[1]Dennis Hardt

1]Institute for Theoretical Physics, University of Cologne, Cologne, 50937,
Germany

2]Division of Physics, Mathematics and Astronomy, California Institute of
Technology, 1200 E California Blvd, Pasadena, 91125, CA, USA

# Active Magnetic Matter: Propelling Ferrimagnetic Domain Walls by Dynamical
Frustration

<EMAIL_ADDRESS>Reza Doostani  Sebastian Diehl  Nina del Ser  Achim
Rosch  [  [

###### Abstract

Active matter encompasses many-particle systems with self-propelling units,
such as flocks of birds or schools of fish. Here, we show how self-propelling
domain walls can be realised in a solid-state system when a ferrimagnet is
weakly driven out of thermal equilibrium by an oscillating field. This
activates the Goldstone mode, inducing a rotation of the antiferromagnetic
$xy$-order in a clockwise or anticlockwise direction, determined by the sign
of the ferromagnetic component. Two opposite directions of rotation meet at a
ferromagnetic domain wall, resulting in ‘dynamical frustration’, with three
main consequences. (i) Domain walls move actively in a direction chosen by
spontaneous symmetry breaking. Their speed is proportional to the square root
of the driving power across large parameter regimes. (ii) In one dimension
(1D), after a quench into the ferrimagnetic phase, this motion and strong
hydrodynamic interactions lead to a linear growth of the magnetic correlation
length over time, much faster than in equilibrium. (iii) The dynamical
frustration makes the system highly resilient to noise. The correlation length
of the weakly driven 1D system can be orders of magnitude larger than in the
corresponding equilibrium system with the same noise level.

###### pacs:

Active matter [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] refers to systems
such as cars on a busy street [4], self-propelling bacteria [5], flocks or
herds of animals [6, 7], systems with active chemical reactions [8], moving
components of the cytoskeleton of living cells [9] or artificial self-
propelled particles (synthetic systems) [10, 3, 11]. They have in common that
the constituents locally use energy to induce, e.g., motion. Thus, detailed
balance is broken on a local level and systems of this kind are intrinsically
far from equilibrium. Such active matter systems show novel collective
behaviours with no equilibrium counterparts. One familiar example is a traffic
jam. Another concerns the synchronisation of the orientation of birds flying
in large flocks. A model introduced by Vicsek et al. in 1995 [7] to describe
flocking birds has been argued to exhibit spontaneous breaking of rotational
symmetry in a two-dimensional system with noise, which is ruled out by the
Mermin-Wagner theorem in equilibrium systems [12]. Furthermore, new types of
phase transitions emerge in such systems [13, 14, 15].

a)

b)

c)

Figure 1: Moving domain wall. a) Schematic picture of a domain wall in a
driven ferrimagnet. The (staggered) in-plane magnetisation rotates in opposite
directions on the left and right side, leading to dynamical frustration. This
can lead to an active motion of the domain wall either to the left or right.
Data from a numerical simulation of a right-moving domain wall are displayed
in panels b) and c). b) shows the magnetisation $m^{z}(x,t)$, while c) shows
the phase $\varphi(x,t)$, describing the orientation of the staggered in-plane
magnetisation. A substantial phase gradient $\partial_{x}\varphi$ is built up
behind the moving domain. Parameters:
$J=1,\Delta=0.8,\delta_{2}=-0.6,\delta_{4}=1,g_{1}=1,g_{2}=0.1,\alpha=0.1,B_{0}=0.15,\omega=3.6$
resulting in a rotation period $T_{\text{rot}}\approx 612$, used to scale the
vertical axis for a system of 40,000 spins.

The goal of this paper is to investigate theoretically a solid-state system
where excitations move actively. Recently, Lindblad operators [16, 17], non-
Hermitian Hamilton operators [18, 19] or ‘active spins’ with unusual dynamics
[20, 21] have been used to obtain quantum analogs of active matter systems.
Here we use a different route. We show that active motion can be realised in a
ferrimagnet, hosting both antiferromagnetic (AFM) order in the $xy$-plane and
ferromagnetic (FM) order in the $z$-direction. This magnet is weakly driven
out of thermal equilibrium by, e.g., external radiation. Our construction
builds on two principles: ‘activation of Goldstone modes’ [22, 23] and
‘dynamical frustration’. We assume that our ferrimagnet [24, 14] is
characterised by a continuous spin-rotation symmetry in the $xy$-plane. An
angle $\varphi$ describes the orientation of the staggered in-plane
magnetisation. Symmetries allow us to consider the following term in the
equation of motion of $\varphi$,

$\displaystyle\partial_{t}\varphi=-\gamma\,m^{z},$ (1)

where $m^{z}$ is the FM magnetisation in the $z$-direction and $\gamma$ is a
prefactor. In thermal equilibrium, $\gamma=0$ and $\varphi=\text{const.}$,
which follows from the fact that a perpetuum mobile is forbidden. This can
also be formally derived from symmetries of the Keldysh action encoding
thermal equilibrium [25]. Out of equilibrium, however, there is no reason why
$\gamma$ should vanish, and indeed one finds that $\gamma$ grows linearly in
the power used to drive the system out of equilibrium [22, 14]. Thus, the
Goldstone mode of the ferrimagnet is activated, $\varphi=\gamma
m^{z}t+\varphi_{0}$. A detailed theory of the activation of Goldstone modes in
helimagnets and single skyrmions is presented in Refs. [22, 23]. In Ref. [14],
it was shown that the coupling $\gamma$ in Eq. (1) drastically changes the
phase transition from an $xy$-AFM phase into the ferrimagnetic phase by
rendering it first-order. Experimentally, rotations of out-of-equilibrium
magnets are, for example, routinely observed in so-called spin-torque
oscillators [26, 27]. Ref. [28] discusses how a rotation of a skyrmion lattice
can be induced by laser pulses. Furthermore, Bose-Einstein condensates of
magnons [29] also lead to oscillating and rotating magnetisation patterns.

Here, we focus on the dynamics of 1D domain walls. Such a domain wall in the
magnetisation $m^{z}$ is shown schematically in Fig. 1a). On the left and
right sides of the domain wall, the (staggered) $xy$-order rotates in opposite
directions, according to Eq. (1). This leads to an effect which we call
‘dynamical frustration’: the in-plane magnetisation directions periodically
point in opposite directions. We will show that for a static domain wall, due
to this frustration no smooth, singularity-free interpolation of the angle
$\varphi(x,t)$ as function of space and time exists across the domain wall.
However, nature can lift this frustration by making the domain wall _move_. We
will indeed show that, across wide parameter regimes, we obtain an active
motion of the domain wall. Here, the direction of motion, left or right,
arises from spontaneous symmetry breaking. We will demonstrate that this
motion has wide-ranging consequences for the dynamics of ordering and the
resilience to noise.

Model –  We consider a 1D classical spin model, $|\bm{S}|=1$, with the
following Hamiltonian

$\displaystyle H=$ $\displaystyle
J\sum_{i}S^{x}_{i}S^{x}_{i+1}+S^{y}_{i}S^{y}_{i+1}-\Delta
S^{z}_{i}S^{z}_{i+1}$
$\displaystyle+\sum_{i}\frac{\delta_{2}}{2}\left.S^{z}_{i}\right.^{2}+\frac{\delta_{4}}{4}\left.S^{z}_{i}\right.^{4}-g_{i}B_{z}(t)S^{z}_{i},$
(2)

where $J,\Delta>0$ impose AFM couplings in the $x$\- and $y$-directions, but
FM interactions for the $z$-component. $\delta_{2}$ and $\delta_{4}>0$
introduce a uniaxial anisotropy. For $\delta_{2}<2J(\Delta-1)$ and $B_{z}=0$,
one obtains a ferrimagnetic ground state with a constant $S_{i}^{z}=\pm
m^{z}_{0}$, where
$m^{z}_{0}=\sqrt{\frac{-\delta_{2}-2J(1-\Delta)}{\delta_{4}}}\geq 0$. The
$g$-factors $g_{i}$ are chosen differently for even and odd sites. Such
staggered $g$ factors naturally occur in magnets with, e.g., a screw axis [30]
and are used here to avoid that $B_{z}(t)$ couples to the conserved
magnetisation only. The dynamics of the system is described by the stochastic
Landau-Lifshitz-Gilbert equation, which includes a phenomenological damping
term $\alpha$ and a corresponding noise term proportional to the temperature
$T$, see methods. We first study the noiseless case, $T=0$.

