---
title: Odd-Parity Magnets Overview
url: https://www.emergentmind.com/topics/odd-parity-magnets-opms
type: topic
---

# Odd-Parity Magnets Overview

Odd-parity magnets (OPMs) are compensated magnetic states whose momentum-space spin polarization is odd under inversion, \( \mathbf{S}(-\mathbf{k})=-\mathbf{S}(\mathbf{k}) \), or equivalently whose nonrelativistic spin splitting satisfies \( \Delta E(\mathbf{k})=-\Delta E(-\mathbf{k}) \) in the absence of spin–orbit coupling in the paradigmatic formulations [2603.05415][2603.07643]. They are complementary to altermagnets, for which the spin splitting is even in momentum, and they support zero net magnetization despite exchange-driven band splitting [2605.28701]. The recent literature treats OPMs not as a single microscopic mechanism but as a symmetry class realized in electronic bands, magnon bands, and even orbital textures, with \(p\)-, \(f\)-, and \(h\)-wave form factors and with collinear, coplanar, or noncoplanar spin textures [2501.02057][2604.18695].

## 1. Definition and conceptual scope

In the contemporary nonrelativistic usage, OPMs differ simultaneously from ferromagnets, collinear antiferromagnets, and altermagnets. Conventional ferromagnets and collinear antiferromagnets are constrained by \(T\) or \(PT\), while altermagnets exhibit even-parity splittings \(s_n(\mathbf{k})=+s_n(-\mathbf{k})\). OPMs instead satisfy
\[
s_n(\mathbf{k})=-s_n(-\mathbf{k}),
\]
so their spin texture is antisymmetric in momentum space [2603.05415]. In helimagnets this appears as odd-parity spin-momentum locking of the electronic spin expectation \(S_n(\mathbf{k})\), while in insulating spin models it appears as odd-wave magnon spin textures [2604.08233][2603.05415].

A central conceptual point is that the literature contains two complementary symmetry narratives. One widely used formulation emphasizes preserved generalized or nonsymmorphic time-reversal operations, such as the spin-translation group
\[
\mathbf{G}_{\rm st}=\{(E|0),(T|\tau)\},
\]
which protects \(s_n(\mathbf{k})=-s_n(-\mathbf{k})\) in a noncollinear compensated magnet [2603.05415]. A more recent reformulation argues that previous studies focused exclusively on non-relativistic spin splitting while overlooking the intrinsically broken \(\mathcal T\) inherent to magnetic order, and proposes that OPMs universally host a hidden Zeeman field rooted in this \(\mathcal T\)-breaking [2603.15147]. The coexistence of these descriptions is an active feature of the field rather than a settled terminological closure.

## 2. Symmetry criteria and classification

The unifying language of recent classification work is the spin-space group. In that notation an operation is written
\[
g=\{XU\parallel R|\tau\},
\]
where \(X=I\) or \(T\), \(U\in SU(2)\), and \(\{R|\tau\}\) is the spatial part; under \(g\), the Bloch spin texture transforms as
\[
S(\lambda Rk)=\lambda\,U\,S(k),
\]
with \(\lambda=+1\) for \(X=I\) and \(\lambda=-1\) for \(X=T\) [2603.07643]. This framework classifies OPMs into three types: type-I collinear, type-II coplanar, and type-III noncoplanar. Representative symmetry elements include \(g_1=\{U_z(\theta)\parallel I\}\), which forces a collinear texture, \(g_2=\{TU_z(\pi)\parallel P\}\), which forces a coplanar texture, and \(g_3=\{T\parallel I\}\), which enforces odd parity under \(\mathbf{k}\to-\mathbf{k}\) [2603.07643].

The same work identifies eight distinct symmetry-driven mechanisms for OPMs and reports 33 OPM candidates from the Magndata database [2603.07643]. A related spin-symmetry analysis likewise identifies eight symmetry-driven cases, organizes OPMs into the same three texture classes, further delineates additional symmetry requirements for \(p\)-wave and \(f\)-wave spin splitting in type-I systems, and reports 48 candidate materials in Magndata together with an intrinsic \(\mathbb{Z}_2\) topology in some OPMs [2510.05512]. A separate antiferromagnetic-exchange program arrives at a different but complementary enumeration: 421 distinct 2D magnetic irreducible representations permitting odd-parity order, 119 minimal microscopic models, and 67 materials for which that theory applies; among those irreducible representations, spin-vector cases appear in 325 \(p\)-wave, 84 \(f\)-wave, and 12 \(h\)-wave examples [2501.02057]. This suggests that candidate counts are framework-dependent, but the repeated emergence of \(p\)-, \(f\)-, and \(h\)-wave OPMs is robust across independent symmetry constructions.

