# Inertial spin waves in ferromagnets and antiferromagnets

Ritwik Mondal<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology (ISM) Dhanbad, IN-826004, Dhanbad, India  Levente Rózsa
<EMAIL_ADDRESS>Fachbereich Physik, Universität Konstanz,
DE-78457 Konstanz, Germany

###### Abstract

Inertial effects in spin dynamics are theoretically predicted to emerge at
ultrashort time scales, but their experimental signatures are often ambiguous.
Here, we calculate the spin-wave spectrum in ferromagnets and two-sublattice
antiferromagnets in the presence of inertial effects. It is shown how
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. Possible ways of
distinguishing between the signatures of inertial dynamics and similar effects
explainable within conventional models are discussed.

August 27, 2024

##  I Introduction

Spin waves or magnons correspond to the elementary excitations of magnetically
ordered systems. The magnon dispersion relation provides extensive information
about the spin configuration and about the microscopic magnetic interactions.
The dispersion relation is possible to probe using a variety of experimental
methods including light, spin-polarized electron or neutron scattering.
Magnons have also been put forward as possible information carriers in
computational architectures [1, 2], where their typically short wavelengths at
the required operational frequencies facilitate their integration with
existing CMOS devices.

Combining a high speed with a long lifetime of magnons is preferred for such
computing applications, but these requirements are often competing with each
other. Very low damping parameters and consequently long lifetimes have been
achieved in ferromagnets or strongly uncompensated ferrimagnets such as
yttrium iron garnet, but the typical excitation frequencies of these materials
lie in the GHz regime [3, 4, 5, 6, 7]. Antiferromagnetic materials are
characterized by THz excitation frequencies due to the exchange enhancement
caused by the coupling between the sublattices. However, the exchange
enhancement also affects the effective damping parameters [8], leading to
short magnon lifetimes.

A possible solution for achieving high frequencies and low damping is offered
by inertial spin dynamics [9, 10, 11]. Including an inertial term in the
Landau–Lifshitz–Gilbert equation [12, 13, 14] describing the time evolution of
the spins leads to a separation between the directions of the magnetic moment
and of the angular momentum, giving rise to spin nutation. Various theoretical
proposals have been put forward for explaining the microscopic origin of the
inertial term [10, 15, 16, 17, 18, 19, 20, 21]. Theoretical and experimental
estimates for the inertial relaxation time $\eta$ (see Eq. (2) below for the
definition used here) range from a few fs to several hundred fs [10, 15, 11,
22, 23, 24], establishing the characteristic time scale on which nutation can
be observed.

The inertial dynamics introduces an additional peak in the ferromagnetic
resonance (FMR) spectrum. While the traditional precession resonance peak is
observable at GHz frequencies, the nutation resonance occurs in the THz range
[25, 26, 27] based on the above estimates for the inertial relaxation time.
The nutation resonance has been observed experimentally in bulk CoFeB and NiFe
[24] and in epitaxial Co films [28]. Despite the enhancement of the excitation
frequency, the effective damping parameter characterizing the linewidth of the
resonance is reduced due to the inertial dynamics [26, 29]. The presence of
spin inertia not only introduces a second resonance peak, but also shifts the
precession resonance frequency to a lower value [30, 31]. While the spins
rotate counterclockwise around their equilibrium direction at the precession
resonance, they rotate clockwise at the nutation resonance [32, 33]. The
opposite rotational sense also induces a sign change in the injected spin
current from a ferromagnet to an adjacent metallic layer between the
precession and the nutation resonances [33]. Compared to ferromagnets, the
interplay between precession and nutation is exchange enhanced in two-
sublattice antiferromagnets, causing an increase in the frequency and
magnitude of the nutation resonance peak while the precession resonance is
suppressed [29]. The high susceptibility in FMR of the nutation resonance
compared to the precession resonance in antiferromagnets is explained by the
profile of the excitation modes: while the two sublattices align almost
antiparallel at the precession resonance, the nutation resonance is dominated
by one of the sublattices, thereby a finite net magnetic moment arises [33].

Away from the center of the Brillouin zone probed by FMR, the inertial
dynamics gives rise to propagating nutation spin waves at finite wave vectors,
which could be detected, e.g., by Brillouin Light Scattering or spin-polarized
electron or neutron scattering. It has been derived in previous theoretical
works [34, 35, 36, 37] that excitation frequencies of nutation spin waves are
enhanced compared to conventional, or precession, spin waves in ferromagnets.
These studies either primarily focused on small wave vectors where
magnetostatic effects dominate [35, 36], or only took isotropic exchange
interactions into account [34, 37]. Nutation spin waves in antiferromagnets
have not been considered so far.

Going beyond the isotropic exchange, the Dzyaloshinsky–Moriya interaction [38,
39] arises in systems with broken inversion symmetry due to spin–orbit
coupling, and leads to non-reciprocal magnon propagation in ferromagnets,
i.e., a different frequency of magnons with wave vectors $\boldsymbol{k}$ and
$-\boldsymbol{k}$ [40, 41, 42]. This asymmetry in the spectrum is widely used
for the experimental determination of the Dzyaloshinsky–Moriya interaction
[43, 44]. This interaction is essential for understanding the stabilization of
non-collinear spin structures such as chiral domain walls, spin spirals and
skyrmions in systems both with ferromagnetic and antiferromagnetic isotropic
exchange interactions [45], as well as for the formation of topological magnon
boundary states [46]. However, the effect of the Dzyaloshinsky–Moriya
interaction on nutation spin waves remains unexplored.

Here, we present a calculation of the magnon frequencies in the presence of
inertial effects within linear spin-wave theory, including the effects of
Heisenberg exchange, Dzyaloshinsky–Moriya interaction and magnetocrystalline
anisotropy terms in the Hamiltonian. The results are discussed for
ferromagnets in Sec. II and for antiferromagnets in Sec. III. The most
pronounced signature of the inertial dynamics is the emergence of high-
frequency nutation magnon bands. In order to account for the difficulties in
the experimental detection of spin waves at such high frequencies, it is also
discussed how the signatures of inertial dynamics could be observed in the
group velocity, the gyromagnetic ratio or the effective damping parameter of
precession magnons due to their hybridization with nutation spin waves.

##  II Ferromagnets

###  II.1 Linear spin-wave theory

We will consider the classical atomistic spin Hamiltonian

$\displaystyle\mathcal{H}$ $\displaystyle=\frac{1}{2}\sum_{i\neq
j}\boldsymbol{S}_{i}\boldsymbol{J}_{ij}\boldsymbol{S}_{j}+\sum_{i}\boldsymbol{S}_{i}\boldsymbol{K}_{i}\boldsymbol{S}_{i}-\sum_{i}\boldsymbol{B}_{i}M_{i}\boldsymbol{S}_{i},$
(1)

where $\boldsymbol{S}_{i}$ is the unit magnetization vector and $M_{i}$ is the
magnitude of the moment at lattice site $i$, the $3\times 3$ tensors
$\boldsymbol{J}_{ij}$ and $\boldsymbol{K}_{i}$ describe interactions between
the spins and on-site magnetocrystalline anisotropy, respectively, while
$\boldsymbol{B}_{i}$ stands for the external magnetic field.

The inertial Landau–Lifshitz–Gilbert (ILLG) equation describing the time
evolution of the spins reads [10, 25, 18]

$\displaystyle\frac{\textrm{d}\boldsymbol{S}_{i}}{\textrm{d}t}$
$\displaystyle=\boldsymbol{S}_{i}\times\left[-\gamma_{i}\boldsymbol{B}^{\rm
eff}_{i}+\alpha_{i}\frac{\textrm{d}\boldsymbol{S}_{i}}{\textrm{d}t}+\eta_{i}\frac{\textrm{d}^{2}\boldsymbol{S}_{i}}{\textrm{d}t^{2}}\right],$
(2)

where $\gamma_{i}$ is the absolute value of the gyromagnetic ratio for
electrons, $\alpha_{i}$ is the Gilbert damping, $\eta_{i}$ is the inertial
relaxation time, and $\boldsymbol{B}^{\rm
eff}_{i}=-\frac{1}{M_{i}}\frac{\partial\mathcal{H}}{\partial\boldsymbol{S}_{i}}$
is the effective magnetic field acting on the spins and the effective
Hamiltonian of the total system is denoted as $\mathcal{H}$.

In the ferromagnetic ground state, we assume that all spins are pointing along
the $\hat{\bm{z}}$ direction. We introduce the variables $\beta_{1i}$ and
$\beta_{2i}$ to describe small deviations from the ground state. The spin
directions are expanded up to second order in these variables as [47]

$\displaystyle\boldsymbol{S}_{i}$
$\displaystyle=\begin{pmatrix}{\beta}_{2i}\\\ -{\beta}_{1i}\\\
1-\dfrac{{\beta}_{1i}^{2}}{2}-\dfrac{{\beta}_{2i}^{2}}{2}\end{pmatrix}.$ (3)

This expansion enables the linearization of the ILLG Eq. (2), see Appendix A
for details. In the following, we consider translationally invariant systems
with a single sublattice, meaning $\gamma_{i}=\gamma$, $\alpha_{i}=\alpha$,
$\eta_{i}=\eta$, $M_{i}=M$, $\boldsymbol{K}_{i}=\boldsymbol{K}$,
$\boldsymbol{J}_{ij}=\boldsymbol{J}\left(\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right)$
and homogeneous external magnetic field $\boldsymbol{B}_{i}=\boldsymbol{B}$.
We introduce the spatial Fourier transforms of the small variables
$\tilde{\beta}_{1,2}\left(\boldsymbol{k}\right)$, and use the circularly
polarized basis defined by
$\tilde{\beta}_{\pm}\left(\boldsymbol{k}\right)=\tilde{\beta}_{2}\left(\boldsymbol{k}\right)\pm{\rm
i}\tilde{\beta}_{1}\left(\boldsymbol{k}\right)$. We also assume a harmonic
time dependence for the variables
$\tilde{\beta}_{\pm}\left(\boldsymbol{k}\right)$, replacing the differential
operator $-\textrm{i}\frac{\textrm{d}}{\textrm{d}t}$ with the frequency
$\omega$, turning the ILLG dynamical equations into a set of algebraic
equations. These may be written as

$\displaystyle\left[D\left(\omega\right)+H_{\textrm{SW}}\left(\boldsymbol{k}\right)\right]\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)=0,$
(4)

with

$\displaystyle\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)=$
$\displaystyle\begin{pmatrix}\tilde{\beta}_{+}\left(\boldsymbol{k}\right)&\tilde{\beta}_{-}\left(\boldsymbol{k}\right)\end{pmatrix}^{T},$
(5) $\displaystyle D\left(\omega\right)=$
$\displaystyle-\eta\omega^{2}+\left(\textrm{i}\alpha+\sigma^{z}\right)\omega,$
(6)

where $\sigma^{z}$ is the Pauli matrix acting on the components of
$\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)$ and the spin-wave
Hamiltonian $H_{\textrm{SW}}\left(\boldsymbol{k}\right)$ is given in Appendix
A. The solutions of the linearized ILLG equation form pairs: for each
eigenvalue $\omega\left(\boldsymbol{k}\right)$ and eigenvector
$\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)$, there is a
corresponding eigenvalue
$\omega\left(-\boldsymbol{k}\right)=-\omega^{*}\left(\boldsymbol{k}\right)$
and eigenvector
$\tilde{\boldsymbol{\beta}}\left(-\boldsymbol{k}\right)=\mathcal{C}\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)$,
where $\mathcal{C}=\sigma^{x}\mathcal{K}$ is the particle-hole symmetry
operator, $\sigma^{x}$ is a Pauli matrix and $\mathcal{K}$ is complex
conjugation. Particle-hole symmetry describes a special structure of the
dynamical equation [48], which is enforced by the commutation relations of
bosonic creation and annihilation operators in the quantum-mechanical case,
and by the requirement that the spin-wave eigenfunctions
$\beta_{1i},\beta_{2i}$ can always be chosen to be real-valued in the
classical model discussed here. Here it is demonstrated that this symmetry is
conserved even in the presence of inertial and damping terms which do not have
a direct quantum-mechanical analogue.

