---
title: Odd-Parity Altermagnetism
url: https://www.emergentmind.com/topics/odd-parity-altermagnetism
type: topic
---

# Odd-Parity Altermagnetism

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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 [2605.21336, 2504.21086].

## 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 $p$-, $f$-, or, in specific nonsymmorphic settings, $h$-wave forms [2510.05512, 2501.02057].

In the broader representation-theoretic treatment of unconventional magnetism, the relevant object is the spin texture $S(k)=(s_x(k),s_y(k),s_z(k))$. When $k$ and $-k$ are related by an effective inversion symmetry in the spin-space group, each nonzero Cartesian component acquires a definite parity $s_i(k)=p_i s_i(-k)$ with $p_i=\pm 1$. Pure odd-parity altermagnetism corresponds to $p_i=-1$ 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 $k$ and $-k$ [2605.21336].

This parity classification is independent of the usual ferromagnet–antiferromagnet dichotomy. Odd-parity altermagnets remain compensated magnets with vanishing net magnetization, but unlike conventional $\mathcal{PT}$-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 [2510.05512].

## 2. Spin-group symmetry criteria

The modern formulation is based on spin-space-group operations
$$
g=\{XU\parallel R\mid \tau\},
$$
where $\{R\mid\tau\}$ acts in real space, $U$ is a proper spin rotation, and $X$ is either the identity or time reversal. In the spin-splitting language, a generic symmetry constraint can be written as
$$
\mathbf d(s_gR_gk)=s_gR_s\,\mathbf d(k),
$$
with $s_g=+1$ for unitary and $s_g=-1$ for antiunitary operations. Odd parity follows when the surviving symmetry maps $k\to -k$ while flipping the allowed spin-splitting components [2510.05512].

Systematic classifications identify collinear, coplanar, and noncoplanar odd-parity cases. One concise criterion is that an antiunitary momentum-inverting symmetry of the $g_5=\{T\parallel I\mid \tau\}$ type enforces $s_i(k)=-s_i(-k)$ 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 $p$- or $f$-wave forms, such as mirrors for $p_x/p_y/p_z$ textures or rotational symmetries for $f$-wave textures [2605.21336, 2510.05512].

A separate but complementary collinear criterion appears in the Floquet literature. In a spin-orbit-free collinear antiferromagnet, preserving $[C_2\parallel P]$ while breaking $[C_2T\parallel E]$ implies $\epsilon(s,k)=\epsilon(-s,-k)$ but არა $\epsilon(s,k)=\epsilon(s,-k)$, 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 [2508.02542, 2507.20705].

Another symmetry language is used for the Haldane-Hubbard route. There the crucial operation is written as $[\bar C_2\parallel \bar E]$ or, in related notation, $[C_2\parallel \bar E]$. In that setting the underlying nonmagnetic time-reversal symmetry is broken by sublattice currents, yet the band relation $E_{k\sigma}=E_{-k,-\sigma}$ 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 [2503.09602, 2507.09906].

## 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 | $U_{2\perp}T$, unique $C_{2x}$, chirality–polarization locking | NiI$_2$ [2504.21086] |
| Sublattice currents / Haldane term | $[\bar C_2\parallel \bar E]$ or $[C_2\parallel \bar E]$, $\gamma_2(-k)=-\gamma_2(k)$ | Haldane-Hubbard [2503.09602], [2507.09906], [2604.25537] |
| Floquet collinear antiferromagnet | break $[C_2T\parallel E]$, preserve $[C_2\parallel P]$ | MnPSe$_3$, honeycomb and dimerized lattices [2508.02542], [2508.18360] |
| Layer/orbital and vdW engineering | $T_{\rm eff}=[\bar E\parallel M_zT]$, or $\Xi=T_{(1,1)}\mathcal T$ | bilayer orbital-order models; sAFM/metal/sAFM [2508.18361], [2602.11251] |

In the spin-spiral multiferroic route, the central quantity is the spin component perpendicular to the spiral plane. For NiI$_2$, the odd-parity condition is expressed as $\langle \sigma_\perp(k)\rangle=-\langle \sigma_\perp(-k)\rangle$, and a minimal itinerant picture yields an effective odd-parity Zeeman field
$$
h_\perp(k)=\frac{k\cdot Q}{2m}.
$$
The same work emphasizes a Landau-type magnetoelectric coupling $F_{\rm coup}\propto \lambda(P\cdot C)$, with $P$ the ferroelectric polarization and $C$ the macroscopic chirality. Because $P$ and chirality are symmetry-locked, electrical switching of $P$ switches the odd-parity spin texture [2504.21086].

