---
title: Floquet Odd-Parity Collinear Magnets
url: https://www.emergentmind.com/topics/floquet-odd-parity-collinear-magnets
type: topic
---

# Floquet Odd-Parity Collinear Magnets

Searching arXiv for recent papers on Floquet odd-parity collinear magnets and closely related odd-parity altermagnetism/magnons.
Floquet odd-parity collinear magnets are periodically driven compensated collinear magnets, typically originating from conventional antiferromagnets, in which light-induced nonequilibrium terms lift spin degeneracy so that the spin splitting becomes odd under momentum inversion. In the notation used for collinear antiferromagnets, the defining relation is $\Delta(s,\mathbf{k})=-\Delta(s,-\mathbf{k})$, in contrast to the even-parity spin splitting of conventional altermagnets. Recent work shows that off-resonant polarized light, especially circularly polarized light (CPL), can convert spin-degenerate collinear antiferromagnets into $p$-wave or $f$-wave odd-parity magnets, and in dimerized lattices can dynamically generate odd-parity $p$-wave altermagnets that further realize Chern-insulating or Weyl-semimetal phases under appropriate drive conditions [2508.02542] [2508.18360].

## 1. Definition and symmetry framework

The starting point is a conventional collinear antiferromagnet with two sublattices related by inversion and with zero net magnetization. In the symmetry language used for Floquet odd-parity collinear magnets, the undriven system typically possesses both $[C_{2}\Vert P]$ and $[C_{2}T\Vert E]$. The first enforces $\varepsilon(s,\mathbf{k})=\varepsilon(-s,-\mathbf{k})$, while the second enforces $\varepsilon(s,\mathbf{k})=\varepsilon(s,-\mathbf{k})$; together they imply spin degeneracy. An altermagnet is then a compensated collinear magnet whose bands are spin-split in momentum space, whereas an odd-parity collinear magnet is defined by a spin splitting odd under $\mathbf{k}\rightarrow-\mathbf{k}$ [2508.02542].

A broader symmetry classification of irradiated 2D collinear antiferromagnets identifies four relevant protecting spin-space-group operations in equilibrium: $[C_{2}\Vert \mathcal{P}]$, $[C_{2}\Vert C_{2z}]$, $[C_{2}\Vert \tau]$, and $[C_{2}\Vert M_{z}]$. Under in-plane driving with $A_{z}=0$, the last two remain exact and therefore do not permit spin splitting, whereas odd-parity spin splitting under irradiation appears when the protecting symmetry is $[C_{2}\Vert \mathcal{P}]$ or $[C_{2}\Vert C_{2z}]$ and is broken by the light-matter coupling [2507.20705].

A persistent misconception in this area is that odd-parity spin splitting is restricted to non-collinear magnetic configurations. Recent Floquet analyses explicitly show that periodic driving provides a route to odd-parity magnetism in conventional collinear antiferromagnets, including systems without intrinsic spin-orbit coupling in the microscopic mechanism emphasized for dimerized lattices [2507.20705].

## 2. Floquet construction and microscopic mechanism

The standard construction uses a time-periodic vector potential $\mathbf{A}(t)$ and Peierls or minimal coupling, written in the cited works as $\mathbf{k}\rightarrow\mathbf{k}+\mathbf{A}(t)$ or $\mathbf{k}\rightarrow\mathbf{k}-e\mathbf{A}(t)/\hbar$. Because the Hamiltonian is periodic in time, Floquet theory introduces Fourier components
$$
H_{n}(\mathbf{k})=\frac{1}{T}\int_{0}^{T}dt\,e^{-in\Omega t}H(\mathbf{k},t),
$$
and in the off-resonant high-frequency regime the leading van Vleck effective Hamiltonian is
$$
H_{\mathrm{eff}}(\mathbf{k})\simeq H_{0}(\mathbf{k})+\frac{1}{\Omega}[H_{-1}(\mathbf{k}),H_{+1}(\mathbf{k})]+\mathcal{O}(\Omega^{-2}).
$$
In the generic formulation for irradiated collinear antiferromagnets, the commutator term is the source of the odd-parity splitting; in model form it appears as a spin-dependent contribution $M_{\Delta}(\mathbf{k})=s_{z}F(\mathbf{k})/\omega$ with $F(\mathbf{k})$ odd in momentum [2508.02542].

The microscopic origin is model dependent but structurally similar across the electronic realizations. In a dimerized lattice, CPL mixes the $\sigma_{x}$ and $\sigma_{y}$ kinetic terms at $\pm\Omega$, and their commutator produces an additional $\sigma_{z}$ term odd in momentum, dynamically emulating spin-orbit-like $p$-wave splitting without intrinsic SOC. In the nearest-neighbor honeycomb model, the same Floquet commutator yields an imaginary next-nearest-neighbor hopping analogous to the Haldane pattern, but acting in a spin-split collinear antiferromagnetic background [2508.18360].

