# Dynamical Switching of Magnetic Topology in Microwave-Driven Itinerant
Magnet

Rintaro Eto  Department of Applied Physics, Waseda University, Okubo,
Shinjuku-ku, Tokyo 169-8555, Japan  Masahito Mochizuki  Department of Applied
Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

###### Abstract

We theoretically demonstrate microwave-induced dynamical switching of magnetic
topology in centrosymmetric itinerant magnets by taking the Kondo-lattice
model on a triangular lattice, which is known to exhibit two types of skyrmion
lattices with different magnetic topological charges of $|N_{\rm sk}|$=1 and
$|N_{\rm sk}|$=2. Our numerical simulations reveal that intense excitation of
a resonance mode with circularly polarized microwave field can switch the
magnetic topology, i.e., from the skyrmion lattice with $|N_{\rm sk}|$=1 to
another skyrmion lattice with $|N_{\rm sk}|$=2 or to a nontopological magnetic
order with $|N_{\rm sk}|$=0 depending on the microwave frequency. This
magnetic-topology switching shows various distinct behaviors, that is,
deterministic irreversible switching, probabilistic irreversible switching,
and temporally random fluctuation depending on the microwave frequency and the
strength of external magnetic field, variety of which is attributable to
different energy landscapes in the dynamical regime. The obtained results are
also discussed in the light of time-evolution equations based on an effective
model derived using perturbation expansions.

##  I Introduction

Figure 1: Schematics of a skyrmion lattice in the itinerant magnet irradiated
with circularly polarized microwave field. The thin solid arrows represent
local magnetizations, whereas the light blue spheres with thick solid arrows
represent itinerant electrons with spins.

Topological magnetisms exemplified by several types of skyrmions Muhlbauer2009
; Yu2010 ; Heinze2011 ; Kezsmarki2015 ; Yu2014 ; Nayak2017 , merons Yu2018 ,
hedgehogs Milde2013 ; Schutte2014 ; Kanazawa2012 ; Fujishiro2019 ;
Ishiwata2020 , and hopfions LiuY2018  are currently attracting enormous
research interest from the viewpoints of both fundamental sciences and
potential applications Seki2016 ; Nagaosa2013 ; Gobel2021 ; Fert2013 ;
Finocchio2016 ; Fert2017 ; Everschor2019 ; Tokura2021 . These nontrivial
magnetic textures with spatially modulated magnetization are often caused by
the Dzyaloshinskii-Moriya interactions Dzyaloshinsky1958 ; Moriya1960a ;
Moriya1960b , which have a relativistic origin and are active when the system
has no spatial inversion symmetry. Therefore, the above topological magnetic
textures are usually hosted in magnets having non-centrosymmetric crystal
structures or magnetic heterostructures having interfaces. In such magnets,
structural symmetries and anisotropies determine the way of magnetization
alignment in magnetic textures through governing the spatial configuration of
Dzyaloshinskii-Moriya vectors Bogdanov1989 ; Bogdanov1994 ; Tanaka2020 .
Consequently, the helicity, chirality, and vorticity of the magnetic textures
are inherently fixed and thus are not variable Nagaosa2013 .

Exchange coupling between itinerant electrons and local magnetizations is
another source of topological magnetisms Hayami2021R ; Hayami2014a ; Ozawa2016
; Ozawa2017a ; Hayami2017 ; Gobel2017 ; Wang2020 ; Hayami2020a ; Hayami2021a ;
Hayami2021b ; Nakazawa2019 ; Hayami2020b ; Hayami2018 ; Okada2018 ;
Okumura2020 ; Shimizu2021a . Recent theoretical studies have predicted
possible emergence of topological magnetisms such as skyrmion lattices
Ozawa2016 ; Ozawa2017a ; Hayami2017 ; Gobel2017 ; Wang2020 ; Hayami2020a ;
Hayami2021a ; Hayami2021b , meron lattices Hayami2021R ; Hayami2021b , and
hedgehog lattices Hayami2021R ; Okumura2020 ; Shimizu2021a  in the Kondo-
lattice model and its effective model which describe localized magnetizations
on a lattice coupled to itinerant electron spins via (anti)ferromagnetic
exchange interactions. In such magnets, spatial modulation of magnetization is
caused by effective long-range interactions among local magnetizations
mediated by conduction electrons [e.g., the Ruderman-Kittel-Kasuya-Yosida
(RKKY) interactions Ruderman1954 ; Kasuya1956 ; Yosida1957 ]. Therefore, in
the case of the Kondo-lattice magnets, even the centrosymmetric systems with
spatial inversion symmetry can host topological magnetisms, despite the
Dzyaloshinskii-Moriya interactions are absent Hayami2021R ; Ozawa2016 ;
Ozawa2017a ; Hayami2017 ; Gobel2017 ; Wang2020 ; Hayami2020a ; Hayami2021a ;
Hayami2021b ; Nakazawa2019 ; Hayami2020b . These magnetic textures are
superpositions of magnetic helices with multiple propagation vectors ($\bm{Q}$
vectors), which are determined by the effective electron-mediated interactions
among magnetizations governed by multiple nesting vectors of the Fermi
surfaces. This fact enables us to have a variety of topological magnetic
textures and their controllability via material variations or by tuning
material parameters Shimizu2021b . Recent experiments have indeed observed
three-dimensional hedgehog lattices in SrFeO3 Ishiwata2020 , triangular
skyrmion lattices in Gd2PdSi3 Kurumaji2019 ; Hirschberger2020a ;
Hirschberger2020b ; Nomoto2020  and Gd3Ru4Al12 Hirschberger2019 ;
Hirschberger2021 , and square skyrmion lattices in GdRu2Si2 Khanh2020 ;
Yasui2020 , all of which are itinerant magnets with centrosymmetric crystal
structure.

In addition to their rich variety, topological magnetic textures in
centrosymmetric metallic magnets have an interesting feature, that is, several
degrees of freedom such as helicity and vorticity remain to be unfrozen
because the Dzyaloshinskii-Moriya interaction is absent. Thereby, their low-
energy excitations and continuous variations are possible, which provide us a
unique opportunity to control and switch the magnetic topology with an
external stimulus such as static magnetic field, electric currents SZLin2013 ,
and microwave magnetic field. It has been revealed that topological magnetisms
such as skyrmions and skyrmion tubes in chiral magnets have peculiar
collective modes or spin-wave modes at microwave frequencies Mochizuki2012 ;
Schwarze2015 ; LinSZ2014 ; McKeever2019 ; LinSZ2019 ; Seki2020 , and they have
turned out to cause interesting physical phenomena and potential device
functions Mochizuki2015 ; Garst2017 ; Lonsky2020 ; Mochizuki2013 ; Okamura2013
; MaF2015 ; WangK2020 ; WangW2015 ; Ikka2018 ; Takeuchi2018 ; SongC2019 ;
YuanHY2019 ; Koide2019 ; Takeuchi2019 ; Miyake2020 ; XingX2020 ; Ser2020 . We
expect that the topological magnetisms in centrosymmetric magnets can also
host interesting dynamical phenomena and functionalities associated with their
microwave-active collective modes.

In this paper, we theoretically demonstrate that magnetic topology in a
centrosymmetric itinerant magnet can be switched dynamically by application of
circularly polarized microwave field by taking the Kondo-lattice model on a
triangular lattice as an example. This model is known to exhibit two distinct
skyrmion-lattice phases [Fig.1] with different skyrmion numbers or magnetic
topological charges $|N_{\rm sk}|$=1 and $|N_{\rm sk}|$=2 Ozawa2017a . We
numerically simulate microwave-driven magnetization dynamics in this model
using a combined technique of the micromagnetic simulation and the kernel
polynomial method. We first find that a single spin-wave mode can be activated
by an in-plane microwave field in all the three phases. We then demonstrate
that by intensely activating this spin-wave mode with circularly polarized
microwave field, the skyrmion lattice with $|N_{\rm sk}|$=1 can be switched to
that with $|N_{\rm sk}|$=2 or a nontopological magnetic state of $N_{\rm
sk}$=0 depending on the microwave frequency. During the switching processes,
we observe emergent topological magnetic patterns characterized by half-
integer skyrmion numbers of $|N_{\rm sk}|$=1/2 and $|N_{\rm sk}|$=3/2 (i.e.,
meron lattices) as transient states. We examine such dynamical transitions for
various initial magnetic configurations in equilibrium and find several
different behaviors, that is, deterministic irreversible switching,
probabilistic irreversible switching, and temporally random fluctuations under
continuous microwave irradiation. This variety of behaviors is attributable to
difference of the energy landscape in the dynamical regime. We also discuss
the obtained results on the basis of an effective model derived using
perturbation expansions Hayami2017 . Note that magnetic frustration is another
important mechanism to realize topological magnetism in centrosymmetric
magnets. The phenomena revealed in this work can also be expected in the
frustrated systems although fine tuning of the exchange interactions is
required to produce topological magnetic textures of frustration origin
Okubo2012 ; Kamiya2014 ; Leonov2015 ; LinSZ2016 ; Hayami2016a ; Hayami2016b ;
LinSZ2018 ; Lohani2019 ; WangZ2021 .