## 3. Microscopic routes to odd-parity magnetism

Several distinct microscopic routes now exist. In the antiferromagnetic-exchange framework, the key secondary order parameter is
\[
\boldsymbol\eta\equiv S_1\times S_2,
\]
which is time-reversal even and inversion odd. The same Landau theory also yields two competing translation-invariant orders, \(\psi\equiv S_1\cdot S_2\) and \(\chi\equiv |S_1|^2-|S_2|^2\), leading respectively to odd-parity spin splitting, nematic order, or scalar odd-parity order related to multiferroicity [2501.02057]. In that setting the effective spin-splitting Hamiltonian is \(H_{\rm spin}=\sum_k c_k^\dagger[g(k)\cdot\sigma]c_k\) with \(g(-k)=-g(k)\).

Single-\(q\) helimagnets provide another analytically controlled route. For a magnetization satisfying
\[
m(r+R)=R_{\hat n}(q\!\cdot\!R)\,m(r),
\]
the Generalized Bloch theorem allows the problem to be solved in the primitive cell through a twisted Hamiltonian \(H_{q,k}\), and magnetic-supercell quantities are recovered by reciprocal-space downfolding [2604.08233]. In the non-relativistic limit, coplanar spiral order always yields odd-parity magnetism: the spin component along the spiral-plane normal obeys \(S_n(-k)=-S_n(k)\), while \(\varepsilon_n(-k)=\varepsilon_n(k)\). First-principles case studies on MnI\(_2\), NiI\(_2\), and MnTe\(_2\) further show that the magnitude of the spin splitting is maximized for states having large odd-orbital (\(p\)-type) character [2604.08233].

More engineered routes broaden the design space. In sAFM/metal/sAFM van der Waals heterostructures, the leading RKKY-type exchange interaction is canceled by the stacking symmetry, exposing a higher-order biquadratic interaction that drives a filling-controlled transition from a collinear phase to an orthogonal \(p\)-wave configuration [2602.11251]. Floquet engineering provides a dynamic route in conventional collinear antiferromagnets: off-resonant circularly polarized light, elliptically polarized light, and bicircular light can induce odd-parity spin splitting, with CPL favoring an \(f\)-wave pattern and EPL or BCL reducing it to a \(p\)-wave structure [2507.20705]. Still more unconventional realizations have been proposed in momentum-space Hatsugai–Kohmoto-like models, where OPMs appear as \(p\)-wave spin-bond order with \(\sigma_k=-\sigma_{-k}\) and vanishing on-site moment [2604.18684], and in loop-current systems, where a spinless lattice model realizes \(p\)-wave orbital magnetism protected by combined translation and time reversal [2604.18695].

## 4. Material platforms and experimental fingerprints

Experimentally, the most direct electronic evidence so far comes from CeNiAsO, NiI\(_2\), and coplanar Fe-based superconductors, while MnI\(_2\), NiI\(_2\), and MnTe\(_2\) serve as first-principles helimagnetic benchmarks [2605.28701][2504.21086][2508.21673][2604.08233].

| Platform | Odd-parity fingerprint | Reported scale |
|---|---|---|
| CeNiAsO | one nodal plane \(k_x=0\); minimal \(H_p(k)=\lambda k_x\sigma_z\) | spin splitting \(\sim 50\!-\!100\,\mathrm{meV}\) |
| NiI\(_2\) | chirality-linked odd \(\langle \sigma_z(k)\rangle\) along \(Y\!-\!\Gamma\!-\!Y'\) | \(\Delta_{\rm NR}\approx 0.2\!-\!1.0\,\mathrm{eV}\) |
| Fe-based superconductors | \(h\)-wave \(g_z(k)\) and finite nonlinear Hall response | FeSe DFT splitting up to \(\Delta\sim0.1\,\mathrm{eV}\) |

