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#  Emergent electric field from magnetic resonances in a one-dimensional
chiral magnet

Kotaro Shimizu  Shun Okumura  Yasuyuki Kato  Yukitoshi Motome  Department of
Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan

###### Abstract

The emergent electric field (EEF) is a fictitious electric field acting on
conduction electrons through the Berry phase mechanism. The EEF is generated
by the dynamics of noncollinear spin configurations and becomes nonzero even
in one dimension. Although the EEF has been studied for several one-
dimensional chiral magnets, most of the theoretical studies were performed in
limited situations with respect to the strength and direction of the magnetic
fields. Furthermore, the effect of edges of the system has not been clarified,
whereas it can be crucial in nano- and micro-scale samples. Here, we perform a
comprehensive theoretical study on the momentum-frequency profile of the EEF
in a one-dimensional chiral magnet using the Landau-Lifshitz-Gilbert equation
while changing the strength and direction of the static and AC magnetic fields
for both bulk and finite-size chains with edges. From the bulk calculations
under the periodic boundary condition, we find that the EEF is resonantly
enhanced at the magnetic resonance frequencies; interestingly, the higher
resonance modes are more clearly visible in the frequency profile of the EEF
response than in the magnetic one. Furthermore, we show that the EEF is
amplified along with the solitonic feature of the spin texture introduced by
the static magnetic field perpendicular to the chiral axis. We also show that
the static magnetic field parallel to the chiral axis drives the EEF in the
field direction, in addition to much slower drift motion in the opposite
direction associated with the Archimedean screw dynamics, suggesting a DC
electric current generation. For the finite-size chains under the open
boundary condition, we find additional resonance modes localized at the edges
of the system that are also more clearly visible in the EEF response than the
magnetic one. Moreover, we show that a substantial EEF is generated from the
edges even in the fully-polarized forced-ferromagnetic phase, although it is
absent in the bulk case. Our results reveal that the emergent electric
phenomena in one-dimensional chiral magnets can be tuned by the magnetic field
and the sample size, and provide not only a good probe of the magnetic
resonances but also a platform for the applications to electronic devices.

##  I Introduction

The Berry phase — a phase factor acquired in an adiabatic motion of quantum
particles [1] brings about a fictitious electromagnetic field [2, 3], which
leads to intriguing quantum transport and optical phenomena for electrons in
solids. For instance, the fictitious magnetic and electric fields arising from
the Berry phase in momentum space bring about a quantum Hall effect [4] and a
quantized charge pumping [5], respectively. Meanwhile, the Berry phase in real
space is also of importance, especially in magnets, where the fictitious
electromagnetic fields, often called the emergent electromagnetic fields, are
generated by noncollinear and noncoplanar spin textures [6, 3, 7, 8, 9]. The
emergent magnetic field arises as a fictitious magnetic flux through a
plaquette when the surrounding spins are noncoplanar, i.e., when the scalar
spin chirality is nonzero, and hence, it requires two- or three-dimensional
noncoplanar spin textures. The typical example is found in magnetic skyrmions
[10, 11, 12, 13], which give rise to the topological Hall effect via the
emergent magnetic field [14, 15, 16, 17, 18]. In contrast, the emergent
electric field (EEF) does not require noncoplanar spin configurations; it
arises from time evolution of noncollinear spin textures, and hence, can be
generated even in one-dimensional systems [19, 20, 21, 22, 6]. Therefore, the
emergent electric phenomena arise in a much wider range of magnetic materials
than the emergent magnetic phenomena.

Quantum phenomena arising from the EEF in magnets have been intensively
studied over the past decade. Experimental detection of the EEF has been done
for field-induced motion of a magnetic domain wall [23], ferromagnetic
resonance in a patterned ferromagnetic film [24], and gyrating motion of a
magnetic vortex [25]. Recently, the EEF has attracted renewed interest in the
dynamics induced by an electric current. It was pointed out that the EEF by a
current-induced motion of swirling spin textures gives rise to an inductance
[26], and such behavior was observed in helical magnets [27] even at room
temperature [28].

The central question that we address in this study is how to enhance the EEF
in magnets by an external magnetic field. To clarify the fundamental behavior
of the EEF, we will focus on a simple model for one-dimensional chiral
magnets. Thus far, the EEF in one-dimensional chiral magnets has been
theoretically studied, mostly by using the collective coordinate method in
continuum space [29, 30, 31]. However, such analytical studies were limited
with respect to the strength and direction of the magnetic fields and
comprehensive studies have not yet to be done, even though the system shows a
variety of spin textures and their magnetic excitations depending on the
fields. In addition, as the EEF is generated by the dynamics of spin textures,
it is important to elucidate the relationship between the EEF and the magnetic
excitations, but a large part of it also remains unexplored. To clarify these
issues, it is desired to systematically study the dependence of the EEF on the
static and dynamical magnetic fields beyond the previous studies.

In addition, we aim at investigating the contributions to the EEF from not
only bulk but also edges of the system, since the latter can be crucial in
experiments for nano- and micro-scale samples. Effect of edges has been
discussed for both static and dynamical properties in magnets. For instance,
in one-dimensional chiral magnets, the magnetization varies discretely as a
function of the magnetic field in finite-size systems with edges [32, 33, 34,
35, 36]. It was also shown that the edges affect the magnetic resonance and
generate additional edge modes [37]. Given the edge effects on the spin
textures and the magnetic excitations, the EEF is expected to be strongly
influenced by the presence of edges as well, but the systematic study has not
yet been conducted.

In this paper, we perform a systematic theoretical study of the EEF in a one-
dimensional chiral magnet. By numerical simulations of the real-space and
real-time spin dynamics based on the Landau-Lifshitz-Gilbert (LLG) equation,
we clarify the momentum-frequency profile of the EEF while changing the
strength and direction of the static and AC magnetic fields for the systems
under the periodic boundary condition (PBC) and the open boundary condition
(OBC). For the bulk response calculated under the PBC, we show that the EEF
becomes prominent at the magnetic resonance frequencies and it is enlarged
along with the solitonic feature enhanced by the perpendicular component of
the static magnetic field to the chiral axis, regardless of the direction of
the AC magnetic field. We find that the higher frequency resonance modes are
more clearly visible in the frequency profile of the EEF response than the
magnetic one. Furthermore, by analyzing the spatiotemporal profiles, we also
show that the parallel component of the static magnetic field drives the EEF
in the field direction due to the magnon propagation, in addition to much
slower drift to the opposite direction due to the Archimedean screw dynamics
[38]. Meanwhile, for the edge contributions calculated under the OBC, we find
additional resonance modes of the EEF localized at the system edges, whose
response can be comparable to or larger than the bulk ones, depending on the
system size, and again more visible than the magnetic response.

In addition, we show that the EEF is generated even in the fully-polarized
forced-ferromagnetic (FFM) phase in the presence of the edges. These
systematic analyses provide comprehensive understanding of the EEF associated
with the magnetic resonance in the one-dimensional chiral magnet, which would
be useful for further exploration of emergent electric phenomena.

The rest of the paper is organized as follows. In Sec. II, we introduce the
model (Sec. II.1), the numerical method (Sec. II.2), and the definitions of
the physical quantities (Sec. II.3) used in the following analyses. In Sec.
III, we show the results for the bulk contribution obtained under the PBC. We
first show the complex admittance of the EEF for various combinations of the
static and AC magnetic fields (Sec. III.1), and then, compare the results with
the dynamical spin susceptibility (Sec. III.2). We also show the
spatiotemporal profiles of the spin textures and the EEF in the resonance
modes (Sec. III.3). In Sec. IV, we show the results for the edge contribution
obtained under the OBC in the similar manner in Sec. III. We discuss the
results in Sec. V. Section VI is devoted to the summary of this paper.

##  II Model and method

In this section, we introduce the model for a one-dimensional chiral magnet
and the numerical method to study the dynamics of the model. In Sec. II.1, we
introduce the model Hamiltonian with the setting of parameters and two types
of boundary conditions. We describe the method based on the LLG equation in
Sec. II.2 and the definitions of the quantities to be measured in Sec. II.3.

Figure 1:  Schematic picture of the setup in this study. The arrows represent
spin configurations in the model in Eq. (1); the color denotes the $S_{z}$
component of spins. The pale curves attached to the arrowheads represent the
trajectories of the spins in the time evolution under the static magnetic
field ${\bf h}^{\rm stat}$ and the AC magnetic field ${\bf h}^{\rm AC}$
depicted in the inset.

###  II.1 Model

In this study, we consider a one-dimensional chiral magnet, which is described
by the time-dependent Hamiltonian given by

$\displaystyle\mathcal{H}(t)=\sum_{l}$ $\displaystyle\bigl{[}-J{\bf
S}_{l}(t)\cdot{\bf S}_{l+1}(t)-D\hat{\bf x}\cdot\left({\bf S}_{l}(t)\times{\bf
S}_{l+1}(t)\right)$ (1) $\displaystyle~{}+{\bf h}(t)\cdot{\bf
S}_{l}(t)\bigr{]},$

where $t$ is time, and ${\bf S}_{l}(t)$ represents the classical spin at site
$l$ and time $t$ with $|{\bf S}_{l}(t)|=1$. The first and second terms in the
square brackets denote the Heisenberg and Dzyaloshinskii-Moriya (DM)
interactions, respectively. The DM vector is taken along the chain direction
parallel to the unit vector $\hat{\bf x}$. Hereafter, we set the energy scale
as $J=1$ and the lattice constant as unity, and take
$D=\tan\left(\frac{\pi}{10}\right)$. This stabilizes a helical spin structure
in the equilibrium state at zero magnetic field, in which the spins rotate in
the $yz$ plane with a period of $20$ lattice sites. The last term in Eq. (1)
describes the Zeeman interaction, where ${\bf h}(t)$ represents a time-
dependent external magnetic field in the unit of $g\mu_{\rm B}$, where $g$ is
the electron g-factor and $\mu_{\rm B}$ is the Bohr magneton. Note that we
take the convention of a positive sign for this term. In the following, we
consider both static and AC magnetic fields, denoted by ${\bf h}^{\rm stat}$
and ${\bf h}^{\rm AC}(t)$, respectively, as

$\displaystyle{\bf h}(t)={\bf h}^{\rm stat}+{\bf h}^{\rm AC}(t).$ (2)

See Fig. 1 for the setup of the model.