To drive the system out of thermal equilibrium, we consider the effect of a
rapidly oscillating magnetic field, $B_{z}(t)=B_{0}\cos(\omega t)$,
$\omega\sim J$. In the ferrimagnetic phase, $0<|m_{0}^{z}|<1$, this
oscillation ‘activates’ the Goldstone mode. The staggered $xy$-order,
described by the angle $\varphi$, begins to precess collectively,

$\varphi(t)=-\omega_{\text{rot}}t+\varphi_{0},\quad\omega_{\text{rot}}\approx\gamma
m^{z},\quad\gamma\propto B_{0}^{2},$ (3)

where the formulas are valid for small $B_{0}$ and a small FM magnetisation.
The fields $m^{z}$ and $\varphi$ are defined by $\langle
S_{i}^{z}\rangle=m^{z}(x)$,
$\langle(S_{i}^{x},S_{i}^{y})\rangle=(-1)^{i}\sqrt{1-(m^{z})^{2}}(\cos(\varphi(x)),\sin(\varphi(x))$.
Here, $x$ is the coarse-grained position variable, with $x=ia$ at the position
of spin ${\bf S}_{i}$. $\langle\dots\rangle$ denotes an average over the
oscillation period of the external field (in plots, however, for simplicity’s
sake we show stroboscopic images instead, see methods). An analytical
calculation of $\omega_{\text{rot}}$ using second order perturbation theory in
$B_{0}$ for arbitrary $m^{z}$ is given in the supplementary material, App. C.
The precise way in which non-equilibrium is implemented is expected to be
irrelevant as long as the $xy$-symmetry remains intact. In an experimental
system one could, e.g., also use a laser to create electronic excitations. In
this case, no staggered $g$-factors are needed and one expects that $\gamma$
is proportional to the power of the laser.

Moving domain walls –  In our system, the $xy$-magnetisation rotates in
clockwise or anticlockwise direction when the FM magnetisation points up or
down, respectively. As argued in the introduction, this leads to dynamical
frustration at the centre of the domain wall where $m^{z}$ changes sign. Fig.
1(b,c) shows the result of a numerical simulation of such a domain wall.
Depending on the initial conditions, the domain wall moves either to the right
or to the left with a constant velocity. Far from the domain wall boundary,
the angle $\varphi$ develops a finite slope which builds up behind the moving
domain wall.

Figure 2: Velocity $v$ of a domain wall as function of the amplitude of the
oscillating field $B_{0}$, for different values of anisotropy $\delta_{2}$ and
damping $\alpha$. Across a wide parameter range, $v$ is linear in $B_{0}$ and
thus proportional to $\sqrt{|\omega_{\text{rot}}|}$, see Eq. (7). Lines:
analytical calculation of the domain wall velocity, Eq. (8), without fitting
parameters. Parameters: $J=1,\Delta=0.8,\delta_{2}=-0.6$ and $-0.8$,
$\delta_{4}=1,g_{1}=1,g_{2}=0.1,\alpha=0.1$ and $0.2,\omega=3.6$ for a system
of 40,000 spins. Numerical errors are smaller than the size of the symbols.

To develop an analytical theory for the velocity of the moving domain wall, we
first consider the limit of vanishing damping, $\alpha\to 0$. For a domain
wall moving with the velocity $v$, we make the ansatz

$\displaystyle m^{z}(x,t)$ $\displaystyle=m^{z}(x-vt),$ (4)
$\displaystyle\varphi(x,t)$ $\displaystyle=\Phi(x-vt)+\tilde{\omega}t,$

see App. D.5 for details. We need only one extra input to solve the problem:
for vanishing damping, the magnetisation $m^{z}$ is conserved,
$\partial_{t}m^{z}+\partial_{x}j^{z}=0$ and the spin (super) current is
proportional to the gradient of the phase [31, 32],

$\displaystyle j^{z}=\rho_{s}\partial_{x}\varphi,$ (5)

where $\rho_{s}\approx Ja^{2}(1-(m_{0}^{z})^{2})$ is the spin stiffness and
$a$ the lattice constant. Substituting in Eq. (4), and integrating the
continuity equation over space from $-x_{0}$ to $x_{0}$, one obtains

$\displaystyle-v\,m^{z}\Bigr{|}_{-x_{0}}^{x_{0}}+\rho_{s}\partial_{x}\Phi\Bigr{|}_{-x_{0}}^{x_{0}}=0.$
(6)

Figure 3: Worldlines of domain walls after a quench from the AFM into the
ferrimagnetic phase driven by a small oscillating field $B_{0}=0.15\,J$.
Worldlines end either when a left-moving and a right-moving domain collide or
when the distance between two domain walls shrinks to zero. Colour: amplitude
of $\partial_{x}\varphi$ in units of $\delta_{\varphi}$, given by the
difference of phase gradients
$\delta_{\varphi}=\partial_{x}\varphi^{\text{left}}-\partial_{x}\varphi^{\text{right}}\approx
0.176$ across a single domain wall, see Fig. 1(b,c). The motion of domain
walls is subject to a long-ranged hydrodynamic interaction mediated by
$\partial_{x}\varphi$. Parameters: same as in Fig. 1, initial state: ordered
$xy$-AFM with a small random $S^{z}_{i}=\pm 0.1$, see methods.

Far from the domain wall, the magnetisation takes its bulk value $m^{z}=\pm
m^{z}_{0}$ and the spins rotate with
$\partial_{t}\varphi=-v\partial_{x}\Phi+\tilde{\omega}=\mp|\omega_{\text{rot}}|$,
Eq. (3). Therefore, we obtain from Eq. (6) directly
$-2vm^{z}_{0}+2\rho_{s}\frac{|\omega_{\text{rot}}|}{v}=0$ or, equivalently,

$\displaystyle
v=\pm\sqrt{\frac{\rho_{s}|\omega_{\text{rot}}|}{m^{z}_{0}}}\approx\pm\sqrt{\rho_{s}\gamma}\quad\text{for
}\alpha\to 0.$ (7)

This is a remarkable result for four reasons: (i) It has been derived without
any knowledge of the shape of the domain wall. All microscopic parameters
entering the equation are known analytically for small $B_{0}$. (ii) The
direction of motion has to be chosen by spontaneous symmetry breaking and is,
for example, determined by small fluctuations of $\varphi$ in the initial
state. (iii) As $\omega_{\text{rot}}$ is linear in $m^{z}_{0}$, the velocity
of domain walls remains finite even for $m^{z}_{0}\to 0$, i.e., when the phase
transition from an $xy$-ordered phase to the ferrimagnetic phase is
approached. Indeed, in a previous analysis [14], we found that this velocity
characterises the field theory of the critical point in higher dimensions.
Finally, (iv), the velocity is _very_ fast. $v$ is proportional to the square
root of $\omega_{\text{rot}}$, making it linear in the amplitude $B_{0}$ of
the oscillating field or, equivalently, proportional to the square root of the
power used to drive our system out of thermal equilibrium. For weak $B_{0}$,
this makes $v$ much larger than, e.g., $\omega_{\text{rot}}\xi_{0}$, where
$\xi_{0}$ is the width of the domain wall. Actively moving phase boundaries
have also been shown to exist in models containing mixtures of active and
passive Brownian particles [11].

