Light-Induced Odd-Parity Magnetism

• The paper attributes light-induced odd-parity magnetism to Floquet-engineered, momentum-odd spin splitting in antiferromagnets using off-resonant optical driving.
• It emphasizes how modified symmetries in systems like Cr2CH2 and MnPSe3 lead to tunable p-wave or f-wave spin textures and topological states.
• Experimental diagnostics include spin-resolved ARPES and transport measurements that uncover nontrivial anomalous Hall and orbital responses.

Light-induced odd-parity magnetism denotes a class of driven magnetic states in which periodic irradiation generates spin splitting that is odd under momentum inversion, typically $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$ or $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$, while the underlying system can remain a compensated antiferromagnet with zero net magnetization (Zhu et al., 4 Aug 2025). In the current literature, the phenomenon is realized mainly through Floquet engineering: circularly polarized, elliptically polarized, bicircular, or phase-locked two-color fields modify effective spin-space-group symmetries and produce $p$-wave or $f$-wave spin textures in collinear, coplanar, and spin-orbital magnets, and even in magnon bands; in several platforms the same mechanism also generates higher-order topology, Chern phases, Weyl phases, or orbital Hall responses (Huang et al., 28 Jul 2025, Zou et al., 18 May 2026, Zhang et al., 29 May 2026).

1. Definition and conceptual scope

In the nonrelativistic compensated-magnet literature, “odd parity” usually refers to the momentum parity of the spin splitting rather than to an inversion-odd real-space magnetic order parameter. The clearest formulation is the driven Cr$_2$CH$_2$ case, where odd parity is defined by $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$; the paper is explicit that this means odd momentum parity of the spin splitting, not odd spatial parity of a real-space magnetic order parameter (Zou et al., 18 May 2026). In this sense, odd-parity magnetism is the odd-$\mathbf{k}$ counterpart of altermagnetic spin splitting: the magnet remains compensated, but its quasiparticle bands acquire a momentum-antisymmetric spin polarization.

A second usage appears in the multipole literature. There, odd-parity magnetism refers to inversion-odd magnetic order parameters such as the layer-staggered spin multipole $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$, or to odd-parity magnetic quadrupole order encoded in antisymmetric spin-orbital polarization in momentum space (Kudo et al., 27 May 2025, Hayami et al., 2021). These usages are related but not identical: one centers on the symmetry of band spin splitting, the other on the symmetry of the underlying magnetic order parameter.

A third extension concerns bosonic excitations. In odd-parity magnonics, the odd object is the magnon-band splitting, $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$, rather than an equilibrium inversion-odd magnetic ground state (Zhang et al., 29 May 2026). The common thread across these branches is that odd parity is diagnosed in momentum space, but the microscopic carrier can be an electronic Bloch band, a spin-orbital composite texture, or a magnon branch.

Context	Defining object	Representative source
Floquet odd-parity electronic magnetism	$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$0 or $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$1	(Zou et al., 18 May 2026, Huang et al., 28 Jul 2025)
Odd-parity multipole magnetism	$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$2	(Kudo et al., 27 May 2025)
Odd-parity magnetic quadrupole order	odd-parity MQ with spin-orbital-momentum locking	(Hayami et al., 2021)
Odd-parity magnons	magnon splitting $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$3	(Zhang et al., 29 May 2026)

2. Symmetry principles and Floquet routes

The dominant electronic mechanism is symmetry-selective Floquet lifting of spin degeneracy. In conventional collinear antiferromagnets, two spin-group symmetries commonly enforce spin degeneracy: $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$4, giving $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$5, and $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$6, giving $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$7. Circularly polarized light breaks $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$8 while preserving $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$9, so the spin degeneracy is lifted but the remaining relation still forces odd-parity spin splitting (Zhu et al., 4 Aug 2025). In dimerized antiferromagnets the same logic is reformulated through the survival of $p$0 after the light field destroys the spinful $p$1 symmetry that had enforced double degeneracy (Liu et al., 25 Aug 2025).