When only the static field is applied, the model in Eq. (1) stabilizes a
conical spin structure, a chiral soliton lattice (CSL) and their mixture, in
addition to the FFM, depending on the field strength and direction [39, 40,
41, 42, 43]. In this study, without loss of generality, we consider the static
magnetic field in the $xz$ plane, i.e., ${\bf h}^{\rm stat}=(h^{\rm
stat}_{x},0,h^{\rm stat}_{z})$. Then, by increasing $h^{\rm stat}_{x}$ with
$h^{\rm stat}_{z}=0$, the spins on the $yz$ plane in the zero-field helical
state cant uniformly in the $x$ direction to form the conical spin structure.
In this case, the magnetic period is unchanged, until the phase transition to
the FFM phase. In contrast, when $h^{\rm stat}_{x}=0$, $h^{\rm stat}_{z}$
introduces a solitonic feature in the spin structure, leading to the CSL. In
this case, the magnetic period increases with $h^{\rm stat}_{z}$ and diverges
at the transition to the FFM state. When both $h^{\rm stat}_{x}$ and $h^{\rm
stat}_{z}$ are nonzero, a complicated spin texture with a mixture of the
conical state and the CSL is stabilized, as exemplified in Fig. 1 . We will
study how the AC field modulates these spin textures and generates the EEF.

In the following calculations, to clarify not only bulk but also edge
contributions, we compare the results for the systems under the PBC and the
OBC. In both cases, we set the system size as $L=10^{3}$ spins, except for the
study of the system size dependence in the OBC case in Sec. IV.2. For the PBC
case, we confirm that $L=10^{3}$ is sufficiently large to study the bulk
contributions. The sum in Eq. (1) is taken for $l=0,1,\cdots,L-1$, where we
impose ${\bf S}_{L}(t)={\bf S}_{0}(t)$ for the PBC case, whereas ${\bf
S}_{L}(t)=0$ for the OBC case.

###  II.2 Landau-Lifshitz-Gilbert equation

We study the real-time dynamics of the model in Eq. (1) by using the LLG
equation given by [44, 45]

$\displaystyle\frac{\partial{\bf S}_{l}(t)}{\partial
t}=\frac{1}{1+\alpha^{2}}$ $\displaystyle\left[-{\bf S}_{l}(t)\times{\bf
h}^{\rm eff}_{l}(t)\right.$ (3) $\displaystyle\ \left.+\alpha{\bf
S}_{l}(t)\times\left({\bf S}_{l}(t)\times{\bf h}^{\rm
eff}_{l}(t)\right)\right],$

where $\alpha$ is the Gilbert damping and ${\bf h}^{\rm eff}_{l}(t)$ is the
mean magnetic field at time $t$ defined by

$\displaystyle{\bf h}^{\rm
eff}_{l}(t)=\frac{\partial\mathcal{H}(t)}{\partial{\bf S}_{l}(t)}$
$\displaystyle=$ $\displaystyle-J\bigl{(}{\bf S}_{l+1}(t)+{\bf
S}_{l-1}(t)\bigr{)}$ (4) $\displaystyle+D\hat{\bf x}\times\bigl{(}{\bf
S}_{l+1}(t)-{\bf S}_{l-1}(t)\bigr{)}+{\bf h}(t).$

Note that the length constraint of $|{\bf S}_{l}(t)|=1$ is deferred only when
taking the derivative of the Hamiltonian with respect to ${\bf S}_{l}(t)$.
Here, we impose ${\bf S}_{-1}(t)={\bf S}_{L-1}(t)$ and ${\bf S}_{-1}(t)=0$ for
the PBC and OBC cases, respectively. We numerically solve Eq. (3) by using the
fourth-order Runge-Kutta method with a time step $\Delta t=0.02$. In the
following calculations, we take $\alpha=0.04$, which is a typical value for
ferromagnetic metals [46, 47, 48, 49].

###  II.3 Physical quantity

The EEF, which is generated by the time evolution of noncollinear spin
textures, is calculated as

$\displaystyle\bar{E}^{\rm em}(t)=\frac{1}{L}\sum_{l}E^{\rm em}_{l}(t),$ (5)

with

$\displaystyle E^{\rm em}_{l}(t)={\bf S}_{l}(t)\cdot\left(\hat{\delta}{\bf
S}_{l}(t)\times\frac{\partial{\bf S}_{l}(t)}{\partial t}\right),$ (6)

from the spin structure obtained by numerically solving the LLG equation in
Eq. (3). In Eq. (6), the spatial derivative $\frac{\partial{\bf
S}_{l}}{\partial x}$ in continuum space [6, 3, 7, 8, 9] is calculated by the
discrete difference $\hat{\delta}{\bf S}_{l}(t)=\frac{1}{2}\left({\bf
S}_{l+1}(t)-{\bf S}_{l-1}(t)\right)$. For the edges in the OBC case, we use
$\hat{\delta}{\bf S}_{0}(t)=-\frac{3}{2}{\bf S}_{0}(t)+2{\bf
S}_{1}(t)-\frac{1}{2}{\bf S}_{2}(t)$ and $\hat{\delta}{\bf
S}_{L-1}(t)=\frac{3}{2}{\bf S}_{L-1}(t)-2{\bf S}_{L-2}(t)+\frac{1}{2}{\bf
S}_{L-3}(t)$ to reduce the discretization errors [50].

We compute the complex admittance of the EEF defined as

$\displaystyle\chi_{E,\mu}(\omega)=\frac{\bar{E}^{\rm
em}(\omega)}{h_{\mu}^{\rm AC}(\omega)},$ (7)

where $\mu=x,y,z$ and the Fourier component of the quantity $\mathcal{O}(t)$
is obtained by

$\mathcal{O}(\omega)=\frac{1}{N_{t}}\sum_{n=0}^{N_{t}-1}\mathcal{O}(t_{n})e^{-i\omega
t_{n}}.$ (8)

Here, we measure the quantity every 50 time steps, and the time at $n$th
measurement and the total number of measurements are denoted by
$t_{n}=50n\Delta t=n$ and $N_{t}$, respectively. We also calculate the
dynamical spin susceptibility defined as

$\displaystyle\chi_{S,\nu\mu}(\omega)=\frac{\Delta\bar{S}_{\nu}(\omega)}{h_{\mu}^{\rm
AC}(\omega)},$ (9)

where $\Delta\bar{\bf S}(t)=\frac{1}{L}\sum_{l}\left[{\bf S}_{l}(t)-{\bf
S}_{l}(0)\right]$ is time variation of the magnetization.

In the actual calculations of $\chi_{E,\mu}(\omega)$ and
$\chi_{S,\nu\mu}(\omega)$ in Secs. III and IV, we apply a pulse magnetic field
${\bf h}^{\rm pulse}(t)$ instead of ${\bf h}^{\rm AC}(t)$ in Eq. (2). This
enables us to obtain the whole spectrum at once, instead of the study of
steady states for each $\omega$. We set

$\displaystyle{\bf h}^{\rm pulse}(t)=\left\\{\begin{array}[]{c}\Delta
h\hat{\bm{\mu}}\quad(0\leq t<1),\\\ 0\quad(1\leq t\leq
6000),\end{array}\right.$ (12)

where $\Delta h=0.002$ and $\hat{\bm{\mu}}$ denotes the unit vector in the
direction of $\mu=x,y,z$. By Fourier transforming the responses, the spectra
of $\chi_{E,\mu}(\omega)$ and $\chi_{S,\nu\mu}(\omega)$ are obtained by Eqs.
(7) and (9) with the substitution of $h_{\mu}^{\rm AC}(\omega)$ by
$h_{\mu}^{\rm pulse}(\omega)$. Meanwhile, for the calculations of the
spatiotemporal profiles in Secs. III.3 and IV.3, we apply an AC magnetic field
with a frequency $\omega$ in the direction of $\hat{\bm{\mu}}$ given by

$\displaystyle{\bf h}^{\rm AC}(t)=\Delta
h\left(1-e^{-t/t_{0}}\right)\sin(\omega t)\hat{\bm{\mu}}\quad(0\leq t\leq
10000),$ (13)

where $t_{0}=50$ is introduced to suppress the initial disturbance by
switching on the field.

##  III Result: bulk properties

In this section, we show the results of the bulk EEF obtained from the
calculations for the system with the PBC. In Sec. III.1, we present the
maximum values of $|\chi_{E,\mu}(\omega)|$ on the plane of $h_{x}^{\rm stat}$
and $h_{z}^{\rm stat}$, and the typical frequency spectra. In Sec. III.2, we
discuss the relation between the EEF and the magnetic excitations by comparing
$\chi_{E,\mu}(\omega)$ and $\chi_{S,\nu\mu}(\omega)$. In Sec. III.3, we show
the spatiotemporal profiles of the spin textures and the EEF of the resonance
modes.

###  III.1 Complex admittance of the EEF

Figure 2:  Maximum values of $|\chi_{E,\mu}(\omega)|$, $\chi_{E,\mu}^{\rm
max}$, calculated for the model in Eq. (1) with the PBC for (a) ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$ ($\mu=x$), (b) ${\bf h}^{\rm AC}\parallel\hat{\bf y}$
($\mu=y$), and (c) ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ ($\mu=z$), as
represented in the insets. The frequency dependence of
$|\chi_{E,\mu}(\omega)|$ along the yellow dashed lines in (a), (b), and (c)
are shown in (d), (e), and (f), respectively: (d) ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$ with $h_{x}^{\rm stat}=0$, (e) ${\bf h}^{\rm
AC}\parallel\hat{\bf y}$ with $h_{x}^{\rm stat}=0$, and (f) ${\bf h}^{\rm
AC}\parallel\hat{\bf z}$ with $h_{x}^{\rm stat}=0.06$. The orange circles and
crosses in (d), (e), and (f) indicate the parameters for which the real-space
behaviors are presented in Fig. 4.

We show the results of the complex admittance $\chi_{E,\mu}(\omega)$ in Eq.
(7) for three different directions of the AC magnetic field: ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$ ($\mu=x$), ${\bf h}^{\rm AC}\parallel\hat{\bf y}$
($\mu=y$), and ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ ($\mu=z$). Let us first
discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf x}$. In Fig. 2(a),
we show the maximum values of $|\chi_{E,x}(\omega)|$, denoted by
$\chi_{E,x}^{\rm max}$, on the plane of $h^{\rm stat}_{x}$ and $h^{\rm
stat}_{z}$. We find that $\chi_{E,x}^{\rm max}$ has a nonzero value in a
region of $0\leq h_{x}^{\rm stat}\lesssim 0.103$ and $0\leq h_{z}^{\rm
stat}\lesssim 0.063$, where the system in the static magnetic field shows a
noncollinear spin texture, i.e., the CSL for $h_{x}^{\rm stat}=0$, conical for
$h_{z}^{\rm stat}=0$, and their mixture otherwise. Within this region,
$\chi_{E,x}^{\rm max}$ increases with increasing $h^{\rm stat}_{z}$, but
decreases with increasing $h^{\rm stat}_{x}$ ; the EEF is maximally generated
at $(h^{\rm stat}_{x},h^{\rm stat}_{z})\simeq(0,0.06)$, which is close to the
phase transition from the CSL to the FFM. Meanwhile, outside this region, the
spins are fully polarized by the magnetic field and $\chi_{E,x}(\omega)$
vanishes, indicating that the FFM phase does not generate the EEF.