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 $\Delta_\sigma=-\Delta_{-\sigma}$ while the full gap
$$
\Delta_{k\sigma}=\Delta_\sigma+\lambda(Z_{A\sigma}+Z_{B\sigma})\gamma_2(k)
$$
inherits odd parity from $\gamma_2(-k)=-\gamma_2(k)$. 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 $k$-odd sublattice imbalance, and consequently a compensated collinear state with odd-parity spin splitting at half filling [2604.25537, 2503.09602].

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 $f$-wave odd-parity pattern; on lower-symmetry or dimerized lattices it produces $p$-wave odd-parity splitting and, under suitable drive conditions, topological phases [2508.02542, 2507.20705, 2508.18360].

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 $T_{\rm eff}=[\bar E\parallel M_zT]$. 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 $p$-wave phase with $\Xi=T_{(1,1)}\mathcal T$ and $\mathcal S=T_{(1,1)}C_{2z}^S$ enforcing an odd $s_z(k)$ texture [2508.18361, 2602.11251].

## 4. Materials and experimental realization

The most direct experimental realization so far is the spin-spiral type-II multiferroic NiI$_2$. Its host lattice is rhombohedral $R\bar 3m$; the magnetic transitions occur at $T_{N,1}\simeq 75$ K and $T_{N,2}\simeq 59.5$ K; and the helimagnetic ground state is an incommensurate proper-screw spiral with $Q\simeq(0.138,0,1.457)$ reciprocal lattice units. The helix retains a single real-space $C_{2x}$ axis, breaks $\mathcal{PT}$ and the parallel spin-space twofold rotation $U_{2\parallel}$, and preserves $U_{2\perp}T$. Non-collinear DFT shows that, without SOC, the bands exhibit momentum-dependent $\langle \sigma_z(k)\rangle$ that is odd along $Y$–$\Gamma$–$Y'$ and zero along $X$–$\Gamma$–$X'$, with the spin polarization reversing when chirality is reversed [2504.21086].

The same study established electrical control of this odd-parity state. Bulk pyroelectric measurements used a poling field of $\pm 20$ kV/m during cooling and yielded an integrated polarization $P\simeq 42\,\mu{\rm C/m^2}$ with a depolarization peak at $59.5$ K. In a 20 nm flake, zero-bias photocurrent appeared below 60 K after field cooling at $\pm 2$ MV/m; repeatable switching at 30 K was achieved with $\pm 12$ MV/m pulses, with coercive field $\simeq 5$ MV/m and saturation $\simeq 10$ MV/m. Circular photogalvanic measurements along $Q_y\parallel[120]$ gave fit coefficients at 30 K of $C\simeq +0.0966$ in one chirality domain and $C\simeq -0.1085$ in the opposite domain, whereas along $P\parallel[100]$ the CPGE coefficient was $C\simeq 0.0068$, about two orders of magnitude smaller. The dichroism was strongest near 680 nm and vanished near $T_{N,2}$, directly linking the signal to the odd-parity magnetic texture [2504.21086].

Floquet first-principles studies broaden the materials landscape to collinear antiferromagnets. For MnPSe$_3$ under right-handed circularly polarized light with $\tilde A=0.2\ {\rm \AA}^{-1}$ and $\hbar\omega=5$ eV, the calculated maximal spin splitting on the top valence bands reaches about 40 meV, and the anomalous Hall conductivity peaks around $\sim 6\,e^2/h$ when the Fermi level is near $-0.1$ eV. Related calculations identify monolayer MnPS$_3$, AFM FeCl$_2$ bilayers, and AFM NiRuCl$_6$ bilayers as light-driven odd-parity platforms at $\hbar\omega=10$ eV and $eA_0/\hbar=0.3\ {\rm \AA}^{-1}$, with $f$-wave or strain-converted $p$-wave patterns depending on light symmetry and lattice symmetry [2508.02542, 2507.20705].

