--- title: Light-Induced Odd-Parity Spin Splitting url: https://www.emergentmind.com/topics/light-induced-odd-parity-spin-splitting type: topic --- # Light-Induced Odd-Parity Spin Splitting Light-induced odd-parity spin splitting is the Floquet generation of a spin-resolved band structure in which the splitting reverses sign under momentum inversion, typically under a relation such as \(E_s(\mathbf{k})=E_{-s}(-\mathbf{k})\), rather than the even-parity pattern characteristic of conventional altermagnets. In the current literature, the effect is realized primarily in compensated collinear antiferromagnets and altermagnetic relatives driven by periodic light fields, especially circularly polarized light (CPL), which remove the symmetry protecting spin degeneracy while preserving a residual symmetry that still forbids net ferromagnetism [2508.02542, 2507.20705]. The resulting momentum-space textures are most commonly \(p\)-wave or \(f\)-wave, and the same mechanism has been connected to Floquet Chern phases, higher-order topology, mixed-parity spin textures, and current-driven static analogs of odd-parity altermagnetism [2508.18360, 2605.19184, 2605.03026, 2503.09602]. ## 1. Symmetry conditions and parity classification The defining symmetry logic is most explicit in collinear \(PT\)-symmetric antiferromagnets. In that setting, two spin-group symmetries protect spin degeneracy: \([C_2||P]\), which enforces \(\varepsilon(s,\mathbf{k})=\varepsilon(-s,-\mathbf{k})\), and \([C_2T||E]\), which enforces \(\varepsilon(s,\mathbf{k})=\varepsilon(s,-\mathbf{k})\). When both are present, the bands are spin degenerate. Odd-parity spin splitting appears when \([C_2T||E]\) is broken while \([C_2||P]\) is preserved, so the degeneracy is lifted but the relation \(\varepsilon(s,\mathbf{k})=\varepsilon(-s,-\mathbf{k})\) remains; the spin splitting is then odd under \(\mathbf{k}\to-\mathbf{k}\) [2508.02542]. This symmetry formulation clarifies the distinction between even- and odd-parity compensated magnets. Even-parity altermagnets retain inversion in a way that constrains the nonrelativistic spin splitting to be even in momentum. Odd-parity magnets instead preserve a relation between opposite spins at opposite momenta, so the sign structure alternates between \(\mathbf{k}\) and \(-\mathbf{k}\). In the Floquet literature on conventional collinear antiferromagnets, this odd-parity condition is written through the driven quasienergies \(E_s(\mathbf{k})\) or \(\epsilon_s(\mathbf{k})\), with \(s=\pm1\), satisfying \(E_s(\mathbf{k})=E_{-s}(-\mathbf{k})\) while generally \(E_s(\mathbf{k})\neq -E_{-s}(\mathbf{k})\) at generic \(\mathbf{k}\) [2507.20705]. A related classification appears in coplanar antiferromagnets. There, the antiunitary symmetry \([\bar{C}_{2z}||\mathcal{T}]\) forces an out-of-plane spin polarization to be odd in momentum, \(\langle s_z(\mathbf{k})\rangle=-\langle s_z(-\mathbf{k})\rangle\), so odd-parity unidirectional spin splitting is symmetry-enforced in equilibrium when such a splitting exists. CPL can also remove that constraint and generate the missing even-parity counterpart, which shows that Floquet driving does not merely activate a preexisting odd-parity sector but can move between parity classes depending on the parent symmetry content [2601.03358]. ## 2. Floquet generation in collinear antiferromagnets The common technical framework is the high-frequency Floquet expansion applied after a Peierls substitution, \(\mathbf{k}\rightarrow \mathbf{k}+\frac{e}{\hbar}\mathbf{A}(t)\). For CPL with \([A_x(t),A_y(t)]=A_0[\cos(\omega t),\,\eta\sin(\omega t)]\), the effective static Hamiltonian takes the form \[ H_{\mathrm{eff}}(\mathbf{k})=H_0(\mathbf{k})+\sum_{n\ge 1}\frac{[H_{-n},H_n]}{n\omega}+O\!