Eq. (7) is valid for $\alpha\to 0$ for finite $\omega_{\text{rot}}$. The
calculation for finite $\alpha$ is more challenging. In the supplementary
material, App. D, we derive an approximate formula for the velocity,

$\displaystyle v\approx$
$\displaystyle\pm\left(\sqrt{\left(\frac{\alpha\rho_{s}}{\xi_{0}\eta}\right)^{2}+\frac{\rho_{s}|\omega_{\text{rot}}|}{m_{0}^{z}}}-\frac{\alpha\rho_{s}}{\xi_{0}\eta}\right)$
(8) $\displaystyle\approx$
$\displaystyle\pm\left\\{\begin{array}[]{cc}\sqrt{\displaystyle\frac{\rho_{s}|\omega_{\text{rot}}|}{m^{z}_{0}}}&\text{for
}\
\displaystyle\frac{|\omega_{\text{rot}}|}{m_{0}^{z}}\gg\alpha^{2}\frac{\rho_{s}}{\xi_{0}^{2}}\\\\[8.53581pt]
\displaystyle\frac{\xi_{0}\eta|\omega_{\text{rot}}|}{2\alpha
m_{0}^{z}}&\text{for }\
\displaystyle\frac{|\omega_{\text{rot}}|}{m_{0}^{z}}\ll\alpha^{2}\frac{\rho_{s}}{\xi_{0}^{2}},\end{array}\right.$
(11)

where $\xi_{0}$ is the width of the domain wall in equilibrium,
$\xi_{0}=\frac{a\sqrt{2J\Delta/\delta_{4}}}{m^{z}_{0}}$, and $\eta$ is a
dimensionless numerical factor, with $\eta=3$ close to the ferrimagnetic phase
transition. For $\omega_{\text{rot}}\to 0$, the domain wall velocity is linear
in $\omega_{\text{rot}}$. In this limit, the domain wall moves during each
rotation by a large distance $\frac{\pi\eta\xi_{0}}{\alpha
m_{0}^{z}}\gg\xi_{0}$ inversely proportional to the (often very small) Gilbert
damping $\alpha$ and the average magnetisation $m_{0}^{z}<1$.

In Fig. 2, we show the velocity of the domain wall obtained from numerical
simulations, together with Eq. (8). Taking into account that there is no
fitting parameter and the approximate nature of the derivation of Eq. (8), the
agreement for $B_{0}\apprle 0.2$ is remarkable. The analytical formula Eq. (8)
does not include the renormalisation of $m_{0}^{z}$ and $\rho_{s}$ for large
$B_{0}$, which may explain small deviations for larger $B_{0}$.

Above a critical strength of the driving field, $B_{0}>B_{c}$, the moving
domain wall solution ceases to exist. Instead, one obtains localised domain
walls. In the supplementary material, App. A.1, we show that topology enforces
in this case the presence of singular configurations of $\varphi(x,t)$ (phase
slips or, equivalently, space-time vortices [33]), which manifest as sudden
bursts of spin excitations occurring periodically once per rotation period,
$T_{\text{rot}}=2\pi/\omega_{\text{rot}}$.

a)

b)

Figure 4:  Buildup of FM correlations after a quench from an AFM-ordered phase
into the ferrimagnetic phase in the absence of noise, $T=0$, see also Fig. 3.
Panel a): Correlation length $\xi=1/n_{\text{DW}}$ as function of time after a
quench from an AFM-ordered phase into the ferrimagnetic phase both for a
driven and a non-driven system, marked by NEQ (non-equilibrium) and EQ
(equilibrium) in the legend, respectively. The inset shows that the short-time
dynamics of the driven and non-driven system is identical, but after a few
rotation periods $T_{\text{rot}}$, the driven system shows a very fast
increase of the correlation length. In the long-time limit, $\xi$ grows
linearly in time ($\sim 0.065\ t$, black dashed line). For comparison, $vt$,
where $v$ is the velocity of a single domain wall, is also shown as a dashed
red line. The linear growth of correlations with time can also be seen
directly from a scaling plot of the equal-time correlation function
$C(x)=\langle S^{z}_{j}S^{z}_{j+x/a}\rangle$ (averaged over $j$), which is
shown (for even $x/a$) in panel b) as function of $x/(vt)$. Here $v$ is the
velocity of a single domain wall. The plot shows that the maximal speed is
$2v$, arising from two domain walls moving in opposite directions. Parameters:
as in Fig. 3, average over 20 initial states in simulations with 500,000 spins
each (15 initial states for the equilibrium states). Error bars denote the
corresponding standard deviation of the mean.

a)

b)

c)

Figure 5: Panels (a,b): Magnetisation as function of $x$ and $t$ after a
quench into the ferrimagnetic phase for two slightly different temperatures,
and thus different noise levels, for systems of 250,000 spins (other
parameters and colour scale as in Fig. 1). Panel c): $m^{z}(x)=S^{z}_{i}$ at
the final timestep of the simulations shown in panels a) and b), for a small
region with 2,000 spins. The lower green curve shows that an equilibrium
system has a much shorter correlation length at similar noise levels.

Dynamics of ordering –  To investigate the dynamics of ordering, we consider a
quench starting from an initial state with long-ranged AFM order perturbed by
a small random component of the magnetisation in the $z$-direction. An example
for the dynamics after such a quench is shown in Fig. 3. The domain walls move
and when two of them meet, they get annihilated. Thus, the correlation length,
defined as the inverse of the domain wall density, $\xi=1/n_{\text{DW}}$,
grows rapidly, see Fig. 4. At intermediate times, the correlation length grows
approximately with the speed of a single domain wall, $\xi\approx vt$. In the
long time limit, $\xi$ also grows linearly in time,

$\displaystyle\xi(t)\approx\mu\,vt\quad\text{for }t\to\infty,$ (12)

where $\mu\lesssim 1$ is a numerical constant. $\mu\,vt$, with $\mu\approx
0.53$ for the chosen parameters, is shown as a dashed black line in Fig. 4 a).

This very efficient and fast growth of the correlation length is not what is
expected from a theory of independently moving and pairwise annihilating
domain walls. Consider a 1D model, where particles with velocity $\pm v$
annihilate each other whenever two of them collide [34]. In such a model,
within time $t$ roughly $N_{R}=N\pm\sqrt{N}$ left-moving and
$N_{L}=N\pm\sqrt{N}$ right-moving particles interact with each other, where
$N\sim vt/d_{0}$ if $d_{0}$ is the initial distance of the particles. After a
time $t$, most of them will have annihilated, but this process leaves
approximately $|N_{R}-N_{L}|\sim\sqrt{N}$ particles in an area of size $vt$.
Thus, in this model the correlation length grows relatively slowly with

$\displaystyle\xi\sim\sqrt{d_{0}vt}\quad\text{for independently moving
domains}$ (13)

for $vt\gg d_{0}$ [34, 35], in contrast to our finding.

Therefore, we conclude that the rapid growth of $\xi$ can only arise from a
strong interaction of the domain walls, which is mediated by the gradients of
the $xy$-order imprinted by dynamical frustration. This is shown by the
following qualitative argument: within a FM domain, the angle $\varphi$ grows
linearly in time, $\varphi\approx\pm\omega_{\text{rot}}t$, with opposite signs
for up and down domains. If domains have the typical size $\xi(t)$, this
implies that the typical gradient of $\varphi$ is of the order of

$\displaystyle|\partial_{x}\varphi|\sim\frac{2|\omega_{\text{rot}}|t}{\xi(t)}.$
(14)

If we now demand that gradients remain bounded for $t\to\infty$, then it is
plausible that $\xi(t)$ has to grow linearly in $t$, which explains our
numerical observation, Eq. (12). This derivation is not rigorous, as it is
possible to construct solutions of, e.g., equally spaced domain walls with a
time-independent distance. However, in our simulations such configurations are
not stable due to their interactions with an inhomogeneous
$\partial_{x}\varphi$-background.