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 $h$-wave spin splitting in FeSe of about 0.1 eV, together with a non-relativistic Edelstein calculation for CeNiAsO [2501.02057]. A later spin-symmetry survey identified 48 candidate materials satisfying odd-parity-magnet criteria, distributed across type-I, type-II, and type-III classes [2510.05512].

## 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 $\chi^{(e)}$ is finite whereas $\sigma_{ij}^{\alpha(o)}=0$. 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 [2605.21336].

The NiI$_2$ experiment added a direct optical probe of the odd sector through CPGE. Its angular dependence,
$$
I_{\rm total}(\phi)=C\sin(2\phi)+L_1\sin(4\phi)+L_2\cos(4\phi)+D,
$$
isolates a helicity-dependent coefficient $C$ 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 [2504.21086].

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 $K$ for one spin and near $K'$ for the other—while each spin channel retains $C_\sigma=1$ and the total Chern number remains $C=2$. Zigzag ribbons develop chiral-symmetry-breaking edge states, armchair ribbons remain inversion symmetric, and the low-frequency Hall conductivity per spin stays quantized at $e^2/h$ [2604.25537].

Floquet platforms reveal several nonequilibrium topological phases. In the honeycomb Floquet odd-parity magnet, the low-energy Dirac masses at $K$ and $K'$ are
$$
H_K=t'(q_x\sigma_x+q_y\sigma_y)+(s m_z-\eta m')\sigma_z,\qquad
H_{K'}=t'(-q_x\sigma_x+q_y\sigma_y)+(s m_z+\eta m')\sigma_z,
$$
with $m'=9[J_1(\tilde A)]^2/\omega$. When $m'>m_z$, the system becomes a Chern insulator with $C=-2$ for right-handed CPL and $C=+2$ for left-handed CPL [2508.02542]. In a dimerized-lattice realization, CPL dynamically generates an odd-parity $p$-wave altermagnet that can become a 2D Chern insulator or a 3D Weyl semimetal under appropriate drive conditions [2508.18360]. 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 [2508.18361].

The topological range broadens further in multilayers. In a bilayer AFM with spin-layer locking, Floquet engineering induces $f$-wave odd-parity altermagnetism and nonequilibrium QAHE phases with tunable Chern numbers up to $C=\pm 8$ in bilayers and $\pm 12$ in trilayers, together with large orbital Hall responses controlled by layer- and valley-dependent band inversions [2603.11483]. A bosonic analogue also exists: exchange-only noncoplanar odd-parity magnets support magnon bands with $p$- and $f$-wave spin textures and a nonrelativistic magnonic thermal Edelstein effect,
$$
\alpha_{\mu\nu}=\frac{1}{VT}\sum_n\int_{\rm BZ}\frac{d^dk}{(2\pi)^d}\,
\tau\, s_n^\mu(k)\,v_n^\nu(k)\,\epsilon_n(k)\,
\Big[-\frac{\partial \rho(\epsilon_n(k))}{\partial \epsilon_n(k)}\Big],
$$
whose angular dependence inherits the underlying partial-wave character [2603.05415].

## 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 $p$- and $f$-wave spin splitting once $[C_2T\parallel E]$ is broken and $[C_2\parallel P]$ is preserved [2508.02542, 2507.20705]. 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 $E_{k\sigma}=E_{-k,-\sigma}$ as a spin-group constraint [2503.09602, 2507.09906]. 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 [2605.21336].

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 [2605.05205]. In rare-earth tetraborides, TbB$_4$ realizes a three-dimensional compensated magnet with odd-parity in-plane $p$- and $f$-wave textures coexisting with an even-parity out-of-plane $d$-wave texture; the odd components arise from a staggered Berry phase generated by scalar spin chirality rather than from SOC [2607.02117]. 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 $p$-wave magnets, a complete spin-group taxonomy of magnetic space groups supporting protected odd-parity splitting, quantitative transport theories including disorder, commensurability and $T\tau$ effects, wafer-scale domain control, clean photogalvanic readout, and integration with CMOS and photonic platforms [2504.21086, 2604.25537, 2510.05512]. 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.

Source: https://www.emergentmind.com/topics/odd-parity-altermagnetism