##  II Model

###  II.1 Kondo-Lattice Model

Figure 2: (a) Ground-state phase diagram of the Kondo-lattice model in Eq. (1)
on the triangular lattice as a function of static magnetic field $H_{z}$
perpendicular to the lattice plane when the microwave magnetic field is absent
(i.e., $\bm{H}(t)=0$) Ozawa2017a . The model parameters are set to be
$t_{1}=-1$, $t_{3}=0.85$, $J_{\rm K}=-0.5$, and $\mu=-3.5$. Successive two
phase transitions among three phases, i.e., the skyrmion-lattice (SkL) phase
with $|N_{\rm sk}|=2$, another SkL phase with $|N_{\rm sk}|=1$, and the
nontopological phase with $N_{\rm sk}=0$ take place at $H_{z}=0.00325$ and
$H_{z}=0.0065$. Several degrees of freedom of magnetic textures such as
helicity and vorticity are not frozen in the centrosymmetric system without
Dzyaloshinskii-Moriya interactions, which results in infinite degeneracy of
magnetic textures (see text). (b), (c) Spatial profiles of local
magnetizations (upper panels) and local scalar spin chiralities (lower panels)
of two degenerate SkLs with $|N_{\rm sk}|=2$, i.e., (b) SkL with $N_{\rm
sk}=-2$ and (c) SkL with $N_{\rm sk}=+2$. (d)-(f) Those of SkLs with $|N_{\rm
sk}|=1$, i.e., (d) Néel-type SkL with $N_{\rm sk}=-1$, (e) Bloch-type SkL with
$N_{\rm sk}=-1$, and (f) antivortex-type SkL with $N_{\rm sk}=+1$.

We consider the Kondo-lattice model on a triangular lattice. The Hamiltonian
is given by

$\displaystyle\mathcal{H}=\mathcal{H}_{\rm KL}+\mathcal{H}_{\rm Zeeman},$ (1)

with

$\displaystyle\mathcal{H}_{\rm
KL}=\sum_{ij\sigma}t_{i,j}\hat{c}^{\dagger}_{i\sigma}\hat{c}_{j\sigma}+J_{\rm
K}\sum_{i,\sigma,\sigma^{\prime}}\hat{c}^{\dagger}_{i\sigma}{\bm{\sigma}}_{\sigma\sigma^{\prime}}\hat{c}_{i\sigma^{\prime}}\cdot{\bm{S}}_{i},$
(2) $\displaystyle\mathcal{H}_{\rm Zeeman}=-\sum_{i}\left[\bm{H}_{\rm
ext}+\bm{H}(t)\right]\cdot{\bm{S}}_{i}.$ (3)

Here $\hat{c}^{\dagger}_{i\sigma}$ $(\hat{c}_{i\sigma})$ denotes the creation
(annihilation) operator of an itinerant electron with spin
$\sigma(=\uparrow,\downarrow)$ on the $i$th site, and ${\bm{S}}_{i}$ denotes
localized magnetization on the $i$th site. The first term of $\mathcal{H}_{\rm
KL}$ represents kinetic energies of itinerant electrons where the nearest-
neighbor hopping $t_{1}$ and the third-nearest-neighbor hopping $t_{3}$ are
set to be $t_{1}=-1$ and $t_{3}=0.85$, respectively. The second term of
$\mathcal{H}_{\rm KL}$ represents the exchange coupling between itinerant
electron spins and local magnetizations where the coupling constant is set to
be $J_{\rm K}=-0.5$. The term $\mathcal{H}_{\rm Zeeman}$ represents the Zeeman
couplings associated with both a static external magnetic field $\bm{H}_{\rm
ext}=(0,0,H_{z})$ and a time-dependent magnetic field $\bm{H}(t)$ acting on
the local magnetizations. For the time-dependent field $\bm{H}(t)$, we
consider a short-period pulse or circularly polarized microwave field in the
present study. Note that we neglect the coupling between the magnetic fields
and the itinerant electrons. In fact, we have examined the effects of the
coupling and have found that consideration of the coupling does not alter the
results even quantitatively. We set the chemical potential $\mu=-3.5$, which
corresponds to the electron filling of 0.2 approximately. The above parameter
values are the same as those used in the previous work Ozawa2017a .

This model is known to exhibit various magnetic orders including topological
ones as superpositions of three magnetic helices. The propagation vectors of
the three helices are $\bm{Q}_{1}=(\pi/3,0)$,
$\bm{Q}_{2}=\hat{R}(2\pi/3)\bm{Q}_{1}$, and
$\bm{Q}_{3}=\hat{R}(4\pi/3)\bm{Q}_{1}$ where $\hat{R}(\phi)$ is an operator to
rotate the vector by the angle $\phi$ around the $z$-axis. A ground-state
phase diagram was studied in Ref. Ozawa2017a  as a function of the strength of
external magnetic field $H_{z}$ when the microwave field $\bm{H}(t)$ is absent
[Fig. 2(a)]. We find that three magnetic phases, i.e., a skyrmion-lattice
phase with $|N_{\rm sk}|=2$, another skyrmion-lattice phase with $|N_{\rm
sk}|=1$ and a nontopological phase with $N_{\rm sk}=0$ successively emerge as
$H_{z}$ increases. Note that the skyrmion-lattice with $|N_{\rm sk}|=2$
emerges even at $H_{z}=0$ in striking contrast to the case of the
Dzyaloshinskii-Moriya magnets in which topological magnetisms usually appear
in the presence of an external magnetic field. More importantly, several
degrees of freedom remain unfrozen in the present system with spatial
inversion symmetry. For example, the helicity and the signs of vorticity are
not frozen for the skyrmion-lattice phase with $|N_{\rm sk}|=1$ in the
presence of the spatial inversion symmetry. Thereby, the magnetic structures
in the present system have infinite degeneracy [see Figs. 2(b)-(f)].

It should be mentioned that the present triangular-lattice system is favorable
for the emergence of the skyrmion lattices with a higher topological number of
$|N_{\rm sk}|=2$. In the centrosymmetric Kondo-lattice system, the skyrmion
lattices emerge as a superposition of three spiral or sinusoidal states of
local magnetizations, which are stabilized by the long-ranged and frustrated
RKKY interactions. The RKKY interactions originate from the coupling between
conduction electrons and local magnetizations and thus are governed by the
electronic structure of conduction electrons, e.g., the Fermi-surface geometry
and the density of states. Consequently, the modulation vectors $\bm{Q}_{\nu}$
determined by the RKKY interactions also depend on the Fermi-surface geometry.
More specifically, these modulation vectors correspond to nesting vectors of
the Fermi surface(s) and, thereby, reflect the symmetry of the lattice
structure. In addition, it has turned out that the skyrmion lattices with
$|N_{\rm sk}|=2$ requires, at least, three magnetization spiral or sinusoidal
states, while the skyrmion lattices with $|N_{\rm sk}|=1$ can be produced only
with two magnetization spiral or sinusoidal states. Hence, the triangular
lattices and the Kagome lattices with triangular or hexagonal symmetries have
more opportunity to host the skyrmion lattices with $|N_{\rm sk}|=2$ as
compared to the simple square lattices.