We plot the spectrum of $|\chi_{E,x}(\omega)|$ in Fig. 2(d) while changing
$h^{\rm stat}_{z}$ at $h^{\rm stat}_{x}=0$, along the yellow dashed line in
Fig. 2(a). $|\chi_{E,x}(\omega)|$ shows several peaks, on top of an almost
$\omega$-independent contribution (the origin will be discussed in Sec.
III.2). The peak frequencies decrease but their intensities increase as
$h^{\rm stat}_{z}$ approaches the critical value for the FFM transition,
$h_{z}^{\rm stat,c}\simeq 0.063$. $|\chi_{E,x}(\omega)|$ takes the maximum
value at the lowest-frequency peak at $\omega\simeq 0.1$, which we denote
$\omega^{\rm bulk}_{1}$. We find the second peak above $\omega^{\rm
bulk}_{2}\gtrsim 0.2$, but the higher ones are difficult to see in the contour
plot [see the $\omega$ profile in Fig. 3(a)]. These peak structures are
related to magnetic resonance, as will be discussed in Sec. III.2.

Next, we discuss $\chi_{E,y}^{\rm max}$ for ${\bf h}^{\rm AC}\parallel\hat{\bf
y}$ shown in Fig. 2(b). The EEF is again maximally generated in the CSL phase
for $h_{x}^{\rm stat}=0$, but at a slightly lower $h_{z}^{\rm stat}\simeq
0.05$ below the critical value for the FFM transition. The intensity is
stronger than that in Fig. 2(a). We find another weaker peak at $(h^{\rm
stat}_{x},h^{\rm stat}_{z})\simeq(0.07,0.04)$, which is a remnant of the peak
for the case of ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ discussed below for
Fig. 2(c). We note that $\chi_{E,y}^{\rm max}$ is strongly reduced in the
conical state at $h^{\rm stat}_{z}=0$ in contrast to the case of ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$ in Fig. 2(a). This is due to twofold rotational
symmetry about the $x$ axis accompanied by time translation by a half period
of ${\bf h}^{\rm AC}$, $\frac{\pi}{\omega}$ [51]. The spectrum of
$|\chi_{E,y}(\omega)|$ for $h^{\rm stat}_{x}=0$ [the yellow dashed line in
Fig. 2(b)] is shown in Fig. 2(e). We can identify three sharp peak structures
within this $\omega$ range, rather more easily than in Fig. 2(d). We note
that, in contrast to the case of Fig. 2(d), there is no $\omega$-independent
contribution, and that the peak intensities are maximized at $h^{\rm
stat}_{z}\simeq 0.05$, and decreased for $h^{\rm stat}_{z}\gtrsim 0.05$.

Lastly, we show the results for ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ in
Figs. 2(c) and 2(f). In this case, the EEF is maximally generated at $(h^{\rm
stat}_{x},h^{\rm stat}_{z})\simeq(0.06,0.045)$, which is close to the
subdominant maximum in Fig. 2(b). Similar to the case of Fig. 2(b),
$\chi_{E,z}^{\rm max}$ is strongly suppressed for $h^{\rm stat}_{z}=0$ by
symmetry [51]. Moreover, in contrast to the cases of Figs. 2(a) and 2(b),
$\chi_{E,z}^{\rm max}$ is zero for $h^{\rm stat}_{x}=0$ due to twofold
rotational symmetry about the $z$ axis. Under these symmetric constraints, the
peak of ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ appears in the intermediate
region near the phase boundary to the FFM state. Figure 2(f) shows
$|\chi_{E,z}(\omega)|$ for $h^{\rm stat}_{x}=0.06$, along the yellow dashed
line through the peak of $\chi_{E,z}^{\rm max}$ in Fig. 2(c). We find several
peaks without an $\omega$-independent background, as in the case of Fig. 2(e).
We note that each peak splits into two, as most clearly seen at the lowest-
frequency peak $\omega^{\rm bulk}_{1}$. This is understood from the magnon
dispersion for the noncoplanar spin state, but the details will be discussed
elsewhere.

###  III.2 Comparison with magnetic excitations

Figure 3:  Comparison between $|\chi_{E,\mu}(\omega)|$ and the imaginary part
of the dynamical spin susceptibility, ${\rm Im}\,\chi_{S,\mu\mu}(\omega)$, for
several values of $h_{z}^{\rm stat}$: (a) ${\bf h}^{\rm AC}\parallel\hat{\bf
x}$ with $h_{x}^{\rm stat}=0$, (b) ${\bf h}^{\rm AC}\parallel\hat{\bf y}$ with
$h_{x}^{\rm stat}=0$, and (c) ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ with
$h_{x}^{\rm stat}=0.06$ for the model in Eq. (1) with the PBC. The data for
the FFM phases are denoted by the dotted lines and scaled for better
visibility.

In this section, we discuss the results of $|\chi_{E,\mu}(\omega)|$ in
comparison with the magnetic excitation spectrum. Figure 3 displays the
comparisons between $|\chi_{E,\mu}(\omega)|$ and the imaginary part of the
dynamical spin susceptibility, ${\rm Im}\,\chi_{S,\mu\mu}(\omega)$, calculated
by Eq. (9) for different field configurations: ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$ ($\mu=x$) and $h^{\rm stat}_{x}=0$ in Fig. 3(a),
${\bf h}^{\rm AC}\parallel\hat{\bf y}$ ($\mu=y$) and $h^{\rm stat}_{x}=0$ in
Fig. 3(b), and ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ ($\mu=z$) and $h^{\rm
stat}_{x}=0.06$ in Fig. 3(c) [on the yellow dashed lines in Figs. 2(a), 2(b),
and 2(c), respectively]. In each figure, the different colors represent the
different values of $h^{\rm stat}_{z}$. The solid lines denote the results in
the noncollinear spin phase where $|\chi_{E,\mu}(\omega)|$ is nonzero in Figs.
2(a)-2(c), while the dashed lines denote the results in the outside, i.e., the
FFM phase where $|\chi_{E,\mu}(\omega)|=0$.

Let us first discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf x}$
and $h^{\rm stat}_{x}=0$. As shown in Fig. 3(a), $|\chi_{E,x}(\omega)|$ and
${\rm Im}\,\chi_{S,xx}(\omega)$ have peaks at the same frequencies, except for
$h^{\rm stat}_{z}=0.07$ where the system is in the FFM phase and
$|\chi_{E,x}(\omega)|$ vanishes. This indicates that the EEF is amplified at
the magnetic resonance frequencies in the noncollinear spin phase. The
resonance frequencies decrease as $h^{\rm stat}_{z}$ increases, which is
consistent with the previous studies for the magnetic resonance in the CSL
phase [52, 53]. Interestingly, the higher-$\omega$ peaks are more clearly
visible in $|\chi_{E,x}(\omega)|$ rather than those in ${\rm
Im}\,\chi_{S,xx}(\omega)$. This indicates that the EEF response has an
advantage to observe the higher-$\omega$ excitations than the magnetic one.
This is presumably due to an additional $\omega$-linear factor from
$\frac{\partial{\bf S}_{l}(t)}{\partial t}$ in Eq. (6); see also Eq. (21) in
Appendix A.

While $|\chi_{E,x}(\omega)|$ accompanies an almost $\omega$-independent
contribution (see below), the peak heights measured from it grow from zero as
$h^{\rm stat}_{z}$ increases from zero, in good correspondence with the growth
of the peaks in ${\rm Im}\,\chi_{S,xx}(\omega)$. We note that the growth rate
of the peaks in $|\chi_{E,x}(\omega)|$ decreases as $h^{\rm stat}_{z}$
approaches the critical field $h_{z}^{\rm stat,c}$, in contrast to the rapid
increase of the growth rate in ${\rm Im}\,\chi_{S,xx}(\omega)$. This feature
is qualitatively understood from the decreases of $\omega_{1}^{\rm bulk}$ and
the vector spin chirality that bridges $\chi_{E,x}(\omega)$ and $\chi_{S,xx}$.
Here, the vector chirality in the ground state is defined as

$\displaystyle\bar{\bf C}^{\rm vc}=\frac{1}{L}\sum_{l=0}^{L-1}{\bf
S}_{l}(0)\times{\bf S}_{l+1}(0).$ (14)

This quantity measures the overall spin noncollinearity, and hence is rapidly
reduced as $h^{\rm stat}_{z}$ approaches $h_{z}^{\rm stat,c}$ where the spin
texture acquires a strong solitonic feature. This and the decrease of
$\omega_{1}^{\rm bulk}$ suppress the growth rate of the peaks in
$|\chi_{E,x}(\omega)|$; see Eq. (21) in Appendix A.

The $\omega$-independent contributions in $|\chi_{E,x}(\omega)|$ is explained
by the consideration of the large-$\omega$ behavior as follows. For larger
$\omega$, we can derive the relation

$\displaystyle\chi_{E,\mu}(\omega)\simeq-\frac{\alpha}{1+\alpha^{2}}\bar{\bf
C}^{\rm vc}\cdot\hat{\bm{\mu}}.$ (15)

See Appendix B for the derivation. Thus, the large-$\omega$ behavior is
independent of $\omega$, and it is given by the overall spin noncollinearity
$\bar{\bf C}^{\rm vc}\cdot\hat{\bf x}$. Since $\bar{\bf C}^{\rm
vc}\cdot\hat{\bf x}$ decreases with increasing $h^{\rm stat}_{z}$, the
$\omega$-independent component of $|\chi_{E,x}(\omega)|$ is reduced, as shown
in the upper panel of Fig. 3(a). We note that this contribution corresponds to
the collective dynamics of spin textures called coherent sliding dynamics
[29]. We will return to this point in Sec. III.3.

Next, we discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf y}$ and
$h^{\rm stat}_{x}=0$. In this case also, $|\chi_{E,y}(\omega)|$ shows peaks
corresponding to the magnetic resonances in ${\rm Im}\,\chi_{S,yy}(\omega)$,
and the higher-$\omega$ modes are more clearly visible in
$|\chi_{E,y}(\omega)|$ than ${\rm Im}\,\chi_{S,yy}(\omega)$, as shown in Fig.
3(b). In contrast to the case with ${\bf h}^{\rm AC}\parallel\hat{\bf x}$,
however, the peak heights in $|\chi_{E,y}(\omega)|$ increase for $0\leq h^{\rm
stat}_{z}\lesssim 0.055$ but decrease for $h^{\rm stat}_{z}\gtrsim 0.055$,
while ${\rm Im}\,\chi_{S,yy}(\omega)$ shows resonance peaks even for $h^{\rm
stat}_{z}=0$ and grows monotonically with $h^{\rm stat}_{z}$. This
nonmonotonic behavior of $|\chi_{E,y}(\omega)|$ is already observed in Fig.
2(b). The decrease of $|\chi_{E,y}(\omega)|$ can be ascribed to the slow
increase of ${\rm Im}\,\chi_{S,yy}(\omega)$ in the lower panel of Fig. 3(b),
in addition to the decreases of $\bar{C}^{\rm vc}_{x}$ and $\omega_{1}^{\rm
bulk}$ discussed above. Furthermore, there is no $\omega$-independent
contributions, since $\bar{C}^{\rm vc}_{y}=0$ in Eq. (14), leading to
$\chi_{E,y}(\omega)\simeq 0$ in Eq. (15) for large $\omega$ 111 The long tails
of $|\chi_{E,y}(\omega)|$ in the large-$\omega$ region are due to the
discretization in the numerical computation of the spatial derivative in Eq.
(6). .