Figure 6: Equal-time correlation function $C(x)=\langle
m^{z}(x_{0}+x)m^{z}(x_{0})\rangle$ in the presence of thermal noise for
systems of the size of 250,000 spins (parameters as in Fig. 1, only even $x/a$
are shown, error bars denote the standard deviation of the mean). The data is
measured at $t\approx 520{,}000/J\approx 850\,T_{\text{rot}}$ after a quench
from an AFM state. The error bars are standard deviation of the mean obtained
by averaging over four runs. With the exception of the curve for the lowest
$T$ (blue), the system has obtained a stationary state at that time, see
supplementary material, App. E.3. Note the logarithmic scale on the $x$-axis,
needed due to a broad distribution of domain sizes, see Fig. 5. Compared to
the undriven system in thermal equilibrium (dashed lines), the correlation
length becomes many orders of magnitude larger for $T\lesssim 0.025$. Inset:
The correlation length $\xi_{\text{max}}$ grows faster than exponential with
$1/T$. For a discussion of error bars in the inset, see supplementary
material, App. E.3.

Microscopically, the necessary interactions are mediated by $\varphi$. More
precisely, the domain walls interact with $\partial_{x}^{2}\varphi$, as can be
seen directly from the relevant Ginzburg-Landau theory, see methods. A
negative $\partial_{x}^{2}\varphi$ leads to $\partial_{x}j^{z}<0$, see Eq.
(5), and thus to an increase of $m^{z}$, which acts as a repulsive force for
an approaching domain with $m^{z}<0$. Here, the main source of large
$\partial_{x}^{2}\varphi$ are cases where two domain walls meet and
annihilate. As can be seen in Fig. 3, each annihiliation event leaves behind a
characteristic trace in $\partial_{x}^{2}\varphi$ where $\partial_{x}\varphi$
changes sign. Close to $x=37{,}000\,a$ in Fig. 3, a left-moving down domain
wall changes its direction of motion when approaching a region with
$\partial_{x}^{2}\varphi<0$. After transforming into a right-moving domain
wall, it finally annihilates with another left-moving domain wall.
Importantly, the interactions mediated by $\partial_{x}^{2}\varphi$ are
strongly retarded, as it takes a very long time proportional to $\xi^{2}$ for
$\partial_{x}^{2}\varphi$ to decay by diffusion. This is much shorter than the
time of order $\xi/v$ needed for the approach of other domain walls, see
supplementary material App. A.3 for a hydrodynamic view of these effects.

Resilience to noise – Finally, we investigate the influence of thermal noise
(described by a random magnetic field ${\boldsymbol{\xi}}_{i}(t)$, see
methods) on our findings. In a 1D system in thermal equilibrium, such a noise
term destroys long-ranged order and induces domain walls with a typical
distance $\xi_{\text{th}}\sim e^{E_{\text{DW}}/{k_{B}T}}$, where
$E_{\text{DW}}$ is the domain wall energy at $T=0$, and the temperature $T$
controls the noise strength. For the parameters chosen in our figures,
$E_{\text{DW}}=0.076\,J$. In the following, we will therefore always use
$T/E_{\text{DW}}$ to specify temperatures. Figs. 5a) and b) show the
magnetisation of a driven ferrimagnet as function of space and time after a
quench from the AFM phase for two different temperatures. As in the noiseless
case studied above, domain walls actively move with a velocity approximately
$10\,\%$ larger than in the noiseless case, see supplementary material, App.
E.4. It is surprising that the velocity is affected so little, taking into
account that the $xy$-correlation length $\xi_{\text{AFM}}$ in the noisy and
driven system becomes tiny, $\xi_{\text{AFM}}\lesssim 30$, see supplementary
material, App. E.1.

Fig. 5a) shows the formation of giant domains with a size exceeding $10^{5}$
spins. This has to be compared to an equilibrium system, where for the same
noise level the correlation length is less than $20$ sites.

In Fig. 6 we show the steady-state correlation function for both the
equilibrium and non-equilibrium systems. In the driven system, the correlation
function decays very slow in space, roughly with a linear slope on a
logarithmic scale, $C(x)=C(0)(1-\ln[x/a]/\ln[\xi_{\text{max}}(T)/a]$, showing
that the system hosts domains with very different sizes. $\xi_{\text{max}}(T)$
can be interpreted as the largest correlation length in the system, see
supplementary material, App. E.3, for details. $\xi_{\text{max}}(T)$ grows
much faster than exponential with $1/T$, see inset of Fig. 6, showing that the
rapid growth is not arising from a thermal process.

How can even a relatively weak oscillating field increase the correlation
length by many orders of magnitude and why is the driven system so resilient
to noise? In the regime investigated here, fluctuations are not small and even
within an ordered domain, they often induce a sign change, see the upper curve
in Fig. 5c). Locally, fluctuations are very similar to the thermal system. The
difference is, however, that the driven system is very efficient in
suppressing the generation of defects and also in healing some of the most
frequently occurring defects. The main origin of this healing effect is the
dynamical frustration of domains rotating in opposite direction. In the
supplementary material, App. B, we discuss and investigate the detailed
microscopic mechanisms behind this observation.

In conclusion, we show that a simple oscillating field transforms a
ferrimagnet into an ‘active magnet’. This is a new state of matter with
properties very different to those of equilibrium systems. Ferrimagnets
recently attracted considerable interest in the field of spintronics [24], as
they combine the advantages of FM and AFM systems. Here, we mainly need the
rotation of an $xy$-order parameter in a driven system to be controlled by the
orientation of a FM component perpendicular to it. The rotation of the
$xy$-order parameter leads to a novel type of ‘dynamical frustration’,
resulting in an efficient propulsion of the domain wall, which now moves
actively either to the left or to the right by spontaneous symmetry breaking.
The net effect is a very fast buildup of FM correlations on huge length scales
and a remarkable robustness of the system to thermal noise due to an efficient
non-equilibrium self-healing mechanism.

A surprising aspect of our results is that very weak perturbations have a
profound effect on the qualitative and quantitative behaviour system. For the
chosen parameters, the speed of rotation of the magnetisation was tiny,
$\omega_{\text{rot}}J\approx\pm 0.01$. This tiny perturbation changes the
dynamics of the system, affects the correlations both on small and large
length scales, see Fig. 6, and can increase the correlation length by orders
of magnitude. Also, in equilibrium systems, tiny perturbations (if they are
‘relevant’ in the renormalisation-group sense) sometimes strongly affect
properties of correlation functions, but only at very long distances. We are
not aware of any equilibrium systems in which small perturbations cause such a
profound change of magnetic correlations on all length scales. Our driven
system is different, as it induces active motion of domain walls.

In our study, we have investigated a minimal model of a driven magnet. We
anticipate that it is possible to activate (approximate) Goldstone modes in a
large class of systems and to use the principle of dynamical frustration to
realise actively moving defects. Given the enormous variety of complex
symmetries and associated topological defects available in solid-state
systems, this might pave the way towards a new field of active magnetic matter
or, more broadly, active quantum matter. An important aspect of this challenge
will be finding experimental realisations of such systems.

### Methods

The dynamics of the ferrimagnetic system, Eq. (2), is obtained from the
(stochastic) Landau-Lifshitz-Gilbert (LLG) equation

$\displaystyle\frac{d{\boldsymbol{S}}_{i}}{dt}={\boldsymbol{S}}_{i}\times\left(\frac{dH}{d{\boldsymbol{S}}_{i}}+{\boldsymbol{\xi}}_{i}(t)\right)+\alpha{\boldsymbol{S}}_{i}\times\frac{d{\boldsymbol{S}}_{i}}{dt},$
(15)

with the dimensionless damping constant $\alpha$. ${\boldsymbol{\xi}}_{i}(t)$
is a Gaussian random noise term with
$\left\langle\xi_{i}(t)\xi_{j}(t^{\prime})\right\rangle=2\alpha
k_{B}T\delta_{i,j}\delta(t-t^{\prime})$ which arises from thermal
fluctuations. $H$ is given in Eq. (2).