\left(\frac{1}{\omega^2}\right), \] and the commutator term is the symmetry-breaking Floquet correction responsible for odd-parity splitting [2508.02542]. CPL is singled out because it breaks the TR-related spin-group symmetry but does not directly couple to spin, so it can preserve the inversion-related spin symmetry needed for a compensated odd-parity state. In the low-symmetry two-sublattice AFM analyzed in "Floquet odd-parity collinear magnets" [2508.02542], right-handed CPL makes the Floquet bands nondegenerate and produces a \(p\)-wave-like odd-parity spin splitting. Restoring \(C_3\) symmetry converts the same mechanism into an \(f\)-wave state on a honeycomb AFM. In that case the Floquet correction is \[ M(\mathbf{k})= 4\sqrt{3}\,\eta\,\frac{J_1(\tilde A)^2}{\omega} \Bigl(\cos\frac{3}{2}k_x-\cos\frac{\sqrt{3}}{2}k_y\Bigr) \sin\frac{\sqrt{3}}{2}k_y \; s_0\sigma_z, \] which discretizes into an effective second-neighbor imaginary hopping \[ H_F = i\delta \sum_{\langle\langle mn\rangle\rangle,\sigma}\nu_{mn}\, c_{m,\sigma}^\dagger c_{n,\sigma}, \qquad \delta=\frac{\sqrt{3}J_1(\tilde A)^2}{\omega}. \] The momentum-space spin splitting then acquires an odd-parity \(f\)-wave pattern rather than the even-parity \(d/g/i\)-wave textures associated with ordinary altermagnets [2508.02542]. The same logic appears in the broader symmetry classification of two-dimensional collinear antiferromagnets. For a hexagonal AFM driven by CPL, \[ H_{\mathrm{eff}}(\mathbf{k})=H_0(\mathbf{k})+\frac{[H_1(\mathbf{k}),H_{-1}(\mathbf{k})]}{\omega}+\mathcal O(\omega^{-2}), \] and the commutator yields \[ \omega H'(\mathbf{k})= 8\sqrt 3\,\eta\,J_1\!\left(\frac{A_0}{\sqrt3}\right)t^2 \prod_i \sin(k'_i/2)\,\sigma_0\tau_z \equiv f(\mathbf{k})\sigma_0\tau_z. \] Because \(f(\mathbf{k})\) is odd in momentum and multiplies \(\tau_z\), the two AFM sublattices acquire opposite light responses. The resulting quasienergies \[ \epsilon_s(\mathbf{k})=\pm\sqrt{\left|J_0\!\left(\frac{A_0}{\sqrt3}\right)\Delta(\mathbf{k})\right|^2+ \left(|m_\zeta|+\frac{s\,f(\mathbf{k})}{\omega}\right)^2} \] show odd-parity spin splitting at generic \(\mathbf{k}\), while spin degeneracy survives on symmetry lines where \(f(\mathbf{k})=0\) [2507.20705]. Not every polarization is effective. In the same hexagonal setting, linearly polarized light gives \(H_1=-H_{-1}\), so \([H_1,H_{-1}]=0\) and the leading Floquet correction vanishes, leaving the bands spin degenerate. By contrast, elliptically polarized light (EPL) and bicircular light (BCL) lower the effective spatial symmetry and can convert an \(f\)-wave pattern into a \(p\)-wave one [2507.20705]. ## 3. Model families and microscopic routes A distinct microscopic realization occurs on dimerized lattices. In "Light-induced odd-parity altermagnets on dimerized lattices" [2508.18360], CPL dynamically converts a collinear \(\mathcal{PT}\)-symmetric antiferromagnet into an odd-parity \(p\)-wave altermagnet. The 2D Floquet Hamiltonian is \[ \mathcal{H}_{\rm eff}(\mathbf k) = J_{0}(A_{0})\Big[(t_+\cos k_x+2t_2\cos k_y)\sigma_x+t_-\sin k_x\,\sigma_y\Big] + M\,\sigma_z s_z -\eta F(A_0,\omega)\sin k_y\cos k_x\,\sigma_z, \] with \[ F(A_0,\omega)=\frac{8J_1^2(A_0)t_2 t_-}{\omega}. \] The last term is the Floquet-generated odd-parity mass. It is helicity-odd, momentum-odd in the spin-splitting channel, and symmetry-constrained by the retained \([C_{2\perp}||\mathcal P]\) operation enforcing \(E(\mathbf{k},s)=E(-\mathbf{k},-s)\). For the type-I dimerization discussed there, the splitting is strongest along \(k_y\) and absent along \(k_x\) [2508.18360]. A different route starts from an even-parity altermagnet with