In Fig. 3(c), we show the results for ${\bf h}^{\rm AC}\parallel\hat{\bf z}$
and $h^{\rm stat}_{x}=0.06$. Similar to the above two cases, the
higher-$\omega$ peaks are more clearly visible for $|\chi_{E,z}(\omega)|$ than
${\rm Im}\,\chi_{S,zz}(\omega)$. The lowest-energy peak in
$|\chi_{E,z}(\omega)|$, which shows a shoulder-like feature reflecting the
splitting mentioned in Sec. III.1, grows with $h^{\rm stat}_{z}$, as in the
case of ${\bf h}^{\rm AC}\parallel\hat{\bf x}$. This is understood from the
increase of $\bar{C}_{z}^{\rm vc}$ with $h^{\rm stat}_{z}$; see again Eq. (21)
in Appendix A. In contrast, the peak heights in ${\rm
Im}\,\chi_{S,zz}(\omega)$ are almost unchanged for $h^{\rm stat}_{z}$ since
the solitonic feature in the spin texture is not well developed in this range
of $h^{\rm stat}_{z}$.

###  III.3 Real-space behavior

Figure 4:  Real-space spin configurations (top and middle) and real-space
distributions of the EEF (bottom) in the steady states under the AC magnetic
field ${\bf h}^{\rm AC}$. (a) and (b), (c) and (d), and (e) and (f) are for
${\bf h}^{\rm AC}\parallel\hat{\bf x}$, ${\bf h}^{\rm AC}\parallel\hat{\bf
y}$, and ${\bf h}^{\rm AC}\parallel\hat{\bf z}$, respectively, at (a)
$(h_{x}^{\rm stat},h_{z}^{\rm stat})=(0,0)$ and $\omega=0.143$, (b)
$(h_{x}^{\rm stat},h_{z}^{\rm stat})=(0,0.06)$ and $\omega=0.103$, (c)
$(h_{x}^{\rm stat},h_{z}^{\rm stat})=(0,0)$ and $\omega=0.143$, (d)
$(h_{x}^{\rm stat},h_{z}^{\rm stat})=(0,0.05)$ and $\omega=0.124$, (e)
$(h_{x}^{\rm stat},h_{z}^{\rm stat})=(0.06,0)$ and $\omega=0.131$, and (f)
$(h_{x}^{\rm stat},h_{z}^{\rm stat})=(0.06,0.04)$ and $\omega=0.123$. All the
frequencies are set at the values for the lowest-$\omega$ resonance mode; the
values are denoted by the orange crosses and circles in Figs. 2(d)–2(f). The
results are shown for $60$ sites at the center of the $1000$-site system with
the PBC. In each figure, the top panel displays the spin configurations by the
arrows with the lines representing the trajectories of the arrowheads during a
single period of ${\bf h}^{\rm AC}$. Meanwhile, the middle panel displays the
spin components $S_{l,x}(t)$, $S_{l,y}(t)$, and $S_{l,z}(t)$ by the red,
green, and blue lines, respectively, and the bottom panel displays the EEF at
each site, $E^{\rm em}_{l}(t)$, during the corresponding period of ${\bf
h}^{\rm AC}$. The color and gray-scale intensities in the middle and bottom
panels, respectively, increase with time evolution in the single period of
${\bf h}^{\rm AC}$. In the middle panel of (a), $S_{l,x}(t)$ is multiplied by
a factor of 300 for better visibility.  Figure 5:  Real-space and real-time
distribution of the $z$ component of spins $S_{l,z}(t)$ (left) and $E^{\rm
em}_{l}(t)$ (right) within 10 periods of ${\bf h}^{\rm AC}$ on the plane of
$l$ and $t$: (a) and (b) correspond to Figs. 4(b) and 4(e), respectively. The
insets show the results for a longer time window.

In this section, we present the spatiotemporal profiles of spins ${\bf
S}_{l}(t)$ and EEF $E_{l}^{\rm em}(t)$ in Eq. (6). Figure 4 displays the
results for the lowest-$\omega$ resonance mode: The data are calculated for
the parameters indicated by the circles and crosses in Figs. 2(d)–2(f). In
each figure of Fig. 4, the top panel shows the spin precession motions, the
middle panel shows the time evolution of each spin component, and the bottom
panel shows the time evolution of $E_{l}^{\rm em}(t)$ during a single period
of ${\bf h}^{\rm AC}$ for $60$ sites at the center of the $1000$-site system
with the PBC.

First, we discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf x}$ and
$h^{\rm stat}_{x}=0$, corresponding to Fig. 2(d). Figure 4(a) is for the
helical state at $(h^{\rm stat}_{x},h^{\rm stat}_{z})=(0,0)$ and
$\omega=0.143$, indicated by the cross in Fig. 2(d). In this case, the spin
dynamics is strongly suppressed, leaving very weak oscillations, as visible in
the enlarged plot of $S_{l,x}(t)$ in the middle panel of Fig. 4(a). We note
that such oscillations are induced not only at the resonance frequency but
also for general $\omega$, and they are called the coherent sliding dynamics
[29]. In this situation, we obtain a small but spatially uniform $E_{l}^{\rm
em}(t)$, as shown in the bottom panel of Fig. 4(a). The spatial average in Eq.
(5), however, can be comparably large to those for the other resonance modes,
which leads to the $\omega$-independent contribution in Fig. 3(a).

Figure 4(b) is also for ${\bf h}^{\rm AC}\parallel\hat{\bf x}$, but at
$(h^{\rm stat}_{x},h^{\rm stat}_{z})=(0,0.06)$ and $\omega=0.103$, indicated
by the cirlcle in Fig. 2(d). In contrast to the above helical case, the spin
dynamics in this CSL state is resonantly activated, especially in the regions
between the solitons with $S_{l,z}(t)\simeq+1$, as shown in the top and middle
panels of Fig. 4(b). In contrast, the EEF is largely generated around the
solitons, as shown in the bottom panel. This trend is understood from the fact
that the vector spin chirality is large near the solitons. The spatially
averaged EEF takes a large value, leading to the resonance peak of
$|\chi_{E,x}(\omega)|$ in Figs. 2(d) and 3(a). We note that in this resonance
with $h^{\rm stat}_{x}=0$, the spin texture is just oscillating around the
original position in real space, and the EEF behaves like a standing wave, as
shown in Fig. 5(a). This is due to twofold rotational symmetry about the $z$
axis with time translation by $\frac{\pi}{\omega}$.

Next, we discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf y}$ and
$h^{\rm stat}_{x}=0$, corresponding to Fig. 2(e). Figure 4(c) is for the
helical state at $h^{\rm stat}_{z}=0$ and $\omega=0.143$, indicated by the
cross in Fig. 2(e). In this case, the spins pointing in the $\pm y$ directions
do not precess and constitute the nodes of oscillating ${\bf S}_{l}(t)$, and
those pointing in the $\pm z$ directions are the antinodes, where the spin
precessions become maximum. The EEF vanishes at the nodes, whereas it is
maximally generated at the antinodes reflecting the large spin precessions.
Note, however, that the average of the EEF is strongly suppressed due to
twofold rotational symmetry about the $x$ axis accompanied by
$\frac{\pi}{\omega}$ time translation of ${\bf h}^{\rm AC}$ [51]. Meanwhile,
Fig. 4(d) is for a CSL state at $h^{\rm stat}_{z}=0.05$ with $\omega=0.124$
corresponding to the circle in Fig. 2(e). In this case, in the regions between
(near) the solitons, the amplitude of the spin precessions is enhanced
(suppressed), while $E_{l}^{\rm em}(t)$ is suppressed (slightly enhanced). The
spatially averaged $E_{l}^{\rm em}(t)$ leads to the sharp resonance peak in
Figs. 2(e) and 3(b). In these cases with $h^{\rm stat}_{x}=0$ also, the spin
texture and the EEF behave like standing waves similar to those in Fig. 5(a)
because of the symmetry.

Finally, we discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf z}$;
here we take $h^{\rm stat}_{x}=0.06$ since $\bar{E}^{\rm em}(t)$ vanishes for
$h^{\rm stat}_{x}=0$, as explained in Sec. III.1. Figure 4(e) is for the
conical state at $h^{\rm stat}_{z}=0$ and $\omega=0.131$, indicated by the
cross in Fig. 2(f). In this case, the spin precessions become larger (smaller)
for $S_{l,z}(t)\simeq 0$ [$S_{l,y}(t)\simeq 0$], since the AC magnetic field
is applied along the $z$ direction. In contrast to the above cases with
$h^{\rm stat}_{x}=0$, the spin texture is driven from right to left, as shown
in the left panel of Fig. 5(b); the velocity is estimated as $v\simeq-0.0035$.
This drift motion is known as the Archimedean screw dynamics [38].
Correspondingly, $E^{\rm em}_{l}(t)$ is largely generated and no longer
behaves like a standing wave. Reflecting the drift motion, the EEF is also
driven from right to left with the same velocity, as shown in the inset of
Fig. 5(b). In addition, we find that the EEF propagates from left to right
with much faster velocity, as shown in the enlarged plot in the main panel of
Fig. 5(b) as well as the bottom panel of Fig. 4(e). In this faster mode, the
EEF propagates from one soliton to the next one during one cycle of time;
hence, the velocity is estimated as
$v^{\prime}\simeq\frac{\omega}{Q_{0}}\simeq 0.42$, where $Q_{0}$ is the
ordering wave number. This is more than 100 times faster than $v$. Such a fast
motion is hardly seen in $S_{l,y}(t)$ and $S_{l,z}(t)$, but discernible in
$S_{l,x}(t)$ in the middle panel of Fig. 4(e). Since the motion in
$S_{l,x}(t)$ corresponds to spin precession due to magnon excitations in the
conical state, the fast propagation of the EEF to the field direction is
associated with the magnon propagation. Thus, the parallel component of the
static magnetic field, $h^{\rm stat}_{x}$, activates two types of propagating
motions of the EEF: the fast motion to the field direction due to the magnon
propagation and the slow drift to the opposite direction due to the
Archimedean screw dynamics of the spin texture.