Figure 1: Sketch of a two-dimensional centered rectangular lattice, describing
a magnetic monolayer on a bcc(110) surface. The lattice constants are
$\sqrt{2}a$ and $a$ along the $z$ and $x$ directions, respectively. The arrows
show the ground state spin orientation along the equilibrium direction $z$,
with a small tilting of spins illustrating the traveling spin wave. The
nearest-neighbor lattice vectors are labeled as $\boldsymbol{\delta}_{1}$ and
$\boldsymbol{\delta}_{2}$, while the next-nearest-neighbor lattice vector is
$\boldsymbol{\delta}_{3}$. The nearest-neighbor coupling coefficients are
denoted by $J$ containing $J^{xx}=J^{yy}$ and $J^{zz}$, while the next-nearest
neighbor Dzyaloshinsky-Moriya vectors are $\boldsymbol{D}$.

Figure 2: Spin-wave dispersion in a ferromagnet without and with the inertia.
(a) Precession and (b) nutation bands along $k_{z}$ for $k_{x}=0$. (c)
Precession and (d) nutation bands along $k_{x}$ for $k_{z}=0$. The parameters
are $J^{zz}=-1.02\times 10^{-21}$ J, $J^{xx}=-0.99\times 10^{-21}$ J,
$D=10^{-22}$ J, $K^{zz}=-10^{-22}$ J, $B^{z}=0$ and $\alpha=0$.

###  II.2 Dispersion relation

As a specific example, we consider the system sketched in Fig. 1, representing
a magnetic monolayer on a bcc(110) surface, for example Fe/W(110) where the
prominent effect of the Dzyaloshinsky–Moriya interaction on the spin-wave
spectrum has been established in previous works [43, 41, 47]. The $z$
direction describing the orientation of the spins is oriented along the long
side of the centered rectangular unit cell, while $x$ is along the
perpendicular in-plane direction. We will consider on-site anisotropy with
only $K^{zz}$ being finite, nearest-neighbor exchange interactions with
$J^{zz}$ and $J^{xx}=J^{yy}$, and next-nearest-neighbor Dzyaloshinsky–Moriya
interactions with $D_{ij}=J_{ij}^{xy}=-J_{ij}^{yx}$. The symmetry of the
system ensures that the Dzyaloshinsky–Moriya vector $\boldsymbol{D}$ is
oriented along the $z$ direction for next-nearest neighbors, and only this
component of the vector enters the linearized ILLG equation. Under these
assumptions, Eq. (4) simplifies to two uncoupled second-order algebraic
equations,

$\displaystyle-\eta\omega^{2}+\left(\textrm{i}\alpha\pm
1\right)\omega+\Omega_{\pm}(\bm{k})=0,$ (7)

with

$\displaystyle\Omega_{\pm}(\bm{k})=$
$\displaystyle\frac{\gamma}{M}\left(-4J^{zz}+2J^{xx}\left[\cos(\bm{k}\cdot\bm{\delta}_{1})+\cos(\bm{k}\cdot\bm{\delta}_{2})\right]\right.$
$\displaystyle\left.-2K^{zz}+MB^{z}\mp
2D\sin(\bm{k}\cdot\bm{\delta}_{3})\right),$ (8)

The dispersion relation for precession and nutation spin waves is given by

$\displaystyle\omega_{\pm\rm Prec}(\bm{k})$
$\displaystyle=\frac{\pm\left(1-a_{\pm}(\bm{k})\right)+\textrm{i}\alpha\left(1-a^{-1}_{\pm}(\bm{k})\right)}{2\eta}\,,$
(9) $\displaystyle\omega_{\pm\rm Nut}(\bm{k})$
$\displaystyle=\frac{\pm\left(1+a_{\pm}(\bm{k})\right)+\textrm{i}\alpha\left(1+a^{-1}_{\pm}(\bm{k})\right)}{2\eta}\,,$
(10)

with

$\displaystyle a_{\pm}(\bm{k})=$
$\displaystyle\sqrt{\frac{\left(1-\alpha^{2}+4b_{\pm}(\bm{k})\right)+\sqrt{\left(1-\alpha^{2}+4b_{\pm}(\bm{k})\right)^{2}+4\alpha^{2}}}{2}}\,,$
(11)

where we introduced the dimensionless factor
$b_{\pm}(\bm{k})=\eta\Omega_{\pm}(\bm{k})$. For $a_{\pm}(\bm{k})\geq 1$, the
imaginary part of all frequencies is positive, meaning that the spin waves
decay over time. It can be shown that this stability condition is satisfied
for $\Omega_{\pm}(\bm{k})\geq 0$ for all $\eta\geq 0$ inertial relaxation
times. In the following, due to the particle-hole symmetry we only show the
solutions with positive real parts, i.e. $\omega_{-\rm Prec}(\bm{k})$ and
$\omega_{+\rm Nut}(\bm{k})$ due to $a_{\pm}(\bm{k})\geq 1$.

These frequencies are illustrated in Fig. 2 for the parameters
$J^{zz}=-1.02\times 10^{-21}$ J, $J^{xx}=J^{yy}=-0.99\times 10^{-21}$ J,
$D=10^{-22}$ J, $K^{zz}=-10^{-22}$ J and $B^{z}=0$, where the damping was set
to $\alpha=0$. From Eqs. (9) and (10) follows $\omega_{\pm\rm
Prec}(\bm{k})+\omega_{\pm\rm Nut}(\bm{k})=\left(\pm
1+\textrm{i}\alpha\right)/\eta$. This means that the relation between the
precession and nutation branches in Figs. 2(a) and (c) and Figs. 2(b) and (d)
is given by

$\displaystyle\omega_{+\rm Nut}(\bm{k})=\omega_{-\rm
Prec}(-\bm{k})+\eta^{-1},$ (12)

where $a_{-}(\bm{k})=a_{+}(-\bm{k})$ was used. The two bands are shifted by a
constant with respect to each other and inverted in $\boldsymbol{k}$ space.
The shift is subtracted from the nutation frequencies for visualization
purposes, otherwise the branches would be further away from each other as
$\eta$ is varied. The inversion is apparent in Figs. 2(c) and (d) where non-
reciprocal spin wave propagation $\omega(\bm{k})\neq\omega(-\bm{k})$ can be
observed. This is caused by the Dzyaloshinsky–Moriya interaction as can be
seen from Eq. (II.2), which creates an asymmetry in the spectrum along the
$k_{x}$ direction, dictated by the symmetry of the system. For the selected
sign of $D$, nutation spin waves have a lower frequency when propagating along
$-k_{x}$, while for precession spin waves have a lower frequency along
$k_{x}$. This follows from the fact that at a given lattice site, precession
and nutation spin waves have opposite circular polarizations, as it is known
for the $\boldsymbol{k}=\boldsymbol{0}$ modes [29]. For $\alpha=0$, the two
frequencies in Eqs. (9) and (10) can be expanded in $b_{\pm}(\bm{k})$ as

$\displaystyle\omega_{-\rm Prec}(\bm{k})$
$\displaystyle\approx\Omega_{-}(\bm{k})-b_{-}(\bm{k})\Omega_{-}(\bm{k})\,,$
(13) $\displaystyle\omega_{+\rm Nut}(\bm{k})$
$\displaystyle\approx\frac{1}{\eta}+\Omega_{+}(\bm{k})-b_{+}(\bm{k})\Omega_{+}(\bm{k}).$
(14)

In this approximation, the original precession frequency
$\Omega_{\pm}(\bm{k})$ is renormalized by the factor $1-b_{\pm}(\bm{k})$ due
to the inertia. While in the $\boldsymbol{k}=\boldsymbol{0}$ point this is
typically small, at high wave vectors $\Omega_{\pm}(\bm{k})$ is dominated by
the exchange terms, and $b_{\pm}(\bm{k})$ is orders of magnitude larger. The
stronger decrease of the excitation frequencies at higher wave vectors leads
to a flattening of the parabolic spin-wave spectrum with increasing inertial
relaxation time $\eta$, as shown in Fig. 2.

The most apparent consequence of the inertial dynamics is the appearance of
the nutation spin-wave band. Since it is shifted by $1/\eta$ with respect to
the precession band, it may be challenging to observe using experimental
techniques calibrated to typical precession frequencies. Therefore, it is
worthwhile to discuss how the signatures of inertial dynamics show up in the
precession band. While the decrease of the spin-wave frequencies between the
inertial and non-inertial cases at the edges of the Brillouin zone can achieve
relatively high values in this model calculation, it is unlikely that inertial
effects can be experimentally demonstrated based on the measurement of the
precession spin-wave band alone. The value of the exchange parameters in such
measurements is determined based on the observed spectrum, and the flattened
dispersion relation in Fig. 2 may also be fitted by renormalizing the nearest-
neighbor exchange and taking interactions with further neighbors into account.
The signatures of inertial dynamics could only be proven if the spin
interactions obtained from the spectrum are compared to static quantities
determined by the spin model parameters but unaffected by the dynamics, such
as magnetization curves or the critical temperature.

The group velocity of precession spin waves is given by the expression

$\displaystyle\boldsymbol{v}_{-{\rm Prec}}\left(\boldsymbol{k}\right)$
$\displaystyle=\frac{\textrm{d}\omega_{-\rm
Prec}(\bm{k})}{\textrm{d}\boldsymbol{k}}=a^{-1}_{-}(\bm{k})\frac{\textrm{d}\Omega_{-}(\bm{k})}{\textrm{d}\boldsymbol{k}}.$
(15)

Figure 3: The calculated group velocity of precession spin waves at several
inertial relaxation times along $k_{x}$ for $k_{z}=0$ with the lattice
constant $a=3$ Å. The parameters are $J^{zz}=-1.02\times 10^{-21}$ J,
$J^{xx}=J^{yy}=-0.99\times 10^{-21}$ J, $D=10^{-22}$ J, $K^{zz}=-10^{-22}$ J,
$B^{z}=0$ and $\alpha=0$.