intrinsic relativistic SOC. In "Dynamical Generation of Higher-order Spin-Orbit Couplings, Topology and Persistent Spin Texture in Light-Irradiated Altermagnets" [2504.00122], periodic light drives convert the even-parity altermagnetic \(\sigma^z\) splitting into odd-parity, light-induced SOCs. The general rule is that an altermagnet with \(k^n\) spin splitting can generate SOCs up to \(k^{n-1}\), and for CPL only the \(k^{n-1}\) correction survives. The explicit sequence given is \(d\)-wave \((n=2)\to k^1\) SOC, \(g\)-wave \((n=4)\to k^3\) SOC for CPL, and \(i\)-wave \((n=6)\to k^5\) SOC for CPL. This does not always amount to a pure odd-parity compensated magnetic phase in the same sense as the Floquet AFM constructions, but it establishes a direct mechanism by which periodic light lowers the momentum order of spin splitting by one power and transfers parity-odd structure into the effective SOC sector [2504.00122]. Phase-locked two-color linearly polarized driving provides a further extension. In "Tunable Odd-Parity Spin Splittings in Altermagnets" [2605.03026], the field \[ {\bf A}(t)=\big(A_1\cos 2\omega t,\; A_2\cos(\omega t+\phi),\;0\big) \] generates a lowest static composite order \(\overline{A_xA_y^2}\propto \cos 2\phi\) with \((P,T)=(-,-)\). The induced order is symmetry-equivalent to a translationally invariant \(P\)-odd loop-current order, and when coupled to a collinear altermagnet it produces a mixed-parity spin texture. In the tetragonal \(D_{4h}\) example, a bare \(k_xk_y\sigma_z\) altermagnetic splitting acquires an additional \(k_y\sigma_z\) component, while the induced odd-parity term scales as \(I^{3/2}\) at weak intensity [2605.03026]. Mixed-parity behavior also appears in driven spin-orbital magnets. In "Mixed-Parity Altermagnetism in Collinear Spin-Orbital Magnets" [2605.05205], CPL at zero staggered potential \(\delta=0\) produces odd-parity altermagnets with \(f\)-wave, \(p_x\)-wave, or \(p_y\)-wave textures depending on the spin-orbital order, whereas finite \(\delta\) yields mixed-parity altermagnetism. This places pure odd parity and mixed parity on the same symmetry-engineering continuum. ## 4. Topological consequences Odd-parity spin splitting generated by light is frequently accompanied by topological band reconstruction. In the \(f\)-wave honeycomb case of [2508.02542], the low-energy valleys are described by \[ H_{K}= t'(q_x\sigma_x+q_y\sigma_y)+(s m_z-\eta m')\sigma_z, \qquad H_{K'}= t'(-q_x\sigma_x+q_y\sigma_y)+(s m_z+\eta m')\sigma_z, \] with \(m'=\frac{9J_1(\tilde A)^2}{\omega}\). When \(m'm_z\), the system becomes an antiferromagnetic Chern insulator with \(\mathcal C=-2\) for RCPL and \(\mathcal C=+2\) for LCPL, together with two chiral edge states [2508.02542]. The dimerized-lattice construction reaches a similar endpoint through Dirac mass inversion. In 2D, once the light-induced mass exceeds the AFM gap, \(\tilde F>M\), the driven odd-parity \(p\)-wave altermagnet becomes a Chern insulator with \(C=2\eta\). In 3D, the same symmetry setting produces Weyl semimetals with odd-parity spin-splitting terms in both \(k_y\) and \(k_z\), and the related Weyl points carry opposite monopole charge under the retained \([C_{2\perp}||\mathcal P]\) symmetry [2508.18360]. The interplay between odd-parity altermagnetism and crystalline higher-order topology is explicit in Cr\(_2\)CH\(_2\). In equilibrium, Cr\(_2\)CH\(_2\) is a 2D AFM HOTI protected by \(\mathcal C_3\), with a bulk gap of about \(0.655\) eV, symmetry indicator \(\chi^{(3)}=\{-2,1\}\), and fractional corner charge \(Q_c^{(3)}=\frac{4e}{3}\). Under CPL, the symmetry \([C_2T||E]\) is broken while \([C_2||P]\) and \([E||C_3]\) remain, producing an odd-parity \(f\)-wave altermagnetic phase with \(\Delta E(\mathbf{k})=-\Delta E(-\mathbf{k})\). The corner states survive for approximately \(eA/\hbar\in[0,0.55]~\text{\AA}^{-1}\), and only after the gap closes near \(eA/\hbar\approx 0.55~\text{\AA}^{-1}\) does the system become an altermagnetic semimetal [2605.19184]. Floquet-generated odd-parity SOCs in irradiated altermagnets also modify topology. The reported Chern-number changes are \(\Delta C=\pm 1\) in the \(d\)-wave case, \(\Delta C=\pm 2\) for CPL-driven cubic SOC in the \(g\)-wave case, and \(\Delta C=\pm 3\) for CPL-driven quintic SOC in the \(i\)-wave case [2504.00122]. ## 5. Materials platforms and experimental diagnostics The materials literature now spans both idealized model systems and first-principles-validated compounds. | Platform | Driving protocol / symmetry setting | Reported outcome | |---|---|---| | Low-symmetry two-sublattice AFM | RCPL; \([C_2T||E]\) broken, \([C_2||P]\) preserved | \(p\)-wave odd-parity collinear magnet [2508.02542] | | Honeycomb AFM; monolayer MnPSe\(_3\) | CPL with \(C_3\) symmetry | \(f\)-wave odd-parity magnet; AFM Chern-insulating regime at high intensity [2508.02542] | | Monolayer MnPS\(_3\), bilayer FeCl\(_2\), bilayer NiRuCl\(_6\) | CPL at \(\hbar\omega=10\) eV and \(eA_0/\hbar=0.3\ \text{\AA}^{-1}\) | \(f\)-wave splitting; chirality reversal flips the sign; EPL/BCL or strain can yield \(p\)-wave [2507.20705] | | Dimerized \(\mathcal{PT}\)-symmetric AFM | CPL on Dirac lattice | Odd-parity \(p\)-wave altermagnet; 2D Chern insulator and 3D Weyl semimetal under appropriate drive [2508.18360] | | Cr\(_2\)CH\(_2\) monolayer | Off-resonant CPL | Odd-parity \(f\)-wave altermagnetic HOTI; corner states persist until gap closure [2605.19184] | The most detailed first-principles confirmation so far is for monolayer MnPSe\(_3\). Under RCPL with \(\tilde A = 0.2\ \text{\AA}^{-1}\) and \(\hbar\omega = 5\ \text{eV}\), the calculations show an \(f\)-wave odd-parity spin-split band structure, spin-resolved Fermi surfaces, anomalous Hall conductivity reaching about \(6\,e^2/h\) near \(E_F\approx -0.1\) eV, and a maximum spin splitting of about \(40\) meV. The spin splitting is tunable with light amplitude and frequency, and uniaxial strain can lower the symmetry and convert the \(f\)-wave state into a \(p\)-wave one [2508.02542]. The broader candidate survey for hexagonal monolayer and bilayer antiferromagnets identifies three categories that support Floquet odd-parity splitting when the required symmetries are present: hexagonal monolayers with Néel-type AFM order, AFM bilayers composed of ferromagnetic monolayers, and AFM bilayers composed of ferrimagnetic monolayers including fully compensated ferrimagnets. For MnPS\(_3\), FeCl\(_2\), and NiRuCl\(_6\), DFT plus Floquet analysis confirms that the undriven materials are spin degenerate, whereas CPL lifts the degeneracy at generic \(\mathbf{k}\) but preserves it along the \(\Gamma\)-M line; FeCl\(_2\) further exhibits explicit conversion to \(p\)-wave splitting under BCL, EPL, or uniaxial strain [2507.20705]. Optical transport can also diagnose odd-parity spin textures that are not themselves light-induced. In NiI\(_2\), the noncollinear spin spiral produces an intrinsic \(p\)-wave magnetic state with odd-parity spin splitting, and nonlinear optical responses separate the inversion-breaking and spin-splitting consequences of that state. The dominant circular-photogalvanic injection component is \(\eta^{yxy}_\mathrm{CP}\), with a peak around \(\sim 4 \times 10^9 \,\text{A/(V}^2\text{s)}\), arising from helicity-selective transitions near 1.27 and 1.35 eV between bands with strong spin-\(z\) splitting. The same