In order to do simulations on the micromagnetic system, we use the GPU-
accelerated open-source software Mumax3 [36]. We simulate systems with sizes
ranging from 20,000 to 500,000 spins with open boundary conditions for
simulations of a single domain wall and periodic boundary conditions for all
other simulations. For quenches from the AFM phase, we use initial states with
$S_{i}^{z}$ drawn from a uniform distribution in the interval of $(-0.1,0.1)$,
$S_{i}^{x}=(-1)^{i}\sqrt{1-(S_{i}^{z})^{2}}$ and $S_{i}^{y}=0$, i.e.,
$\varphi=0$.

In all figures displaying the time evolution, we show stroboscopic images,
obtained at integer multiples of the driving period $2\pi/\omega$.

For our analytical arguments, we mainly use the LLG Eq. (15) either directly
or evaluated in a continuum limit, see supplementary material, App. C and D,
for details. Alternatively, one can analyse the appropriate (noisy) Ginzburg-
Landau theory describing the ferrimagnet close to the transition to the AFM
[14]. This Ginzburg Landau theory is formulated in the continuum limit for FM
magnetisation $m^{z}(x,t)$ and the angle $\varphi(x,t)$ parametrising the
orientation of the AFM $xy$-order. A Taylor expansion in $m^{z}$, derivatives
of $\varphi$ and the number of gradients gives to leading order

$\displaystyle\alpha\dot{\varphi}=-\alpha\gamma m^{z}$
$\displaystyle+\rho_{s}\partial_{x}^{2}\varphi+\dot{m}^{z}+\xi_{\varphi},$
$\displaystyle\alpha\dot{m}^{z}=-rm^{z}-$ $\displaystyle
u(m^{z})^{3}+D\partial_{x}^{2}m^{z}{}$ (16) $\displaystyle-$ $\displaystyle
K\partial_{x}^{2}\varphi-\dot{\varphi}+\xi_{m},$

where $\xi_{\varphi/m}$ describe Gaussian white noise with
$\langle\xi_{\varphi/m}\rangle=0$ and
$\langle\xi_{\varphi/m}\xi_{\varphi/m}\rangle=2\alpha
T\delta(t-t^{\prime})\delta(x-x^{\prime})$. The noise strength is fixed by the
damping terms ($\sim\alpha$) and temperature. The equilibrium system is
obtained by setting $\gamma=K=0$. All parameters of Eq. (Methods) are known
analytically with the exception of $K$, with $r=\delta_{2}+2J(1-\Delta)$,
$u=\delta_{4}$, $\rho_{s}=Ja^{2}$, $D=J\Delta\,a^{2}$. $\gamma$, responsible
for the rotation of the $xy$-order, is proportional to $B_{0}^{2}$ for small
$B_{0}$, and is computed in the supplementary material, App. C. Note that the
effective field theory cannot be used to quantitatively describe the space-
time vortices discussed in the supplementary material, App. A.1, as the
magnetisation reaches $\pm 1$ at the vortex core, which is outside of the
range of validity of Eq. (Methods).

From the first line in Eq. (Methods), one recovers Eq. (1) in the
ferrimagnetic phase for $\partial_{x}^{2}\varphi=0$ and stationary $m^{z}$.
Similarly, for $\alpha\to 0$ one obtains a continuity equation for $m_{z}$,
with the current given by Eq. (5).

Acknowledgements

We acknowledge useful discussions with Romain Daviet, Gerhard Gompper, Joachim
Krug, and Carl Zelle. We thank the German Research Foundation for financial
support via CRC 1238 (Project-ID 277146847, project C04) and CRC/TRR 183
(Project-ID 277101999, project A01). Furthermore, we thank the Regional
Computing Center of the University of Cologne (RRZK) for providing computing
time on the DFG-funded (Funding number: INST 216/512/1FUGG) High Performance
Computing (HPC) system CHEOPS as well as technical support.

Data availability The supporting data and codes for this article are available
from Zenodo [37].

Author contributions D.H., N.d.S and A.R. developed the analytical theory with
N.d.S deriving the theory of domain wall velocities at finite $\alpha$. D.H.
and R.D. performed simulations using the effective Ginzburg Landau theory
(D.H.) and Mumax (R.D.). S.D. and A.R. designed the study. All authors
contributed to the manuscript.

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### Supplementary Material to ‘Active Magnetic Matter: Propelling Domain Walls
by Dynamical Frustration’

##

###  Appendix A Topology of Dynamically Frustrated Domain Walls

####  A.1 Localised Domain Walls and Their Topology

In this section, we discuss the properties of not-moving domain walls. We
observe those primarily when the strength of the driving oscillating field is
large. Moving domain walls only exist below a critical field $B_{c}$, see Fig.
2 of the main text. For $B<B_{c}$, we obtain a coexistence of moving and and
localised domain walls, while for $B_{0}>B_{c}$ only localised single domain
walls exist. For smaller $B_{0}$, however, more and more fine-tuning of
initial conditions is required to obtain localised solutions.

An example of such a localised domain wall is shown in Fig. S1. One sees that
the domain wall periodically emits very sharply defined instanton-like bursts
of magnetisation. This is also a consequence of the dynamical frustration: as
the sense of rotation of $\varphi$ is opposite on both sides of the domain,
tension necessarily builds and gets suddenly released. Close to the crosses in
Fig. S1, the magnetisation reaches $\pm 1$ and the phase describing the
orientation of spins in the $xy$-plane becomes ill-defined. Below, we will
show that topology enforces singularities in $\varphi$.

To see this, it is useful to compute the winding number of the angle
$\varphi(x,t)$ along a contour $C$ in space-time,
$W=\frac{1}{2\pi}\oint_{C}\boldsymbol{\nabla}_{q}\varphi\,d\boldsymbol{q}$
with $\boldsymbol{q}=(t,x)$,
$\boldsymbol{\nabla}_{q}=(\partial_{t},\partial_{x})$. Choosing for $C$ a
rectangular-shaped contour (dashed white line in Fig. S1b)) extending over $N$
oscillation periods $T_{\text{rot}}=\frac{2\pi}{\omega_{\text{rot}}}$ in the
time direction, we find for large $N$ the winding $W=\pm 2N$, with opposite
signs for down-up and up-down domain walls. Here we used that
$\partial_{t}=-|\omega_{\text{rot}}|$ for an up-domain and
$\partial_{t}=|\omega_{\text{rot}}|$ for a down domain. Note that the spatial
parts of the contour cancel each for a periodic solution with
$\mathbf{S}_{i}(t)=\mathbf{S}_{i}(t+NT_{\text{rot}})$. Thus, topology enforces
that $\varphi(x,t)$ must have singularities, or space-time vortices [33], with
a net winding number of two per oscillation period. In the context of
superconductivity, these space-time vortices are called phase slips.
Alternatively, one can also characterise the resulting magnetisation profile
as a fractionally charged defect [38] in space-time.

