (Ryu et al., 2 Jul 2026), and orbital-loop-current systems for which the orbital Hall conductivity, nonlocal transport, magneto-optical probes, edge or interface orbital accumulation, and circular-dichroism ARPES have been suggested as probes (Li et al., 20 Apr 2026).

6. Stability, reduced dimensionality, and collective excitations

A central stability issue is incommensuration. In inversion-asymmetric antiferromagnets with odd-parity spin-polarization patterns, the same non-symmorphic symmetry that allows a $\bm k\to-\bm k$01-wave spin texture also allows a non-relativistic Lifshitz invariant in the Ginzburg–Landau free energy,

$\bm k\to-\bm k$02

or equivalently

$\bm k\to-\bm k$03

Because the lower quadratic eigenvalue becomes $\bm k\to-\bm k$04, the first instability is shifted away from $\bm k\to-\bm k$05. The consequence is that a continuous transition directly into a commensurate odd-parity antiferromagnet is generally disfavored; such phases are likely to be preceded by an incommensurate phase or to emerge through a first-order transition (Lee et al., 8 Aug 2025).

In one dimension the same fragility appears as a commensurability condition. In a bosonized Kondo-lattice model for a $\bm k\to-\bm k$06-wave magnet, the Dzyaloshinskii–Moriya interaction pins a chiral spin background with $\bm k\to-\bm k$07 and preserves the combined time-reversal and half-translation symmetry $\bm k\to-\bm k$08. Odd-parity single-particle signatures survive only when $\bm k\to-\bm k$09, realized at quarter filling. At this commensurate fixed point, backscattering gaps out one helicity sector and the electronic spectral function acquires a pronounced $\bm k\to-\bm k$10-wave character. Away from commensurate filling, the relevant backscattering disappears and the odd-parity character is lost (Eikeland et al., 24 Jun 2026).

The odd-parity concept extends beyond electrons to collective modes. Minimal spin models with only isotropic Heisenberg and biquadratic exchange realize magnon bands with

$\bm k\to-\bm k$11

including vector $\bm k\to-\bm k$12-wave odd-parity magnons and collinear $\bm k\to-\bm k$13- and $\bm k\to-\bm k$14-wave antialtermagnets. In the $\bm k\to-\bm k$15-wave antialtermagnetic case, the magnon spin polarization is restricted to a global axis, yet remains odd in momentum and produces a nonrelativistic thermal Edelstein effect,

$\bm k\to-\bm k$16

whose two-lobed angular dependence directly reflects the partial-wave structure of the spin-polarized magnon bands (Neumann et al., 5 Mar 2026).

Taken together, these developments show that non-relativistic odd-parity magnetism is not a single symmetry label attached to one family of noncollinear metals. It is a broad symmetry-governed framework spanning exchange-driven antiferromagnets, helimagnets, orbital and current-loop states, Floquet-engineered collinear systems, van der Waals heterostructures, topological band phases, and magnonic excitations, with stability and observability controlled as much by commensurability, lattice symmetry, and orbital character as by the magnetic order itself.