system also supports pure spin photocurrents, with charge and spin flow directions exchanged between linear and circular excitation [2603.25516]. ## 6. Conceptual boundaries, related mechanisms, and extensions A recurrent misconception is that odd-parity nonrelativistic spin splitting requires noncollinear magnetism. That view is contradicted by the Floquet constructions in ordinary collinear antiferromagnets, where CPL, EPL, or BCL can remove spin degeneracy and produce \(p\)-wave or \(f\)-wave odd-parity textures without introducing noncollinearity [2508.02542, 2507.20705]. A second misconception is that any periodic drive suffices. In fact, the symmetry of the drive is decisive: in the hexagonal AFM model, linearly polarized light gives \([H_1,H_{-1}]=0\) and no leading spin splitting, whereas CPL is effective precisely because it breaks the TR-related symmetry but preserves the inversion-related one required for compensated odd parity [2507.20705]. The Floquet mechanism also has a static current-driven analog. In the Haldane-Hubbard model and its bipartite generalizations, sublattice currents break \(\mathcal T\) in a nonmagnetic crystal structure and generate a \(\tau^3\)-type sublattice imbalance. When onsite repulsion drives opposite ferromagnetic moments on the two sublattices at half filling, that imbalance is converted into a nonrelativistic odd-parity spin splitting, and in the weak-altermagnet regime the phase can remain topological as an ALM Chern insulator [2503.09602]. The two-color linearly polarized protocol of [2605.03026] makes this correspondence explicit by generating a static \((P,T)=(-,-)\) composite order symmetry-equivalent to a translationally invariant \(P\)-odd loop-current order. The distinction between light-induced generation and light-based readout is also essential. "Electrical switching of an unconventional odd parity magnet" [2504.21086] concerns equilibrium odd-parity spin splitting in the spin-spiral multiferroic NiI\(_2\), not a Floquet-generated state. There, zero-bias photocurrent and CPGE serve as optical probes of an electrically switchable chirality-locked odd-parity texture. This establishes a complementary experimental paradigm: Floquet light can create odd-parity spin splitting, but optical helicity can also reveal an odd-parity spin texture that already exists in equilibrium [2504.21086]. The same parity-engineering principles extend beyond electronic quasiparticles. "Odd-Parity Magnons" [2605.31411] shows that CPL can generate odd-parity magnon band splitting in collinear antiferromagnets through a Floquet Aharonov–Casher mechanism. The effective first-order Floquet correction is a chiral three-spin term that becomes a momentum-odd \(D(\mathbf{k})\) in spin-wave theory, enabling \(p\)-wave and \(f\)-wave magnon splitting as well as light-driven topological magnon transitions in bilayers [2605.31411]. Taken together, these results define light-induced odd-parity spin splitting as a symmetry-engineered Floquet phenomenon rather than a single material-specific effect. The central ingredients are now well identified: a compensated magnetic or spin-active parent state, a drive that breaks the symmetry enforcing spin degeneracy, a residual symmetry that relates \((\mathbf{k},s)\) to \((-\mathbf{k},-s)\), and a lattice harmonic content capable of supporting the required odd momentum form factor. Under those conditions, periodic light can generate momentum-odd nonrelativistic spin splitting, reshape its angular character between \(p\)- and \(f\)-wave sectors, and couple it directly to Chern, Weyl, or higher-order topology. Source: https://www.emergentmind.com/topics/light-induced-odd-parity-spin-splitting