###  II.2 Global Symmetry in Skyrmion Lattices

Let us discuss the degeneracy of topological magnetic textures in the present
centrosymmetric system in more detail. We first consider the cases without
external magnetic field. The local magnetizations $\bm{S}_{i}$ for the
skyrmion-lattice phases with $|N_{\rm sk}|=1$ and $|N_{\rm sk}|=2$ are given
by,

$\displaystyle{\bm{S}}_{i}^{N_{\rm Sk}=\pm 1}$
$\displaystyle\propto\sum_{\nu=1}^{3}\left(\begin{array}[]{c}\sin\mathcal{Q}_{\nu}\cos\phi_{\nu}\\\
\lambda_{\rm v}\sin\mathcal{Q}_{\nu}\sin\phi_{\nu}\\\
\cos\mathcal{Q}_{\nu}\end{array}\right),$ (7)
$\displaystyle{\bm{S}}_{i}^{N_{\rm Sk}=-2}$
$\displaystyle\propto\left(\begin{array}[]{c}\cos\mathcal{Q}_{1}\\\
\cos\mathcal{Q}_{2}\\\ \cos\mathcal{Q}_{3}\end{array}\right),$ (11)
$\displaystyle{\bm{S}}_{i}^{N_{\rm Sk}=+2}$
$\displaystyle\propto\left(\begin{array}[]{c}\cos\mathcal{Q}_{1}\\\
\cos\mathcal{Q}_{3}\\\ \cos\mathcal{Q}_{2}\end{array}\right),$ (15)

where

$\displaystyle\mathcal{Q}_{\nu}={\bm{Q}}_{\nu}\cdot{\bm{r}}_{i}+\theta_{\nu},\quad\phi_{\nu}=2\pi(\nu-1)/3.$
(16)

Here the angles $\theta_{\nu}$ ($\nu$=1,2,3) represent phase shifts, and
$\Theta=\sum_{\nu=1}^{3}\theta_{\nu}$ is an internal degree of freedom called
phason Nakazawa2019 ; Hayami2020b . The variable $\lambda_{\rm v}(=\pm 1)$ is
called vorticity, and $\lambda_{\rm v}=+1(-1)$ corresponds to $N_{\rm
sk}=-1(+1)$. Because the variation of $\Theta$ is accompanied by a change in
energy, we set $\theta_{\nu}$=0 hereafter. These formulae represent that the
skyrmion-lattice with $|N_{\rm sk}|=1$ is a superposition of the three
helices, while that with $|N_{\rm sk}|=2$ is a superposition of the three
cosine waves.

These two skyrmion lattices break different symmetries. Specifically, the
skyrmion lattices with $|N_{\rm sk}|=1$ break the U(1) symmetry associated
with the in-plane rotational invariance and thus have a degree of freedom
called helicity. The variation of magnetic texture upon the helicity shift by
$\phi_{1}$ is given, e.g., by,

$\displaystyle{\bm{S}}_{i}^{N_{\rm Sk}=\pm 1}$
$\displaystyle\propto\hat{R}_{z}(\phi_{1})\sum_{\nu=1}^{3}\left(\begin{array}[]{c}\sin\mathcal{Q}_{\nu}\cos\phi_{\nu}\\\
\lambda_{\rm v}\sin\mathcal{Q}_{\nu}\sin\phi_{\nu}\\\
\cos\mathcal{Q}_{\nu}\end{array}\right)$ (20)

where $\hat{R}_{\gamma}(\varphi)$ is an operator to rotate the vector by the
angle $\varphi$ around the $\gamma$-axis. On the other hand, the skyrmion
lattices with $|N_{\rm sk}|=2$ break the SO(3) symmetry. The variation of
magnetic texture upon the SO(3)-invariant rotational operations by angles
$\theta_{2}$ and $\phi_{2}$ is given, e.g., by,

$\displaystyle{\bm{S}}_{i}^{N_{\rm Sk}=-2}$
$\displaystyle\propto\hat{R}_{x}(\theta_{2})\hat{R}_{z}(\phi_{2})\left(\begin{array}[]{c}\cos\mathcal{Q}_{1}\\\
\cos\mathcal{Q}_{2}\\\ \cos\mathcal{Q}_{3}\end{array}\right).$ (24)

Note that the energy of the skyrmion lattice with $|N_{\rm sk}|=1$ does not
change upon the variation of $\phi_{1}$, whereas that with $|N_{\rm sk}|=2$
does not change upon the variations of $\theta_{2}$ and $\phi_{2}$.

Then we consider the effects of external magnetic field. When a magnetic field
is applied, some of the symmetries mentioned in the above discussion would be
violated. The U(1) symmetry around the $z$-axis in the skyrmion lattice with
$|N_{\rm sk}|=1$ becomes absent when the external magnetic field has in-plane
components. Therefore, the magnetic structures of the $|N_{\rm sk}|=1$
skyrmion lattice do not have any global symmetries when irradiated with
circularly polarized microwave field. In addition, the SO(3) symmetry in the
skyrmion lattice with $|N_{\rm sk}|=2$ is partially violated by the external
magnetic field, whereas the U(1) symmetry around the magnetic field remains.
More specifically, the $|N_{\rm sk}|=2$ skyrmion lattice has the U(1) symmetry
around the total magnetic field at every moment, and thus its symmetry axis
temporally varies. We also note that a magnetic field also modulates a degree
of freedom associated with phasons $\Theta$.

##  III Method

We simulate time evolution of the local magnetizations in the Kondo-lattice
system by numerically solving the Landau-Lifshitz-Gilbert (LLG) equation. The
LLG equation is given by,

$\displaystyle\frac{d{\bm{S}}_{i}}{dt}=-{\bm{S}}_{i}\times{\bm{H}}^{\rm
eff}_{i}+\frac{\alpha_{\rm G}}{S}{\bm{S}}_{i}\times\frac{d{\bm{S}}_{i}}{dt},$
(25)

where $\alpha_{\rm G}(=0.05)$ is the dimensionless Gilbert-damping constant,
and $S(=1)$ is the saturation magnetization. The effective magnetic field
${\bm{H}}^{\rm eff}_{i}$ acting on the local magnetization at the $i$th site
is calculated by,

$\displaystyle{\bm{H}}^{\rm
eff}_{i}=-\frac{\partial\Omega}{\partial{\bm{S}}_{i}}+{\bm{H}}_{\rm
ext}+{\bm{H}}(t).$ (26)

Here $\Omega$ is the grand canonical potential of $\mathcal{H}_{\rm KL}$,
which is given by,

$\displaystyle\Omega=\int\rho(\varepsilon,\\{{\bm{S}}_{i}\\})F(\varepsilon-\mu)\;d\varepsilon,$
(27)

with

$\displaystyle\rho(\varepsilon,\\{{\bm{S}}_{i}\\})=\frac{1}{2N}\sum_{k=1}^{2N}\delta(\varepsilon-\varepsilon_{k}(\\{{\bm{S}}_{i}\\})).$
(28)

Here $F(\varepsilon-\mu)$ is the free energy of the system, and
$\rho(\varepsilon,\\{{\bm{S}}_{i}\\})$ is the density of state of conduction
electrons for a given set of the local magnetizations $\\{{\bm{S}}_{i}\\}$. To
calculate $\Omega$ and its magnetization-derivatives
$\partial\Omega/\partial{\bm{S}}_{i}$, we adopt the kernel polynomial method,
which is based on the Chebyshev polynomial expansion of $\Omega$ and the
automatic differentiation Motome1999 ; Weisse2006 ; Barros2013 ; Wang2016 ;
Ozawa2017b ; Chern2018 ; Wang2018 .

All the simulations are performed at zero temperature with no thermal
fluctuations in order to demonstrate that the microwave application can solely
induce the magnetic topological switching and to capture the physics of this
field-induced phenomenon. For the simulations, a lattice with $N=36^{2}$
sites, on which periodic boundary conditions are imposed, is adopted. We use
324 correlated random vectors Wang2018 ; Tang2012  for simulating relaxation
dynamics to obtain initial magnetic configurations through minimizing the
energy, whereas we adopt complete orthonormal basis states for the simulations
of microwave-induced dynamics. We use Chebyshev polynomials up to the 2000th-
order for the expansion of $\Omega$ and adopt the fourth-order Runge-Kutta
method with a time slice of $\Delta t=4$ to solve the LLG equation in Eq.
(25). The spatiotemporal dynamics of local magnetizations $\bm{S}_{i}$ and
local scalar spin chiralities $C_{i}$ are computed. The spin chirality $C_{i}$
is calculated by,

$\displaystyle
C_{i}={\bm{S}}_{i}\cdot{\bm{S}}_{i+\hat{a}}\times{\bm{S}}_{i+\hat{a}+\hat{b}}+{\bm{S}}_{i}\cdot{\bm{S}}_{i+\hat{a}+\hat{b}}\times{\bm{S}}_{i+\hat{b}},$
(29)

where $\hat{a}$ and $\hat{b}$ are the primitive lattice vectors of triangular
lattice. We also compute time profiles of the net magnetization $\bm{S}$ and
the skyrmion number $N_{\rm sk}$, which are respectively calculated by Kim2020
,

$\displaystyle\bm{S}=\frac{1}{N}\sum_{i=1}^{N}\bm{S}_{i},$ (30) $\displaystyle
N_{\rm sk}=\frac{1}{4\pi N_{\rm m}}\sum_{i=1}^{N}$
$\displaystyle\left[2\tan^{-1}\left(\frac{{\bm{S}}_{i}\cdot{\bm{S}}_{i+\hat{a}}\times{\bm{S}}_{i+\hat{a}+\hat{b}}}{1+{\bm{S}}_{i}\cdot{\bm{S}}_{i+\hat{a}}+{\bm{S}}_{i+\hat{a}}\cdot{\bm{S}}_{i+\hat{a}+\hat{b}}+{\bm{S}}_{i+\hat{a}+\hat{b}}\cdot{\bm{S}}_{i}}\right)\right.$
$\displaystyle+\left.2\tan^{-1}\left(\frac{{\bm{S}}_{i}\cdot{\bm{S}}_{i+\hat{a}+\hat{b}}\times{\bm{S}}_{i+\hat{b}}}{1+{\bm{S}}_{i}\cdot{\bm{S}}_{i+\hat{a}+\hat{b}}+{\bm{S}}_{i+\hat{a}+\hat{b}}\cdot{\bm{S}}_{i+\hat{b}}+{\bm{S}}_{i+\hat{b}}\cdot{\bm{S}}_{i}}\right)\right].$
(31)

Here $N_{\rm m}(=27)$ is the number of magnetic unit cells where one unit cell
contains 48 sites. Note that the sizes of magnetic unit cells are common for
all the magnetic patterns which appear in the present simulations because they
are all constituted with three magnetic helices with the same wave vectors
$\bm{Q}_{\nu}$ ($\nu=1,2,3$) determined by the Fermi-surface nesting.

