Odd-Parity Altermagnetism

• Odd-Parity Altermagnetism is a class of compensated magnets that exhibit inversion-odd, momentum-dependent spin splitting through symmetry-protected mechanisms.
• It leverages spin-space-group classifications to distinguish itself from even-parity and hybrid systems, enabling anisotropic, nodal, and topology-bearing spin textures.
• Experimental and theoretical studies employ methods like Floquet engineering, sublattice current models, and multiferroic chirality to reveal unique optical, transport, and topological phenomena.

to=arxiv_search.search 天天中彩票不ార్గెట్ code variant_json {"query":"odd-parity altermagnetism odd-parity magnetism altermagnetism", "max_results": 10} to=arxiv_search.search 彩神争霸网站 code variant_json {"query":"(Song et al., 29 Apr 2025) OR (Zhu et al., 4 Aug 2025) OR (Luo et al., 20 May 2026) OR (Yu et al., 3 Jan 2025) odd parity altermagnetism", "max_results": 10} to=arxiv_search.search 派奖中 code variant_json {"query":"Electrical switching of an unconventional odd parity magnet", "max_results": 5} Odd-parity altermagnetism denotes a compensated magnetic state with zero net magnetization but finite non-relativistic spin splitting whose momentum dependence is inversion-odd. In its standard band-theoretic form, one writes $E_{k,\uparrow/\downarrow}=E_0(k)\pm \Delta(k)$ with $\Delta(-k)=-\Delta(k)$, or equivalently, for the relevant spin component, $s(k)=-s(-k)$. Within spin-space-group classifications, it is the odd-parity branch of unconventional magnetism. The terminology is not fully uniform: several theory papers use “odd-parity altermagnetism,” whereas the NiI$_2$ study explicitly adopts “odd-parity p-wave magnets,” treating them as a closely related but distinct class of compensated magnets with non-relativistic spin splitting (Luo et al., 20 May 2026, Song et al., 29 Apr 2025).

1. Definition, parity, and relation to altermagnetism

The basic distinction from canonical altermagnetism is parity under momentum inversion. Even-parity altermagnets satisfy $\Delta(-k)=\Delta(k)$ and are commonly associated with $d$-, $g$-, or $i$-wave form factors. Odd-parity altermagnets satisfy $\Delta(-k)=-\Delta(k)$ and are correspondingly associated with odd harmonics such as $p$-, $\Delta(-k)=-\Delta(k)$0-, or, in specific nonsymmorphic settings, $\Delta(-k)=-\Delta(k)$1-wave forms (Luo et al., 7 Oct 2025, Yu et al., 3 Jan 2025).

In the broader representation-theoretic treatment of unconventional magnetism, the relevant object is the spin texture $\Delta(-k)=-\Delta(k)$2. When $\Delta(-k)=-\Delta(k)$3 and $\Delta(-k)=-\Delta(k)$4 are related by an effective inversion symmetry in the spin-space group, each nonzero Cartesian component acquires a definite parity $\Delta(-k)=-\Delta(k)$5 with $\Delta(-k)=-\Delta(k)$6. Pure odd-parity altermagnetism corresponds to $\Delta(-k)=-\Delta(k)$7 for all nonzero components. This separates it from even-parity altermagnetism, hybrid-parity magnets with component-dependent parities, and unconstrained-parity magnets in which no symmetry relates $\Delta(-k)=-\Delta(k)$8 and $\Delta(-k)=-\Delta(k)$9 (Luo et al., 20 May 2026).

This parity classification is independent of the usual ferromagnet–antiferromagnet dichotomy. Odd-parity altermagnets remain compensated magnets with vanishing net magnetization, but unlike conventional $s(k)=-s(-k)$0-symmetric antiferromagnets they exhibit non-relativistic spin-split bands. Unlike ferromagnets, the splitting is not a uniform exchange offset. It is symmetry-structured in momentum space and can therefore be anisotropic, nodal, and topology-bearing (Luo et al., 7 Oct 2025).