At the Hamiltonian level, these works use a Peierls-substituted time-periodic tight-binding model and the high-frequency Floquet expansion

$p$2

or equivalent van Vleck/Magnus forms. The commutator term is the central source of odd-parity magnetism: it encodes virtual photon absorption and emission and generates symmetry-selective effective hoppings or mass terms that are odd in momentum. In honeycomb and dimerized models this correction can take the form of a Haldane-like imaginary next-nearest-neighbor hopping or an inversion-odd mass term, thereby tying odd-parity spin splitting directly to Floquet-induced topology (Zhu et al., 4 Aug 2025, Liu et al., 25 Aug 2025).

A second optical route does not rely on circular polarization. In altermagnets driven by a phase-locked two-color linearly polarized field,

$p$3

the time-averaged cubic composite

$p$4

transforms as $p$5. Its coupling to the intrinsic altermagnetic order produces an induced odd-parity spin-splitting component; microscopically this appears as

$p$6

so an even-parity $p$7 altermagnetic splitting becomes a tunable mixed-parity $p$8 texture (Yu, 4 May 2026). Closely related spin-orbital models show that off-resonant CPL induces purely odd-parity altermagnetism at zero staggered potential and mixed-parity altermagnetism at finite staggered potential, with the parity class fixed by which mirror and inversion-related spin symmetries survive in the Floquet Hamiltonian (Zhuang et al., 6 May 2026).

3. Materials platforms and representative realizations

The most developed material-specific realization is the Cr$p$9CH$f$0 monolayer. In equilibrium it is a stable A-type antiferromagnet, the AFM state is lower in energy than the FM state by about $f$1 eV, the phonon spectrum has no imaginary modes, and the bulk gap is about $f$2 eV. Topologically it is a $f$3-protected higher-order topological insulator with $f$4 and fractional corner charge $f$5. Under circularly polarized light, at $f$6 eV and $f$7, the Floquet bands develop an $f$8-wave odd-parity altermagnetic splitting governed by the effective symmetry $f$9, while the corner states remain intact over a broad driving window (Zou et al., 18 May 2026).

A broader symmetry program targets conventional 2D collinear antiferromagnets. For systems preserving either $_2$0 or $_2$1, but not the symmetry classes $_2$2 or $_2$3, Floquet irradiation can universally induce odd-parity spin splitting. First-principles plus Floquet calculations verify this in MnPS$_2$4 monolayer, FeCl$_2$5 bilayer, and NiRuCl$_2$6 bilayer, using $_2$7 eV and $_2$8; under circularly polarized light the induced texture is $_2$9-wave, while elliptically polarized light, bicircular light, or uniaxial strain convert it to $_2$0-wave (Huang et al., 28 Jul 2025).

A closely related honeycomb realization is monolayer MnPSe$_2$1. Under right-handed circularly polarized light with $_2$2 and $_2$3 eV, first-principles Floquet calculations show $_2$4-wave odd-parity spin splitting, a maximal splitting of about $_2$5 meV, and anomalous Hall conductivity reaching about $_2$6 when the Fermi level is tuned near $_2$7 eV. Because the threefold-related symmetry $_2$8 survives under the drive, the splitting is $_2$9-wave; uniaxial strain is proposed as a route to a $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$0-wave odd-parity phase (Zhu et al., 4 Aug 2025).

Light-induced odd-parity magnetism is not limited to these Floquet-engineered cases. Coplanar single-$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$1 helimagnets provide an equilibrium reference class in which odd-parity spin textures already exist, with $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$2. Generalized-Bloch-theorem calculations on MnI$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$3, NiI$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$4, and metallic MnTe$\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$5 show $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$6-wave order for commensurate $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$7 spirals and $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$8-wave order for more generic $\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})$9, providing a static baseline for future photoinduced control schemes (Larsen et al., 9 Apr 2026).