The group velocity is presented in Fig. 3 for the considered model system. Due
to the inertial dynamics, $\boldsymbol{v}_{-{\rm
Prec}}\left(\boldsymbol{k}\right)$ is rescaled by the factor
$a^{-1}_{-}(\bm{k})$. The group velocity vanishes in the minimum of the spin-
wave spectrum, which is shifted along the $k_{x}$ direction due to the
Dzyaloshinsky–Moriya interaction as discussed above. The position of the
minimum is independent of $\eta$. The magnitude of $\boldsymbol{v}_{-{\rm
Prec}}$ is reduced for increased inertial relaxation times $\eta$ and further
away from the minimum of the spectrum. As it was explained above for the
dispersion relation, the experimental determination of the group velocities in
only the precession band is probably insufficient for concluding on the
strength of inertial effects, since a similar decrease could also be explained
by a different set of interaction parameters.

It is worthwhile to compare the results here to the calculations in Refs. [34]
and [37], to which the formalism here is equivalent if only isotropic exchange
interactions are considered, and the Fourier transforms are expanded in
$\boldsymbol{k}$ in the long-wavelength limit. In Ref. [34] it was stated that
nutation waves have higher frequencies and speeds compared to conventional
precession spin waves. While we can confirm the higher frequencies of the
nutation spin waves by the constant shift of $1/\eta$ at a fixed wave vector
(the non-reciprocity vanishes in the absence of the Dzyaloshinsky–Moriya
interaction) in agreement with Ref. [37], nutation waves have the exact same
group velocity given by Eq. (16) as precession waves. A higher speed for the
nutation waves can only be observed if different wave vectors are compared
between the two bands, as it was performed in Ref. [34].

Figure 4: The effective gyromagnetic ratio $\gamma_{\textrm{eff,Prec}}$ and
effective damping parameter $\alpha_{\rm eff,Prec}$ from Eq. (16) and Eq. (17)
calculated for several inertial relaxation times along $k_{z}$ for $k_{x}=0$.
The values are divided by $\gamma$ and $\alpha$ observable for free
precession. The parameters are $J^{zz}=-1.02\times 10^{-21}$ J,
$J^{xx}=J^{yy}=-0.99\times 10^{-21}$ J, $D=10^{-22}$ J, $K^{zz}=-10^{-22}$ J
and $B^{z}=0$.

A closer inspection of the dispersion relation of precession spin waves may
shed further light on the consequences of inertial dynamics. For an external
magnetic field applied along the $\bm{\hat{z}}$ direction, the effective
gyromagnetic ratio is computed as

$\displaystyle\gamma_{\rm eff}(\bm{k})=\frac{\textrm{d}\omega_{-\rm
Prec}(\bm{k})}{\textrm{d}B^{z}}$
$\displaystyle=a^{-1}_{-}(\bm{k})\frac{\textrm{d}\Omega_{-}(\bm{k})}{\textrm{d}B^{z}}$
$\displaystyle=\frac{\gamma}{\sqrt{1+4b_{-}(\bm{k})}}$ (16)

where the last formula holds for $\alpha=0$. This expression is illustrated in
Fig. 4. While $\gamma_{\rm eff}$ corresponds to the free gyromagnetic ratio in
the absence of inertia, it displays a wave-vector-dependent reduction as
$\eta$ increases. The renormalization factor $a^{-1}\left(\bm{k}\right)$ is
the same as for the group velocity in Eq. (15). Note that $\gamma_{\rm eff}$
is the same for the nutation band when $a^{-1}_{-}(\bm{k})$ is replaced by
$a^{-1}_{+}(\bm{k})$ because of the constant shift and the inversion in
$\boldsymbol{k}$ space described above.

However, a renormalization of the gyromagnetic ratio with the wave vector is
not unique to inertial dynamics. It can also be observed in ferromagnets with
biaxial anisotropy, see Appendix A for the calculation; however, the
renormalization shows a different qualitative dependence on $\boldsymbol{k}$.
In biaxial ferromagnets in the absence of inertia, the gyromagnetic factor is
enhanced at short wave vectors and then converges to $\gamma$ on the scale
$ka\propto\sqrt{\left(K^{xx}-K^{yy}\right)/J}$, where $K^{xx}-K^{yy}$ is the
anisotropy energy between the intermediate and the hard axes. In contrast, in
the inertial case the gyromagnetic ratio is always smaller than its free
value, and it typically displays a decrease in a broad range in reciprocal
space, occurring on the characteristic scale of
$J_{\bm{0}}-J_{\bm{k}}\approx\frac{M}{\gamma}\eta$, which is away from the
center of the Brillouin zone for typical values of the exchange interaction
and of the inertial relaxation time. Therefore, the experimental determination
of the gyromagnetic ratio in a wide range of wave vectors should make it
possible to distinguish between anisotropic and inertial effects.

The effective damping parameter of the spin waves is defined as

$\displaystyle\alpha_{\textrm{eff},-\textrm{Prec}}(\bm{k})=\frac{\textrm{Im}\>\omega_{-\rm
Prec}(\bm{k})}{\textrm{Re}\>\omega_{-\rm
Prec}(\bm{k})}=a_{-}^{-1}(\bm{k})\alpha;$ (17)

for nutation spin waves $a_{-}^{-1}(\bm{k})$ has to be replaced by
$a_{+}^{-1}(\bm{k})$. Since the dependence on the wave vector is precisely the
same as for the effective gyromagnetic ratio in Eq. (16), Fig. 4 also provides
an illustration of the wave-vector-dependence of the effective damping. The
analysis also follows similar lines: while the effective damping also depends
on the wave vector in biaxial ferromagnets, it is only enhanced in the
vicinity of the $\boldsymbol{k}=\boldsymbol{0}$ point, while inertial effects
reduce the effective damping and yield a broad variation in the Brillouin
zone. Although the Gilbert damping $\alpha$ is material dependent while
$\gamma$ is more or less constant between magnetic systems, the different
behavior in the Brillouin zone should make it possible to distinguish between
inertial and anisotropic effects on the effective damping. Finally, we remark
that the enhanced robustness of nutation waves compared to precession waves
proposed in Ref. [34] only holds if effective damping parameters at different
wave vectors are compared. The same value for the effective damping between
precession and nutation waves and its decrease at higher wave vectors agrees
with the results of Ref. [37], where the quality factor
$Q=1/(2\alpha_{\textrm{eff}})$ was calculated.

##  III Antiferromagnets

###  III.1 Linear spin-wave theory

Here, we consider two-sublattice antiferromagnets. In the ground state, the
magnetic moments on the two sublattices $A$ and $B$ point antiparallel,
assumed to lie along the $+\hat{\boldsymbol{z}}$ and $-\hat{\boldsymbol{z}}$
directions. Introducing the variables $\beta_{1i}$, $\beta_{2i}$ on sublattice
$i\in A$ and $\beta_{1j}$, $\beta_{2j}$ on sublattice $j\in B$, the spin
directions are expanded as

$\displaystyle\bm{S}_{i}$ $\displaystyle=\begin{pmatrix}{\beta}_{2i}\\\
-{\beta}_{1i}\\\
1-\dfrac{{\beta}_{1i}^{2}}{2}-\dfrac{{\beta}_{2i}^{2}}{2}\end{pmatrix},$ (18)
$\displaystyle\bm{S}_{j}$ $\displaystyle=\begin{pmatrix}{\beta}_{2j}\\\
{\beta}_{1j}\\\
-1+\dfrac{{\beta}_{1j}^{2}}{2}+\dfrac{{\beta}_{2j}^{2}}{2}\end{pmatrix}.$ (19)

Following the same procedure as for ferromagnets, in Fourier space and the
circularly polarized basis we arrive again at the linearized equation of
motion Eq. (4). In this case, the eigenvectors contain four components,

$\displaystyle\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)=$
$\displaystyle\begin{pmatrix}\tilde{\beta}_{+A}\left(\boldsymbol{k}\right)&\tilde{\beta}_{-A}\left(\boldsymbol{k}\right)&\tilde{\beta}_{+B}\left(\boldsymbol{k}\right)&\tilde{\beta}_{-B}\left(\boldsymbol{k}\right)\end{pmatrix}^{T},$
(20) $\displaystyle D\left(\omega\right)=$
$\displaystyle\textrm{diag}\left(D_{+A}\left(\omega\right),D_{-A}\left(\omega\right),D_{+B}\left(\omega\right),D_{-B}\left(\omega\right)\right),$
(21)

with

$\displaystyle D_{\pm A/B}=$
$\displaystyle-\eta_{A/B}\omega^{2}+\left(\textrm{i}\alpha_{A/B}\pm
1\right)\omega.$ (22)

The derivation and the expression for $H_{\textrm{SW}}\left(\bm{k}\right)$ are
given in Appendix B.

Figure 5: Sketch of a two-dimensional centered rectangular lattice with
nearest-neighbor antiferromagnetic coupling. The lattice constants are
$\sqrt{2}a$ and $a$ along the $z$ and $x$ directions, respectively. The arrows
show the ground state spin orientation along the equilibrium direction $z$,
with a small tilting of spins illustrating the traveling spin wave. The
nearest-neighbor lattice vectors are labeled as $\boldsymbol{\delta}_{1}$ and
$\boldsymbol{\delta}_{2}$. The nearest-neighbor coupling coefficients are
denoted by $J$ with $J^{xx}=J^{yy}$ and $J^{zz}$, while the nearest-neighbor
Dzyaloshinsky-Moriya vectors are $\boldsymbol{D}$.

The equation of motion stil satisfies particle-hole symmetry, implying
$\omega\left(-\boldsymbol{k}\right)=-\omega^{*}\left(\boldsymbol{k}\right)$
for the eigenvalues and
$\tilde{\boldsymbol{\beta}}\left(-\boldsymbol{k}\right)=\mathcal{C}\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)$
for the eigenvectors. Now we consider the antiferromagnetic limit, where the
two sublattices are equivalent, i.e., $\gamma_{A}=\gamma_{B}=\gamma$,
$\alpha_{A}=\alpha_{B}=\alpha$, $\eta_{A}=\eta_{B}=\eta$, $M_{A}=M_{B}$,
$\boldsymbol{K}_{A}=\boldsymbol{K}_{B}$, and
$\boldsymbol{J}_{\boldsymbol{k}AB}=\boldsymbol{J}_{\boldsymbol{k}BA}$. The
external magnetic field is set to zero ($B^{z}=0$), and the $A$ and $B$
sublattices are assumed to form a Bravais lattice together. In this case and
for $\alpha=0$, the eigenvalues
$\omega\left(-\boldsymbol{k}\right)=\omega\left(\boldsymbol{k}\right)$ and the
eigenvectors
$\tilde{\boldsymbol{\beta}}\left(-\boldsymbol{k}\right)=\mathcal{T}\tilde{\boldsymbol{\beta}}\left(\boldsymbol{k}\right)$
are divided into pairs between wave vectors $\boldsymbol{k}$ and
$-\boldsymbol{k}$ by the effective time-reversal symmetry
$\mathcal{T}=\tau^{x}\mathcal{K}$, where $\tau^{x}$ is a Pauli matrix
exchanging the two sublattices. $\mathcal{T}$ satisfies certain natural
requirements on time reversal, namely that it is broken if the system has a
finite net magnetic moment due to the inequivalence of the sublattices or
because of the application of an external field, and it does not hold in the
presence of dissipation $\alpha$.

