Nutation Spin Waves: Inertial Magnetic Dynamics

• Nutation spin waves are collective magnetic excitations driven by the inertial nutation of magnetization, introducing a high-frequency branch beyond conventional precession.
• Theoretical models enhance the Landau–Lifshitz–Gilbert equation with a second-order time derivative, yielding a gapped nutation mode with distinct dispersion and hybridization features.
• Experiments in ultrafast spectroscopy and nanostructures confirm inertial nutation dynamics, highlighting their role in topological magnonics and advanced magnetic switching.

Nutation spin waves, or the paper’s “nutation waves,” are collective magnetic excitations that arise when magnetization inertia becomes important at very short time scales. In ordinary ferromagnetic dynamics, the magnetization mostly precesses around an effective field as described by the Landau–Lifshitz–Gilbert equation, but when one includes an inertial term the magnetization can also perform a fast nutation motion—a kind of cycloidal wobble or gyroscopic nutation on top of precession. In a spatially extended magnetic system, this local inertial wobble can propagate through exchange coupling as a new wave mode, giving a high-frequency nutational branch in addition to the usual low-frequency precessional branch; in current theory this branch appears in ferromagnets, antiferromagnets, ferrimagnets, spin spirals, magnetostatic thin films, and topological magnonic models, while experiment has established the underlying local inertial mode and rotating-frame nutation dynamics (Makhfudz et al., 2019, De et al., 2024).

1. Inertial origin and the separation of magnetization from angular momentum

In the standard description of ferromagnetic dynamics, the magnetization vector $\mathbf{M}$ is treated as being locked to the total angular momentum $\mathbf{L}$ of the spin system. This is only an approximation valid when inertial effects are negligible. If the magnetic system has a finite moment of inertia, then after a sudden perturbation the magnetization and angular momentum can transiently decouple: the angular momentum begins to precess immediately, while the magnetization cannot instantly follow because it has inertia. That mismatch produces a second motion, analogous to the nutation of a gyroscope (De et al., 2024).

The standard Landau–Lifshitz–Gilbert equation contains a precessional torque from $\mathbf{H}_{\mathrm{eff}}$ and a damping term proportional to $d\mathbf{M}/dt$, but it has no second-order time derivative, so it cannot produce inertia or nutation. The inertial Landau–Lifshitz–Gilbert equation adds the second-order term. In one formulation,

$$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$

and in another,

$$\partial_{t}\boldsymbol{S}_{i} = \boldsymbol{S}_{i}\times\left( -\gamma \boldsymbol{B}_{\rm eff,i} +\alpha\,\partial_{t}\boldsymbol{S}_{i} +\eta\,\partial_{tt}\boldsymbol{S}_{i} \right).$$

The new term proportional to $d^2\mathbf M/dt^2$ or $\partial_{tt}\boldsymbol{S}_i$ is the key new ingredient: it creates a second characteristic resonance besides the usual precession resonance and gives rise to the nutational branch (Titov et al., 2021, Cherkasskii et al., 2024).

A concise statement of the separation between $\mathbf{M}$ and $\mathbf{L}$ is

$\mathbf{L}$0

When $\mathbf{L}$1, the second term vanishes and $\mathbf{L}$2. For finite $\mathbf{L}$3 and time-dependent dynamics, they are no longer parallel. The magnetization nutates and precesses, while the angular momentum mainly precesses and does not exhibit the same nutational motion (De et al., 2024).

2. Collective nutation waves and their dispersion structure

For a ferromagnetic spin chain with exchange coupling, the inertial dynamics plus exchange converts a local nutation into a propagating collective mode, the nutation wave. For a 1D isotropic Heisenberg chain, the spatially extended inertial equation takes the form

$\mathbf{L}$4

with

$\mathbf{L}$5

Linearization yields two physically distinct branches: a gapless spin-wave branch and a gapped nutation-wave branch. The spin-wave dispersion is

$\mathbf{L}$6

whereas the nutation-wave dispersion is

$\mathbf{L}$7

with gap

$\mathbf{L}$8

In this theory the nutation wave is a massive excitation, unlike the spin wave, and the one-particle excitations are interpreted as “nutatons” (Makhfudz et al., 2019).