Numerically, we find that two vortices with identical winding numbers are
located a bit to the left and to the right side of the domain wall. The space-
time vortices send out sharp pulses of spin excitations, exactly spaced by an
oscillation period $T_{\text{rot}}$. At the vortex core, the spins point in
the $z$-direction, $m^{z}=\pm 1$. Unexpectedly, the vortex cores are not
points in space-time, but show an unusual elongated shape. This is shown in
Fig. S1c), which is a zoom of Fig. S1a) to the core of a vortex. Using the
discretised data underlying Fig. S1b), we calculate the local winding number
from four neighbouring data points of $\varphi$ on the space-time lattice by
checking whether the four phase differences (projected on interval from $-\pi$
to $\pi$) add up to $-2\pi$, $0$, or $2\pi$. Black and white dots in Fig. S1c)
denote vortices with winding number $\pm 1$. Several of them align in a line
but the net winding is always $-1$, as enforced by topology.

a)b)

c)d)

Figure S1: Non-moving domain wall obtained for a larger oscillating field.
Panel a) shows the magnetisation $m^{z}(x,t)$ and panel b) the phase
$\varphi(x,t)$, describing the orientation of the staggered in-plane
magnetisation. For a non-moving domain wall, topology enforces the emergence
of two space-time vortices (or, equivalently, phase slips) per rotation period
(white crosses in panel a)). The magnetisation reaches $\pm 1$ at the
(elongated) vortex cores and triggers the emission of sharply defined pulses
in both directions. Panel c) shows a zoom to the centre of such an elongated
vortex core. Here, white and black dots denote the position of space-time
vortices with winding number $\pm 1$. In the shown example, there are 4 white
and 5 black dots. Thus, the total winding number is $-1$. Panel d) directly
shows $m^{z}(x)$ for $t=nT_{\text{rot}}/12$, $n=0,\dots,11$. For better
visibility, the curves are shifted relative to each other. The thin dashed
lines denote $m^{z}=1$. At the location of the vortices (brown curve in the
middle, at $x=17{,}976\,a$), the magnetisation reaches $1$. Parameters:
$J=1,\Delta=0.8,\delta_{2}=-0.6,\delta_{4}=1,g_{1}=1,g_{2}=0.1,\alpha=0.1,B_{0}=0.42,\omega=3.6$
with $T_{\text{rot}}\approx 81$ for a system of 40,000 spins.

####  A.2 Topology of a Moving Domain Wall

Above, we have shown that a localised domain wall will necessarily produce
periodically singular spin configurations characterised by a space-time vortex
(phase-slip) in $\varphi$. Thus, any smooth, non-singular $xy$-spin
configuration will enforce a motion of the domain wall.

Here we discuss constraints arising from topology for such a domain wall. We
can use that far from the domain wall, the magnetisation is constant in space
and also $\partial_{x}^{2}\varphi=0$. Furthermore, the $xy$-magnetisation on
the right and left side rotate with opposite speeds,
$\varphi_{R/L}=\pm\omega_{\text{rot}}t$. Thus we can make the ansatz

$\displaystyle\varphi_{R/L}(x,t)=\varphi_{R/L}^{0}+(\partial_{x}\varphi_{R/L})x\pm\omega_{\text{rot}}t,$
(S1)

which is valid sufficiently far away from the domain wall on the left and
right sides of it. Next, we calculate the winding number for a long loop
oriented parallel to a moving domain wall with velocity $v$. Such a loop in
distance $d$ can, e.g., be parameterised by $(t,x)=(\lambda
NT_{\text{rot}},vNT_{\text{rot}}\lambda+d)$ for $0\leq\lambda<1$,
$(t,x)=(NT_{\text{rot}},vNT_{\text{rot}}+(3-2\lambda)d$ for $1\leq\lambda<2$,
$(t,x)=(NT_{\text{rot}}(3-\lambda),vNT_{\text{rot}}(3-\lambda)-d)$ for
$2<\lambda<3$, and $(t,x)=(0,d(2\lambda-7))$ for $3<\lambda<4$, closing the
loop with $T_{\text{rot}}=\frac{2\pi}{|\omega_{\text{rot}}|}$,
$N\in\mathbb{Z},N\gg 1$. A smooth spin configuration can only be obtained for
a vanishing winding number,

$\displaystyle
W=\frac{N}{2\pi}\left(2\omega_{\text{rot}}T_{\text{rot}}+vT_{\text{rot}}(\partial_{x}\varphi_{R}-\partial_{x}\varphi_{L})\right)=0\quad\Rightarrow\quad\partial_{x}\varphi_{R}-\partial_{x}\varphi_{L}=-\frac{\omega_{\text{rot}}}{v}.$
(S2)

where $\omega_{\text{rot}}$ is the rotation speed on the right side of the
domain wall, see above. The wavelength corresponding to the phase changes is
given by $\frac{2\pi}{\lambda_{R/L}}=\partial_{x}\varphi_{R,L}$. Therefore,
equation (S2) can also be written in the more intuitive form

$\displaystyle|v|=\frac{|\Delta\lambda|}{T_{\text{rot}}},\quad\frac{1}{\Delta\lambda}=\frac{1}{\lambda_{L}}-\frac{1}{\lambda_{R}}.$
(S3)

Topology enforces that the domain wall has to move by $\Delta\lambda$ within a
rotation period. The difference of gradients to the left and right side of the
domain wall has to be inversely proportional to the velocity of the domain
wall, as long as $\varphi(x,t)$ configurations are non-singular. This result
is based only on topology and on properties far away from the domain wall. As
the difference of gradients has to be finite, topology enforces a finite
domain wall velocity $v$ for vortex-free $\varphi(x,t)$ solutions. To
determine the value of $v$, one needs an extra equation, see main text and
App. D.

a)

b)

Figure S2: Magnetisation as a function of $x$ and $t$ starting from
ferrimagnetic initial conditions for the given temperature. Panel b) shows a
zoom-in on the region marked by a black dashed line in panel a). Note that for
the same noise level, an equilibrium system has a correlation length of only
20 sites. Two types of defects are visible. The blue lines in panel b) are
pairs of parallel moving domains with a width of up to a few hundred sites.
They do not affect the long-ranged order. The simulation also shows a single
defect of a different type: a pair of domain walls moving in opposite
directions, which is able to revert the magnetisation. Parameters:
$J=1,\Delta=0.8,\delta_{2}=-0.6,\delta_{4}=1,\alpha=0.1,g_{1}=1,g_{2}=0.1,\omega=3.6,B_{0}=0.15$,
$T_{\text{rot}}=612$, $T=0.328\,E_{\text{DW}}$ for a system of 250,000 spins.

####  A.3 Hydrodynamics of Winding Numbers

The discussion in the preceding two sections shows that the presence or
absence of space-time vortices strongly affects the dynamics of the system. As
space-time vortices are rare for small driving amplitudes,
$\partial_{x}\varphi$ is an approximately conserved quantity. Therefore, one
can introduce the topological charge density

$\displaystyle\rho_{\text{top}}=\frac{\partial_{x}\varphi}{2\pi}.$ (S4)

For smooth $\varphi(x,t)$ and periodic boundary conditions, the topological
charge $Q_{\text{top}}=\int\rho_{\text{top}}\,dx$ is a conserved winding
number which encodes how frequently the AFM magnetisation winds in the
$xy$-plane. The integer $Q_{\text{top}}$ changes by $\pm 1$ at the location of
a space-time vortex but such events are absent for, e.g., the parameters
chosen in Fig. 3 of the main text.

Using this quantity, we can reformulate some of our findings. Taking the
derivative of the first line of Eq. (Methods) in the method section, we find
that the topological current defined by
$\partial_{t}\rho_{\text{top}}+\partial_{x}j_{\text{top}}=0$, defined by
$\partial_{t}\rho_{\text{top}}+\partial_{x}j_{\text{top}}=0$ for non-singular
configurations of $\varphi(x,t)$, is given by

$\displaystyle
j_{\text{top}}=\frac{\gamma}{2\pi}m^{z}-\frac{\rho_{s}}{\alpha}\partial_{x}\rho_{\text{top}}-\frac{1}{2\pi\alpha}\left(\dot{m}^{z}+\xi_{\varphi}\right).$
(S5)

Far from a domain wall, where $\partial_{t}m^{z}=\partial_{x}^{2}\varphi=0$,
we therefore obtain a topological current
$j_{\text{top}}\approx\frac{\gamma}{2\pi}m^{z}$ of opposite sign to the left
and right side of the domain wall. This necessarily leads to an accumulation
of topological charge close to the domain wall. For a moving domain wall, this
explains the accumulation of $\partial_{x}\varphi$ associated with the motion
of the domain wall, see Fig. 1c) of the main text. For the localised domain
wall, the extra topological charge is removed by phase-slips (space-time
vortices) in a quantised way.