##  IV Results

###  IV.1 Spin-Wave Modes

Figure 3: (a)-(c) Microwave-absorption spectra for in-plane microwave magnetic
fields in respective magnetic phases, i.e., (a) the skyrmion-lattice phase
with $|N_{\rm sk}|=2$, (b) the skyrmion-lattice phase with $|N_{\rm sk}|=1$,
and (c) the nontopological phase with $N_{\rm sk}=0$. (d) $H_{z}$-dependence
of resonance frequency $\omega_{\rm res}$ for spin-wave modes active to an in-
plane microwave field (corresponding to peak positions of the microwave-
absorption spectra).

First, we study microwave-active resonance modes in each magnetic phase by
numerically calculating the dynamical magnetic susceptibilities,

$\displaystyle\chi_{\gamma}(\omega)=\frac{\Delta
S_{\gamma}(\omega)}{H_{\gamma}(\omega)}\quad\quad(\gamma=x,y,z),$ (32)

where $H_{\gamma}(\omega)$ and $\Delta M_{\gamma}(\omega)$ are Fourier
components of the time-dependent magnetic field $\bm{H}(t)$ and those of the
time-profile of total magnetization $\Delta\bm{S}(t)=\bm{S}(t)-\bm{S}(0)$.
Here we particularly focus on the resonance modes active to the in-plane
polarized microwave field and thus set $\gamma=x$. To calculate these
quantities, we adopt a spatially uniform short-time pulse of magnetic field
for $\bm{H}(t)$, which is given by,

$\displaystyle\bm{H}(t)=\left\\{\begin{aligned} &(H_{\rm
pulse},0,0)\quad&0\leq t\leq 1\\\ &0\quad&{\rm others}\end{aligned}\right.$
(33)

where $t=(t_{1}/\hbar)\tau$ is the dimensionless time with $\tau$ and $t_{1}$
respectively being the real time and the nearest-neighbor hopping integral. We
compute time evolutions of local magnetizations $\bm{S}_{i}(t)$ and their sum
$\bm{S}(t)$ after applying this pulse field to the system. The usage of the
short-time pulse is advantageous because the Fourier components
$H_{\gamma}(\omega)$ become constant being independent of $\omega$ up to the
first order of $\omega\Delta t$ for a sufficiently short duration $\Delta t$
(i.e., $\omega\Delta t\ll 1$). The Fourier components are calculated as

$\displaystyle H_{\gamma}(\omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{\Delta t}\;H_{\rm pulse}e^{i\omega t}dt=\frac{H_{\rm
pulse}}{i\omega}\left(e^{i\omega\Delta t}-1\right)$ (34) $\displaystyle\sim$
$\displaystyle H_{\rm pulse}\Delta t.$

As a result, we obtain the relationship $\chi_{\gamma}(\omega)\propto\Delta
M_{\gamma}(\omega)$.

In Figs. 3(a)-(c), we present the calculated microwave absorption spectra,
i.e., imaginary part of the dynamical magnetic susceptibility Im$\chi_{x}$,
for (a) the skyrmion-lattice phase with $|N_{\rm sk}|$=2, (b) the skyrmion-
lattice phase with $|N_{\rm sk}|$=1, and (c) the nontopological phase with
$N_{\rm sk}$=0. Each of the spectra has a single peak, indicating the
existence of a single resonance mode in each phase. The mode in the skyrmion-
lattice phase with $|N_{\rm sk}|$=1 is a rotational mode in which all the
skyrmions constituting the skyrmion lattice rotate uniformly in the
counterclockwise fashion. It is known that skyrmion lattices in the
Dzyaloshinskii-Moriya magnets without inversion symmetry exhibit two rotation
modes with opposite rotation senses (i.e., counterclockwise and clockwise) at
different frequencies Mochizuki2012 , whereas skyrmion lattices stabilized by
frustrated exchange interactions in centrosymmetric Heisenberg magnets exhibit
a counterclockwise mode only Leonov2015 . The situation in our centrosymmetric
metallic magnets with the RKKY interactions resembles the latter case. Figure
3(d) presents the resonance frequency $\omega_{\rm res}$ as a function of
external magnetic field $H_{z}$. We find that a relation $\omega=H_{z}$ (i.e.,
$\omega=g\mu_{\rm B}H_{z}/\hbar$ in dimensionfull units) holds in the phases
with $|N_{\rm sk}|$=1 and $N_{\rm sk}$=0.

###  IV.2 Microwave-Induced Dynamics

Figure 4: Simulated time profiles of the net magnetization
$\bm{S}=(S_{x},S_{y},S_{z})$ and the skrmion number $N_{\rm sk}$ in the Kondo-
lattice model under irradiation with circularly polarized microwave field with
(a) $\omega$=0.005 and (b) $\omega$=0.01 when $H_{z}=0.005$. For both cases,
we start with a skyrmion-lattice configuration with $N_{\rm sk}=-1$ as an
initial state, which is the ground state at $H_{z}=0.005$. (c),(d) Snapshots
of the temporally varying spatial profiles of local magnetizations
$\bm{S}_{i}$ in the microwave-induced nonequilibrium steady phases, i.e., (c)
the nontopological phase $N_{\rm sk}=0$ and (d) the skyrmion-lattice phase
with $N_{\rm sk}=-2$ at selected moments indicated by inverted triangles in
(a) and (b), respectively. (e), (f) Spatial profiles of the local scalar spin
chiralities $C_{i}$ in each nonequilibrium steady phase, which do not change
temporally. Figure 5: Nonequilibrium phase diagram in plane of the microwave
frequency $\omega$ and the strength of static perpendicular magnetic field
$H_{z}$ under irradiation with circularly polarized microwave field of
$H^{\omega}$=0.01. The Gilbert-damping coefficient is set to be $\alpha_{\rm
G}$=0.05 for the simulations. Equilibrium phase diagram in the absence of
microwave field (i.e., $H^{\omega}$=0) is also presented at the left end.
(b)-(d) Schematics of the energy landscapes for different switching behaviors,
i.e., (b) the deterministic irreversible switching, (c) the probabilistic
irreversible switching, and (d) the temporally random fluctuation.

Next we simulate the magnetization dynamics under irradiation with circularly
polarized microwave field, which is given by,

${\bm{H}}(t)=H^{\omega}\beta(t)(\cos\omega t,\sin\omega t,0),$ (35)

with

$\beta(t)=\tanh\left(\frac{t}{\tau_{\rm d}}\right).$ (36)

Here the time-dependent prefactor $\beta(t)$ with $\tau_{\rm d}=2\pi/\omega$
is introduced to avoid unexpected artifacts due to impact-force effects
through gradually increasing the microwave amplitude.

Figures 4(a) and (b) present simulated time profiles of the net magnetization
$\bm{S}=(S_{x},S_{y},S_{z})$ and the skrmion number $N_{\rm sk}$ in a system
irradiated with circularly polarized microwave field with (a) $\omega$=0.005
and (b) $\omega$=0.01 when $H_{z}=0.005$. We start the simulation with a
skyrmion-lattice configuration with $N_{\rm sk}=-1$ as an initial state for
both cases, which is the ground state at $H_{z}=0.005$. We observe a
microwave-induced switching of the magnetic topology from $N_{\rm sk}=-1$ to
$N_{\rm sk}=0$ in Fig. 4(a), whereas that from $N_{\rm sk}=-1$ to $N_{\rm
sk}=-2$ in Fig. 4(b). The phase with $N_{\rm sk}=-1$ and the phase with
$N_{\rm sk}=0$ in respective cases appear as nonequilibrium steady phases
where the net magnetizations show steady oscillations.