The same logic extends to bosonic spin excitations. For magnons, a circularly polarized electric field produces an Aharonov-Casher phase on magnon hopping, and the first-order Floquet-Magnus term generates a three-spin scalar-chirality interaction. After projection onto the collinear Néel background, that term reduces to an effective next-nearest-neighbor Dzyaloshinskii-Moriya interaction, which is the odd-parity mass responsible for $p$- or $f$-wave magnon splitting [2605.31411].

## 3. Wave-form classification: $p$-wave and $f$-wave sectors

Two symmetry-distinct odd-parity patterns dominate the electronic literature. In lattices without a protecting threefold rotational symmetry, the induced splitting is $p$-wave, represented in the minimal rhombic model by
$$
\Delta_{p}(\mathbf{k})=\sum_{i=1}^{3}A_{i}\sin(\mathbf{k}\cdot\delta_{i}),
$$
with coefficients scaling as $J_{1}(eA_{0}/\hbar)^{2}/\omega$. In honeycomb systems with $C_{3}$ symmetry, the induced splitting is $f$-wave; in the minimal model it takes the explicit form
$$
\Delta_{f}(\mathbf{k})=
4\sqrt{3}\,\eta\,\frac{[J_{1}(\tilde A)]^{2}}{\omega}
\bigl[\cos(3k_{x}/2)-\cos(\sqrt{3}k_{y}/2)\bigr]\sin(\sqrt{3}k_{y}/2),
$$
which yields the characteristic six-petal pattern in the spin-resolved Fermi surface [2508.02542].

A more general symmetry analysis shows that the wave form can be controlled by the polarization state of the drive. In hexagonal collinear antiferromagnets, CPL preserves the higher rotational symmetry and produces an $f$-wave basis, while elliptically polarized light (EPL) or bicircular light (BCL) lowers the effective point-group symmetry so that the commutator term transforms as a $p$-wave basis $(k_{x},k_{y})$. Reversing the helicity $\eta\rightarrow-\eta$ flips the sign of the induced splitting, and reducing the crystal symmetry from $D_{6}$ to $D_{2}$ or $C_{2}$, for example by uniaxial strain, also enforces a $p$-wave component [2507.20705].

The candidate materials identified for 2D collinear antiferromagnets fall into three categories:

| Category | Driven odd-parity response | Representative materials |
|---|---|---|
| I | Monolayer hexagonal Néel AFMs; CPL $\rightarrow f$-wave, EPL/BCL $\rightarrow p$-wave | MnPS$_3$, MnPSe$_3$, Fe$_2$O$_3$, MnS, MnSe, MnTe, MnBi$_2$Te$_4$ |
| II | AFM bilayers of two identical FM monolayers; CPL $\rightarrow f$-wave | RuO$_2$, VSe$_2$, NbSe$_2$, TaS$_2$, FeCl$_2$, FeBr$_2$, MnO$_2$ |
| III | AFM bilayers of ferri- or uncompensated ferrimagnets; CPL $\rightarrow f$-wave | NiRuCl$_6$ |

This classification is restricted by the requirement that the two sublattices are not connected by $[C_{2}\Vert\tau]$ or $[C_{2}\Vert M_{z}]$, but do admit either $[C_{2}\Vert C_{2z}]$ or $[C_{2}\Vert\mathcal{P}]$ in equilibrium [2507.20705].

## 4. Dimerized lattices and Floquet odd-parity altermagnets

A particularly explicit realization is the PT-symmetric collinear antiferromagnet on a 2D dimerized square lattice. Its static Bloch Hamiltonian is
$$
H_{0}(\mathbf{k})=
(t_{+}\cos k_{x}+2t_{2}\cos k_{y})\,\sigma_{x}\otimes s_{0}
+t_{-}\sin k_{x}\,\sigma_{y}\otimes s_{0}
+M\,\sigma_{z}\otimes s_{z},
$$
with $t_{+}=t_{0}+t_{1}$, $t_{-}=t_{0}-t_{1}$, inter-dimer hopping $t_{2}$, and staggered exchange $M>0$. This model has spinful PT symmetry $PT=\sigma_{x}s_{y}K$ with $(PT)^{2}=-1$, hence Kramers degeneracy in the undriven state [2508.18360].