2. Spin-group symmetry criteria

The modern formulation is based on spin-space-group operations

$s(k)=-s(-k)$1

where $s(k)=-s(-k)$2 acts in real space, $s(k)=-s(-k)$3 is a proper spin rotation, and $s(k)=-s(-k)$4 is either the identity or time reversal. In the spin-splitting language, a generic symmetry constraint can be written as

$s(k)=-s(-k)$5

with $s(k)=-s(-k)$6 for unitary and $s(k)=-s(-k)$7 for antiunitary operations. Odd parity follows when the surviving symmetry maps $s(k)=-s(-k)$8 while flipping the allowed spin-splitting components (Luo et al., 7 Oct 2025).

Systematic classifications identify collinear, coplanar, and noncoplanar odd-parity cases. One concise criterion is that an antiunitary momentum-inverting symmetry of the $s(k)=-s(-k)$9 type enforces $_2$0 for all spin components. More specialized combinations yield type-I collinear, type-II coplanar, and type-III noncoplanar odd-parity textures. The same framework also specifies the additional crystalline constraints needed for particular $_2$1- or $_2$2-wave forms, such as mirrors for $_2$3 textures or rotational symmetries for $_2$4-wave textures (Luo et al., 20 May 2026, Luo et al., 7 Oct 2025).

A separate but complementary collinear criterion appears in the Floquet literature. In a spin-orbit-free collinear antiferromagnet, preserving $_2$5 while breaking $_2$6 implies $_2$7 but არა $_2$8, which directly yields odd-parity spin splitting. This mechanism is central to light-induced odd-parity collinear magnets and shows that noncollinearity is not a prerequisite (Zhu et al., 4 Aug 2025, Huang et al., 28 Jul 2025).

Another symmetry language is used for the Haldane-Hubbard route. There the crucial operation is written as $_2$9 or, in related notation, $\Delta(-k)=\Delta(k)$0. In that setting the underlying nonmagnetic time-reversal symmetry is broken by sublattice currents, yet the band relation $\Delta(-k)=\Delta(k)$1 still survives as a composite spin-group constraint. This is important because it shows that physical time reversal need not be present for odd-parity altermagnetism to occur; an effective spin-group relation is sufficient (Lin, 12 Mar 2025, Zeng et al., 14 Jul 2025).

3. Microscopic routes and model realizations

Several microscopic mechanisms now realize the same odd-parity condition through different symmetry backbones.

Route	Key symmetry ingredient	Representative platform
Spin-spiral multiferroic	$\Delta(-k)=\Delta(k)$2, unique $\Delta(-k)=\Delta(k)$3, chirality–polarization locking	NiI$\Delta(-k)=\Delta(k)$4 (Song et al., 29 Apr 2025)
Sublattice currents / Haldane term	$\Delta(-k)=\Delta(k)$5 or $\Delta(-k)=\Delta(k)$6, $\Delta(-k)=\Delta(k)$7	Haldane-Hubbard (Lin, 12 Mar 2025, Zeng et al., 14 Jul 2025, Zeng et al., 28 Apr 2026)
Floquet collinear antiferromagnet	break $\Delta(-k)=\Delta(k)$8, preserve $\Delta(-k)=\Delta(k)$9	MnPSe$d$0, honeycomb and dimerized lattices (Zhu et al., 4 Aug 2025, Liu et al., 25 Aug 2025)
Layer/orbital and vdW engineering	$d$1, or $d$2	bilayer orbital-order models; sAFM/metal/sAFM (Zhuang et al., 25 Aug 2025, Kim et al., 11 Feb 2026)

In the spin-spiral multiferroic route, the central quantity is the spin component perpendicular to the spiral plane. For NiI$d$3, the odd-parity condition is expressed as $d$4, and a minimal itinerant picture yields an effective odd-parity Zeeman field

$d$5

The same work emphasizes a Landau-type magnetoelectric coupling $d$6, with $d$7 the ferroelectric polarization and $d$8 the macroscopic chirality. Because $d$9 and chirality are symmetry-locked, electrical switching of $g$0 switches the odd-parity spin texture (Song et al., 29 Apr 2025).

In the Haldane-Hubbard route, odd parity is generated by combining compensated collinear order with an inversion-odd Haldane term. In the correlated formulation, the spin-dependent gap obeys $g$1 while the full gap

$g$2

inherits odd parity from $g$3. The sublattice-current formulation makes the same point in real-space language: complex next-nearest-neighbor hoppings generate opposite circulating currents on the two sublattices, a $g$4-odd sublattice imbalance, and consequently a compensated collinear state with odd-parity spin splitting at half filling (Zeng et al., 28 Apr 2026, Lin, 12 Mar 2025).