4. Topological consequences and collective excitations

Odd-parity Floquet magnetism often appears together with nontrivial topology because the same light-generated commutator terms act both as spin-splitting fields and as topological masses. In Cr$\mathbf{k}$0CH$\mathbf{k}$1, the driven odd-parity altermagnetic phase coexists with $\mathbf{k}$2-protected higher-order topology. The system remains insulating for $\mathbf{k}$3 at $\mathbf{k}$4 eV, corner-localized in-gap states remain sharply localized, and only at about $\mathbf{k}$5 does the gap close and the system evolve into an altermagnetic semimetallic state; even at $\mathbf{k}$6, the characteristic $\mathbf{k}$7-wave spin texture survives (Zou et al., 18 May 2026).

In honeycomb collinear antiferromagnets, the Floquet correction can be written as a Haldane-like next-nearest-neighbor hopping. The low-energy Dirac masses are

$\mathbf{k}$8

with $\mathbf{k}$9. When $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$0, the gap closes; for $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$1, the $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$2-wave odd-parity magnet becomes an antiferromagnetic Chern insulator with $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$3 for right-handed CPL and $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$4 for left-handed CPL, confirmed by two chiral edge states traversing the gap (Zhu et al., 4 Aug 2025).

Dimerized collinear antiferromagnets provide a second topological route. There the CPL-generated odd-parity $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$5-wave mass

$\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$6

competes with the staggered exchange mass. In 2D, when the Floquet mass exceeds the antiferromagnetic mass, the system enters a Chern-insulating phase with total Chern number $\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$7; in stacked 3D versions, the same mechanism produces Weyl semimetals with symmetry-related Weyl nodes and Berry-curvature monopoles (Liu et al., 25 Aug 2025).

Layered platforms can amplify these effects. In the bilayer VSi$\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$8N$\hat M_{\rm O}=\sum_j \hat S_{j1}-\hat S_{j2}$9 proposal, CPL converts hidden spin-layer locking into $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$0-wave odd-parity altermagnetism and drives a nonequilibrium QAHE with tunable Chern numbers up to $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$1. The same band inversions strongly reshape the orbital Hall effect: the model equilibrium plateau is $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$2, a sub-resonant driven state reaches $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$3, and topological phases display values such as $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$4 and $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$5 (Tian et al., 12 Mar 2026).

The odd-parity idea also extends to bosonic spin excitations. In collinear antiferromagnets, circularly polarized light generates a scalar spin-chirality term that reduces to a next-nearest-neighbor Dzyaloshinskii–Moriya interaction in the ordered background. The resulting magnon splitting is governed by

$\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$6

so $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$7 and the magnon branches acquire odd-parity splitting. In bilayer A-type antiferromagnets, this can drive a topological magnon transition with $\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$8, chiral edge modes, and an abrupt jump in magnon thermal Hall conductivity (Zhang et al., 29 May 2026).

5. Methods, diagnostics, and measurable signatures

The standard computational workflow combines first-principles electronic structure, Wannierization, Peierls-substituted light coupling, and high-frequency Floquet theory. Cr$\Delta(\mathbf{k})=-\Delta(-\mathbf{k})$9CH$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$00 is treated with VASP using PBE within GGA+$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$01, $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$02 eV on Cr $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$03 orbitals, a 550 eV cutoff, $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$04 $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$05-mesh, and $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$06 \AA\ vacuum; edge states are obtained with a Green-function approach and corner states from finite triangular flakes before the Floquet analysis is applied to the Wannier Hamiltonian (Zou et al., 18 May 2026). Equilibrium helimagnets are analyzed differently: the generalized Bloch theorem reduces single-$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$07 spiral calculations to the primitive crystallographic cell and reconstructs supercell-resolved spin textures through reciprocal-space downfolding, which is particularly useful for long-period or nearly incommensurate odd-parity magnets (Larsen et al., 9 Apr 2026).