Although the spatially averaged EEF $\bar{E}^{\rm em}(t)$ vanishes in the case
with $h^{\rm stat}_{z}=0$, it becomes nonzero for nonzero $h^{\rm stat}_{z}$,
as a solitonic feature is induced in the spin texture. Such results are shown
in Fig. 4(f) for $h^{\rm stat}_{z}=0.04$ and $\omega=0.123$ corresponding to
the circle in Fig. 2(f). In this case also, the spin texture and the EEF is
driven to left associated with the Archimedean screw dynamics, while the EEF
also propagates to right with much faster velocity associated with the magnon
propagation, similar to Fig. 5(b).

##  IV Result: Edge contributions

Figure 6:  Maximum value of $|\chi_{E,\mu}(\omega)|$, $\chi_{E,\mu}^{\rm
max}$, calculated for the model in Eq. (1) with the OBC for (a) ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$ ($\mu=x$), (b) ${\bf h}^{\rm AC}\parallel\hat{\bf y}$
($\mu=y$), and (c) ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ ($\mu=z$). The
system size is $L=1000$. The frequency dependence of $|\chi_{E,\mu}(\omega)|$
along the yellow dashed lines in (a), (b), and (c) are shown in (d), (e), and
(f), respectively: (d) ${\bf h}^{\rm AC}\parallel\hat{\bf x}$ with $h^{\rm
stat}_{x}=0$, (e) ${\bf h}^{\rm AC}\parallel\hat{\bf y}$ with $h^{\rm
stat}_{x}=0$, and (f) ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ with $h^{\rm
stat}_{x}=0.06$. The orange circles in (d), (e), and (f) indicate the
parameters for which the real-space behaviors are presented in Fig. 8.

Thus far, we have shown the results for the bulk contributions in the system
with the PBC. In this section, we turn our attention to the contributions from
the edges of the system by employing the OBC. In Sec. IV.1, we first present
the results of $|\chi_{E,\mu}(\omega)|$ to show the edge contributions to the
EEF through additional resonance modes. In Sec. IV.2, we discuss the
difference between the bulk resonance modes and the additional modes based on
the spectra of $|\chi_{E,\mu}(\omega)|$ and ${\rm
Im}\,\chi_{S,\mu\mu}(\omega)$, including the system size dependence. In Sec.
IV.3, we show the spatiotemporal profiles of the spin textures and the EEF for
the additional resonance modes to explicitly show that these modes are
localized at the edges of the system.

###  IV.1 Complex admittance of the EEF

As in Sec. III.1, we show the results of the complex admittance
$\chi_{E,\mu}(\omega)$ for the system with the OBC. First, we discuss the
results for ${\bf h}^{\rm AC}\parallel\hat{\bf x}$ shown in Figs. 6(a) and
6(d). Similar to the PBC case, in the noncollinear spin phase,
$\chi_{E,x}^{\rm max}$ increases as $h^{\rm stat}_{x}$ decreases and $h^{\rm
stat}_{z}$ increases, showing the maximum at $(h^{\rm stat}_{x},h^{\rm
stat}_{z})\simeq(0,0.06)$. Notably, however, $\chi_{E,x}^{\rm max}$ becomes
nonzero also in the FFM phase outside of this region, being comparably large
near the phase boundary at $(h^{\rm stat}_{x},h^{\rm
stat}_{z})\simeq(0.04,0.055)$. This contribution originates purely from the
edges of the system with the OBC since it was absent for the PBC case in Fig.
2(a). In addition, we find an additional resonance mode in the noncollinear
spin phase, as shown in the spectrum of $|\chi_{E,x}(\omega)|$ in Fig. 6(d)
along the yellow dashed line in Fig. 6(a); besides several peaks on top of an
almost $\omega$-independent contribution already present for the PBC case in
Fig. 2(d), $|\chi_{E,x}(\omega)|$ shows a sharp peak at a lower $\omega$. The
resonance frequency of this additional mode, which we denote $\omega^{\rm
edge}$, increases as $h^{\rm stat}_{z}$ increases, in contrast to the bulk
resonance frequencies $\omega_{n}^{\rm bulk}$. This is the edge mode reported
in the previous study [37]. We will show that the intensity at $\omega^{\rm
edge}$ depends on the system size and that the spin dynamics and the EEF
associated with this resonance mode are localized near the edges of the system
in the following sections.

The situation is similar for the case of ${\bf h}^{\rm AC}\parallel\hat{\bf
y}$ shown in Figs. 6(b) and 6(e); we obtain an additional resonance mode at
$\omega^{\rm edge}$, in addition to nonzero contributions even in the FFM
phase. In contrast, for ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ in Figs. 6(c)
and 6(f), instead of the additional resonance peak, we find a sharp dip at
$\omega^{\rm edge}$ on top of the long tail of $\omega_{1}^{\rm bulk}$. This
is due to the fact that $\chi_{E,z}(\omega)$ from the edges has an opposite
sign to that for the bulk. We also note that $|\chi_{E,z}(\omega)|$ shows a
sharper peak in the FFM phase, compared to the other cases in Figs. 6(d) and
6(e).

###  IV.2 Comparison with magnetic excitations

Figure 7:  Comparison between $|\chi_{E,\mu}(\omega)|$ and the imaginary part
of the dynamical spin susceptibility, ${\rm Im}\,\chi_{S,\mu\mu}(\omega)$, for
several values of $h_{z}^{\rm stat}$: (a) ${\bf h}^{\rm AC}\parallel\hat{\bf
x}$ with $h_{x}^{\rm stat}=0$, (b) ${\bf h}^{\rm AC}\parallel\hat{\bf y}$ with
$h_{x}^{\rm stat}=0$, and (c) ${\bf h}^{\rm AC}\parallel\hat{\bf z}$ with
$h_{x}^{\rm stat}=0.06$ for the model in Eq. (1) with the OBC. The solid and
dashed (dotted and dashed-dotted) lines denote the data in the noncollinear
spin (FFM) phase with $L=1000$ and $L=500$, respectively. The data for the FFM
phases are scaled for better visibility.

Next, we compare the results of $|\chi_{E,\mu}(\omega)|$ with ${\rm
Im}\,\chi_{S,\mu\mu}(\omega)$ for the system with the OBC, as in Sec. III.2.
Let us first discuss the results for ${\bf h}^{\rm AC}\parallel\hat{\bf x}$
and $h^{\rm stat}_{x}=0$ shown in Fig. 7(a); the solid and dashed lines denote
the results with $L=1000$ and $L=500$, respectively, in the noncollinear spin
phase for $h^{\rm stat}_{z}\lesssim 0.063$, while the dotted and dashed-dotted
lines denote the results with $L=1000$ and $L=500$, respectively, in the FFM
phase for $h^{\rm stat}_{z}\gtrsim 0.063$. In the noncollinear spin phase,
both $|\chi_{E,x}(\omega)|$ and ${\rm Im}\,\chi_{S,xx}(\omega)$ exhibit a
resonance peak at $\omega^{\rm edge}\simeq 0.06-0.08$, in addition to the bulk
resonance peaks at higher $\omega^{\rm bulk}_{n}$ and the $\omega$-independent
contribution already observed in the PBC case. We will show explicitly that
this additional contribution originates from the localized modes at the edges
in the next section. Accordingly, the edge contribution shows conspicuous
system size dependences: The peak heights for $L=1000$ become about twice as
small as those for $L=500$, while the bulk ones do not change largely. As
$|\chi_{E,x}(\omega)|$ describes the response averaged over the system, the
result indicates that the edge contribution is almost system size independent,
while the bulk one is proportional to the system size. Notably, the intensity
of the edge resonance peak of $|\chi_{E,x}(\omega)|$ increases with $h^{\rm
stat}_{z}$ and can be stronger than the bulk ones, even though that of ${\rm
Im}\,\chi_{S,xx}(\omega)$ is much weaker than the bulk ones, as shown in the
lower panel of Fig. 7(a). We also find that $|\chi_{E,x}(\omega)|$ shows a
broad peak in the FFM phase for $h^{\rm stat}_{z}\gtrsim 0.063$, in contrast
to the PBC case.

Similar results are obtained for ${\bf h}^{\rm AC}\parallel\hat{\bf y}$, as
shown in Fig. 7(b). Meanwhile, for ${\bf h}^{\rm AC}\parallel\hat{\bf z}$,
$|\chi_{E,z}(\omega)|$ shows a dip at $\omega^{\rm edge}$ as shown in Fig.
7(c), as a result of a cancellation between the long tail of the bulk response
at $\omega^{\rm bulk}_{1}$ and the edge resonance contribution with the
opposite sign. This is confirmed by the observation that the dip becomes
almost twice shallower for $L=1000$ compared to that for $L=500$. We note that
the edge contribution is much weaker in both $|\chi_{E,z}(\omega)|$ and ${\rm
Im}\,\chi_{S,zz}(\omega)$ compared to the previous two cases due to the less
solitonic feature of the spin texture. In contrast, the additional peaks in
the FFM phase are much sharper than the previous ones.

###  IV.3 Real-space behavior

Figure 8:  Real-space spin configurations (top and middle) and real-space
distributions of the EEF (bottom) in the steady states under the AC magnetic
field ${\bf h}^{\rm AC}$. (a), (b), and (c) are for ${\bf h}^{\rm
AC}\parallel\hat{\bf x}$, ${\bf h}^{\rm AC}\parallel\hat{\bf y}$, and ${\bf
h}^{\rm AC}\parallel\hat{\bf z}$, respectively, at (a) $(h^{\rm
stat}_{x},h^{\rm stat}_{z})=(0,0.06)$ and $\omega=0.077$, (b) $(h^{\rm
stat}_{x},h^{\rm stat}_{z})=(0,0.06)$ and $\omega=0.077$, and (c) $(h^{\rm
stat}_{x},h^{\rm stat}_{z})=(0.06,0.04)$ and $\omega=0.061$. All the
frequencies are set at the values for the edge mode at $\omega^{\rm edge}$;
the values are denoted by the orange circles in Figs. 6(d)–6(f). The results
are shown for each of the $60$ sites at the left and right edges of the
$1000$-site system with the OBC. The notations are common to those in Fig. 4.
Figure 9:  Real-space and real-time distribution of the $z$ component of spins
$S_{l,z}(t)$ (left) and $E^{\rm em}_{l}(t)$ (right) within 10 periods of ${\bf
h}^{\rm AC}$ on the plane of $l$ and $t$: (a) and (b) correspond to Figs. 8(a)
and 8(c), respectively. The results are shown for 60 sites at the left edge,
center, and right edge of the 1000-site system.