The accumulation of topological charge and its interplay with the domain wall
motion is also the main source of interaction effects. Consider two right- and
left-moving domain walls which annihilate. Each of them leaves a trace of
positive and negative topological charge behind them, see Fig. 3 of the main
text. After annihilation, this imbalance of $\rho_{\text{top}}$ can decay by
diffusion, but this takes a very long time of order $T_{d}=\alpha
L_{d}^{2}/\rho_{s}$, where $L_{d}$ is the typical distance of domain walls.
For $L_{d}\gg\rho_{s}/(\alpha v)$, this time scale is much larger than
$L_{d}/v$, the typical time needed for the next domain walls to arrive at the
collision point. While the speed of the domain wall is not affected by a
constant $\rho_{\text{top}}$, interactions with gradients
$\partial_{x}\rho_{\text{top}}$ are strong. This can be understood
qualitatively from the fact that phase gradients also induce a spin current,
Eq. (5) in the main text. Thus, extra magnetisation $m_{z}$ accumulates
locally whenever $\partial_{x}\rho_{\text{top}}$ is finite. An excess of ‘up’
magnetisation then suppresses incoming down-domains.

###  Appendix B Resilience of the Ordered Phase to Noise

As discussed in the main text, even when the magnet is driven by a very weak
oscillating field, its correlation length grows by many orders of magnitude
and ordered domains show a remarkable resilience to sizable noise. There are
three main reasons for this: (i) The formation of a domain wall with the
opposite direction of rotation is suppressed by the dynamical frustration of
the domain wall. (ii) The system has an efficient self-healing mechanism and
is stable with respect to pairs of domain walls moving in a parallel
direction. (iii) Pairs of domain walls moving in antiparallel directions are
very rare, as their nucleation is strongly suppressed by an effective
attractive interaction of the domain walls.

In Fig. S2, we show a simulation of 250,000 spins, lasting over
$3{,}100\,T_{\text{rot}}\approx 1.9\cdot 10^{6}J^{-1}$, or more than $10^{6}$
oscillation periods of the external magnetic field $B_{z}(t)$. The simulations
are in a regime where the correlation length is larger than the system size.
At the same noise level, the equilibrium system has a correlation length of
less than 20 sites.

An enlarged view, Fig. S2b), of the area within the dashed rectangle in Fig.
S2a) shows two types of defects. Type I defects are pairs of domain walls
which move parallel to each other with velocity $\pm v$, where $v$ is the
single-domain velocity. They are visible as blue streaks in Fig. S2b). They
occur relatively frequently and can have a width of up to a few hundred sites.
Nevertheless, they do not destroy the long-ranged order and are often
eliminated when they meet one another.

More important are type II defects, which can be viewed as a pair of domain
walls moving in opposite directions. They occur so rarely that there is only a
single such event in our very long simulation. At this defect, a spin-down
domain (light blue) is nucleated within a spin-up (orange) domain, which is
rapidly growing to macroscopic scales with velocity $2v$. At higher noise
level, these events are more frequent, see Fig. 5b) of the main text.

While the driven system has a very efficient self-healing mechanism for type I
defects, this is not the case for type II defects, which efficiently nucleate
a phase with reversed magnetisation.

a)b)

c)d)

Figure S3: Simulation of a type II defect in the absence of thermal noise. As
an initial state, we consider a spin-down domain of width $L$ embedded in a
spin-up domain. The orientation of the $xy$-order is chosen according to Eq.
(S6), with $\alpha=0.4$ and $\delta_{0}=0.17$. The left and right panels show
$m^{z}(x,t)$ and $\varphi(x,t)$, respectively, for $L=200$ (panels a) and b))
and $L=220$ (panels c) and d)). The smaller domain shrinks and is self-
annihilating, while the larger domain continues to grow. Parameters: same as
in Fig. S2 at $T=0$ for systems of 20,000 spins.

As type II defects are so rare, there must be a mechanism which suppresses
their creation. We will now argue that this is due to an effective attraction
of domain walls moving in opposite directions, which is able to eliminate type
II defects as long as their width is small.

To show that, we have performed a set of simulations of a system without
noise. We have initialised the system with a down-domain of width $L$ embedded
in an up-domain. To trigger a motion in opposite direction, we choose as an
initial condition for the orientation of the staggered $xy$-order,

$\displaystyle\varphi=\left(\frac{L}{4}-\frac{x^{2}}{L}\right)\alpha\delta_{0}\quad\text{
for}\quad-\frac{L}{2}<x<\frac{L}{2},$ (S6)

and $\varphi=0$ for $|x|\geq L/2$. Therefore, the gradient
$\partial_{x}\varphi$ jumps at the domain wall boundaries by
$\alpha\delta_{0}$. Here $\delta_{0}=0.17$ is the jump of the gradient
obtained for a single moving domain wall in the long-time limit, see Fig. 1c)
of the main text. The amplitude of the jump is directly related to the initial
velocity of the domain wall, see Eq. (S2). The profile and signs are chosen
such that for $\alpha>0$, a motion of the left domain wall to the left and of
the right domain wall to the right are expected. In Fig. S3, two such domains
are shown. The larger one, Fig. S3(c,d), with a width of 220 sites, indeed
grows in size over time. In contrast, the smaller one, Fig. S3(a,b), with a
width of 200 sites, initially expands, but then starts to shrink and finally
vanishes after $6\,T_{\text{rot}}$. In Fig. S4, we denote in red the
parameters for which we obtain shrinking domains and in black those for which
we find growing ones. The data shows that for small $L$, all domain walls are
self-annihilating. Thermal fluctuations have to conspire for a long time in a
spatial interval of $O(100)$ sites to create the initial conditions for a
growing type II defect. It is therefore extremely difficult and rare to create
a type II defect which grows linearly in time, as shown in Fig. S2. This
suggests an exponentially small rate by which such defects are created.

The origin of the collapse of smaller domains is an effective attractive
interaction mediated by the inhomogeneous $\varphi(x,t)$ profile. In the
centre of the domain, $\partial_{x}^{2}\varphi<0$. Accordingly, spin currents
are flowing towards the centre,
$\partial_{x}j_{z}\approx\rho_{s}\partial_{x}^{2}\phi<0$, see Eq. (5) of the
main text. Thus, the system would like to increase the magnetisation in the
center, $\partial_{t}m^{z}\approx-\partial_{x}j_{z}>0$ which favours a
shrinking of the spin-down domain in the center. As
$\partial_{x}^{2}\varphi\propto\frac{1}{L}$ this mechanism is most effective
for small domains, which explains why only domains larger than a critical size
can expand.

In conclusion, the correlation length of the driven system is so large because
frequently occurring type I defects (parallel moving domain walls) are self-
healing. At the same time, the much more dangerous type II defects (anti-
parallel moving domain walls) are strongly suppressed by an attractive
interaction mediated by $\partial_{x}^{2}\varphi$. Only domains exceeding
$O(100)$ sites are able to grow for the chosen parameters.

Figure S4: Phase diagram of type II defects as a function of $L$ and $\alpha$,
which parametrise the initial width of the domain wall and the initial profile
of $\varphi(x)$, see Eq. (S6). In the red region, we find solutions where the
domain always shrinks and is self-annihilating, see Fig. S3(a,b). Only the
domains in the black region with a large initial width are able to grow, see
Fig. S3(c,d). The fact that only very large domains can grow explains why
growing domains are rarely created by thermal noise. Parameters: same as in
Fig. S3.