CeNiAsO is currently the clearest prototype \(p\)-wave OPM. Below \(T_{N2}\approx7.4\,\mathrm K\), the Ce moments form a coplanar, noncollinear \(2\times1\) antiferromagnetic order in the \(ab\) plane. Symmetry permits the minimal low-energy form \(H_p(k)=\lambda k_x\sigma_z\), implying \(S_z(k_x)=-S_z(-k_x)\) and \(S_z=0\) on the nodal plane \(k_x=0\). Spin-resolved ARPES resolves \(a_1/a_2\) and \(\beta_1/\beta_2\) splittings of \(\sim50\!-\!100\,\mathrm{meV}\), with \(S_z(k)\) reversing sign across the nodal plane; below \(T_{N1}=9.4\,\mathrm K\), the magnetoresistance is two-fold, the bulk \(\Delta R/R\) reaches \(\sim35\%\) at \(6\,\mathrm T\), and FIB devices reach \(\sim48\%\) at \(9\,\mathrm T\), enabling field-induced switching between high-resistance and low-resistance states through domain selection [2605.28701].

NiI\(_2\) realizes the helimagnetic and multiferroic route. Noncollinear DFT on a NiI\(_2\) monolayer model of the bulk helix finds large spin splitting \(\Delta_{\rm NR}\approx0.2\!-\!1.0\,\mathrm{eV}\) along \(Y\!-\!\Gamma\!-\!Y'\) even with SOC turned off, while \(\langle\sigma_z(k)\rangle\) vanishes along \(X\!-\!\Gamma\!-\!X'\) and is antisymmetric along the propagation axis [2504.21086]. Zero-bias photocurrent and circular photogalvanic measurements then directly connect the sign of the odd-parity spin texture to spiral chirality and ferroelectric polarization: below \(T_{N2}\approx59.5\,\mathrm K\), the CPGE coefficient is \(C\approx+0.0966\) for one chirality and \(C\approx-0.1085\) for the opposite chirality along \(Q_y\), while along the ferroelectric axis it is negligible, \(C\approx0.0068\) [2504.21086]. The same improper-ferroelectric coupling permits voltage-based switching of the chirality and hence of the odd-parity spin polarization.

In Fe-based superconductors with coplanar magnetic order, low-energy modeling and DFT identify an \(h\)-wave OPM. The leading spin-splitting form factor is
\[
h(\mathbf{k})=(k_x^2-k_y^2)k_xk_y\sin k_z,
\]
or, in lattice form for FeSe,
\[
g_z(k)\propto \sin k_x\sin k_y(\cos k_x-\cos k_y)\sin k_z.
\]
DFT fits indicate splittings of order a few meV in LaFeAsO for realistic \(\Delta\sim0.2\,\mu_B\), while in FeSe the splitting can reach \(\Delta\sim0.1\,\mathrm{eV}\) [2508.21673][2501.02057]. A related and experimentally important point is that these OPMs support finite out-of-plane Berry curvature and a nonlinear anomalous Hall effect even though the linear Hall effect vanishes [2508.21673].

## 5. Magnons and odd-parity bosonic excitations

Odd-parity magnetism is not restricted to electronic quasiparticles. In insulating noncoplanar compensated magnets, minimal spin Hamiltonians built only from isotropic Heisenberg bilinear and biquadratic terms produce magnon bands with odd-wave spin textures and no relativistic interactions [2603.05415]. Three minimal examples are particularly important: a vector \(p\)-wave OPM on the kagome lattice, a \(p\)-wave antialtermagnet with collinear \(s_n^z(\mathbf{k})=-s_n^z(-\mathbf{k})\), and an \(f\)-wave antialtermagnet on stacked triangular layers with three nodal planes and collinear magnon spin along the \([111]\) axis. These models show that antialtermagnetism can arise in the magnon sector even when the ground state is noncoplanar [2603.05415].