The spatiotemporal profiles of ${\bf S}_{l}(t)$ and $E^{\rm em}_{l}(t)$ for
the additional mode at $\omega^{\rm edge}$ are shown in Fig. 8. Here, we
display the spin textures and the EEF for each of the 60 sites at the left and
right edges of the system with $L=1000$. Figure 8(a) shows the result for
${\bf h}^{\rm AC}\parallel\hat{\bf x}$ in the CSL state with $(h^{\rm
stat}_{x},h^{\rm stat}_{z})=(0,0.06)$ and $\omega=0.077$, indicated by the
orange circle in Fig. 6(d). In this case, as shown in the top and the middle
panels of Fig. 8(a), we find that the spin dynamics with $\omega^{\rm edge}$
is localized around the edges, where the spins are twisted in the ground
state, known as the chiral surface twist [55]. Hence, the additional mode
found in Sec. IV.1 is the edge mode. Note that the EEF $E_{l}^{\rm em}(t)$ at
both edges has the same sign and its amplitude is about ten times larger than
that in the bulk, leading to a comparably large contribution to the bulk
responses shown in Fig. 7(a). Figure 9(a) shows the time evolution of
$S_{l,z}(t)$ and $E^{\rm em}_{l}(t)$. Both patterns obey twofold rotational
symmetry about the $z$ axis with time translation by $\frac{\pi}{\omega}$ with
respect to the center of the system, but $E^{\rm em}_{l}(t)$ appears to
propagate from the edges to the inside as shown in the right panel, while
$S_{l,z}(t)$ on the left does not clearly show such a behavior.

We observe similar spatiotemporal profiles of the edge mode for ${\bf h}^{\rm
AC}\parallel\hat{\bf y}$, as shown in Fig. 8(b). Meanwhile, as already
discussed in Sec. IV.2, the edge mode for ${\bf h}^{\rm AC}\parallel\hat{\bf
z}$ is less significant as shown in Fig. 8(c). In this case, the EEF
propagates from left to right similar to the PBC case, even near the edges, as
shown in the right panel of Fig. 9(b), while the pattern near the right edge
looks complicated.

##  V Discussion

###  V.1 Bulk contribution

Through the calculations for the PBC in Sec. III, we showed that the EEF is
enhanced at the magnetic resonance. An interesting finding is that the EEF
resonance peaks are more clearly visible compared to the magnetic ones for
higher frequency modes. Thus, the measurement of the EEF could be a good probe
of the high-frequency magnetic resonances. In addition, the EEF is amplified
by the solitonic feature of the spin textures which can be controlled by the
static magnetic field as well as the direction of the AC magnetic field. In
particular, it is strongly enhanced when ${\bf h}^{\rm stat}$, ${\bf h}^{\rm
AC}$, and the chiral axis are orthogonal to each other, as exemplified in
Figs. 2(b) and 2(e). As the total voltage generated by the EEF is proportional
to the system size, the solitonic noncollinear spin structure is a promising
platform for exploring the emergent electric phenomena, compared to magnetic
domain walls [23].

Besides the total EEF, the spatiotemporal profile of the EEF in Sec. III.3 is
important for electronic transport phenomena. When $h^{\rm stat}_{x}=0$, the
EEF behaves like a standing wave, and hence, only an AC electric current is
expected for electrons coupled to the spin texture. When $h^{\rm stat}_{x}$
becomes nonzero, however, the EEF begins to propagate in the $x$ direction
along with the magnon propagation, in addition to the slower drift to the
opposite direction due to the Archimedean screw dynamics. These motions of the
EEF give rise to a DC electric current as well as the AC one, and hence, the
noncollinear spin texture works as a generator of the AC and DC electric
currents through the EEF, which can be tuned by the static and AC magnetic
fields. The current direction as well as the velocity may depend on how the
electrons couple to the spin texture.

###  V.2 Edge contribution

From the calculations for the OBC in Sec. IV, we showed that there is a
sizable contribution to the EEF from the edges of the system, which can be
comparable to or larger than the bulk one depending on the system size. It is
worth noting that the edge resonances of the EEF are more conspicuous than the
magnetic ones. Thus, the EEF is also a good probe of the edge modes, in
addition to the higher-frequency bulk modes.

In CrNb3S6, which is one of the candidate materials for the monoaxial chiral
helimagnets, the lattice spacing between neighboring spins along the chiral
axis is about $0.6$ nm and the magnetic modulation period of the CSL phase
ranges from $50$ to $100$ nm depending on the magnetic field [56]. The
situation is similar to that in our calculations, suggesting that a large edge
contribution of the EEF comparable to the bulk one is expected for this
material even in micro-meter samples. We note that while the magnetic
resonances were measured for CrNb3S6, the edge contribution has not been
clearly observed [57]. It is thus worth measuring the EEF to identify the edge
contributions. For this purpose, measurements for samples with different sizes
would be useful since the edge contribution is independent of the system size
in contrast to the bulk one proportional to the system size.

###  V.3 Order estimate

Finally, let us estimate the amplitude of the EEF expected in real materials.
Experimentally, the noncollinear spin phase including the CSL has been
observed, e.g., in CrNb3S6 [56], MnNb3S6 [58], strained Cu2OSeO3 [59], and
Yb(Ni1-xCux)3Al9 [60]. The actual value of the EEF depends on the energy scale
of the magnetic interactions and the length scale set by the lattice constant
in each material. By assuming the typical values of the Heisenberg interaction
$J$ and the lattice constant being $1$ meV and $1$ nm, respectively, the time
unit $t=1$, the frequency $\omega=1$, the magnetic field $h=1$, and the EEF
$\bar{E}^{\rm em}=1$ correspond to $\sim 0.66$ psec, $\sim 241$ GHz, $\sim
8.6$ T, and $\sim 5\times 10^{5}$ V/m, respectively. In our results, we
typically obtained $|\chi_{E,\mu}(\omega)|\sim 0.01$ at the lowest-energy
resonance frequency $\omega_{1}^{\rm bulk}\sim 0.1\sim 24.1$ GHz, and hence,
the AC magnetic field with $|{\bf h}^{\rm AC}|\sim 0.01\sim 860$ Oe can
generate the total EEF of $\bar{E}^{\rm em}\sim 50$ V/m. This leads to the
electric voltage $\sim 500$ $\mu$V for a sample with $10$ $\mu$m length scale,
which could be experimentally measurable. Note that the resonance frequency
observed in CrNb3S6 is in good agreement with our results [57].

Meanwhile, our spatiotemporal profile of the EEF revealed that the local value
of $E_{l}^{\rm em}(t)$ reaches $\sim 10^{-2}$, which corresponds to the
electric field of $\sim 5$ KV/m. Thus, even though the total EEF is relatively
small, electrons coupled to the noncollinear spin texture would be largely
influenced by such a strong local electric field. This may bring about
intriguing electric responses, such as the AC and DC electric currents
mentioned in Sec. V.1. Note that the local EEF takes such a large value even
in the helical and the conical states where the solitonic feature is absent.

We note that the generated EEF depends on the value of the Gilbert damping
$\alpha$ in Eq. (3). The EEF associated with the resonance dynamics is roughly
proportional to $1/\alpha$. This is because the peak intensity of
$\chi_{E,\mu}(\omega)$ is approximately proportional to
$\chi_{S,\mu\mu}(\omega)$ [see Eq. (21)] and ${\rm
Im}\,\chi_{S,\mu\mu}(\omega)$ is proportional to $1/\alpha$. Meanwhile, the
EEF by the coherent sliding dynamics, which gives the $\omega$-independent
contribution, is proportional to $\alpha$, as implied by Eq. (15) for small
$\alpha$. The value of $\alpha$ depends on materials and the realistic
estimate is not an easy task, while we used $\alpha=0.04$ as a typical value
for ferromagnetic metals [46, 47, 48, 49].

##  VI Summary

To summarize, we have theoretically studied the EEF in a one-dimensional
chiral magnet by using numerical simulations based on the LLG equation. In the
system with the PBC, we clarified that the bulk contribution of
$|\chi_{E,\mu}(\omega)|$ consists of the resonance peaks reflecting the
magnetic resonances and the $\omega$-independent contribution arising from the
coherent sliding dynamics. We showed that the peak height of
$|\chi_{E,x}(\omega)|$ increases with the solitonic feature of the spin
textures; it is maximally enhanced when the static magnetic field, the AC one,
and the chiral axis are perpendicular to each other. Meanwhile, the
$\omega$-independent contribution decreases with the vector spin chirality in
the direction of the AC magnetic field; it vanishes when the AC magnetic field
is perpendicular to the chiral axis. Comparing the spectra of
$|\chi_{E,\mu}(\omega)|$ with those of ${\rm Im}\,\chi_{S,\mu\mu}(\omega)$, we
revealed that the higher-$\omega$ resonance peaks are more clearly visible in
$|\chi_{E,\mu}(\omega)|$ rather than in ${\rm Im}\,\chi_{S,\mu\mu}(\omega)$,
since the EEF includes an additional $\omega$-linear factor coming from the
time derivative of the spin configuration. In addition to the bulk averaged
responses, by directly investigating the spatiotemporal profiles of the EEF
and the spin texture, we revealed that the local EEF associated with the
resonance dynamics can take a considerable value even when the total EEF
vanishes. The amplitude of the EEF is maximized around the solitons where the
vector spin chirality is maximized. Furthermore, we showed that the EEF
behaves like a standing wave when the static magnetic field is perpendicular
to the chiral axis. In contrast, when the static field has a nonzero component
along the chiral axis, the EEF is driven to the field direction by the magnon
propagation, in addition to the slower drift to the opposite direction due to
the Archimedean screw dynamics of the spin texture. This works as a generator
of AC and DC electric currents.

In addition to the bulk contribution, we have studied the effect of edges of
the system under the OBC. We found the additional resonance mode in the EEF
spectra at a lower frequency, corresponding to the edge mode in the magnetic
excitations. The frequency of the additional edge mode increases with the
solitonic feature of the spin texture, in contrast to the bulk resonance
frequencies. We also found that the peak intensity of the edge mode can be
greater than those of the bulk modes; the former is almost system size
independent, while the latter is inversely proportional to the system size.
From the spatiotemporal profiles for the edge mode, we found that the local
EEF near the edges can be more than one order of magnitude larger than deep
inside the bulk. Furthermore, we found that the EEF can be generated even in
the FFM phase in contrast to the PBC case.

Our findings unveil the systematic changes of the EEF for the static and AC
magnetic fields in a one-dimensional chiral magnet. They would pave the way to
enhance the EEF and explore intriguing electronic and magnetic functionalities
in chiral magnets. To reveal such functionalities explicitly, further studies
including electrons coupled to the chiral magnetic textures are desired. While
the present study was limited to the one-dimensional case, various topological
spin textures have been found in two- and three-dimensional magnets, e.g.,
skyrmions [10, 11, 12, 13], Bloch points [61, 62, 63] (or equivalently,
magnetic hedgehogs [6, 64, 65]), and hopfions [66, 67, 68]. It is also
intriguing to extend our study to the emergent electric phenomena in these
topologically-nontrivial spin textures.