###  III.2 Dispersion relation

Figure 6: Spin-wave dispersion relation in the antiferromagnet without and
with the inertia, computed based on Eq. (23). Solid and dashed lines indicate
the $+$ and $-$ branches of the dispersion relation. (a) Precession and (b)
nutation bands along $k_{z}$ for $k_{x}=0$. (c) Precession and (d) nutation
bands along $k_{x}$ for $k_{z}=0$. The parameters are $J^{zz}=1.02\times
10^{-21}$ J, $J^{xx}=0.99\times 10^{-21}$ J, $D=-10^{-22}$ J,
$K^{zz}=-10^{-22}$ J, $B^{z}=0$ and $\alpha=0$.

For illustration, we again consider the centered rectangular lattice sketched
in Fig. 5. Such a local antiferromagnetic order have been observed in
monolayers of Mn [49] and Cr [50] on W(110), although on longer length scales
they are modulated to form spin spirals. We only consider exchange
($J^{zz},J^{xx}=J^{yy}$) and Dzyaloshinsky–Moriya
($D_{ij}=J_{ij}^{xy}=-J_{ij}^{yx}$) interactions between the nearest neighbors
belonging to different sublattices, and a uniaxial on-site anisotropy term
$K_{A/B}^{zz}$. In this model, the linearized equations of motion decouple
into two two-by-two systems of equations containing the components
$\left(\tilde{\beta}_{+A}\left(\boldsymbol{k}\right),\tilde{\beta}_{-B}\left(\boldsymbol{k}\right)\right)$
and
$\left(\tilde{\beta}_{-A}\left(\boldsymbol{k}\right),\tilde{\beta}_{+B}\left(\boldsymbol{k}\right)\right)$,
respectively. Denoting the first pair by $+$ and the second pair by $-$, we
obtain the following secular equation for the eigenfrequencies:

$\displaystyle\mathcal{A}_{\pm}\omega^{4}(\bm{k})+\mathcal{B}_{\pm}\omega^{3}(\bm{k})+\mathcal{C}_{\pm}\omega^{2}(\bm{k})+\mathcal{D}_{\pm}\omega(\bm{k})+\mathcal{E}_{\pm}(\bm{k})=0,$
(23)

where the following parameters have been defined:

$\displaystyle\mathcal{A}_{\pm}=$ $\displaystyle\eta_{A}\eta_{B},$ (24)
$\displaystyle\mathcal{B}_{\pm}=$
$\displaystyle\left[\pm(\eta_{A}-\eta_{B})-{\rm
i}\left(\alpha_{B}\eta_{A}+\alpha_{A}\eta_{B}\right)\right],$ (25)
$\displaystyle\mathcal{C}_{\pm}=$
$\displaystyle\left[-1-\alpha_{A}\alpha_{B}-(\Omega_{AA}\eta_{B}+\Omega_{BB}\eta_{A})\mp{\rm
i}(\alpha_{A}-\alpha_{B})\right],$ (26) $\displaystyle\mathcal{D}_{\pm}=$
$\displaystyle\left[\mp\left(\Omega_{AA}-\Omega_{BB}\right)+{\rm
i}\left(\Omega_{AA}\alpha_{B}+\Omega_{BB}\alpha_{A}\right)\right],$ (27)
$\displaystyle\mathcal{E}_{\pm}(\bm{k})=$
$\displaystyle\Omega_{AA}\Omega_{BB}-W_{\pm AB}(\bm{k})W_{\mp BA}(\bm{k}),$
(28) $\displaystyle\Omega_{AA}=$
$\displaystyle\frac{\gamma_{A}}{M_{A}}\left(4J^{zz}-2K_{A}^{zz}+M_{A}B^{z}\right),$
(29) $\displaystyle\Omega_{BB}=$
$\displaystyle\frac{\gamma_{B}}{M_{B}}\left(4J^{zz}-2K_{B}^{zz}-M_{B}B^{z}\right),$
(30) $\displaystyle W_{\pm AB}(\bm{k})=$
$\displaystyle\frac{\gamma_{A}}{M_{A}}\left(2J^{xx}\left[\cos(\bm{k}\cdot\bm{\delta}_{1})+\cos(\bm{k}\cdot\bm{\delta}_{2})\right]\right.$
$\displaystyle\left.\mp
2D\left[\sin(\bm{k}\cdot\bm{\delta}_{2})-\sin(\bm{k}\cdot\bm{\delta}_{1})\right]\right),$
(31) $\displaystyle W_{\pm BA}(\bm{k})=$
$\displaystyle\frac{\gamma_{B}}{M_{B}}\left(2J^{xx}\left[\cos(\bm{k}\cdot\bm{\delta}_{1})+\cos(\bm{k}\cdot\bm{\delta}_{2})\right]\right.$
$\displaystyle\left.\pm
2D\left[\sin(\bm{k}\cdot\bm{\delta}_{2})-\sin(\bm{k}\cdot\bm{\delta}_{1})\right]\right).$
(32)

We present the analytical solution of Eq. (23) only for two identical
sublattices, and set the damping and the external field to zero. This leads to
the simplification $\Omega=\Omega_{AA}=\Omega_{BB}$ and $W_{\pm}=W_{\pm
AB}=W_{\mp BA}$. The dispersion relation is given by

$\displaystyle\omega_{\pm\textrm{Prec}}^{2}\left(\boldsymbol{k}\right)=$
$\displaystyle\frac{1+2b}{2\eta^{2}}\left(1-\sqrt{1-\frac{4\eta^{2}}{\left(1+2b\right)^{2}}\left(\Omega^{2}-W^{2}_{\pm}\left(\boldsymbol{k}\right)\right)}\right),$
(33) $\displaystyle\omega_{\pm\textrm{Nut}}^{2}\left(\boldsymbol{k}\right)=$
$\displaystyle\frac{1+2b}{2\eta^{2}}\left(1+\sqrt{1-\frac{4\eta^{2}}{\left(1+2b\right)^{2}}\left(\Omega^{2}-W^{2}_{\pm}\left(\boldsymbol{k}\right)\right)}\right)$
(34)

for precession and nutation spin waves, respectively. Here $b=\eta\Omega$ was
introduced, analogously to $b_{\pm}\left(\bm{k}\right)$ in the ferromagnetic
case. We will only consider the
$\omega_{\pm\textrm{Prec,Nut}}\left(\boldsymbol{k}\right)\geq 0$ solutions due
to particle-hole symmetry.

The spin-wave frequencies are shown in Fig. 6 for two equivalent sublattices
with the parameters $J^{zz}=1.02\times 10^{-21}$ J, $J^{xx}=J^{yy}=0.99\times
10^{-21}$ J, $D=-10^{-22}$ J, $K^{zz}=-10^{-22}$ J and in the absence of
damping. In Fig. 6(c) and (d) it can be seen that the individual branches
$\omega_{+\textrm{Prec}}\left(\boldsymbol{k}\right),\omega_{-\textrm{Prec}}\left(\boldsymbol{k}\right),\omega_{+\textrm{Nut}}\left(\boldsymbol{k}\right)$,
and $\omega_{-\textrm{Nut}}\left(\boldsymbol{k}\right)$ are non-reciprocal due
to the presence of the Dzyaloshinsky–Moriya interaction. However, the branches
are connected by the reciprocal symmetry $\mathcal{R}=\tau^{x}$, ensuring
$\omega_{+\textrm{Prec}}\left(\boldsymbol{k}\right)=\omega_{-\textrm{Prec}}\left(-\boldsymbol{k}\right)$
and
$\omega_{+\textrm{Nut}}\left(\boldsymbol{k}\right)=\omega_{-\textrm{Nut}}\left(-\boldsymbol{k}\right)$.
In contrast to the time-reversal symmetry $\mathcal{T}=\mathcal{R}\mathcal{K}$
introduced above, reciprocal symmetry requires certain spatial symmetries
beyond the equivalence of the two sublattices. These are satisfied by the
present system as discussed in Appendix A. Note that reciprocal symmetry also
holds for finite damping $\alpha$, unlike time-reversal symmetry.

While in the ferromagnetic case the sum of the precession and nutation
frequencies was fixed in the same spin-wave branch while
$\omega_{+\textrm{Prec}}\left(\boldsymbol{k}\right)$ and
$\omega_{+\textrm{Nut}}\left(\boldsymbol{k}\right)$ had opposite signs, in the
antiferromagnetic case the sum of the squared frequencies is a constant as a
function of wave vector,

$\displaystyle\omega^{2}_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right)+\omega^{2}_{\pm\textrm{Nut}}\left(\boldsymbol{k}\right)=\frac{1+2b}{\eta^{2}}.$
(35)

This leads to a considerably different spectrum. While in the ferromagnet the
precession and nutation bands are only shifted with respect to each other and
inverted in $\boldsymbol{k}$ space, in the antiferromagnet the nutation
frequencies are smaller for higher precession frequencies, meaning that they
decrease when moving away from the center of the Brillouin zone. This can also
be seen when expanding the frequencies in the parameter $b$, yielding

$\displaystyle\omega_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right)\approx$
$\displaystyle\sqrt{\frac{\Omega^{2}-W^{2}_{\pm}\left(\boldsymbol{k}\right)}{1+2b}},$
(36) $\displaystyle\omega_{\pm\textrm{Nut}}\left(\boldsymbol{k}\right)\approx$
$\displaystyle\frac{\sqrt{1+2b}}{\eta}-\frac{\eta\left(\Omega^{2}-W^{2}_{\pm}\left(\boldsymbol{k}\right)\right)}{2\left(1+2b\right)^{\frac{3}{2}}}.$
(37)

While the ordinary precession frequencies
$\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)=\sqrt{\Omega^{2}-W^{2}_{\pm}\left(\boldsymbol{k}\right)}$
are decreased by a factor of $\left(1+2b\right)^{-\frac{1}{2}}$, the nutation
frequency is around $\sqrt{1+2b}/\eta$, which is enhanced compared to the
ferromagnetic case by $\sqrt{1+2b}$, as it was already discussed in Ref. [29]
in the $\boldsymbol{k}=\boldsymbol{0}$ case. While the precession band is
almost linear in wave vector except very close to the minimum where the
anisotropy leads to a quadratic dispersion, the nutation band behaves as
$\propto-\eta\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}$
at low wave vectors, which describes a broad quadratic dispersion with a small
negative curvature, as is also visible in Fig. 6(b) and (d). Note that the
nutation bands are shifted down by $1/\eta$ in the figure as in the
ferromagnetic case, which here does not lead them to coincide with the
precession bands, but it does make it possible to visualize them in the same
frequency range regardless of the value of $\eta$.

Figure 7: Group velocity of spin waves in antiferromagnets along $k_{z}$ for
$k_{x}=0$, computed based on Eq. (23). The solid and dashed lines are the
group velocities for precession waves and nutation waves, respectively. The
used parameters are $J^{zz}=1.02\times 10^{-21}$ J, $J^{xx}=0.99\times
10^{-21}$ J, $D=-10^{-22}$ J, $K^{zz}=-10^{-22}$ J, $B^{z}=0$, $\alpha=0$, and
lattice constant $a=3$ Å.