Although the skyrmion number $N_{\rm sk}$ is constant, the spatial magnetic
configurations in these nonequilibrium steady phases under irradiation with
microwave field vary periodically in time. Figures 4(c) and (d) present a
series of snapshots of the temporally varying local magnetizations
$\bm{S}_{i}$ for (c) the nontopological phase $N_{\rm sk}=0$ and (d) the
skyrmion-lattice phase with $N_{\rm sk}=-2$ under irradiation with the
microwave field at selected moments. In fact, as long as the skyrmion number
is constant, these magnetic configurations are connected to each other by
certain rotational operations. For example, the four magnetic configurations
with $N_{\rm sk}=-2$ shown in Fig. 4(d) are all connected to each other via
the SO(3)-invariant rotational operations by angles $\theta_{2}$ and
$\phi_{2}$ represented by Eq. (24). Note that the local scalar spin
chiralities $C_{i}$ are time-independent in contrast to the local
magnetizations $\bm{S}_{i}$. In Figs. 4(e) and (f), we present the spatial
profiles of $C_{i}$ in the microwave-induced $N_{\rm sk}=0$ and $N_{\rm
sk}=-2$ phases, respectively, which do not change temporally.

Then we study nonequilibrium magnetic phases after sufficient a duration of
the microwave irradiation by varying the microwave frequency $\omega$ and the
applied static magnetic field $H_{z}$. Figure 5 presents the obtained
nonequilibrium phase diagram in plane of $\omega$ and $H_{z}$ for a system
under continuous irradiation with microwave field given by Eq. (35). We trace
time-evolutions of the magnetizations for a sufficiently long duration up to
$t$=48000 at most. Here we take the microwave amplitude $H^{\omega}=0.01$ for
the simulations. Note that when the microwave field is absent (i.e.,
$H^{\omega}$=0), the system exhibits ground-state phases shown in Fig. 1(a)
where the three magnetic phases with different skyrmion numbers $|N_{\rm
sk}|$=2, $|N_{\rm sk}|$=1, and $N_{\rm sk}$=0 successively emerge as $H_{z}$
increases. We select a ground-state magnetic configuration as an initial state
for the time-evolution simulations at a given field strength of $H_{z}$. We
present this ground-state phase diagram also in Fig. 5 for a reference. We
also note that in the static limit of $\omega$=0, all these phases turn into
the nontopological phase with $N_{\rm sk}$=0 in the presence of static in-
plane magnetic field of $H^{\omega}$=0.01. Thus, the phases on the
$H_{z}$-axis are all assigned to $N_{\rm sk}$=0.

To discuss the phase diagram in Fig. 5, we should first note that a circularly
polarized microwave field generates an effective static magnetic field
$\pm\omega\bm{e}_{z}$ perpendicular to the polarization plane Miyake2020 ;
Mochizuki2018 . The amplitude is equal to $\omega$ (i.e.,
$\hbar\omega/g\mu_{\rm B}$ in real units), while the sign is determined by the
sense of the circular polarization, i.e., positive (negative) for the
clockwise (counterclockwise) polarization. In the present case, the sign is
negative because the microwave field circulating in counterclockwise sense is
applied. Thereby, a static component of the total magnetic field acting on the
system is $H_{z}^{\rm tot}=H_{z}-\omega$. This means that the application of
this microwave field effectively work to shift the system towards a low-field
regime in the equilibrium phase diagram. Thus, the nonequilibrium phase with
$N_{\rm sk}$=0 in the low-frequency regime tends to change into the skyrmion-
lattice phase with $|N_{\rm sk}|$=1 and further into the skyrmion-lattice
phase with $|N_{\rm sk}|$=2 as $\omega$ increases. Indeed, the skyrmion-
lattice phase with $|N_{\rm sk}|$=2 appears in the right area of the phase
diagram where $\omega$ is large, whereas the skyrmion-lattice phase with
$|N_{\rm sk}|$=1 can emerge when $\omega$ is intermediate in the areas
referred to as “the probabilistic irreversible switching” regime and “the
temporally random fluctuation” regime in the phase diagram.

Importantly, the ways of the emergence of the $|N_{\rm sk}|$=1 phase are
distinct from those of the $|N_{\rm sk}|$=2 phase and the $N_{\rm sk}$=0
phase. The latter two phases emerge in a deterministic and irreversible way
under irradiation with microwave field. In contrast, the $|N_{\rm sk}|$=1
phase emerges in a probabilistic but irreversible way when $H_{z}$ is low,
whereas either the $|N_{\rm sk}|$=1 phase or the $|N_{\rm sk}|$=2 phase
randomly emerges in a temporally fluctuating manner when $H_{z}$ is high.
These peculiar behaviors might be attributable to the energy landscapes
characterized by the energy minima and the energy barriers. We expect an
energy landscape in Fig. 5(b) for the areas where the system enters the
$|N_{\rm sk}|$=2 phase or the $N_{\rm sk}$=0 phase in a deterministic and
irreversible manner. On the other hand, we expect energy landscapes in Figs.
5(c) and (d) for the probabilistic-irreversible-switching regime and the
temporally-random-fluctuation regime, respectively, in which the energy minima
are nearly degenerate. For the probabilistic irreversible regime, the energy
barrier is rather high [see Fig. 5(c)], with which the system becomes settled
down once it falls into one of the minima, resulting in the probabilistic
irreversible switching. In contrast, we expect a small energy barrier in the
temporally fluctuating regime where the system fluctuates between the two
minima under irradiation with the microwave field.

Figure 6: (a) Snapshots of the local magnetizations $\bm{S}_{i}$ (left panel)
and the scalar spin chiralities $C_{i}$ (right panel) for the dynamical
magnetic pattern with a half-integer skyrmion number of $N_{\rm sk}=-1/2$
emerging in the transient process in Fig. 4(a). (b) Those for the dynamical
magnetic pattern with $N_{\rm sk}=-3/2$ emerging in the transient process in
Fig. 4(b).

Interestingly, we find dynamical formations of topological magnetic patterns
with half-integer skyrmion numbers during the switching processes. We present
snapshots of their spatial configurations in Figs. 6(a) and (b). Specifically,
Fig. 6(a) shows the dynamical magnetic pattern with $N_{\rm sk}=-1/2$ emerging
during the process of magnetic-topology switching from $N_{\rm sk}=-1$ to
$N_{\rm sk}=0$ in Fig. 4(a), whereas Fig. 6(b) shows that with $N_{\rm
sk}=-3/2$ emerging during the switching from $N_{\rm sk}=-1$ to $N_{\rm sk}=0$
in Fig. 4(b). Topological magnetic textures having a half-integer topological
charge are referred to as merons or antimerons, and their crystallized states
have been an issue of intensive research Yu2018 ; Hayami2021b ; WangZ2021 ;
LinSZ2015 ; Kharkov2017 ; Gobel2019 . Clarifications of the observed dynamical
topological magnetic patterns in the transient processes are left for future
studies.

##  V Discussion

Figure 7: Simulated time profiles of the net magnetization
$\bm{S}=(S_{x},S_{y},S_{z})$ and the skrmion number $N_{\rm sk}$ in the
effective BBQ model under irradiation with circularly polarized microwave
field with (a) $\omega$=0.2 and (b) $\omega$=0.4 when $H_{z}=0.2$. For both
cases, we start with a skyrmion lattice with $N_{\rm sk}=-1$ as an initial
state, which is the ground state at $H_{z}=0.2$. The skyrmion number remains
constant to be $N_{\rm sk}=-1$ even after a sufficient duration, indicating
that the switching of magnetic topology does not occur in contrast to the case
of the Kondo-lattice model. (c),(d) Snapshots of the local magnetizations
$\bm{S}_{i}$ (left panels) and the scalar spin chiralities $C_{i}$ (right
panels) in the microwave-induced nonequilibrium steady phases at selected
moments. The magnetic configuration in (c) corresponds to the bimeron crystal
with $N_{\rm sk}=-1$, whereas that in (d) corresponds to the antiskyrmion
crystal with $N_{\rm sk}=-1$

In this section, we discuss our results in the light of an effective model
introduced in literature Ozawa2016 ; Hayami2017 ; Akagi2012 ; Hayami2014b ,
which is called the effective bilinear-biquadratic (BBQ) model. This model is
derived from the original Kondo-lattice model in Eq. (2) using the
perturbation expansions when the hopping term dominates the Kondo-coupling
term. We argue that effective three-body interactions originating from the
third-order perturbation processes might be of essential importance for the
observed microwave-induced switching of magnetic topology in the
centrosymmetric itinerant magnets.