The magnon spin is obtained from the paraunitary Bogoliubov transformation of the linear spin-wave Hamiltonian, and its odd-wave character immediately produces nonequilibrium responses. In particular, the thermal Edelstein effect is
\[
M_\mu=\sum_\nu \chi_{\mu\nu}(-\nabla_\nu T),
\]
with \(\chi_{\mu\nu}\) determined by the magnon spin polarization, magnon velocity, magnon energy, and the derivative of the Bose function [2603.05415]. In the \(p\)-wave antialtermagnet only \(\chi_{zx,y}\neq0\), giving a two-lobed \(p\)-orbital angular pattern; the vector \(p\)-wave model instead yields an in-plane response, while the \(f\)-wave case would produce a six-fold anisotropy [2603.05415].

A second bosonic route appears in the Haldane–Hubbard model. There, topological spin-flip excitons in the paramagnetic phase condense into a collinear Néel state, and the resulting magnon bands acquire odd-parity \(f\)-wave splitting [2606.05837]. In the topological antiferromagnetic phase the two magnon bands carry Chern number \(C_{\mathrm{mag},\pm}=-2\); an electronic band-gap closing enforces a magnon gap closing and changes the odd-parity magnon topology [2606.05837]. This establishes odd-parity magnons as both a symmetry phenomenon and a topological one.

## 6. Response functions, topology, and active directions

The signature response of electronic OPMs is the non-relativistic Edelstein effect. In a coplanar single-\(q\) helimagnet, an electric field shifts the Fermi surface, \(\delta k\sim E\), and the antisymmetry \(S_n(-k)=-S_n(k)\) produces a net nonequilibrium spin, summarized in the chain \(E\to\delta k\to\delta S\) [2604.08233]. The same Generalized Bloch theorem framework extends directly to Kubo-type linear and nonlinear response functions without ever constructing large magnetic supercells [2604.08233]. In van der Waals heterostructures, the orthogonal \(p\)-wave phase exhibits a gate-tunable Edelstein tensor \(\chi_{zy}\), and the response remains qualitatively unchanged even for Rashba SOC as large as \(\lambda_{\rm SO}/t=0.3\) [2602.11251].

At the same time, OPMs do not exhibit a universal Edelstein response. In the Fe-based \(h\)-wave OPMs, the Edelstein effect vanishes without SOC because \(S_x(\mathbf{k})=S_y(\mathbf{k})=0\) exactly and mirror symmetries force \(\chi^E_{iz}=0\); only after adding atomic SOC do in-plane \(p\)-wave spin components and a finite in-plane Edelstein response appear [2508.21673]. This is an important correction to the common expectation that odd parity in momentum space automatically implies current-induced spin polarization.

Topological and adjacent extensions are now central. One spin-symmetry analysis shows that OPMs can exhibit an intrinsic \(\mathbb{Z}_2\) topology [2510.05512]. A different proposal argues that OPMs are an ideal platform for topological superconductivity because the large non-relativistic spin splitting coexists with a hidden Zeeman field and can support chiral or unidirectional Majorana boundary modes in the presence of \(s\)-wave pairing [2603.15147]. Beyond spin, loop-current models demonstrate \(p\)-wave orbital magnetism with zero macroscopic orbital magnetization but finite orbital Hall conductivity [2604.18695]. Quantum-geometric mechanisms add yet another layer: on the bilayer Lieb lattice, the quantum metric enhances ferroic odd-parity multipole fluctuations, and an on-site Hubbard interaction drives their RPA condensation into long-range odd-parity multipole order [2505.20907].

Three misconceptions are now difficult to sustain. First, OPMs are not limited to noncollinear magnets: they can occur in collinear, coplanar, and noncoplanar orders, and can even be induced dynamically in conventional collinear antiferromagnets [2603.07643][2507.20705]. Second, OPMs are not restricted to \(p\)-wave spin textures: \(f\)-wave and \(h\)-wave forms are explicit in both symmetry classifications and material realizations [2501.02057][2508.21673]. Third, OPMs are not exclusively electronic-spin phenomena: magnons, orbital moments, and odd-parity multipole fluctuations all realize closely related symmetry structures [2603.05415][2604.18695][2505.20907].

The field has therefore moved from a narrow definition of exchange-driven odd spin splitting to a broader program of symmetry-guided design. Current directions emphasize primitive-cell treatments of helimagnets, electrically controlled heterostructures, multiferroic switching, odd-parity magnons, and field-free topological superconductivity, while the precise relation between generalized time-reversal descriptions and hidden-\(\mathcal T\)-breaking descriptions remains a central conceptual issue [2604.08233][2603.15147].

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