The group velocity in the antiferromagnetic case is

$\displaystyle\boldsymbol{v}_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right)$
$\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{1-\frac{4\eta^{2}}{\left(1+2b\right)^{2}}\left(\Omega^{2}-W^{2}_{\pm}\right)}}\frac{1}{\sqrt{1+2b}}\boldsymbol{v}^{(0)}_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right),$
(38)
$\displaystyle\boldsymbol{v}_{\pm\textrm{Nut}}\left(\boldsymbol{k}\right)$
$\displaystyle\approx$
$\displaystyle-\frac{1}{\sqrt{1-\frac{4\eta^{2}}{\left(1+2b\right)^{2}}\left(\Omega^{2}-W^{2}_{\pm}\right)}}\frac{2\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\eta}{\left(1+2b\right)^{\frac{3}{2}}}\boldsymbol{v}^{(0)}_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right),$
(39)

where
$\boldsymbol{v}^{(0)}_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right)=\frac{1}{2\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)}\frac{\partial\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}}{\partial\boldsymbol{k}}$
is the group velocity in the absence of inertia.

The group velocity is shown in Fig. 7 along the $k_{z}$ direction. Compared to
the ferromagnetic case in Fig. 3, the slope of the group velocity is higher at
low wave vector, indicative of the high curvature of the dispersion relation.
The leading correction of
$\boldsymbol{v}_{\pm\textrm{Prec}}\left(\boldsymbol{k}\right)$ due to the
inertial dynamics is a renormalization by a factor of
$\left(1+2b\right)^{-\frac{1}{2}}$, similarly to the frequency itself. Since
this renormalization is independent of the wave vector, it would not be
possible to detect it experimentally, unless the interaction parameters are
known from independent measurements as mentioned in the ferromagnetic case.
The first term on the right-hand side of Eq. (38) describes a further decrease
of the group velocity, which only becomes relevant at high wave vectors where
$\eta^{2}\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}$ is
relatively large. The group velocity of nutation spin waves points along the
opposite direction compared to precession spin waves, and here it is
considerably lower in magnitude for all wave vectors because of the flat
dispersion relation.

Figure 8: Effective gyromagnetic ratio for spin waves in antiferromagnets
along $k_{x}$ for $k_{z}=0$, computed based on Eq. (23). The solid and dashed
lines are the gyromagnetic ratios for precession waves and nutation waves,
respectively. The used parameters are $J^{zz}=1.02\times 10^{-21}$ J,
$J^{xx}=0.99\times 10^{-21}$ J, $D=-10^{-22}$ J, $K^{zz}=-10^{-22}$ J, and
$\alpha=0$.

Similarly to the ferromagnetic case, deriving the effective gyromagnetic
ratios and damping parameters from the dispersion relation could provide
signatures for the experimental detection of the inertial dynamics if only the
frequency range of the precession spin waves is accessible. The gyromagnetic
ratios are expressed as

$\displaystyle\gamma_{\textrm{eff,Prec}\pm}\left(\boldsymbol{k}\right)=-\gamma_{\textrm{eff,Nut}\pm}\left(\boldsymbol{k}\right)\approx$
$\displaystyle\pm\frac{1}{1+2b}\left(1+\frac{2\eta^{2}\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}}{\left(1+2b\right)^{2}}\right)\gamma,$
(40)

while the effective damping parameters read

$\displaystyle\alpha_{\textrm{eff,Prec}}\left(\boldsymbol{k}\right)\approx$
$\displaystyle\frac{\Omega-\frac{\eta\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}}{\left(1+2b\right)}}{\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\left(1+2b\right)^{\frac{1}{2}}}\alpha,$
(41)
$\displaystyle\alpha_{\textrm{eff,Nut}}\left(\boldsymbol{k}\right)\approx$
$\displaystyle\left[\frac{1+b}{\left(1+2b\right)^{\frac{3}{2}}}+\frac{3-5b}{2\left(1+2b\right)^{\frac{7}{2}}}\eta^{2}\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}\right]\alpha;$
(42)

see Appendix B for the idea of the derivation. These expressions are
linearized in the small parameter
$\eta^{2}\left(\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)\right)^{2}\ll 1$,
which should be valid at low wave vectors for typical interaction and inertial
parameters.

Figure 9: Effective damping for spin waves in antiferromagnets along $k_{z}$
for $k_{x}=0$, computed based on Eq. (23). The solid and dashed lines
represent effective damping for precession and nutation modes, respectively.
The used parameters are $J^{zz}=1.02\times 10^{-21}$ J, $J^{xx}=0.99\times
10^{-21}$ J, $D=-10^{-22}$ J, $K^{zz}=-10^{-22}$ J, $B^{z}=0$, and
$\alpha=0.05$.

The gyromagnetic ratios are illustrated in Fig. 8. Note that the two spin-wave
branches $\omega_{\pm}$ are described by opposite signs of the gyromagnetic
ratio, i.e., one is increasing in external magnetic field while the other is
decreasing. This is a consequence of the antiferromagnetic alignment of the
spins irrespective of the inertial dynamics. Only the positive gyromagnetic
ratio is shown in Fig. 8. Due to the finite value of $\eta$, the effective
gyromagnetic ratio of precession spin waves in Eq. (40) is decreased at zero
wave vector stronger than in ferromagnets, since $b_{\textrm{AFM}}\gg
b_{\textrm{FM}}\left(\bm{k}=\bm{0}\right)$. However, at increasing wave vector
the gyromagnetic ratio increases, in contrast to the ferromagnetic case. The
relative change in $\gamma_{\textrm{eff,Prec}}$ between the center and the
boundary of the Brillouin zone is smaller in antiferromagnets than in
ferromagnets due to a decreased prefactor. While the dependence on the wave
vector is caused entirely by inertial effects in this model, a biaxial
anisotropy may enhance the gyromagnetic ratio at low wave vectors even without
inertia, as already discussed for the ferromagnetic case. According to Eq.
(40), the gyromagnetic ratio for the nutation band is inverted in
$\boldsymbol{k}$ compared to the precession band, when the branches
$\gamma_{\textrm{eff,Prec}+}\left(\boldsymbol{k}\right)$ and
$\gamma_{\textrm{eff,Prec}-}\left(\boldsymbol{k}\right)$ with positive
gyromagnetic ratios are considered, as is also shown in Fig. 8.

The effective damping parameters are shown in Fig. 9. Already in the absence
of inertia, $\alpha_{\textrm{eff}}$ is increased by a factor of
$\Omega/\omega_{\pm}^{(0)}\left(\boldsymbol{k}\right)$ with respect to
$\alpha$, which constitutes an exchange enhancement $\propto\sqrt{J/K}$ at low
wave vectors and decreases away from the center of the Brillouin zone. This
effect is qualitatively similar to the case of biaxial ferromagnets mentioned
above. The leading correction due to inertial dynamics is once again the
renormalization by the constant factor $\left(1+2b\right)^{-\frac{1}{2}}$,
decreasing the effective damping. However, the renormalization by a factor
independent of the wave vector is not possible to detect experimentally if the
value of the Gilbert damping is not known in advance. Expanding the effective
damping in $\eta$ leads to a sub-leading correction which leads to a stronger
decrease away from the center of the Brillouin zone; however, based on Fig. 9
this correction may be difficult to discern since the shape is already
qualitatively similar without inertia. Nutation spin waves are not influenced
by the exchange enhancement [29], and their effective damping parameters are
reduced compared to the Gilbert damping $\alpha$ by a factor of
$\left(1+b\right)/\left(1+2b\right)^{\frac{3}{2}}$. The dependence on the wave
vector only appears when expanding up to second order in $\eta$, leading to a
weak increase at higher wave vectors if $b$ is sufficiently small.

##  IV Conclusions

We calculated the spin-wave dispersion relation in ferromagnets and
antiferromagnets in the linear spin-wave approximation in the presence of an
inertial term in the equation of motion. The inertial term doubles the number
of spin-wave bands due to the appearance of nutation spin waves at high
frequency, and also decreases the conventional precession spin-wave
frequencies due to the hybridization between the bands. The decrease of the
precession spin-wave frequencies becomes more pronounced at higher wave
vectors. In uniaxial ferromagnets, the nutation band is shifted by $\eta^{-1}$
compared to the precession band, and is inverted in reciprocal space. If the
spin-wave propagation is non-reciprocal, for example due to the presence of
the Dzyaloshinsky–Moriya interaction, then the preferential direction for
spin-wave propagation is inverted between precession and nutation spin waves.
In antiferromagnets with two identical sublattices, the sum of the squares of
the precession and nutation spin waves is independent of the wave vector. This
leads to the fact that the dispersion relation of the nutation waves is
quadratic at low wave vectors in contrast to the typical linear dispersion of
antiferromagnetic precession spin waves, and the frequency of nutation spin
waves decreases at higher wave vectors. The group velocity of spin waves is
reduced as a consequence of inertial effects, and nutation spin waves in
antiferromagnets propagate in the opposite direction compared to their wave
vectors.

While the direct detection of nutation spin waves would provide the most
direct evidence of inertial effects on spin dynamics, analyzing the precession
band alone may also provide indications for inertial motion. Since the
renormalization of spin-wave frequencies and the group velocity may also be
explained by a different set of spin-model parameters, these parameters could
be fixed by the measurement of static properties such as the critical
temperature which is not affected by the inertia. Alternatively, the decrease
of the effective gyromagnetic ratio or the effective damping parameter with
increasing wave vector could hint towards the role of inertial dynamics.
Although a similar wave-vector-dependence may also be caused by anisotropy
terms, those modify the gyromagnetic ratio and the effective damping typically
at lower frequencies compared to where inertial effects are the most
pronounced. Such signatures are expected to be more pronounced in ferromagnets
than in antiferromagnets, since in the latter the frequencies of precession
spin waves and all derived quantities are renormalized by a factor which
depends very weakly on the wave vector. Such a constant renormalization cannot
be determined from the investigation of spin waves unless the interaction
parameters, the gyromagnetic ratio or the damping is known from independent
measurements. The comparison between precession and nutation spin waves is
more intriguing in antiferromagnets: while the effective damping is identical
in the two spin-wave bands in ferromagnets, in antiferromagnets nutation spin
waves do not experience an exchange enhancement of the effective damping.

## Acknowledgments

The authors would like to thank Mikhail Cherkasskii for fruitful discussions.
We gratefully acknowledge the funding from the Swedish Research Council via
Grant No. 2019-06313 and from the National Research, Development, and
Innovation Office of Hungary under Project No. K131938.