The Hamiltonian of the effective BBQ model is given by,

$\mathcal{H}=\mathcal{H}_{\rm BBQ}+\mathcal{H}_{\rm Zeeman},$ (37)

with

$\mathcal{H}_{\rm
BBQ}=\sum_{\nu=1}^{3}\left[-J\left|{\bm{S}}_{{\bm{Q}}_{\nu}}\right|^{2}+\frac{K}{N}\left|{\bm{S}}_{{\bm{Q}}_{\nu}}\right|^{4}\right].$
(38)

The first term of Eq. (38) represents the effective interactions originating
from the second-order perturbation processes, which is often referred to as
the RKKY interactions. On the other hand, the second term represents parts of
the contributions from the fourth-order perturbation processes Akagi2012 ;
Hayami2014b . Note that contributions from the odd-order perturbation
processes usually vanish when magnetic fields are absent because of the time-
reversal symmetry, whereas they should appear in the presence of magnetic
fields. For the Zeeman-coupling term $\mathcal{H}_{\rm Zeeman}$, we consider
both the static perpendicular magnetic field $\bm{H}_{\rm ex}=(0,0,H_{z})$ and
the circularly polarized microwave field with amplitude of $H^{\omega}$. When
the microwave field is absent (i.e., $H^{\omega}=0$), this effective BBQ model
is known to host two different skyrmion-lattice phases with $|N_{\rm sk}|=1$
and $|N_{\rm sk}|=2$ as well as the nontopological phase with $N_{\rm sk}=0$
in the ground state depending on the strength of external magnetic field
$H_{z}$.

The coupling constants $J$ and $K$ in Eq. (38) depend on the electronic
structures of conduction electrons such as Fermi surfaces and density of
states, which are governed by the lattice structure, the transfer integrals,
and the electron filling. The values of $J$ and $K$ can be evaluated from the
original Kondo-lattice model by the perturbation-expansion calculations in
principle. Through the second-order perturbation expansions, we obtain the
following formula for the coupling constant $J$,

$J=J_{\rm K}^{2}\sum_{\bm{q}\in{\rm
BZ}}\chi_{0}(\bm{q})e^{i\bm{q}\cdot\bm{r}_{1}}.$ (39)

Here $\chi_{0}(\bm{q})$ is the bare susceptibility of conduction electrons,
and $\bm{r}_{1}$ denotes the Bravis vectors of the triangular lattice. For the
Kondo-lattice model in Eq. (2) and the parameters used in the present work
(i.e., $t_{1}=-1$, $t_{3}=0.85$, $J_{\rm K}=-0.5$, and $\mu=-3.5$), we
evaluate the value as $J\sim 0.0035t_{1}$, which is much smaller than unity
and thus supports the validity of the perturbational treatment. We can also
evaluate the coupling constant $K$ microscopically from the Kondo-lattice
model. However, to get a general insight into the effective BBQ model, we
treat the model rather phenomenologically by regarding $J$, $K$, $H_{z}$ and
$H^{\omega}$ as parameters and by setting the constant $J$ as energy units
(i.e., $J=1$) in the following discussion.

Now we examine the microwave-induced magnetization dynamics in this effective
BBQ model by deriving a time-evolution equation for the Fourier components of
magnetization $\bm{S}_{\bm{q}}$. The equation is given by,

$\displaystyle\frac{d\bm{S}_{\bm{q}}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[\mathcal{H}_{\rm BBQ}+\mathcal{H}_{\rm
Zeeman},\bm{S}_{\bm{q}}\right]_{-}$
$\displaystyle=-\sum_{\nu=1}^{3}\left(-J+\frac{K}{N}|\bm{S}_{\bm{Q}_{\nu}}|^{2}\right)\left(\bm{S}_{\bm{Q}_{\nu}}\times\bm{S}_{-\bm{Q}_{\nu}+\bm{q}}-\bm{S}_{\bm{Q}_{\nu}+\bm{q}}\times\bm{S}_{-\bm{Q}_{\nu}}\right)+\bm{S}_{\bm{q}}\times\left[\bm{H}_{\rm
ext}+\bm{H}^{\prime}(t)\right].$ (40)

By numerically solving the derived equation, we examine two cases with
$\omega=0.005$ and $\omega=0.01$, for which we have respectively observed the
switching of magnetic topology in the original Kondo-lattice model, i.e., the
switching from $N_{\rm sk}=-1$ to $N_{\rm sk}=0$ [Fig. 4(a)] and the switching
from $N_{\rm sk}=-1$ to $N_{\rm sk}=-2$ [Fig. 4(b)]. We set the parameter
values as $J=1$, $K=0.5$, $H_{z}=0.2$ and $H^{\omega}=0.4$. Surprisingly, the
switching of magnetic topology is not observed for both cases [see Figs. 7(a)
and (b)].

The failure of the effective BBQ model in reproduction of the topology
switching may be ascribed to ingredients which are incorporated in the
original Kondo-lattice model but missing in the effective BBQ model. One
missing ingredient is effective interactions originating from the third-order
perturbation processes. The effective BBQ model contains only terms bilinear
and biquadratic with respect to $\bm{S}_{\bm{Q}_{\nu}}$ and
$\bm{S}_{-\bm{Q}_{\nu}}$, which come from the second-order and fourth-order
perturbation processes, respectively. We infer that the other-order terms
might contribute to the topology switching. Among those terms, the lowest-
order terms, i.e., the third-order terms are most likely, which are three-body
interactions with respect to $S_{\bm{Q}_{\nu}}$ and $S_{-\bm{Q}_{\nu}}$ and
thus have no time-reversal symmetry. Therefore, the third-order terms are
forbidden and should vanish in the absence of magnetic field. However, they
are allowed to appear once a magnetic field is applied, although they are not
incorporated in the effective BBQ model even under a magnetic field.
Contributions of the third-order terms to the magnetization dynamics are
described by the following Heisenberg equation of motion for the Fourier
component $\bm{S}_{\bm{q}}$,

$\frac{d\bm{S}_{\bm{q}}}{dt}=\frac{i}{\hbar}\left[F^{(3)},\bm{S}_{\bm{q}}\right]_{.}$
(41)

There are five kinds of third-order terms $F^{(3)}$ as derived in Ref.
Hayami2017 , and they turn out to contribute to the time evolution of
${\bm{S}}_{\bm{q}}$ as shown in the appendix. Another missing ingredient is
contributions from momenta apart from $\bm{Q}_{\nu}$. The effective BBQ model
contains only the Fourier components of magnetization $\bm{S}_{\bm{Q}_{\nu}}$
with $\bm{Q}_{\nu}$ being the modulation wavevectors. However, the bilinear
and biquadratic interactions originally have components of other wavevectors.
These neglected components of the interactions may play a key role in the
topology switching. The issue requires further investigations and is left for
future studies.

In the meanwhile, the counterclockwise circularly polarized microwave field
considered in the present study generates an effective static magnetic field
perpendicular to the polarization plane $-\omega\bm{e}_{z}$ Miyake2020 ;
Mochizuki2018 . Therefore, a perpendicular component of the total magnetic
field is $H^{\rm tot}_{z}=H_{z}-\omega$. In the case of Fig. 7(a), the
effective static component $H^{\rm tot}_{z}$ vanishes (i.e., $H^{\rm
tot}_{z}=0$) because we set $H_{z}=\omega=0.2$. Under this circumstance, the
bimeron crystal with $|N_{\rm sk}|=1$ Kharkov2017 ; Gobel2019  appears as a
nonequilibrium steady state after a sufficient duration of the microwave
irradiation [Fig. 7(c)]. Note that the effective BBQ model in Eq. (38)
exhibits the skyrmion-lattice phase with $|N_{\rm sk}|=1$ in the equilibrium
case when both the static and microwave magnetic fields are absent (i.e.,
$H_{z}$=$H^{\omega}$=0). On the other hand, the effective static component
$H^{\rm tot}_{z}$ is negative (i.e., $H^{\rm tot}_{z}=-0.2$) because we set
$H_{z}=0.2$ and $\omega=0.4$. We observe the antiskyrmion lattice with
$|N_{\rm sk}|=1$ [Fig. 7(d)] after a sufficient duration of the microwave
irradiation, in which the core magnetizations point upwards.

##  VI Conclusion

In summary, we have theoretically proposed possible microwave-induced
switching of magnetic topology in centrosymmetric itinerant magnets by
numerically analyzing the magnetization dynamics in the Kondo-lattice model on
a triangular lattice using a combined method of the micromagnetic simulation
and the kernel polynomial expansion technique. We have demonstrated that the
intense excitation of spin-wave mode with circularly polarized microwave field
can switch the skyrmion lattice with $|N_{\rm sk}|$=1 into that with $|N_{\rm
sk}|$=2 or the nontopological magnetic state with $N_{\rm sk}$=0 depending on
the microwave frequency. During these switching processes, transient
topological magnetic patterns with half-integer skyrmion numbers of $|N_{\rm
sk}|$=1/2 and $|N_{\rm sk}|$=3/2 were observed. These fractionalizations of
magnetic topological charges in the dynamical regime are an issue of interest,
which should be clarified in future studies. We have found several different
switching behaviors under continuous microwave irradiation, that is,
deterministic irreversible switching, probabilistic irreversible switching,
and temporally random fluctuation depending on the microwave frequency and the
strength of external magnetic field, which is attributable to difference of
the energy landscape in the dynamical regime. We have also examined the
effective BBQ model derived from the perturbation expansions of the Kondo-
lattice model and have found that this model fails to reproduce the dynamical
switching of magnetic topology. The failure of the effective model containing
only the even-order terms conversely indicates that contributions from the
odd-order perturbation processes, which break the time-reversal symmetry, are
important to understand the observed magnetic topology switching. Here we
emphasize that various unfrozen degrees of freedom inherent in centrosymmetric
magnets are sources of rich magnetic textures and their controllability with
external parameters. Our work will open a new research field to manipulate
magnetic topologies in centrosymmetric magnets.