The same inertial doubling appears in more general linear spin-wave theory. In ferromagnets and two-sublattice antiferromagnets, precession and nutation spin waves hybridize with each other, leading to the renormalization of the frequencies, the group velocities, the effective gyromagnetic ratios and the effective damping parameters. In ferromagnets, the nutation mode is shifted by approximately $\mathbf{L}$9 relative to the precession band. In antiferromagnets, the secular equation becomes quartic, the precession and nutation frequencies satisfy a constant relation for the sum of squared frequencies, and the nutation branch has a broad, weakly negative quadratic curvature at small $\mathbf{H}_{\mathrm{eff}}$0 (Mondal et al., 2022).

For non-collinear order, a general linear spin-wave framework valid for arbitrary non-collinear spin structures shows that the high-frequency nutational spin-wave band is not fixed to a ferromagnetic-like shape. In planar and conical spin spirals, the curvature of the high-frequency nutational spin-wave band changes sign as the spiral period is decreased when passing from the ferromagnetic to the antiferromagnetic limit, and conditions can be identified where the curvature changes sign and asymptotical flat bands are formed (Cherkasskii et al., 2024).

3. Ferromagnetic films, electrodynamics, and propagation geometry

A Maxwell–ILLG treatment of a ferromagnet in a strong applied field derives dispersion relations describing the propagation of nutation spin waves in an arbitrary direction relative to the applied magnetic field. In that framework, the inertia of magnetization causes the hybridization of electromagnetic waves and nutation spin waves in ferromagnets, hybrid nutation spin waves emerge, and the redshift of frequencies of precession spin waves is initiated, which transforms to precession-nutation spin waves. These effects depend sharply on the direction of wave propagation relative to the applied magnetic field: the waves propagating parallel to the applied field are circularly polarized, while the waves propagating perpendicular to that field are elliptically polarized (Titov et al., 2021).

For propagation perpendicular to the applied field, the literature distinguishes several branches, including precession magnetostatic, precession electromagnetic, precession-nutation magnetostatic, nutation electromagnetic, and hybrid nutation branches. For propagation parallel to the field, the hybridization pattern changes, a precession-nutation electromagnetic branch appears, and the inertia-induced redshift is organized differently. The physical interpretation given is that ordinary spin waves are no longer purely precessional; they acquire nutational character and a frequency redshift (Titov et al., 2021).

In the magnetostatic limit for an in-plane magnetized ferromagnetic thin film, inertia effects result in nutation surface spin waves, which propagate at terahertz frequencies. These nutation surface spin waves have group velocity smaller than that of conventional precession spin waves and are backward spin waves, with

$\mathbf{H}_{\mathrm{eff}}$1

In the numerical example reported for the conventional surface branch, the upper limit is lowered by about $\mathbf{H}_{\mathrm{eff}}$2, corresponding to a shift of about $\mathbf{H}_{\mathrm{eff}}$3. For the backward-volume branch with $\mathbf{H}_{\mathrm{eff}}$4, $\mathbf{H}_{\mathrm{eff}}$5, $\mathbf{H}_{\mathrm{eff}}$6, and $\mathbf{H}_{\mathrm{eff}}$7, the lower limit is shifted downward by about $\mathbf{H}_{\mathrm{eff}}$8, while the upper limit is shifted downward by about $\mathbf{H}_{\mathrm{eff}}$9 (Cherkasskii et al., 2021).