Floquet engineering provides a universal collinear route. Circularly polarized light, and in later work elliptically polarized and bicircular light, generate an effective commutator term in the high-frequency expansion that breaks the time-reversal-related spin-group symmetry while preserving the inversion-related one. On the honeycomb lattice this produces a Haldane-like imaginary next-nearest-neighbor term and an $g$5-wave odd-parity pattern; on lower-symmetry or dimerized lattices it produces $g$6-wave odd-parity splitting and, under suitable drive conditions, topological phases (Zhu et al., 4 Aug 2025, Huang et al., 28 Jul 2025, Liu et al., 25 Aug 2025).

Layered constructions supply a distinct nonrelativistic mechanism. One proposal stacks two noncentrosymmetric monolayers in an interlayer antiferromagnetic configuration and applies an in-plane layer-flip operation. There odd-parity spin splitting originates from orbital orders rather than SOC and is protected by an effective time-reversal symmetry $g$7. A separate van der Waals proposal uses an sAFM/metal/sAFM trilayer in which the leading RKKY exchange cancels by symmetry, exposing a higher-order biquadratic interaction that stabilizes an orthogonal $g$8-wave phase with $g$9 and $i$0 enforcing an odd $i$1 texture (Zhuang et al., 25 Aug 2025, Kim et al., 11 Feb 2026).

4. Materials and experimental realization

The most direct experimental realization so far is the spin-spiral type-II multiferroic NiI$i$2. Its host lattice is rhombohedral $i$3; the magnetic transitions occur at $i$4 K and $i$5 K; and the helimagnetic ground state is an incommensurate proper-screw spiral with $i$6 reciprocal lattice units. The helix retains a single real-space $i$7 axis, breaks $i$8 and the parallel spin-space twofold rotation $i$9, and preserves $\Delta(-k)=-\Delta(k)$0. Non-collinear DFT shows that, without SOC, the bands exhibit momentum-dependent $\Delta(-k)=-\Delta(k)$1 that is odd along $\Delta(-k)=-\Delta(k)$2–$\Delta(-k)=-\Delta(k)$3–$\Delta(-k)=-\Delta(k)$4 and zero along $\Delta(-k)=-\Delta(k)$5–$\Delta(-k)=-\Delta(k)$6–$\Delta(-k)=-\Delta(k)$7, with the spin polarization reversing when chirality is reversed (Song et al., 29 Apr 2025).

The same study established electrical control of this odd-parity state. Bulk pyroelectric measurements used a poling field of $\Delta(-k)=-\Delta(k)$8 kV/m during cooling and yielded an integrated polarization $\Delta(-k)=-\Delta(k)$9 with a depolarization peak at $p$0 K. In a 20 nm flake, zero-bias photocurrent appeared below 60 K after field cooling at $p$1 MV/m; repeatable switching at 30 K was achieved with $p$2 MV/m pulses, with coercive field $p$3 MV/m and saturation $p$4 MV/m. Circular photogalvanic measurements along $p$5 gave fit coefficients at 30 K of $p$6 in one chirality domain and $p$7 in the opposite domain, whereas along $p$8 the CPGE coefficient was $p$9, about two orders of magnitude smaller. The dichroism was strongest near 680 nm and vanished near $\Delta(-k)=-\Delta(k)$00, directly linking the signal to the odd-parity magnetic texture (Song et al., 29 Apr 2025).