Under normally incident CPL,
$$
\mathbf{A}(t)=A_{0}(\cos\Omega t,\eta\sin\Omega t,0),
$$
the leading Floquet correction produces the effective Hamiltonian
$$
H_{\mathrm{eff}}(\mathbf{k})=
J_{0}(A_{0})\bigl[(t_{+}\cos k_{x}+2t_{2}\cos k_{y})\sigma_{x}+t_{-}\sin k_{x}\sigma_{y}\bigr]
+M\sigma_{z}s_{z}
-\eta F(A_{0},\Omega)\sin k_{y}\cos k_{x}\sigma_{z},
$$
with
$$
F(A_{0},\Omega)=\frac{8J_{1}^{2}(A_{0})t_{2}t_{-}}{\Omega}.
$$
The induced term
$$
f(\mathbf{k})\sigma_{z}\equiv-\eta F(A_{0},\Omega)\sin k_{y}\cos k_{x}\sigma_{z}
$$
satisfies $f(-\mathbf{k})=-f(\mathbf{k})$ and therefore has odd inversion parity, identified in the paper as a $p$-wave form [2508.18360].

Because the underlying dimerized lattice supports a Dirac band structure, the driven state is not merely spin-split but topological under suitable parameters. The same analysis extends to stacked dimerized layers, where arbitrary-incidence CPL generates two odd-parity terms, $F_{y}\sin k_{y}\cos k_{x}$ and $F_{z}\sin k_{z}\cos k_{x}$ in the $\sigma_{z}$ channel, enabling Weyl semimetals in three dimensions [2508.18360].

## 5. Floquet-induced topology

The topological content of Floquet odd-parity collinear magnets is one of their defining features. In the honeycomb $f$-wave model, the Floquet term acts as a valley- and spin-dependent Dirac mass. Expanding around $K$ and $K'$, the masses are $M_{K,s}=s\,m_{z}-\eta m'$ and $M_{K',s}=s\,m_{z}+\eta m'$, and when $m'$ exceeds $m_{z}$ the total Chern number jumps from $0$ to $-2$ for right-handed CPL. Numerical ribbon calculations confirm the corresponding chiral edge states and quantized Hall conductivity $\sigma_{xy}=C(e^{2}/h)$ [2508.02542].

In the dimerized $p$-wave altermagnet, the transition is controlled by the Floquet scale relative to the staggered exchange. The CPL-induced $\sigma_{z}$ shift breaks spinful PT, and when $|F(A_{0},\Omega)|>M$ the gap closes and reopens. Near the two Dirac points $\mathbf{k}_{D,\pm}=(0,\pi\mp\alpha)$ with $\alpha=\arccos(t_{+}/2t_{2})$, two cones flip as the effective mass changes sign, producing a total Chern number $C=2\eta$ in the 2D phase diagram [2508.18360].

The 3D extension yields a Weyl semimetal rather than a fully gapped Chern insulator. Weyl nodes appear in the $k_{x}=0$ plane when
$$
\tilde t_{x}+\tilde t_{y}\cos k_{y}+\tilde t_{z}\cos k_{z}=0,
\qquad
sM+F_{y}\sin k_{y}+F_{z}\sin k_{z}=0,
$$
which gives at least four Weyl points, with inversion-related nodes carrying opposite chirality [2508.18360].

Later work broadened the topological landscape. In a 2D collinear antiferromagnetic multilayer with layer-dependent coupling to CPL, Floquet engineering induces $f$-wave odd-parity altermagnetism and a nonequilibrium quantum anomalous Hall effect with tunable Chern numbers up to $C=\pm8$, attributed to layer- and valley-dependent band inversions [2603.11483]. In the Cr$_2$CH$_2$ monolayer, equilibrium higher-order topology characterized by the symmetry indicator $\chi^{(3)}=\{-2,1\}$ survives under CPL while the system develops an odd-parity $f$-wave altermagnetic state; the corner states persist up to approximately $eA/\hbar\approx0.55\,\mathrm{\AA}^{-1}$, beyond which the gap closes and an altermagnetic semimetal emerges [2605.19184].

## 6. First-principles realizations and experimental signatures

First-principles calculations provide direct material support for the Floquet picture. In monolayer MnPSe$_3$, a DFT plus Wannier plus Floquet treatment with right-handed CPL at $\hbar\omega=5\,\mathrm{eV}$ and $\tilde A=0.2\,\mathrm{\AA}^{-1}$ gives spin splitting of the two top valence bands by up to $40\,\mathrm{meV}$, an $f$-wave sixfold Fermi-surface pattern, and anomalous Hall conductivity reaching approximately $6\,e^{2}/h$ when the Fermi energy is tuned just below the gap [2508.02542].