##  Appendix A Derivation of the linearized ILLG equation for ferromagnets

The Hamiltonian Eq. (1) in the harmonic approximation may be written as

$\displaystyle\mathcal{H}$
$\displaystyle=E_{0}+H_{\textrm{lin}}+H_{\textrm{SW}},$ (43)

with

$\displaystyle E_{0}$ $\displaystyle=\frac{1}{2}\sum_{i\neq
j}J_{ij}^{zz}+\sum_{i}K_{i}^{zz}-\sum_{i}B_{i}^{z}M_{i}\,$ (44)

the ground-state energy,

$\displaystyle H_{\textrm{lin}}=$ $\displaystyle\frac{1}{2}\sum_{i\neq
j}\left(J_{ij}^{xz}\beta_{2i}+J_{ij}^{zx}\beta_{2j}-J_{ij}^{yz}\beta_{1i}-J_{ij}^{zy}\beta_{1j}\right)+\sum_{i}\left(2K_{i}^{xz}\beta_{2i}-2K_{i}^{yz}\beta_{1i}\right)-\sum_{i}M_{i}\left(B_{i}^{x}\beta_{2i}-B_{i}^{y}\beta_{1i}\right)\,$
(45)

the term linear in the expansion variables $\beta_{1i}$ and $\beta_{2i}$, and

$\displaystyle H_{\textrm{SW}}=$ $\displaystyle\frac{1}{2}\sum_{i\neq
j}\left[-J_{ij}^{zz}\left(\dfrac{{\beta}_{1i}^{2}}{2}+\dfrac{{\beta}_{2i}^{2}}{2}+\dfrac{{\beta}_{1j}^{2}}{2}+\dfrac{{\beta}_{2j}^{2}}{2}\right)+J_{ij}^{xx}\beta_{2i}\beta_{2j}+J_{ij}^{yy}\beta_{1i}\beta_{1j}-J_{ij}^{xy}\beta_{2i}\beta_{1j}-J_{ij}^{yx}\beta_{1i}\beta_{2j}\right]$
$\displaystyle+\sum_{i}\left[K_{i}^{xx}\beta_{2i}^{2}+K_{i}^{yy}\beta_{1i}^{2}-K_{i}^{zz}\left(\beta_{2i}^{2}+\beta_{1i}^{2}\right)\right]-\sum_{i}2K_{i}^{xy}\beta_{1i}\beta_{2i}+\sum_{i}B_{i}^{z}\left(\dfrac{{\beta}_{1i}^{2}}{2}+\dfrac{{\beta}_{2i}^{2}}{2}\right)M_{i}\,.$
(46)

the spin-wave Hamiltonian. The term $H_{\textrm{lin}}$ must vanish if the
ferromagnetic state is equilibrium, leading to the conditions

$\displaystyle\sum_{j}J_{ij}^{xz}+2K_{ij}^{xz}-B_{i}^{x}M_{i}=0,$ (47)
$\displaystyle\sum_{j}J_{ij}^{yz}+2K_{ij}^{yz}-B_{i}^{y}M_{i}=0,$ (48)

when using the identity $J_{ij}^{\alpha\beta}=J_{ji}^{\beta\alpha}$ following
from the definition of the Hamiltonian including a double summation over
lattice sites $i$ and $j$.

The effective field in Eq. (2) is computed as

$\displaystyle\bm{B}^{\rm eff}_{i}$
$\displaystyle=-\frac{1}{M_{i}}\frac{\partial\mathcal{H}}{\partial\boldsymbol{S}_{i}}=-\frac{1}{M_{i}}\frac{\partial\mathcal{H}}{\partial{\beta}_{2i}}\hat{\bm{x}}+\frac{1}{M_{i}}\frac{\partial\mathcal{H}}{\partial{\beta}_{1i}}\hat{\bm{y}}\,.$
(49)

Using this expression, the linearized ILLG equation may be written as

$\displaystyle\frac{\mathrm{d}\beta_{2i}}{\mathrm{d}t}=$
$\displaystyle\frac{\gamma_{i}}{M_{i}}\sum_{j}\left(-J_{ij}^{zz}\beta_{1i}+J_{ij}^{yy}\beta_{1j}-J_{ij}^{yx}\beta_{2j}\right)$
$\displaystyle+\frac{\gamma_{i}}{M_{i}}\left(-2K_{i}^{zz}\beta_{1i}+2K_{i}^{yy}\beta_{1i}-2K_{i}^{xy}\beta_{2i}\right)+\gamma_{i}B_{i}^{z}\beta_{1i}+\alpha_{i}\frac{\mathrm{d}\beta_{1i}}{\mathrm{d}t}+\eta_{i}\frac{\mathrm{d}^{2}\beta_{1i}}{\mathrm{d}t^{2}},$
(50) $\displaystyle\frac{\mathrm{d}\beta_{1i}}{\mathrm{d}t}=$
$\displaystyle-\frac{\gamma_{i}}{M_{i}}\sum_{j}\left(-J_{ij}^{zz}\beta_{2i}+J_{ij}^{xx}\beta_{2j}-J_{ij}^{xy}\beta_{1j}\right)$
$\displaystyle-\frac{\gamma_{i}}{M_{i}}\left(-2K_{i}^{zz}\beta_{2i}+2K_{i}^{xx}\beta_{2i}-2K_{i}^{xy}\beta_{1i}\right)-\gamma_{i}B_{i}^{z}\beta_{2i}-\alpha_{i}\frac{\mathrm{d}\beta_{2i}}{\mathrm{d}t}-\eta_{i}\frac{\mathrm{d}^{2}\beta_{2i}}{\mathrm{d}t^{2}}.$
(51)

For translationally invariant systems, Eqs. (A) and (51) may be rewritten in a
block-diagonal form in $\boldsymbol{k}$ after Fourier transformation,

$\displaystyle\frac{\mathrm{d}\tilde{\beta}_{2}\left(\boldsymbol{k}\right)}{\mathrm{d}t}=$
$\displaystyle\frac{\gamma}{M}\left(-J_{\boldsymbol{k}=\boldsymbol{0}}^{zz}+J_{\boldsymbol{k}}^{yy}-2K^{zz}+2K^{yy}+MB^{z}\right)\tilde{\beta}_{1}\left(\boldsymbol{k}\right)+\frac{\gamma}{M}\left(-J_{\boldsymbol{k}}^{yx}-2K^{xy}\right)\tilde{\beta}_{2}\left(\boldsymbol{k}\right)+\alpha\frac{\mathrm{d}\tilde{\beta}_{1}\left(\boldsymbol{k}\right)}{\mathrm{d}t}+\eta\frac{\mathrm{d}^{2}\tilde{\beta}_{1}\left(\boldsymbol{k}\right)}{\mathrm{d}t^{2}},$
(52)
$\displaystyle\frac{\mathrm{d}\tilde{\beta}_{1}\left(\boldsymbol{k}\right)}{\mathrm{d}t}=$
$\displaystyle-\frac{\gamma}{M}\left(-J_{\boldsymbol{k}}^{xy}-2K^{xy}\right)\tilde{\beta}_{1}\left(\boldsymbol{k}\right)-\frac{\gamma}{M}\left(-J_{\boldsymbol{k}=\boldsymbol{0}}^{zz}+J_{\boldsymbol{k}}^{xx}-2K^{zz}+2K^{xx}+MB^{z}\right)\tilde{\beta}_{2}\left(\boldsymbol{k}\right)-\alpha\frac{\mathrm{d}\tilde{\beta}_{2}\left(\boldsymbol{k}\right)}{\mathrm{d}t}-\eta\frac{\mathrm{d}^{2}\tilde{\beta}_{2}\left(\boldsymbol{k}\right)}{\mathrm{d}t^{2}},$
(53)

with the definitions

$\displaystyle\tilde{\beta}_{1(2)}\left(\boldsymbol{k}\right)$
$\displaystyle=\frac{1}{\sqrt{N}}\sum_{\bm{R}_{i}}e^{-{\rm
i}\bm{k}\cdot\bm{R}_{i}}\beta_{1i(2i)},$ (54)
$\displaystyle\boldsymbol{J}_{\boldsymbol{k}}$
$\displaystyle=\sum_{\bm{R}_{i}-\bm{R}_{j}}e^{-{\rm
i}\bm{k}\cdot\left(\bm{R}_{i}-\bm{R}_{j}\right)}\boldsymbol{J}_{ij}.$ (55)

Switching to the circularly polarized basis and performing Fourier
transformation in time results in Eq. (4) in the main text, with the spin-wave
Hamiltonian $H_{\textrm{SW}}\left(\bm{k}\right)$ given by

$\displaystyle H_{\textrm{SW}}\left(\bm{k}\right)=$
$\displaystyle\begin{pmatrix}\Omega_{+}\left(\bm{k}\right)&W_{+}\left(\bm{k}\right)\\\
W_{-}\left(\bm{k}\right)&\Omega_{-}\left(\bm{k}\right)\end{pmatrix},$ (56)
$\displaystyle\Omega_{\pm}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma}{2M}\left(-2J_{\boldsymbol{k}=\boldsymbol{0}}^{zz}+J_{\boldsymbol{k}}^{xx}+J_{\boldsymbol{k}}^{yy}\mp\textrm{i}J_{\boldsymbol{k}}^{yx}\pm\textrm{i}J_{\boldsymbol{k}}^{xy}-4K^{zz}+2K^{xx}+2K^{yy}+2MB^{z}\right),$
(57) $\displaystyle W_{\pm}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma}{2M}\left(J_{\boldsymbol{k}}^{xx}-J_{\boldsymbol{k}}^{yy}\mp\textrm{i}J_{\boldsymbol{k}}^{yx}\mp\textrm{i}J_{\boldsymbol{k}}^{xy}+2K^{xx}-2K^{yy}\mp\textrm{i}4K^{xy}\right).$
(58)

The relations
$\Omega_{\pm}^{*}\left(\boldsymbol{k}\right)=\Omega_{\mp}\left(-\boldsymbol{k}\right)$
and
$W_{\pm}^{*}\left(\boldsymbol{k}\right)=W_{\mp}\left(-\boldsymbol{k}\right)$
are satisfied by definition, and they enforce the particle-hole symmetry
mentioned in the main text.

To calculate the dependence of the gyromagnetic ratio on the wave vector in
biaxial ferromagnets, we keep the off-diagonal terms
$W_{\pm}\left(\bm{k}\right)$ of $H_{\textrm{SW}}$ in Eq. (57). In the absence
of an inertial term, this yields the dispersion relation

$\displaystyle\omega^{(0)}_{\rm
Prec}(\bm{k})=\frac{1}{1+\alpha^{2}}\left(\frac{\Omega_{-}-\Omega_{+}+\textrm{i}\alpha\left(\Omega_{-}+\Omega_{+}\right)}{2}+\sqrt{\left(1-\alpha^{2}\right)\left(\frac{\Omega_{-}+\Omega_{+}}{2}\right)^{2}+\textrm{i}\alpha\frac{\Omega^{2}_{-}-\Omega^{2}_{+}}{2}-W_{+}W_{-}}\right),$
(59)

the gyromagnetic ratio

$\displaystyle\gamma^{(0)}_{\rm
eff}(\bm{k})=\frac{\gamma}{\sqrt{1-\frac{4W_{+}(\bm{k})W_{-}(\bm{k})}{\left(\Omega_{-}(\bm{k})+\Omega_{+}(\bm{k})\right)^{2}}}}$
(60)

for $\alpha=0$, and the effective damping parameter

$\displaystyle\alpha^{(0)}_{\rm
eff}(\bm{k})=\frac{\alpha}{\sqrt{1-\frac{4W_{+}(\bm{k})W_{-}(\bm{k})}{\left(\Omega_{-}(\bm{k})+\Omega_{+}(\bm{k})\right)^{2}}}}.$
(61)

While Eq. (60) predicts a decrease of the gyromagnetic ratio with
$\boldsymbol{k}$ similarly to Eq. (16), quantitatively the formulae depend
differently on the wave vector, as discussed in the main text. The same
conclusion applies to the effective damping, since Eq. (61) shows the same
dependence on the wave vector as Eq. (60), similarly to Eqs. (17) and (16) in
the inertial case.