4. Antiferromagnets, ferrimagnets, and anisotropic nutational inertia

In two-sublattice magnets, the inertial extension of the dynamics introduces not only high-frequency nutation modes but also inter-sublattice structure. For antiferromagnets and ferrimagnets with inter- and intra-sublattice nutation coupling, the poles of the susceptibility are obtained from a quartic equation, yielding two precession modes and two nutation modes. The reported qualitative trends are specific: the precession resonance frequencies and effective damping decrease with intra-sublattice nutation coupling, while they increase with inter-sublattice nutation in an antiferromagnet; the THz nutation resonance frequencies decrease with both the intra- and inter-sublattice nutation couplings; and ferrimagnets show two nutation modes with distinct frequencies, unlike antiferromagnets (Mondal et al., 2021).

A related linear-response analysis compares ferromagnetic, antiferromagnetic, and ferrimagnetic resonance within the inertial Landau–Lifshitz–Gilbert equation. Precession and nutation resonance peaks are identified, and the interplay between precession and nutation is found to be the most prominent in antiferromagnets, where the timescale of the exchange-driven sublattice dynamics is comparable to inertial relaxation times. The conclusion drawn is that antiferromagnetic resonance techniques should be better suited for the search for intrinsical inertial spin dynamics on ultrafast timescales than ferromagnetic resonance (Mondal et al., 2020).

Normal metal–ferromagnet heterostructures add another layer of structure because the dynamical RKKY interaction generates retarded, spin-space-anisotropic interactions. In that setting the generalized LLG equation contains an inertia tensor

$d\mathbf{M}/dt$0

and for the ferromagnetic case studied in the local limit

$d\mathbf{M}/dt$1

The nutation is therefore transverse only, $d\mathbf{M}/dt$2, rather than isotropic. A major prediction is a third ferromagnetic resonance peak, beyond the usual two associated with ordinary precession and isotropic nutation, and that resonance frequency is tunable by applying an external magnetic field (Johnsen et al., 2024).

5. Experimental demonstrations and spectroscopic signatures

The most direct experimental evidence for the inertial degree of freedom underlying nutation spin waves comes from ultrafast magnetization dynamics in a thin $d\mathbf{M}/dt$3 film. In a pump–probe time-resolved magneto-optical Kerr effect experiment, a $d\mathbf{M}/dt$4, $d\mathbf{M}/dt$5 pump pulse excites the sample and a delayed $d\mathbf{M}/dt$6 probe pulse measures the Kerr rotation. The Kerr signal contains a conventional ferromagnetic precession in the GHz range and a much faster oscillation at about $d\mathbf{M}/dt$7. By comparison with atomistic spin dynamics simulations, this observation cannot be explained by the well-known Landau–Lifshitz–Gilbert equation of motion but can be attributed to inertial contributions leading to nutation of the magnetization vector around its angular momentum. From the observed nutation frequency $d\mathbf{M}/dt$8, the paper extracts a nutation timescale $d\mathbf{M}/dt$9 for the samples (De et al., 2024).

That experiment is important precisely because it does not directly observe a propagating nutation spin wave in the usual spatially resolved sense. Instead, it demonstrates the local, coherent inertial mode in a thin film: a nutation of the magnetization at a single excitation point or uniform-mode level. The physical relevance to nutation spin waves is that the same second-order time derivative that produces the local nutation also modifies spin-wave dispersion in extended systems (De et al., 2024).

A different experimental realization appears in a single YIG nanodisc driven into a deeply nonlinear ferromagnetic resonance regime. There, a strong microwave tone establishes a large-angle periodic trajectory, and a second weak microwave field resonantly excites small oscillations around that periodic trajectory. The reported resonances occur symmetrically around $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$0, at about $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$1 and $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$2 for $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$3, giving $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$4. The paper emphasizes that these are not ordinary spin waves about equilibrium; they are nutation modes or nutation spin waves of a time-periodic nonlinear state in the rotating frame (Li et al., 2019).

The neutron-scattering signature predicted for exchange-mediated nutation waves is a new gapped peak in the scattering structure factor $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$5 at $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$6. A plausible implication is that experimental access to nutation spin waves spans distinct spectroscopic limits: $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$7 or near-uniform inertial resonances in Kerr, FMR, and AFMR measurements, rotating-frame nutation modes in nonlinear spectroscopy, and finite-$$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$8 gapped peaks in inelastic neutron scattering (Makhfudz et al., 2019).