###### Acknowledgements.

The authors thank J. Kishine, M. Mochizuki, Y. Shimamoto, and Y. Togawa for
fruitful discussions. This research was supported by Grant-in-Aid for
Scientific Research Grants (Nos. JP18K03447, JP19H05822, JP19H05825,
JP21J20812, and No. JP22K13998), JST CREST (Nos. JP-MJCR18T2 and JP-MJCR19T3),
and the Chirality Research Center in Hiroshima University and JSPS Core-to-
Core Program, Advanced Research Networks. K.S. was supported by the Program
for Leading Graduate Schools (MERIT-WINGS). Parts of the numerical
calculations were performed in the supercomputing systems in ISSP, the
University of Tokyo.

##  Appendix A $\chi_{E,\mu}(\omega)$ on resonance

In this Appendix, we discuss $\chi_{E,\mu}(\omega)$ at resonance frequencies
analytically in the continuum limit. In continuous space, the EEF at position
$x$ and time $t$ in Eq. (6) is written as

$\displaystyle E^{\rm em}(x,t)=\frac{\partial{\bf S}(x,t)}{\partial
t}\cdot{\bf C}^{\rm vc}(x,t),$ (16)

where ${\bf C}^{\rm vc}(x,t)$ is the vector spin chirality given by

$\displaystyle{\bf C}^{\rm vc}(x,t)={\bf S}(x,t)\times\frac{\partial{\bf
S}(x,t)}{\partial x}.$ (17)

The Fourier component of the EEF is obtained as

$\displaystyle E^{\rm em}(q,\omega)$ $\displaystyle=$ $\displaystyle-i\int
dq^{\prime}\int d\omega^{\prime}~{}\omega^{\prime}$ (18)
$\displaystyle\qquad{\bf S}(q^{\prime},\omega^{\prime})\cdot{\bf C}^{\rm
vc}(q-q^{\prime},\omega-\omega^{\prime}).$

Then, at a resonance frequency, the total EEF, $\bar{E}^{\rm
em}(\omega)=E^{\rm em}(q=0,\omega)$, is well approximated by

$\displaystyle\bar{E}^{\rm em}(\omega)\simeq-i\omega\int dq~{}{\bf
S}(q,\omega)\cdot{\bf C}^{\rm vc}(-q,\omega=0).$ (19)

For the noncollinear spin textures discussed in the main text, ${\bf C}^{\rm
vc}(q,0)$ is dominated by the $q=mQ$ components, where $m$ is an integer and
$Q$ is the ordering wave number. Thus, $\chi_{E,\mu}(\omega)$ for the
resonance dynamics is approximately given by

$\displaystyle\chi_{E,\mu}(\omega)$ $\displaystyle\simeq$
$\displaystyle-i\omega\sum_{m}\frac{{\bf S}(mQ,\omega)\cdot{\bf C}^{\rm
vc}(-mQ,\omega=0)}{h_{\mu}^{\rm AC}(\omega)}.$ (20)

In the helical state with $h^{\rm stat}_{x}=h^{\rm stat}_{z}=0$, ${\bf C}^{\rm
vc}(q,\omega=0)$ consists of only the $q=0$ component ($m=0$). Thus, when
$h^{\rm stat}_{x}$ and $h^{\rm stat}_{z}$ are small and the modulation from
the helical spin texture is weak, Eq. (20) is further approximated by

$\displaystyle\chi_{E,\mu}(\omega)\simeq-i\omega\chi_{S,\nu\mu}(\omega)\bar{C}^{\rm
vc}_{\nu},$ (21)

where $\chi_{S,\nu\mu}(\omega)$ is the dynamical spin susceptibility in Eq.
(9) and $\bar{C}^{\rm vc}_{\nu}$ is the $\nu$ component of the vector spin
chirality for the ground state given by

$\displaystyle\bar{\bf C}^{\rm vc}=\frac{1}{L}\int dx~{}{\bf C}^{\rm
vc}(x,t=0).$ (22)

In Eq. (21), $\chi_{E,\mu}(\omega)$ is linear in $\omega$. This explains why
the resonance peaks with high frequencies appear more clearly in
$\chi_{E,\mu}(\omega)$ than in $\chi_{S,\mu\nu}(\omega)$, as found in Sec.
III.2.

##  Appendix B $\chi_{E,\mu}(\omega)$ in large-$\omega$ region

In this Appendix, we show that $\chi_{E,\mu}(\omega)$ in the case of large
$\omega$ can be understood from the vector spin chirality, as in Eq. (15). For
simplicity, we consider the continuum limit again. By plugging Eq. (3) into
Eq. (6), we obtain

$\displaystyle E^{\rm em}(x,t)=\frac{1}{1+\alpha^{2}}\left[-{\bf
S}(x,t)\times{\bf h}^{\rm eff}(x,t)\right.$ $\displaystyle\left.+\alpha{\bf
S}(x,t)\times\left({\bf S}(x,t)\times{\bf h}^{\rm
eff}(x,t)\right)\right]\cdot\left({\bf S}(x,t)\times\frac{\partial{\bf
S}(x,t)}{\partial x}\right)$ $\displaystyle=-\frac{1}{1+\alpha^{2}}{\bf
h}^{\rm eff}(x,t)\cdot\left(\frac{\partial{\bf S}(x,t)}{\partial x}+\alpha{\bf
S}(x,t)\times\frac{\partial{\bf S}(x,t)}{\partial x}\right).$ (23)

Since ${\bf h}^{\rm eff}(x,t)-{\bf h}(t)$ is always perpendicular to
$\frac{\partial{\bf S}(x,t)}{\partial x}$, the total EEF is calculated as

$\displaystyle\bar{E}^{\rm em}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{L}\int_{0}^{L}dx~{}E^{\rm em}(x,t)$ (24)
$\displaystyle=$ $\displaystyle-\frac{1}{L(1+\alpha^{2})}\left[{\bf
h}(t)\cdot\int_{0}^{L}dx~{}\frac{\partial{\bf S}(x,t)}{\partial x}\right.$
$\displaystyle\left.+\alpha\int_{0}^{L}dx~{}{\bf h}^{\rm
eff}(x,t)\cdot\left({\bf S}(x,t)\times\frac{\partial{\bf S}(x,t)}{\partial
x}\right)\right].$

The first term in the square bracket of Eq. (24) vanishes in the PBC case,
while it remains in the order of $1/L$ in the OBC case. Hence, in the
following, we focus on the second term in Eq. (24). Note that it vanishes when
$\alpha=0$, indicating that the dissipative dynamics is essential for
generating the bulk EEF [29].

For sufficiently large $\omega$, the spins hardly follow the AC magnetic
field, and hence, it is a good approximation to replace ${\bf S}(x,t)$ with
the ground-state spin configuration at $t=0$, ${\bf S}(x,0)$. In this
approximation, ${\bf h}^{\rm eff}(x,t)$ can be decomposed as

$\displaystyle{\bf h}^{\rm eff}(x,t)={\bf h}^{\rm AC}(t)+{\bf h}^{\rm
eff}(x,0).$ (25)

By substituting Eq. (25) into Eq. (24), the EEF is given by

$\displaystyle\bar{E}^{\rm em}(t)$ $\displaystyle\simeq$
$\displaystyle-\frac{\alpha}{L(1+\alpha^{2})}{\bf h}^{\rm
AC}(t)\cdot\int_{0}^{L}dx~{}{\bf S}(x,0)\times\frac{\partial{\bf
S}(x,0)}{\partial x}$ (26) $\displaystyle=$
$\displaystyle-\frac{\alpha}{1+\alpha^{2}}{\bf h}^{\rm AC}(t)\cdot\bar{\bf
C}^{\rm vc}.$

Note that the contribution from ${\bf h}^{\rm eff}(x,0)$ vanishes as
$\bar{E}^{\rm em}(0)=0$ and ${\bf h}^{\rm AC}(0)=0$. Equation (26) leads to
Eq. (15).

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1(3)/4-\langle\mathcal{Z}\rangle_{s(t)}e^{-x}$. Thus we have $\partial_{x}\ln
H^{s(t)}=-1$. To match the asymptotic behavior on the two limits, we have
$\phi=\pi/2$ and $\kappa\sqrt{\delta_{\theta}}\ln\mathcal{Z}^{s(t),typ}+\pi=0$
[18]. This gives us the scaling form

$\mathcal{Z}^{s(t),typ}\sim\exp(-\frac{K}{\sqrt{\delta_{\theta}}})\quad
K=\frac{\pi}{\kappa},$ (90)

when $(\theta_{1},\theta_{2})$ approaches the critical point
$(\theta_{1}^{\ast},\theta_{2}^{\ast})\in\mathcal{C}$ from the fuzzy phase
side.

When the parameters $\theta_{1,2}$ sit exactly at a point
$(\theta_{1}^{\ast},\theta_{2}^{\ast})$ on the critical boundary, the
ballistic velocity of $\ln\mathcal{Z}^{s(t),typ}_{k}$ is zero. However, there
can still be a sub-ballistic decay toward $-\infty$ [18]. Since the wavefront
$\ln\mathcal{Z}^{s(t),typ}_{k}$ changes slowly with $k$, we use the same
conjecture in Refs. [18, 28] that within the regime
$1\ll|x|\ll|\ln\mathcal{Z}^{s(t),typ}|$, the form of $G^{s(t)}_{k}(x)$ is
similar to the saturated solution in the fuzzy phase. Thus we consider the
ansatz $H^{s(t)}_{k}(x)\sim\alpha^{s(t)}e^{-\lambda_{k}(x-f_{k})}$, with
$f_{k}$ the location of wavefront. Notice that now $\lambda$ has an imaginary
part and it depends on $k$. The position of the wavefront is determined by
[18]

$f_{k}\sim-\frac{\pi}{\left|\imaginary\lambda_{k}\right|}.$ (91)

Using the linearized recursive relation of $G_{k}(x)$, we get

$\displaystyle\begin{pmatrix}\alpha^{s}\\\ \\\
\alpha^{t}\end{pmatrix}e^{-\Delta_{k}(x)}=\begin{pmatrix}\frac{1}{4}\left(\frac{9}{4}a\right)^{1-\lambda_{k}}b^{\lambda_{k}}&\frac{3}{4}\left(\frac{1}{12}a\right)^{1-\lambda_{k}}b^{\lambda_{k}}\\\
\\\
\frac{3}{4}b^{1-\lambda_{k}}\left(\frac{1}{12}a\right)^{\lambda_{k}}&\frac{1}{4}b^{1-\lambda_{k}}\left(\frac{9}{4}a\right)^{\lambda_{k}}\end{pmatrix}\begin{pmatrix}\alpha^{s}\\\
\\\ \alpha^{t}\end{pmatrix},$ (92)

where we defined
$\Delta_{k}(x)=\lambda_{k+1}x-\lambda_{k}x-\lambda_{k+1}f_{k+1}+\lambda_{k}f_{k}$.
At large enough $k$, the imaginary part of $\lambda$ should decay to $0$ since
the wavefront goes to $-\infty$. At the same time, the speed of $f_{k}$ also
decays to $0$ due to the sub-ballistic motion of the wavefront. Using these
facts, we can approximate $\Delta_{k}$ by191919Notice that since
$1\ll|x|\ll\min(|\ln\mathcal{Z}^{s,typ}|,|\ln\mathcal{Z}^{t,typ}|)$, the part
of $\Delta_{k}$ that depends on $x$ can be ignored. $-(f_{k+1}-f_{k})/2$. Then
we get

$\imaginary\lambda=\left.\frac{\sqrt{2-B}}{\sqrt{\partial^{2}_{\lambda}B}}\right|_{\lambda=1/2,\mathcal{C}}\sqrt{|f_{k+1}-f_{k}|}.$
(93)

Combining Eq. (91) and Eq. (93), we get

$f_{k}\sim-k^{1/3},$ (94)

and thus

$\ln\mathcal{Z}^{s(t),typ}\sim-k^{1/3}$ (95)

on the critical contour $\mathcal{C}$.