##  Appendix B Derivation of the linearized ILLG equation for antiferromagnets

Using the definition of the spins on the two sublattices in Eqs. (18) and
(19), expansion of the Hamiltonian for an antiferromagnet has the following
form:

$\displaystyle E_{0}=$ $\displaystyle\sum_{i\in A,j\in
B}-J_{ij}^{zz}+\sum_{i\in A}K_{i}^{zz}+\sum_{j\in B}K_{j}^{zz}-\sum_{i\in
A}B_{i}^{z}M_{i,A}-\sum_{j\in B}-B_{j}^{z}M_{j,B}\,,$ (62) $\displaystyle
H_{\textrm{lin}}=$ $\displaystyle\sum_{i\in A,j\in
B}\left(-J_{ij}^{xz}\beta_{2i}+J_{ij}^{zx}\beta_{2j}+J_{ij}^{yz}\beta_{1i}+J_{ij}^{zy}\beta_{1j}\right)+\sum_{i\in
A}\left(2K_{i}^{xz}\beta_{2i}-2K_{i}^{yz}\beta_{1i}\right)+\sum_{j\in
B}\left(-2K_{j}^{xz}\beta_{2j}-2K_{j}^{yz}\beta_{1j}\right)$
$\displaystyle-\sum_{i\in
A}M_{i,A}\left(B_{i}^{x}\beta_{2i}-B_{i}^{y}\beta_{1i}\right)-\sum_{j\in
B}M_{j,B}\left(B_{i}^{x}\beta_{2i}+B_{i}^{y}\beta_{1i}\right)\,,$ (63)
$\displaystyle H_{\textrm{SW}}=$ $\displaystyle\sum_{i\in A,j\in
B}\left[J_{ij}^{zz}\left(\dfrac{{\beta}_{1i}^{2}}{2}+\dfrac{{\beta}_{2i}^{2}}{2}+\dfrac{{\beta}_{1j}^{2}}{2}+\dfrac{{\beta}_{2j}^{2}}{2}\right)+J_{ij}^{xx}\beta_{2i}\beta_{2j}-J_{ij}^{yy}\beta_{1i}\beta_{1j}+J_{ij}^{xy}\beta_{2i}\beta_{1j}-J_{ij}^{yx}\beta_{1i}\beta_{2j}\right]$
$\displaystyle+\sum_{i\in
A}\left(K_{i}^{xx}\beta_{2i}^{2}+K_{i}^{yy}\beta_{1i}^{2}-K_{i}^{zz}\left(\beta_{2i}^{2}+\beta_{1i}^{2}\right)-2K_{i}^{xy}\beta_{1i}\beta_{2i}\right)$
$\displaystyle+\sum_{j\in
B}\left(K_{j}^{xx}\beta_{2j}^{2}+K_{j}^{yy}\beta_{1j}^{2}-K_{j}^{zz}\left(\beta_{2j}^{2}+\beta_{1j}^{2}\right)+2K_{j}^{xy}\beta_{1j}\beta_{2j}\right)$
$\displaystyle+\sum_{i\in
A}B_{i}^{z}\left(\dfrac{{\beta}_{1i}^{2}}{2}+\dfrac{{\beta}_{2i}^{2}}{2}\right)M_{i,A}+\sum_{j\in
B}-B_{j}^{z}\left(\dfrac{{\beta}_{1j}^{2}}{2}+\dfrac{{\beta}_{2j}^{2}}{2}\right)M_{j,B}$
(64)

The linearized equations of motion are calculated similarly to the
ferromagnetic case. The effective fields acting on each sublattice
$\bm{B}^{\rm eff,A}_{i}$ and $\bm{B}^{\rm eff,B}_{j}$ are determined and
inserted into the ILLG equation. Translational invariance is assumed with
$\gamma_{i,A/B}=\gamma_{A/B}$, $\alpha_{i,A/B}=\alpha_{A/B}$,
$\eta_{i,A/B}=\eta_{A/B}$, $M_{i,A/B}=M_{A/B}$,
$\boldsymbol{K}_{i/j}=\boldsymbol{K}_{A/B}$ and
$\boldsymbol{J}_{ij}=\boldsymbol{J}\left(\boldsymbol{R}_{i}-\boldsymbol{R}_{j}\right)$,
enabling diagonalization in lattice sites via Fourier transformation. In the
circularly polarized basis and after Fourier transformation in time, the
equations of motion are formally identical to Eq. (4), as mentioned in the
main text. The spin-wave Hamiltonian reads

$\displaystyle H_{\textrm{SW}}\left(\bm{k}\right)=$
$\displaystyle\begin{pmatrix}\Omega_{+AA}\left(\boldsymbol{k}\right)&W_{+AA}\left(\bm{k}\right)&\Omega_{+AB}\left(\boldsymbol{k}\right)&W_{+AB}\left(\bm{k}\right)\\\
W_{-AA}\left(\bm{k}\right)&\Omega_{-AA}\left(\boldsymbol{k}\right)&W_{-AB}\left(\bm{k}\right)&\Omega_{-AB}\left(\boldsymbol{k}\right)\\\
\Omega_{+BA}\left(\boldsymbol{k}\right)&W_{+BA}\left(\bm{k}\right)&\Omega_{+BB}\left(\boldsymbol{k}\right)&W_{+BB}\left(\bm{k}\right)\\\
W_{-BA}\left(\bm{k}\right)&\Omega_{-BA}\left(\boldsymbol{k}\right)&W_{-BB}\left(\bm{k}\right)&\Omega_{-BB}\left(\boldsymbol{k}\right)\end{pmatrix},$
(65)

containing the coefficients

$\displaystyle\Omega_{\pm AA}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{A}}{2M_{A}}\left(J_{\boldsymbol{k}AA}^{xx}+J_{\boldsymbol{k}AA}^{yy}\mp\textrm{i}J_{\boldsymbol{k}AA}^{yx}\pm\textrm{i}J_{\boldsymbol{k}AA}^{xy}-2J_{\boldsymbol{k}=\boldsymbol{0}AA}^{zz}+2J_{\boldsymbol{k}=\boldsymbol{0}AB}^{zz}-4K_{A}^{zz}+2K_{A}^{xx}+2K_{A}^{yy}+2M_{A}B^{z}\right),$
(66) $\displaystyle W_{\pm AA}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{A}}{2M_{A}}\left(J_{\boldsymbol{k}AA}^{xx}-J_{\boldsymbol{k}AA}^{yy}\mp\textrm{i}J_{\boldsymbol{k}AA}^{yx}\mp\textrm{i}J_{\boldsymbol{k}AA}^{xy}+2K_{A}^{xx}-2K_{A}^{yy}\mp\textrm{i}4K_{A}^{xy}\right),$
(67) $\displaystyle\Omega_{\pm BB}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{B}}{2M_{B}}\left(J_{\boldsymbol{k}BB}^{xx}+J_{\boldsymbol{k}BB}^{yy}\pm\textrm{i}J_{\boldsymbol{k}BB}^{yx}\mp\textrm{i}J_{\boldsymbol{k}BB}^{xy}-2J_{\boldsymbol{k}=\boldsymbol{0}BB}^{zz}+2J_{\boldsymbol{k}=\boldsymbol{0}BA}^{zz}-4K_{B}^{zz}+2K_{B}^{xx}+2K_{B}^{yy}-2M_{B}B^{z}\right),$
(68) $\displaystyle W_{\pm BB}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{B}}{2M_{B}}\left(J_{\boldsymbol{k}BB}^{xx}-J_{\boldsymbol{k}BB}^{yy}\pm\textrm{i}J_{\boldsymbol{k}BB}^{yx}\pm\textrm{i}J_{\boldsymbol{k}BB}^{xy}+2K_{B}^{xx}-2K_{B}^{yy}\pm\textrm{i}4K_{B}^{xy}\right),$
(69) $\displaystyle\Omega_{\pm AB}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{A}}{2M_{A}}\left(-J_{\boldsymbol{k}AB}^{yy}+J_{\boldsymbol{k}AB}^{xx}\mp\textrm{i}J_{\boldsymbol{k}AB}^{yx}\mp\textrm{i}J_{\boldsymbol{k}AB}^{xy}\right),$
(70) $\displaystyle W_{\pm AB}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{A}}{2M_{A}}\left(J_{\boldsymbol{k}AB}^{yy}+J_{\boldsymbol{k}AB}^{xx}\mp\textrm{i}J_{\boldsymbol{k}AB}^{yx}\pm\textrm{i}J_{\boldsymbol{k}AB}^{xy}\right),$
(71) $\displaystyle\Omega_{\pm BA}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{B}}{2M_{B}}\left(-J_{\boldsymbol{k}BA}^{yy}+J_{\boldsymbol{k}BA}^{xx}\pm\textrm{i}J_{\boldsymbol{k}BA}^{yx}\pm\textrm{i}J_{\boldsymbol{k}BA}^{xy}\right),$
(72) $\displaystyle W_{\pm BA}\left(\bm{k}\right)=$
$\displaystyle\frac{\gamma_{B}}{2M_{B}}\left(J_{\boldsymbol{k}BA}^{yy}+J_{\boldsymbol{k}BA}^{xx}\pm\textrm{i}J_{\boldsymbol{k}BA}^{yx}\mp\textrm{i}J_{\boldsymbol{k}BA}^{xy}\right).$
(73)

Particle-hole symmetry is enforced by the coefficients satisfying
$\Omega_{\pm}^{*}\left(\boldsymbol{k}\right)=\Omega_{\mp}\left(-\boldsymbol{k}\right)$
and
$W_{\pm}^{*}\left(\boldsymbol{k}\right)=W_{\mp}\left(-\boldsymbol{k}\right)$
for all sublattice indices.

Assuming that the sublattices are identical and the external field is set to
zero, the spin-wave Hamiltonian simplifies to

$\displaystyle H_{\textrm{SW}}\left(\bm{k}\right)=$
$\displaystyle\begin{pmatrix}\Omega\left(\boldsymbol{k}\right)&W\left(\bm{k}\right)&\Omega^{{}^{\prime}}\left(\boldsymbol{k}\right)&W^{{}^{\prime}}\left(\bm{k}\right)\\\
W^{*}\left(-\bm{k}\right)&\Omega\left(-\boldsymbol{k}\right)&W^{{}^{\prime}*}\left(-\bm{k}\right)&\Omega^{{}^{\prime}*}\left(-\boldsymbol{k}\right)\\\
\Omega^{{}^{\prime}*}\left(\boldsymbol{k}\right)&W^{{}^{\prime}}\left(-\bm{k}\right)&\Omega\left(-\boldsymbol{k}\right)&W^{*}\left(-\bm{k}\right)\\\
W^{{}^{\prime}*}\left(\bm{k}\right)&\Omega^{{}^{\prime}}\left(-\boldsymbol{k}\right)&W\left(\bm{k}\right)&\Omega\left(\boldsymbol{k}\right)\end{pmatrix},$
(74)

where
$\Omega\left(\boldsymbol{k}\right)=\Omega^{*}\left(\boldsymbol{k}\right)$ and
$W\left(\bm{k}\right)=W\left(-\bm{k}\right)$. Time-reversal symmetry with
$\mathcal{T}=\tau^{x}\mathcal{K}$ also requires
$W^{{}^{\prime}}\left(\bm{k}\right)=W^{{}^{\prime}*}\left(\bm{k}\right)$ and
$\Omega^{{}^{\prime}}\left(\bm{k}\right)=\Omega^{{}^{\prime}}\left(-\bm{k}\right)$.
These conditions are satisfied if the two sublattices together form a Bravais
lattice, e.g. the centered rectangular lattice consisting of two primitive
rectangular sublattices on the bcc(110) surface discussed in the main text.
Such effective time-reversal symmetries in antiferromagnetic systems are
always protected by a crystal symmetry, since the physical time reversal
inverts the spin direction on both sublattices, and the crystal symmetry is
required for exchanging the two sublattices in order to get back to the
initial configuration.