Floquet first-principles studies broaden the materials landscape to collinear antiferromagnets. For MnPSe$\Delta(-k)=-\Delta(k)$01 under right-handed circularly polarized light with $\Delta(-k)=-\Delta(k)$02 and $\Delta(-k)=-\Delta(k)$03 eV, the calculated maximal spin splitting on the top valence bands reaches about 40 meV, and the anomalous Hall conductivity peaks around $\Delta(-k)=-\Delta(k)$04 when the Fermi level is near $\Delta(-k)=-\Delta(k)$05 eV. Related calculations identify monolayer MnPS$\Delta(-k)=-\Delta(k)$06, AFM FeCl$\Delta(-k)=-\Delta(k)$07 bilayers, and AFM NiRuCl$\Delta(-k)=-\Delta(k)$08 bilayers as light-driven odd-parity platforms at $\Delta(-k)=-\Delta(k)$09 eV and $\Delta(-k)=-\Delta(k)$10, with $\Delta(-k)=-\Delta(k)$11-wave or strain-converted $\Delta(-k)=-\Delta(k)$12-wave patterns depending on light symmetry and lattice symmetry (Zhu et al., 4 Aug 2025, Huang et al., 28 Jul 2025).

Material surveys now extend well beyond individual examples. The antiferromagnetic-exchange framework identified 67 materials in the Magndata database for which its theory applies and presented DFT evidence for an $\Delta(-k)=-\Delta(k)$13-wave spin splitting in FeSe of about 0.1 eV, together with a non-relativistic Edelstein calculation for CeNiAsO (Yu et al., 3 Jan 2025). A later spin-symmetry survey identified 48 candidate materials satisfying odd-parity-magnet criteria, distributed across type-I, type-II, and type-III classes (Luo et al., 7 Oct 2025).

5. Topology, optics, and transport

Odd-parity altermagnetism has unusually rich transport phenomenology because the odd spin texture reshapes Berry geometry, optical selection rules, and current-induced spin polarization. A key symmetry result is that pure odd-parity altermagnets generically allow a non-relativistic Edelstein effect while forbidding the time-reversal-odd magnetic spin Hall conductivity. In the notation of linear response, the Edelstein susceptibility $\Delta(-k)=-\Delta(k)$14 is finite whereas $\Delta(-k)=-\Delta(k)$15. This contrasts with even-parity altermagnets, for which the magnetic spin Hall response is allowed and the Edelstein response vanishes, and with hybrid-parity systems, where both can coexist (Luo et al., 20 May 2026).

The NiI$\Delta(-k)=-\Delta(k)$16 experiment added a direct optical probe of the odd sector through CPGE. Its angular dependence,

$\Delta(-k)=-\Delta(k)$17

isolates a helicity-dependent coefficient $\Delta(-k)=-\Delta(k)$18 that reverses with chirality. Because the odd-parity spin polarization is antisymmetric along the helix propagation direction, the CPGE does not simply diagnose ferroelectric switching; it resolves the reversal of the momentum-odd spin texture itself (Song et al., 29 Apr 2025).

In correlated topological settings, odd-parity altermagnetism can reconstruct local topology without changing the global invariant. In the Haldane-Hubbard ALM-CI phase, the Berry curvature becomes spin- and valley-selective—peaking near $\Delta(-k)=-\Delta(k)$19 for one spin and near $\Delta(-k)=-\Delta(k)$20 for the other—while each spin channel retains $\Delta(-k)=-\Delta(k)$21 and the total Chern number remains $\Delta(-k)=-\Delta(k)$22. Zigzag ribbons develop chiral-symmetry-breaking edge states, armchair ribbons remain inversion symmetric, and the low-frequency Hall conductivity per spin stays quantized at $\Delta(-k)=-\Delta(k)$23 (Zeng et al., 28 Apr 2026).

Floquet platforms reveal several nonequilibrium topological phases. In the honeycomb Floquet odd-parity magnet, the low-energy Dirac masses at $\Delta(-k)=-\Delta(k)$24 and $\Delta(-k)=-\Delta(k)$25 are

$\Delta(-k)=-\Delta(k)$26

with $\Delta(-k)=-\Delta(k)$27. When $\Delta(-k)=-\Delta(k)$28, the system becomes a Chern insulator with $\Delta(-k)=-\Delta(k)$29 for right-handed CPL and $\Delta(-k)=-\Delta(k)$30 for left-handed CPL (Zhu et al., 4 Aug 2025). In a dimerized-lattice realization, CPL dynamically generates an odd-parity $\Delta(-k)=-\Delta(k)$31-wave altermagnet that can become a 2D Chern insulator or a 3D Weyl semimetal under appropriate drive conditions (Liu et al., 25 Aug 2025). In an orbital-order bilayer route, the same odd-parity logic instead produces quantum spin Hall phases with helical edge states and quantized spin Hall conductance (Zhuang et al., 25 Aug 2025).