The most direct diagnostic is momentum-resolved spectroscopy of the driven spin texture. Several studies explicitly point to spin-resolved ARPES or pump-probe tr-ARPES as the natural probe of opposite splittings at $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$08 and $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$09, of odd-in-$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$10 constant-energy maps, and of the $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$11-wave or $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$12-wave angular form of the Floquet bands (Zou et al., 18 May 2026). Transport is a second major route. In MnPSe$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$13, the driven anomalous Hall conductivity reaches about $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$14 near $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$15 eV; in VSi$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$16N$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$17, the driven orbital Hall conductivity and QAHE plateaus provide a combined orbitronic-topological fingerprint of the odd-parity Floquet state (Zhu et al., 4 Aug 2025, Tian et al., 12 Mar 2026).

Finite-geometry observables are especially important when odd-parity magnetism coexists with topology. Cr$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$18CH$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$19 retains corner-localized states under driving, while honeycomb and dimerized Floquet odd-parity magnets exhibit chiral edge states in their Chern phases (Zou et al., 18 May 2026, Zhu et al., 4 Aug 2025). In spin-orbital magnets, the electrically driven spin-resolved orbital Edelstein effect,

$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$20

is proposed as a complementary probe of the parent spin-orbital order underlying the odd- and mixed-parity Floquet states (Zhuang et al., 6 May 2026). For magnons, neutron scattering, Brillouin light scattering, THz pump-probe spectroscopy, and thermal Hall measurements are the natural probes of odd-parity band nonreciprocity and its topological consequences (Zhang et al., 29 May 2026).

6. Conceptual boundaries, controversies, and open directions

A central boundary condition is that light does not generically produce odd parity. In coplanar antiferromagnets whose reference symmetry is $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$21, circularly polarized light breaks the odd-parity constraint and dynamically generates the missing even-parity counterpart, with $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$22 and a $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$23-wave texture rather than an odd-parity phase (Zhu et al., 6 Jan 2026). A related distinction appears in the $E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$24 Ising spin-order framework: there, CPL induces even-parity ferromagnetic or altermagnetic splitting, while odd-parity spin splitting is produced instead by parity-breaking electric fields (Yu et al., 12 Mar 2026). Light-induced odd-parity magnetism is therefore symmetry-selective, not a universal consequence of optical driving.

A second recurring issue is the distinction between a Floquet-reconstructed electronic symmetry class and a genuinely new microscopic magnetic order parameter. The Cr$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$25CH$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$26 analysis is explicit on this point: the evidence concerns Floquet-induced lifting of band degeneracy and odd-parity spin splitting, not a time-dependent spin-dynamics or self-consistent nonequilibrium calculation proving a new real-space magnetic order. The safer interpretation is a reclassification of the antiferromagnetic electronic state into an odd-parity altermagnetic Floquet phase (Zou et al., 18 May 2026). More broadly, several Floquet studies work in an off-resonant or high-frequency regime and do not provide detailed heating, dissipation, or pulse-duration analyses, even when topological transitions are mapped in detail (Liu et al., 25 Aug 2025, Tian et al., 12 Mar 2026).

The outstanding opportunity is the size of the candidate space. Spin-group analysis identifies 48 candidate materials for static odd-parity magnets spanning collinear, coplanar, and noncoplanar orders (Luo et al., 7 Oct 2025), while exchange-driven non-symmorphic antiferromagnet theory identifies 67 materials in the Magndata database for odd-parity spin splitting induced by antiferromagnetic exchange (Yu et al., 3 Jan 2025). Together with the demonstrated Floquet routes in Cr$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$27CH$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$28, MnPSe$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$29, MnPS$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$30, FeCl$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$31, NiRuCl$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$32, VSi$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$33N$E_s(\mathbf{k})=E_{-s}(-\mathbf{k})$34, and odd-parity magnon platforms, this suggests that light-induced odd-parity magnetism is best viewed as a symmetry-engineering program: a way of converting compensated magnets into momentum-odd spin-split phases whose parity, angular harmonics, and topological content are controlled by the residual symmetries of the driven Hamiltonian.