##  VII Appendix

When the electron hoppings dominate the Kondo exchange coupling in the Kondo-
lattice model, we can derive effective interactions among the local
magnetizations from this model by using the perturbation expansion technique.
In the main text, we have argued that the effective interactions originating
from the third-order perturbation processes can contribute to the time-
evolutions of the Fourier components $S_{\bm{q}}$. In Ref. Hayami2017 , the
effective interactions from the third-order perturbation processes have been
derived, which have turned out to be three-body interactions among the Fourier
components ${\bm{S}}_{\bm{Q}_{\nu}}$ and ${\bm{S}}_{-\bm{Q}_{\nu}}$ with three
different ${\bm{Q}}_{\nu}$ vectors. The third-order contributions are given
by,

$\displaystyle F^{(3)}$
$\displaystyle=F^{(3)}_{1}+F^{(3)}_{2}+F^{(3)}_{3}+F^{(3)}_{4}+F^{(3)}_{5},$
(42)

with

$\displaystyle F^{(3)}_{1}$ $\displaystyle=-2\frac{J_{\rm
K}^{3}}{\sqrt{N}}\sum_{\nu}(C_{1}-C_{2})\left[S_{{\bm{Q}}_{\nu}}^{z}\left(S_{\bm{0}}^{x}S_{-{\bm{Q}}_{\nu}}^{x}+S_{\bm{0}}^{y}S_{-{\bm{Q}}_{\nu}}^{y}\right)+{\rm
h.c.}\right],$ (43) $\displaystyle F^{(3)}_{2}$ $\displaystyle=-2\frac{J_{\rm
K}^{3}}{\sqrt{N}}\sum_{\nu}(C_{3}-C_{4})\left[S_{{\bm{0}}}^{z}\left(S_{{\bm{Q}}_{\nu}}^{x}S_{-{\bm{Q}}_{\nu}}^{x}+S_{{\bm{Q}}_{\nu}}^{y}S_{-{\bm{Q}}_{\nu}}^{y}\right)+{\rm
h.c.}\right],$ (44) $\displaystyle F^{(3)}_{3}$ $\displaystyle=-2\frac{J_{\rm
K}^{3}}{\sqrt{N}}\sum_{\nu}(C_{5}-C_{6})S_{{\bm{0}}}^{z}S_{{\bm{Q}}_{\nu}}^{z}S_{-{\bm{Q}}_{\nu}}^{z},$
(45) $\displaystyle F^{(3)}_{4}$ $\displaystyle=-2\frac{J_{\rm
K}^{3}}{\sqrt{N}}(D_{1}-D_{2})\left[S_{{\bm{Q}}_{1}}^{z}\left(S_{{\bm{Q}}_{2}}^{x}S_{{\bm{Q}}_{3}}^{x}+S_{{\bm{Q}}_{2}}^{y}S_{{\bm{Q}}_{3}}^{y}\right)+{\rm
h.c.}\right]$ $\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow{\bm{Q}}_{2},\
{\bm{Q}}_{2}\rightarrow{\bm{Q}}_{3},\ {\bm{Q}}_{3}\rightarrow{\bm{Q}}_{1})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow{\bm{Q}}_{3},\
{\bm{Q}}_{2}\rightarrow{\bm{Q}}_{1},\ {\bm{Q}}_{3}\rightarrow{\bm{Q}}_{2}),$
(46) $\displaystyle F^{(3)}_{5}$ $\displaystyle=-2\frac{J_{\rm
K}^{3}}{\sqrt{N}}(D_{3}-D_{4})\left[S_{{\bm{Q}}_{1}}^{z}S_{{\bm{Q}}_{2}}^{z}S_{-{\bm{Q}}_{3}}^{z}+{\rm
h.c.}\right].$ (47)

Here the coefficients $C_{\nu}$ and $D_{\nu}$ ($\nu=1,2,3,4$) are calculated
by the convolution of Green’s functions. Detailed formulas of these
coefficients are given in Ref. Hayami2017 .

Equations of the time evolution of ${\bm{S}}_{\bm{q}}$ due to the third-order
contributions are given by,