A wider first-principles survey using Wannier-based tight-binding models examined MnPS$_3$, FeCl$_2$ bilayer, and NiRuCl$_6$ under $\hbar\omega=10\,\mathrm{eV}$ and light intensity $eA_{0}/\hbar=0.3\,\mathrm{\AA}^{-1}$. All are spin-degenerate without light or SOC, but under right-handed CPL exhibit clear odd-parity spin splitting: MnPS$_3$ shows splitting up to approximately $50\,\mathrm{meV}$ at $K$ with six $f$-wave lobes, FeCl$_2$ bilayer shows approximately $45\,\mathrm{meV}$ and converts from $f$-wave to twofold $p$-wave under EPL or BCL, and fully compensated ferrimagnetic NiRuCl$_6$ shows approximately $60\,\mathrm{meV}$ near $E\approx-0.1\,\mathrm{eV}$ [2507.20705].

The dimerized-lattice construction specifies complementary drive conditions: the frequency must be off-resonant with $\Omega\gg$ bandwidth, numerically chosen as $\Omega\approx9\,t_{0}$; the amplitude should be of order unity, $A_{0}\sim1$–$2$, so that $J_{1}(A_{0})\neq0$ and $F(A_{0},\Omega)>M$; and the polarization should be strictly circular to break time-reversal symmetry while preserving $[C_{2\perp}\Vert P]$, which enforces $E(\mathbf{k},\uparrow)=E(-\mathbf{k},\downarrow)$ [2508.18360].

Proposed probes are correspondingly specific. Angle-resolved photoemission should observe spin-split bands satisfying $E(\mathbf{k},\uparrow)=E(-\mathbf{k},\downarrow)$ in the odd-parity phases. In 2D Chern-insulating regimes, transport should show Hall conductance $\sigma_{xy}=2\,e^{2}/h$ controlled by CPL handedness. In 3D Weyl phases, negative magnetoresistance and Fermi arcs are expected, with Fermi arcs on surfaces normal to $y$ or $z$ but spin-split on $x$-normal surfaces. Additional proposals include time- and spin-resolved ARPES, photo-induced anomalous Hall measurements, magnetic circular dichroism, and second-harmonic generation of spin currents [2508.18360] [2508.02542].

## 7. Extensions: magnons, excitons, and mixed-parity regimes

The electronic Floquet odd-parity collinear magnet has rapidly become the template for a broader odd-parity program. In magnonics, a complete spin-point-group classification has been formulated for odd-parity magnon splitting in 2D collinear antiferromagnets. There the leading odd-parity splitting types are again $p$- and $f$-wave, and circularly polarized light or loop currents provide the effective time-reversal-breaking mechanism. In bilayer systems, dynamical modulation can drive a topological magnon phase transition accompanied by chiral edge modes and an abrupt jump in the magnon thermal Hall conductivity [2605.31411].

In the Haldane-Hubbard setting, topological excitons in the paramagnetic phase condense at $\Gamma$ and drive the system into a collinear Néel state identified as an odd-wave magnet. In the ordered phase the low-energy collective modes become odd-parity magnons with characteristic $f$-wave splitting, and the magnon topology changes when the electronic bandgap closes. That work explicitly notes potential realizations in Floquet-driven materials and cold atomic gases [2606.05837].

Other developments loosen the dichotomy between even- and odd-parity responses. In a two-sublattice two-orbital hexagonal model, circularly polarized light induces pure odd-parity altermagnetism at zero staggered potential $\delta=0$, but for finite $\delta$ yields mixed-parity spin splitting; in that construction, the odd-parity limit is a $p$-wave Floquet altermagnet near $\Gamma$ [2605.05205]. A distinct route uses phase-locked two-color linearly polarized light to induce a static $(P,T)=(-,-)$ order symmetry-equivalent to a translationally invariant $P$-odd loop-current order, which in a collinear altermagnet produces a controllable mixed-parity spin texture rather than a purely odd-parity one [2605.03026].

Taken together, these results indicate that Floquet odd-parity collinear magnets are best understood not as a single model system but as a symmetry-governed nonequilibrium class. Their essential ingredients are a compensated collinear magnetic background, a light field that removes the symmetry enforcing $\varepsilon(s,\mathbf{k})=\varepsilon(s,-\mathbf{k})$ while retaining the relation between opposite spins at opposite momenta, and a commutator-generated odd-in-momentum Floquet term. This suggests a unifying viewpoint in which electronic altermagnets, odd-parity magnons, higher-order topological antiferromagnets, and loop-current-assisted mixed-parity states are different realizations of the same driven symmetry principle.

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