6. Topological extensions, switching physics, and interpretation

Recent theory extends nutation spin waves from an additional band to a route for topological magnonics. In a honeycomb ferromagnet, spin inertia creates a doubled spectrum containing precessional and nutational magnons. The inertial term alone doubles the spectrum but does not, by itself, produce a topological gap between the two sectors; for that, one needs an interaction that breaks angular-momentum conservation. In the cited model this role is played by the pseudodipolar interaction $$\frac{d\mathbf M}{dt} = -\gamma\,\mathbf M \times \left( \mathbf H_{\rm eff} +\frac{\alpha}{M_S}\frac{d\mathbf M}{dt} +\frac{h}{M_S}\frac{d^2\mathbf M}{dt^2} \right),$$9. When precessional and nutational bands overlap, the pseudodipolar term hybridizes them and a gap opens; slab geometries then show chiral edge states, including a chiral boundary mode in the gap between the precessional and nutational bands (Ghosh et al., 6 Mar 2026).

Nutation can also act as a functional dynamical stage in switching. In ferromagnets, linearly polarized fields align the magnetization perpendicular to the external field, enabling $$\partial_{t}\boldsymbol{S}_{i} = \boldsymbol{S}_{i}\times\left( -\gamma \boldsymbol{B}_{\rm eff,i} +\alpha\,\partial_{t}\boldsymbol{S}_{i} +\eta\,\partial_{tt}\boldsymbol{S}_{i} \right).$$0 switching, while circularly polarized fields in the $$\partial_{t}\boldsymbol{S}_{i} = \boldsymbol{S}_{i}\times\left( -\gamma \boldsymbol{B}_{\rm eff,i} +\alpha\,\partial_{t}\boldsymbol{S}_{i} +\eta\,\partial_{tt}\boldsymbol{S}_{i} \right).$$1 plane can produce $$\partial_{t}\boldsymbol{S}_{i} = \boldsymbol{S}_{i}\times\left( -\gamma \boldsymbol{B}_{\rm eff,i} +\alpha\,\partial_{t}\boldsymbol{S}_{i} +\eta\,\partial_{tt}\boldsymbol{S}_{i} \right).$$2 switching. In antiferromagnets, external fields with frequencies higher than the nutation frequency align the order parameter parallel to the field direction, while for lower frequencies it is oriented perpendicular to the field. The switching frequency increases with the magnetic field strength, and it deviates from the Larmor frequency, making it possible to outpace precessional switching in high magnetic fields (Winter et al., 2022).

A recurring interpretive issue is whether inertial signatures can be distinguished unambiguously from conventional renormalizations of exchange or anisotropy. The spin-wave literature explicitly notes that possible ways of distinguishing between the signatures of inertial dynamics and similar effects explainable within conventional models must be discussed carefully, and that flattening of the dispersion, reduced group velocity, or wave-vector-dependent $$\partial_{t}\boldsymbol{S}_{i} = \boldsymbol{S}_{i}\times\left( -\gamma \boldsymbol{B}_{\rm eff,i} +\alpha\,\partial_{t}\boldsymbol{S}_{i} +\eta\,\partial_{tt}\boldsymbol{S}_{i} \right).$$3 and damping are not by themselves decisive. The strongest indicator remains a separate high-frequency nutation branch or a direct nutation resonance with the polarization, symmetry, or hybridization properties predicted by inertial theory (Mondal et al., 2022).

Within condensed-matter magnetism, therefore, “nutation spin waves” denotes the propagating or collective expression of inertial spin dynamics. The term does not simply mean any precessional modulation, nor does every observation of magnetic nutation amount to a full dispersion measurement. The field is currently organized around a consistent hierarchy: local inertial nutation, uniform nutation resonance, propagating nutation spin waves, and hybrid precession–nutation or topological magnon branches built from the same second-order spin dynamics (De et al., 2024)