The topological range broadens further in multilayers. In a bilayer AFM with spin-layer locking, Floquet engineering induces $\Delta(-k)=-\Delta(k)$32-wave odd-parity altermagnetism and nonequilibrium QAHE phases with tunable Chern numbers up to $\Delta(-k)=-\Delta(k)$33 in bilayers and $\Delta(-k)=-\Delta(k)$34 in trilayers, together with large orbital Hall responses controlled by layer- and valley-dependent band inversions (Tian et al., 12 Mar 2026). A bosonic analogue also exists: exchange-only noncoplanar odd-parity magnets support magnon bands with $\Delta(-k)=-\Delta(k)$35- and $\Delta(-k)=-\Delta(k)$36-wave spin textures and a nonrelativistic magnonic thermal Edelstein effect,

$\Delta(-k)=-\Delta(k)$37

whose angular dependence inherits the underlying partial-wave character (Neumann et al., 5 Mar 2026).

6. Taxonomy, misconceptions, and open problems

Three recurrent misconceptions are not supported by the current literature. First, odd-parity altermagnetism is not confined to noncollinear magnets: Floquet constructions show that collinear antiferromagnets can host odd-parity $\Delta(-k)=-\Delta(k)$38- and $\Delta(-k)=-\Delta(k)$39-wave spin splitting once $\Delta(-k)=-\Delta(k)$40 is broken and $\Delta(-k)=-\Delta(k)$41 is preserved (Zhu et al., 4 Aug 2025, Huang et al., 28 Jul 2025). Second, physical time reversal is not a universal requirement: some odd-parity magnets preserve a nonsymmorphic or effective antiunitary symmetry, while others explicitly break the nonmagnetic parent’s time-reversal symmetry through sublattice currents or light yet still obey $\Delta(-k)=-\Delta(k)$42 as a spin-group constraint (Lin, 12 Mar 2025, Zeng et al., 14 Jul 2025). Third, odd-parity altermagnetism is not the full landscape of unconventional magnetism: hybrid-parity and unconstrained-parity classes are now established, and their response tensors differ systematically from the pure odd-parity case (Luo et al., 20 May 2026).

This broader taxonomy is already visible in neighboring phases. In spin-orbital models, circularly polarized light can convert an even-parity collinear antiferromagnet into a pure odd-parity phase at zero staggered potential, but into a mixed-parity phase at finite staggered potential, with even and odd components coexisting in the same spin splitting (Zhuang et al., 6 May 2026). In rare-earth tetraborides, TbB$\Delta(-k)=-\Delta(k)$43 realizes a three-dimensional compensated magnet with odd-parity in-plane $\Delta(-k)=-\Delta(k)$44- and $\Delta(-k)=-\Delta(k)$45-wave textures coexisting with an even-parity out-of-plane $\Delta(-k)=-\Delta(k)$46-wave texture; the odd components arise from a staggered Berry phase generated by scalar spin chirality rather than from SOC (Ryu et al., 2 Jul 2026). This suggests that odd-parity altermagnetism is best regarded as one sector of a parity-resolved spin-texture classification rather than as an isolated binary alternative to canonical altermagnetism.

Open problems are correspondingly structural rather than merely materials-specific. Current papers emphasize the search for room-temperature odd-parity $\Delta(-k)=-\Delta(k)$47-wave magnets, a complete spin-group taxonomy of magnetic space groups supporting protected odd-parity splitting, quantitative transport theories including disorder, commensurability and $\Delta(-k)=-\Delta(k)$48 effects, wafer-scale domain control, clean photogalvanic readout, and integration with CMOS and photonic platforms (Song et al., 29 Apr 2025, Zeng et al., 28 Apr 2026, Luo et al., 7 Oct 2025). A parallel issue is energetic stabilization: some routes rely on Floquet driving, some on multiferroic chirality, some on sublattice currents, and others on higher-order interlayer exchange or orbital-phase engineering. The diversity of mechanisms indicates that odd-parity altermagnetism is not a single microscopic phenomenon but a symmetry class with multiple nonrelativistic realizations.