$\displaystyle\frac{d{\bm{S}}_{\bm{q}}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[F_{1}^{(3)},{\bm{S}}_{\bm{q}}\right]_{-}$
$\displaystyle=2\frac{J_{\rm
K}^{3}}{N}\sum_{\nu}(C_{1}-C_{2})\left[\left({\bm{S}}_{{\bm{q}}+{\bm{Q}}_{\nu}}\times{\bm{e}}_{z}\right)\left(S^{x}_{\bf
0}S^{x}_{-{\bm{Q}}_{\nu}}+S^{y}_{\bf 0}S^{y}_{-{\bm{Q}}_{\nu}}\right)\right.$
$\displaystyle\ \ \ \ \
\left.+\left(\begin{array}[]{c}-S_{{\bm{Q}}_{\nu}}^{z}\left(S^{y}_{\bf
0}S^{z}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{z}_{{\bm{q}}}S^{y}_{-{\bm{Q}}_{\nu}}\right)\\\
\ \ S_{{\bm{Q}}_{\nu}}^{z}\left(S^{x}_{\bf
0}S^{z}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{z}_{{\bm{q}}}S^{x}_{-{\bm{Q}}_{\nu}}\right)\\\
-S_{{\bm{Q}}_{\nu}}^{z}\left(S^{x}_{\bf
0}S^{y}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{y}_{{\bm{q}}}S^{x}_{-{\bm{Q}}_{\nu}}\right)+S_{{\bm{Q}}_{\nu}}^{z}\left(S^{y}_{\bf
0}S^{x}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{x}_{{\bm{q}}}S^{y}_{-{\bm{Q}}_{\nu}}\right)\end{array}\right)\right]$
(51) $\displaystyle\ \ \ \ \ +({\bm{Q}}_{\nu}\rightarrow-{\bm{Q}}_{\nu}),$
(52) $\displaystyle\frac{d{\bm{S}}_{\bm{q}}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[F_{2}^{(3)},{\bm{S}}_{\bm{q}}\right]_{-}$
$\displaystyle=2\frac{J_{\rm
K}^{3}}{N}\sum_{\nu}(C_{3}-C_{4})\left[\left({\bm{S}}_{{\bm{q}}}\times{\bm{e}}_{z}\right)\left(S^{x}_{{\bm{Q}}_{\nu}}S^{x}_{-{\bm{Q}}_{\nu}}+S^{y}_{{\bm{Q}}_{\nu}}S^{y}_{-{\bm{Q}}_{\nu}}\right)\right.$
$\displaystyle\ \ \ \ \ \left.+\left(\begin{array}[]{c}-S_{\bf
0}^{z}\left(S^{y}_{{\bm{Q}}_{\nu}}S^{z}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{z}_{{\bm{q}}+{\bm{Q}}_{\nu}}S^{y}_{-{\bm{Q}}_{\nu}}\right)\\\
\ \ S_{\bf
0}^{z}\left(S^{x}_{{\bm{Q}}_{\nu}}S^{z}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{z}_{{\bm{q}}+{\bm{Q}}_{\nu}}S^{x}_{-{\bm{Q}}_{\nu}}\right)\\\
-S_{\bf
0}^{z}\left(S^{x}_{{\bm{Q}}_{\nu}}S^{y}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{y}_{{\bm{q}}+{\bm{Q}}_{\nu}}S^{x}_{-{\bm{Q}}_{\nu}}\right)+S_{\bf
0}^{z}\left(S^{y}_{{\bm{Q}}_{\nu}}S^{x}_{{\bm{q}}-{\bm{Q}}_{\nu}}+S^{x}_{{\bm{q}}+{\bm{Q}}_{\nu}}S^{y}_{-{\bm{Q}}_{\nu}}\right)\end{array}\right)\right]$
(56) $\displaystyle\ \ \ \ \ +({\bm{Q}}_{\nu}\rightarrow-{\bm{Q}}_{\nu}),$
(57) $\displaystyle\frac{d{\bm{S}}_{\bm{q}}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[F_{3}^{(3)},{\bm{S}}_{\bm{q}}\right]_{-}$
$\displaystyle=2\frac{J_{\rm
K}^{3}}{N}\sum_{\nu}(C_{5}-C_{6})\left(\begin{array}[]{c}\ \
{\bm{S}}^{x}_{{\bm{q}}}{\bm{S}}^{z}_{{\bm{Q}}_{\nu}}{\bm{S}}^{z}_{-{\bm{Q}}_{\nu}}+{\bm{S}}^{z}_{\bf
0}{\bm{S}}^{x}_{{\bm{q}}+{\bm{Q}}_{\nu}}{\bm{S}}^{z}_{-{\bm{Q}}_{\nu}}+{\bm{S}}^{z}_{\bf
0}{\bm{S}}^{z}_{{\bm{Q}}_{\nu}}{\bm{S}}^{x}_{{\bm{q}}-{\bm{Q}}_{\nu}}\\\
-{\bm{S}}^{y}_{{\bm{q}}}{\bm{S}}^{z}_{{\bm{Q}}_{\nu}}{\bm{S}}^{z}_{-{\bm{Q}}_{\nu}}-{\bm{S}}^{z}_{\bf
0}{\bm{S}}^{y}_{{\bm{q}}+{\bm{Q}}_{\nu}}{\bm{S}}^{z}_{-{\bm{Q}}_{\nu}}-{\bm{S}}^{z}_{\bf
0}{\bm{S}}^{z}_{{\bm{Q}}_{\nu}}{\bm{S}}^{y}_{{\bm{q}}-{\bm{Q}}_{\nu}}\\\
0\end{array}\right),$ (61) $\displaystyle\frac{d{\bm{S}}_{\bm{q}}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[F_{4}^{(3)},{\bm{S}}_{\bm{q}}\right]_{-}$
$\displaystyle=2\frac{J_{\rm
K}^{3}}{N}(D_{1}-D_{2})\left[\left({\bm{S}}_{{\bm{q}}+{\bm{Q}}_{1}}\times{\bm{e}}_{z}\right)\left(S^{x}_{{\bm{Q}}_{2}}S^{x}_{{\bm{Q}}_{3}}+S^{y}_{{\bm{Q}}_{2}}S^{y}_{{\bm{Q}}_{3}}\right)\right.$
$\displaystyle\ \ \ \ \
\left.+\left(\begin{array}[]{c}-S_{{\bm{Q}}_{1}}^{z}\left(S^{y}_{{\bm{Q}}_{2}}S^{z}_{{\bm{q}}+{\bm{Q}}_{3}}+S^{z}_{{\bm{q}}+{\bm{Q}}_{2}}S^{y}_{{\bm{Q}}_{3}}\right)\\\
\ \
S_{{\bm{Q}}_{1}}^{z}\left(S^{x}_{{\bm{Q}}_{2}}S^{z}_{{\bm{q}}+{\bm{Q}}_{3}}+S^{z}_{{\bm{q}}+{\bm{Q}}_{2}}S^{x}_{{\bm{Q}}_{3}}\right)\\\
-S_{{\bm{Q}}_{1}}^{z}\left(S^{x}_{{\bm{Q}}_{2}}S^{y}_{{\bm{q}}+{\bm{Q}}_{3}}+S^{y}_{{\bm{q}}+{\bm{Q}}_{2}}S^{x}_{{\bm{Q}}_{3}}\right)+S_{{\bm{Q}}_{1}}^{z}\left(S^{y}_{{\bm{Q}}_{2}}S^{x}_{{\bm{q}}+{\bm{Q}}_{3}}+S^{x}_{{\bm{q}}+{\bm{Q}}_{2}}S^{y}_{{\bm{Q}}_{3}}\right)\end{array}\right)\right]$
(65) $\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow{\bm{Q}}_{2},\
{\bm{Q}}_{2}\rightarrow{\bm{Q}}_{3},\ {\bm{Q}}_{3}\rightarrow{\bm{Q}}_{1})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow{\bm{Q}}_{3},\
{\bm{Q}}_{2}\rightarrow{\bm{Q}}_{1},\ {\bm{Q}}_{3}\rightarrow{\bm{Q}}_{2})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow-{\bm{Q}}_{1},\
{\bm{Q}}_{2}\rightarrow-{\bm{Q}}_{2},\ {\bm{Q}}_{3}\rightarrow-{\bm{Q}}_{3})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow-{\bm{Q}}_{2},\
{\bm{Q}}_{2}\rightarrow-{\bm{Q}}_{3},\ {\bm{Q}}_{3}\rightarrow-{\bm{Q}}_{1})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow-{\bm{Q}}_{3},\
{\bm{Q}}_{2}\rightarrow-{\bm{Q}}_{1},\ {\bm{Q}}_{3}\rightarrow-{\bm{Q}}_{2}),$
(66) $\displaystyle\frac{d{\bm{S}}_{\bm{q}}}{dt}$
$\displaystyle=\frac{i}{\hbar}\left[F_{5}^{(3)},{\bm{S}}_{\bm{q}}\right]_{-}$
$\displaystyle=2\frac{J_{\rm K}^{3}}{N}(D_{3}-D_{4})\left(\begin{array}[]{c}\
\
{\bm{S}}^{x}_{{\bm{q}}+{\bm{Q}}_{1}}{\bm{S}}^{z}_{{\bm{Q}}_{2}}{\bm{S}}^{z}_{-{\bm{Q}}_{3}}+{\bm{S}}^{z}_{{\bm{Q}}_{1}}{\bm{S}}^{x}_{{\bm{q}}+{\bm{Q}}_{2}}{\bm{S}}^{z}_{-{\bm{Q}}_{3}}+{\bm{S}}^{z}_{{\bm{Q}}_{1}}{\bm{S}}^{z}_{{\bm{Q}}_{2}}{\bm{S}}^{x}_{{\bm{q}}-{\bm{Q}}_{3}}\\\
-{\bm{S}}^{y}_{{\bm{q}}+{\bm{Q}}_{1}}{\bm{S}}^{z}_{{\bm{Q}}_{2}}{\bm{S}}^{z}_{-{\bm{Q}}_{3}}-{\bm{S}}^{z}_{{\bm{Q}}_{1}}{\bm{S}}^{y}_{{\bm{q}}+{\bm{Q}}_{2}}{\bm{S}}^{z}_{-{\bm{Q}}_{3}}-{\bm{S}}^{z}_{{\bm{Q}}_{1}}{\bm{S}}^{z}_{{\bm{Q}}_{2}}{\bm{S}}^{y}_{{\bm{q}}-{\bm{Q}}_{3}}\\\
0\end{array}\right)$ (70) $\displaystyle\ \ \ \ \
+({\bm{Q}}_{1}\rightarrow{\bm{Q}}_{2},\ {\bm{Q}}_{2}\rightarrow{\bm{Q}}_{3},\
{\bm{Q}}_{3}\rightarrow{\bm{Q}}_{1})$ $\displaystyle\ \ \ \ \
+({\bm{Q}}_{1}\rightarrow{\bm{Q}}_{3},\ {\bm{Q}}_{2}\rightarrow{\bm{Q}}_{1},\
{\bm{Q}}_{3}\rightarrow{\bm{Q}}_{2})$ $\displaystyle\ \ \ \ \
+({\bm{Q}}_{1}\rightarrow-{\bm{Q}}_{1},\
{\bm{Q}}_{2}\rightarrow-{\bm{Q}}_{2},\ {\bm{Q}}_{3}\rightarrow-{\bm{Q}}_{3})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow-{\bm{Q}}_{2},\
{\bm{Q}}_{2}\rightarrow-{\bm{Q}}_{3},\ {\bm{Q}}_{3}\rightarrow-{\bm{Q}}_{1})$
$\displaystyle\ \ \ \ \ +({\bm{Q}}_{1}\rightarrow-{\bm{Q}}_{3},\
{\bm{Q}}_{2}\rightarrow-{\bm{Q}}_{1},\ {\bm{Q}}_{3}\rightarrow-{\bm{Q}}_{2}).$
(71)

##  VIII Acknowledgement

This work is supported by Japan Society for the Promotion of Science KAKENHI
(Grant No. 16H06345, No. 19H00864, No. 19K21858, and No. 20H00337), CREST, the
Japan Science and Technology Agency (Grant No. JPMJCR20T1), a Research Grant
in the Natural Sciences from the Mitsubishi Foundation, and a Waseda
University Grant for Special Research Projects (Project No. 2020C-269 and No.
2021C-566). A part of the numerical simulations was performed at the
Supercomputer Center of the Institute for Solid State Physics in the
University of Tokyo.

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