For the monolayer on the bcc(110) surface discussed in the main text, any two
lattice sites can be exchanged by a $180^{\circ}$ rotation around the out-of-
plane $y$ axis, which implies $J_{ij}^{xy}=-J_{ij}^{yx}$. The mirror symmetry
on the $yz$ plane leads to $K^{xy}=0$. Due to these conditions,
$H_{\textrm{SW}}\left(\bm{k}\right)$ is a real matrix (in particular,
$\Omega^{{}^{\prime}}\left(\boldsymbol{k}\right)=\Omega^{{}^{\prime}*}\left(\boldsymbol{k}\right)$
and $W\left(\bm{k}\right)=W^{*}\left(\bm{k}\right)$ since the other two
coefficients are real anyways as mentioned above), and the equation of motion
satisfies reciprocal symmetry with $\mathcal{R}=\tau^{x}$, as discussed in the
main text.

To calculate the effective gyromagnetic ratios and damping parameters, we
treat the damping $\alpha$ and the magnetic field $B^{z}$ perturbatively in
Eq. (23). For identical sublattices this yields

$\displaystyle\eta^{2}\omega^{4}\left(\boldsymbol{k}\right)-\left(1+2b\right)\omega^{2}\left(\boldsymbol{k}\right)+\Omega^{2}-W_{\pm}^{2}\left(\boldsymbol{k}\right)+\textrm{i}2\alpha\left(\Omega\omega^{\prime}\left(\boldsymbol{k}\right)-\eta\left(\omega^{\prime}\left(\boldsymbol{k}\right)\right)^{3}\right)\mp
2\gamma\omega^{\prime}\left(\boldsymbol{k}\right)B^{z}=0,$ (75)

where $\omega^{\prime}\left(\boldsymbol{k}\right)$ is the eigenfrequency
without damping or magnetic field. Following the procedure in Ref. [29], the
eigenfrequencies are extended up to linear order in $B^{z}$ and $\alpha$,
leading to Eqs. (40)-(42) in the main text.

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By construction $U^{\top}\eta\,{\hskip 0.85358pt}U=\eta$ so that total
probabilities are conserved in the sense that
$(\Phi,\Phi)_{\Omega,\Sigma_{t}}$ is constant, in concordance with Lemma 6.8.
It is then not difficult to see that

$\left.\frac{d\langle
X\rangle_{\Psi,t}}{dt}\right|_{t=T}=\langle{\mathcal{L}}_{P}X\rangle_{\Psi,T}\,,$

as predicted by Corollary 6.9 (note that the supervector $P$ acts on the
components $(x,x_{a},\chi)$ of $X$ according to
$(x^{\prime},(x^{\prime}+Mx)_{a},\chi^{\prime})$).

Finally we can consider the evolution of a set of probabilities. For further
simplicity we now take the case $\eta=\delta$ (we could always change basis to
achieve this). To discuss standard probabilities, we need to introduce some
choice of a polyhedral convex cone containing the convex cone of states (see
[66] for a related construction). For that recall the indicator functions of
Example 5.10

$\displaystyle X_{1}$ $\displaystyle=\tfrac{1}{4}+\tfrac{\sqrt{8}{\hskip
0.85358pt}\theta^{1}+i\theta^{1}\theta^{2}}{12}\,,\qquad\qquad\quad\>X_{2}=\tfrac{1}{4}+\tfrac{-\sqrt{2}\theta^{1}+\sqrt{6}\theta^{2}+i\theta^{1}\theta^{2}}{12}\,,$
$\displaystyle X_{3}$
$\displaystyle=\tfrac{1}{4}+\tfrac{-\sqrt{2}\theta^{1}-\sqrt{6}\theta^{2}+i\theta^{1}\theta^{2}}{12}\,,\qquad\;\;X_{4}=\tfrac{1}{4}+\tfrac{-3i\theta_{1}\theta_{2}}{12}\,.$

We next write a state function in terms of probabilities corresponding to
expectations of these indicators. Note that, unlike states, state functions
can have vanishing body. From Equation (7.3) we see that any state function
with vanishing body $\phi=0$ here has indicator expectations

$\langle X_{i}\rangle=\frac{1}{4}\,,\quad i=1,\ldots,4\,.$

Otherwise, for $\phi\neq 0$, we may first normalize $\Phi$ along a fixed $t=T$
laboratory such that its star-square has unit body

(7.5)
$(\Phi,\Phi)_{T}:=\big{(}\Phi{\filledstar_{\scriptscriptstyle\nabla}}\Phi\big{)}_{0}=\phi^{2}+\phi_{a}\delta^{ab}\phi_{b}+\varphi^{2}=1\,,$

which simplifies Equation (7.3). Calling $p_{i}:=\langle X_{i}\rangle\in[0,1]$
we then have

$\displaystyle p_{1}$
$\displaystyle=\tfrac{1}{4}+\phi\left(\tfrac{\sqrt{8}{\hskip
0.85358pt}\phi_{1}-\varphi}{6}\right)\,,\qquad\qquad\quad\>p_{2}=\tfrac{1}{4}+\phi\left(\tfrac{-\sqrt{2}\phi_{1}+\sqrt{6}\phi_{2}-\varphi}{6}\right)\,,$
$\displaystyle p_{3}$
$\displaystyle=\tfrac{1}{4}+\phi\left(\tfrac{-\sqrt{2}\phi_{1}-\sqrt{6}\phi_{2}-\varphi}{6}\right)\,,\qquad\;\;p_{4}=\tfrac{1}{4}+\tfrac{\phi\varphi}{2}\,.$

Since $\phi\neq 0$, we may solve $(\phi_{1},\phi_{2},\varphi)$ in terms of
$(\phi,p_{1},p_{2},p_{3})$

$\phi_{1}=\frac{2p_{1}-p_{2}-p_{3}}{\sqrt{2}\phi},\quad\phi_{2}=\frac{\sqrt{3}(p_{2}-p_{3})}{\sqrt{2}\phi},\quad\varphi=\frac{3-4(p_{1}+p_{2}+p_{3})}{2\phi}\,.$

Finally we can find $\phi$ by plugging the previous list of expressions for
$\phi_{a},\varphi$ in (7.5). While the resulting quartic equation itself has
four solutions, we choose the root

$\phi=\frac{\sqrt{1+2\sqrt{-\Delta_{\rm ellipsoid}}}}{\sqrt{2}}\,,$

where $\frac{\Delta_{\rm
ellipsoid}}{6}:=(p_{1}+p_{2}+p_{3})^{2}-(p_{1}+p_{2}+p_{3}+p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})+\frac{1}{3}$.
This makes sense only when $\Delta_{\rm ellipsoid}\leq 0$, implying that the
supersymplectic probabilities are constrained to lie inside the ellipsoid of
Equation (5.21)in $\mathbb{R}_{\geq 0}^{3}$.

It remains to study the evolution of probabilities. Returning to Equation
(7.4) and calling ${\mathfrak{c}}(t):=\cos\big{(}\frac{(t-T)H}{2}\big{)}$,
${\mathfrak{s}}(t):=\sin\big{(}\frac{(t-T)H}{2}\big{)}$, we obtain

$\phi_{1}(t)={\hskip 0.85358pt}{\hskip 0.85358pt}{\hskip
0.85358pt}\>\>{\mathfrak{c}}(t)\phi_{1}(T)+{\hskip
0.85358pt}{\mathfrak{s}}(t)\phi_{2}(T)\,,$
$\phi_{2}(t)=-{\mathfrak{s}}(t)\phi_{1}(T)+{\mathfrak{c}}(t)\phi_{2}(T)\,.$

Thus

$\displaystyle\begin{pmatrix}p_{1}(t)\\\\[4.2679pt] p_{2}(t)\\\\[4.2679pt]
p_{3}(t)\\\\[4.2679pt]
p_{4}(t)\end{pmatrix}=\begin{pmatrix}\frac{1+2{\mathfrak{c}}(t)}{3}&\frac{1-{\mathfrak{c}}(t)+\sqrt{3}{\mathfrak{s}}(t)}{3}&\frac{1-{\mathfrak{c}}(t)-\sqrt{3}{\mathfrak{s}}(t)}{3}&0\\\
\frac{1-{\mathfrak{c}}(t)-\sqrt{3}{\mathfrak{s}}(t)}{3}&\frac{1+2{\mathfrak{c}}(t)}{3}&\frac{1-{\mathfrak{c}}(t)+\sqrt{3}{\mathfrak{s}}(t)}{3}&0\\\
\frac{1-{\mathfrak{c}}(t)+\sqrt{3}{\mathfrak{s}}(t)}{3}&\frac{1-{\mathfrak{c}}(t)-\sqrt{3}{\mathfrak{s}}(t)}{3}&\frac{1+2{\mathfrak{c}}(t)}{3}&0\\\\[2.84526pt]
0&0&0&1\end{pmatrix}\begin{pmatrix}p_{1}(T)\\\\[4.2679pt]
p_{2}(T)\\\\[4.2679pt] p_{3}(T)\\\\[4.2679pt] p_{4}(T)\end{pmatrix}\,.$

$\Delta_{\rm cone}\leq 0$$\Delta_{\rm ellipsoid}\leq 0$$p_{1},p_{2},p_{3}\geq
0,\>\>p_{1}+p_{2}+p_{3}\leq 1$$p_{1}$$p_{2}$$p_{3}$ Figure 1. Dynamics
corresponds to a path inside the orange ellipsoid

For probabilities to evolve to probabilities, examining the above transition
matrix we see that the initial probabilities must obey the conical constraint

$\Delta_{\rm
cone}:=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}-2(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})\leq
0\,.$

This is automatically satisfied for supersymplectic probability distributions,
videlicet those that satisfy $\Delta_{\rm ellipsoid}\leq 0$ or lie inside the
ellipsoid of Equation (5.21), as depicted in Figure 1. While the above
transition matrix is not Markov (its entries in any row or column sum to unity
without being individually strictly non-negative) its eigenvalues do have
magnitudes bounded by unity. It thus defines a Markov-like stochastic process.

## Acknowledgements

S.C. acknowledges a U.C. Davis Dean’s Summer Fellowship. A.W. acknowledges
support from the Royal Society of New Zealand via Marsden Grant 19-UOA-008.
A.W. was also supported by the Simons Foundation Collaboration Grant for
Mathematicians ID 686131. C.Y. acknowledges support of the Scientific and
Technological Research Council of Turkey (TUBITAK) 2214-A Grant No.
1059B141900615.

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