So far we have argued that the spin-sharpening transition at $p=1$ obtained by
tuning the gate parameter $\theta_{1,2}$ may still be in the glass
universality class. It would also be interesting to understand the critical
behaviors when we fix $\theta_{1}$ and $\theta_{2}$ and let $p$ approach $1$.
We leave this question in future work.

$m^{\prime}s$ input $s,s$ $s,t$ $t,s$ $t,t$ $s$,$s$ $(0,s)$ $(0,s)$ $(0,s)$
$(0,s)$ $s$,$t$ $(0,t)$ $(0,t)$ $(0,t)$ $(0,t)$ $t$,$s$ $(0,t)$ $(0,t)$
$(0,t)$ $(0,t)$ $t$,$t$
$(\frac{4(1-c_{-})}{9(1-c_{+})}(\mathcal{Z}_{1}+\mathcal{Z}_{2}),s)$
$(\frac{1-c_{+}}{1-c_{-}}(\frac{\mathcal{Z}_{1}}{12}+\frac{4\mathcal{Z}_{2}}{9}),t)$
$(\frac{1-c_{+}}{1-c_{-}}(\frac{4\mathcal{Z}_{1}}{9}+\frac{\mathcal{Z}_{2}}{12}),t)$
$(12\frac{1-c_{-}}{1-c_{+}}(\mathcal{Z}_{1}+\mathcal{Z}_{2}),s)$

Table 3: Values of the node output parameters $(\mathcal{Z},\eta)$, to linear
order in $\mathcal{Z}_{1,2}$, at $p=1$. The top row labels the types (singlet
$s$ or triplet $t$) of the two input density matrices to the node, while the
leftmost column labels the two mid-curcuit measurement outcomes (also singlet
$s$ or triplet $t$), where the first outcome corresponds to the inner qubit
pair and the second corresponds to the outer qubit pair in Fig. 3. We again
use the shorthand notation $c_{i}=\cos(2\theta_{i})$, $i=1,2$, and
$c_{\pm}=\cos(2(\theta_{1}\pm\theta_{2}))$. Here we assume that
$\theta_{1}+\theta_{2}\notin\pi\mathbb{Z}$ and
$\theta_{1}-\theta_{2}\notin\pi\mathbb{Z}$ (otherwise the system sharpens
immediately).

##  Appendix G Alternative measurement outcome distributions

While real quantum measurements are distributed according to the Born rule, it
is sometimes valuable to consider alternative (unphysical) distributions that
may be more analytically or numerically tractable, while still shedding light
on certain features of the ensemble of quantum trajectories [9, 58].

One such distributions is given by forced measurements [18], where an output
string is chosen uniformly at random (among allowed outcomes, i.e. those that
have $p({\mathbf{m}})>0$) and corresponding projectors are then applied to the
circuit. This corresponds to replacing $p({\mathbf{m}})\mapsto{\rm const}>0$
for all $\mathbf{m}$ with $p(\mathbf{m})>0$. More generally, one can study the
distribution $p({\mathbf{m}})\mapsto
p(\mathbf{m})^{n}/\sum_{\mathbf{m}^{\prime}}p(\mathbf{m}^{\prime})^{n}$, for
any value of $n$; $n=0$ gives forced measurements, $n=1$ gives the Born rule,
and $n>1$ gives modified distributions that concentrate more weight to the
more likely trajectories in the Born distribution. $n$ plays the role of a
“replica number” in statistical-mechanical treatments of the MIPT.

We can study spin sharpening in these modified trajectory ensembles by
introducing the family of order parameters

$\displaystyle r_{n}$
$\displaystyle=\sum_{\mathbf{m}}\frac{(\sigma(\mathbf{m})+\tau(\mathbf{m}))^{n}}{\sum_{\mathbf{m}^{\prime}}(\sigma(\mathbf{m}^{\prime})+\tau(\mathbf{m}^{\prime}))^{n}}\frac{\sigma(\mathbf{m})^{2}+\tau(\mathbf{m})^{2}}{(\sigma(\mathbf{m})+\tau(\mathbf{m}))^{2}}$
$\displaystyle=\frac{\sum_{\mathbf{m}}(\sigma(\mathbf{m})+\tau(\mathbf{m}))^{n-2}(\sigma(\mathbf{m})^{2}+\tau(\mathbf{m})^{2})}{\sum_{\mathbf{m}}(\sigma(\mathbf{m})+\tau(\mathbf{m}))^{n}}.$
(96)

In Eq. (96), we are particularly interested in the cases of $n=0$ and $n\geq
2$. When $n=0$, it means that all trajectories have the same probability as
long as $\sigma+\tau\neq 0$. This corresponds to all mid-circuit measurements
being forced, a case that can be efficiently simulated by the pool method (see
Refs. [18, 28]).

Another tractable case is when $n\geq 2$ and $n$ is integer. Then, Eq. (96)
can be expressed purely in terms of the variables

$x_{l}\equiv\sum_{\mathbf{m}}\sigma(\mathbf{m})^{l}\tau(\mathbf{m})^{n-l},$
(97)

for $0\leq l\leq n$. Specifically, we have

$r_{n}=\frac{\sum_{l=0}^{n-2}\binom{n-2}{l}(x_{l}+x_{n-l})}{\sum_{l=0}^{n}\binom{n}{l}x_{l}}.$
(98)

Using the recursive relations in Eq. (12), we can efficiently express the
variables $\\{x_{l}\\}_{l=0}^{n}$ for a $k$-layer tree from those of a
$(k-1)$-layer tree with a quadratic recursion, and thus get $r_{n}$ for in
principle arbitrarily large values of the tree depth $k$.

####  G.0.1 Forced measurements ($n=0$)

In the tree model with only forced measurement, the dynamics can be
efficiently simulated by the pool method as mentioned before. We find that
when all allowed trajectories ($\sigma+\tau\neq 0$) have equal probability,
the dynamics flows to a sharp state no matter the choice of $\theta_{1}$ and
$\theta_{2}$. This suggests that within all allowed trajectories, the spin-
sharp ones are prevalent; a fuzzy phase must require some over-weighting of
the less-common fuzzy trajectories in the ensemble.

####  G.0.2 Non-Born-rule measurements ($n\geq 2$)

For all integer $n\geq 2$, we find similar behavior up to the numerically
accessible $n\approx 10$. This allows us to focus on $n=2$ to illustrate the
phenomenology. Our results for the order parameter $r_{2}$ at very large $k$
are shown in Fig. 9. Here we observe that, although most of the phase diagram
is in a sharp phase, there are also very clear fuzzy phases. This behavior
holds for all numerically accessible $n$, which suggests that the most likely
(i.e., highest-Born-probability) trajectories in the ensemble are fuzzy in
these parts of parameter space.

Figure 9: Simulation of $SU(2)$-symmetric tree with $n=2$-replica measurement
outcome distribution at $p=1$. We use the efficient recursion method described
in the main text and iterate up to a well-converged value of $k=400$.

Interestingly, the phase boundaries in this case are strongly asymmetric under
the reflection $\theta_{i}\mapsto\pi-\theta_{i}$, unlike the symmetric contour
we identified for the Born-rule ($n=1$) measurements, Eq. (14). The existence
of this asymmetric pattern in the $n\geq 2$ measurements raises the natural
question of whether hints of such a pattern can also be seen in the real
measurement case. To check this, we antisymmetrize the real-measurement data
(obtained for $k=6$, i.e. 128 system qubits) about the $\theta_{1}=\pi/2$
axis, defining $\Delta
r(\theta_{1},\theta_{2})=r(\pi-\theta_{1},\theta_{2})-r(\theta_{1},\theta_{2})$;
results are shown in Fig. 10(a). Interestingly, this reveals a very similar
anti-symmetric pattern as for the $n=2$ measurements.

We further check the dependence of this asymmetry on the tree depth $k$ in
Fig. 10(b), where we plot $\max_{\theta_{1,2}}\Delta r$ as a function of
$1/k$. The results, while finite-size limited, are suggestive of a persistent
asymmetry in the $k\to\infty$ limit, indicating that this is a real feature of
the steady state and is not transient.

Figure 10: (a) Spin sharpening order parameter $r$ for Born-rule measurement
($n=1$) at $p=1$, from Fig. 5(a), antisymmetrized about the $\theta_{1}=\pi/2$
axis (dashed line): $\Delta
r(\theta_{1},\theta_{2})=r(\pi-\theta_{1},\theta_{2})-r(\theta_{1},\theta_{2})$.
(b) Maximum of $\Delta r$ vs $1/k$ suggests a persistent asymmetry in $r$ in
the $k\to\infty$ limit.

In summary, we have that:

  1. 1.

Almost all trajectories (by number, not by Born probability) are sharp across
the phase diagram, as shown by the $n=0$, forced-measurement result;

  2. 2.

The Born-rule average across trajectories becomes fuzzy in roughly circular
regions, as shown by the $n=1$ data in Fig. 5 and the analytical phase
boundary Eq. (14);

  3. 3.

The most likely (i.e. highest Born rule probability) trajectories become fuzzy
in roughly semi-cricular regions, as shown by the $n=2$ data in Fig. 9; these
do not match the $n=1$ phase boundary: in parts of the sharp phase the most
likely trajectories become fuzzy, and vice versa.

This is evidence of additional interesting structure in the ensemble of
trajectories that goes beyond the Born-averaged spin sharpening. We leave
further investigations of these structures to future work.