###  Appendix C Activation of the Goldstone Mode

In this section we show how driving a ferrimagnetic chain with a weak
oscillating magnetic field activates the rotational Goldstone mode in the
$xy$-order of the spins. We will calculate analytically the rotation frequency
of the AFM $xy$-order, $\omega_{\text{rot}}$, for small amplitudes of
$B_{z}(t)$. This serves as a particular example of a general phenomenon,
expected to occur in any driven magnetic system as long as the drive breaks
enough symmetries. Recent studies of other magnetic textures with ‘activated’
translational and rotational Goldstone modes include helimagnets, skyrmions
and skyrmion lattices [22, 39, 23, 28]. A simplified version of the analysis
presented here, valid near the critical point $m_{0}=0$, has been published in
[14].

Our starting point is a ferrimagnetic chain ground state. On account of the
chain’s periodicity, we have $\mathbf{S}_{i}=\mathbf{S}_{i+2n}$ for any
$n\in\mathbb{Z}$. We expect this assumption to remain valid once the system is
driven, as long as the drive is weak enough that no finite-$k$ instabilities
are excited [22]. Thus, it is sufficient to define two species of spin, even
and odd, with $\mathbf{S}_{e}=\mathbf{S}_{2n}$,
$\mathbf{S}_{o}=\mathbf{S}_{2n+1}$, and solve for their coupled dynamics. We
parametrise $\mathbf{S}_{e/o}$ using spherical polar angles $\theta_{e/o}$ and
$\phi_{e/o}$,

$\mathbf{S}_{e}(t)=\begin{pmatrix}\sin(\theta_{e})\cos(\phi_{e})\\\
\sin(\theta_{e})\sin(\phi_{e})\\\
\cos(\theta_{e})\end{pmatrix},\quad\mathbf{S}_{o}(t)=\begin{pmatrix}\sin(\theta_{o})\cos(\pi+\phi_{o})\\\
\sin(\theta_{o})\sin(\pi+\phi_{o})\\\ \cos(\theta_{o})\end{pmatrix}.$ (S7)

Note that $\varphi_{i}$ in the main text refers to $\phi_{e}$ for even $i$ and
$\phi_{o}$ for odd $i$. In the absence of a drive, $\theta_{e/o}$ and
$\phi_{e/o}$ are obtained by solving the Euler-Lagrange equations derived from
the static Hamiltonian, Eq. 2 with $B_{0}=0$. We find
$\theta_{e/o}=\theta_{0}=\arccos(m_{0})$, where
$m_{0}=\sqrt{\frac{-\delta_{2}-2J(1-\Delta)}{\delta_{4}}}$, and
$\phi_{e}=\phi_{0}$, $\phi_{o}=\phi_{0}+\pi$. Here, the constant $\phi_{0}$
can take any value, as it corresponds to the rotational Goldstone mode of the
system.

Turning on the oscillating $B$-field, $B_{0}>0$, we now expect
$\theta_{e/o}(t),\phi_{e/o}(t)$ to become functions of time. Assuming
$B_{0}/J\ll 1$, we can expand them perturbatively around their equilibrium
values,

$\displaystyle\theta_{e/o}(t)$
$\displaystyle=\theta_{0}+\epsilon\theta^{(1)}_{e/o}(t)+\epsilon^{2}\theta^{(2)}_{e/o}+\mathcal{O}(\epsilon^{2}),$
(S8) $\displaystyle\phi_{e/o}(t)$
$\displaystyle=\phi_{0}+\epsilon\phi^{(1)}_{e/o}(t)+\epsilon^{2}\omega_{\text{rot},e/o}t+\mathcal{O}(\epsilon^{2}).$

Here, $\epsilon$ is a bookkeeping parameter we introduce for doing
perturbation theory, which we also place in front of the driving field term,
$B_{z}(t)\to\epsilon B_{0}\cos(\omega t)$. Generically, we expect the first
order response to be purely oscillatory, with $\theta^{(1)}_{e/o}(t)$,
$\phi^{(1)}_{e/o}(t)\sim e^{\pm i\omega t}$ oscillating at the same frequency
as the driving frequency. By the same logic, at second order, the response
should either be oscillatory again, with twice the frequency, $e^{\pm 2i\omega
t}$, or DC, $e^{0i\omega t}$. More interestingly, the
$\mathcal{O}(\epsilon^{2})$ response can also grow _linearly_ in $t$ if the
mode in question is a Goldstone mode of the system. Thus, we add the
$\omega_{\text{rot},e/o}t$ terms, which physically describe a rotation in time
of the $xy$-order around the $S^{z}_{i}$-axis, to our ansatz Eq. S8 at second
order in $\epsilon$. The goal of our calculation is to obtain an analytical
expression for the rate of rotation $\omega_{\text{rot},o/e}$ and to find out
how it depends on the system and drive parameters.

To this end, we first cross the LLG, Eq. 15 in the methods section, with
$\mathbf{S}_{e/o}$, and then project it onto
$\frac{\partial\mathbf{S}_{e/o}}{\partial\theta_{e/o}}$,
$\frac{\partial\mathbf{S}_{e/o}}{\partial\phi_{e/o}}$, thus obtaining four
coupled equations of motion for $\theta_{e/o},\phi_{e/o}$,

$\displaystyle\dot{\phi}_{e/o}\sin(\theta_{e/o})-\alpha\dot{\theta}_{e/o}$
$\displaystyle=\frac{\delta H}{\delta\theta_{e/o}},$ (S9)
$\displaystyle-\sin(\theta_{e/o})\dot{\theta}_{e/o}-\alpha\dot{\phi}_{e/o}\sin^{2}(\theta_{e/o})$
$\displaystyle=\frac{\delta H}{\delta\phi_{e/o}}.$ (S10)

Due to the $\mathbf{S}_{i}=\mathbf{S}_{i+2n}$ symmetry, we only need to
consider a sub-Hamiltonian consisting of any two adjacent sites in the
infinite chain Eq. 2 in the main text. We denote this two-site Hamiltonian as
$\mathcal{H}_{\text{2 sites}}$. Splitting it into its static and driven
components, $\mathcal{H}_{\text{2
sites}}=\mathcal{H}_{\text{stat}}+\mathcal{H}_{\text{drive}}$, we have

$\displaystyle\mathcal{H}_{\text{stat}}$
$\displaystyle=J\Bigg{[}-2\big{(}\sin(\theta_{e})\sin(\theta_{o})\cos(\phi_{e}-\phi_{o})+\Delta\cos(\theta_{e})\cos(\theta_{o})\big{)}\Bigg{]}$
(S11)
$\displaystyle+\frac{1}{4}\Bigg{[}\left(\delta_{2}+\frac{\delta_{4}}{2}\right)\left(\cos(2\theta_{e})+\cos(2\theta_{o})\right)+\frac{\delta_{4}}{8}\left(\cos(4\theta_{e})+\cos(4\theta_{o})\right)\Bigg{]},$
$\displaystyle\mathcal{H}_{\text{drive}}$ $\displaystyle=-\epsilon
B_{1}(t)\left(g_{e}\cos(\theta_{e})+g_{o}\cos(\theta_{o})\right).$

Note that for an undriven static system,
$\dot{\theta}_{e/o}=\dot{\phi}_{e/o}=0$, so that Eq. S9 and S10 are equivalent
to energy minimisation, $\frac{\partial
H}{\partial\theta_{e/o}}=\frac{\partial H}{\partial\phi_{e/o}}=0$. We now
substitute the spin parametrisation Eq. S7, together with the perturbative
expansion Eq. S8, into Eq. S9 and S10, and solve perturbatively in powers of
$\epsilon$. At linear order in $\epsilon$, the
$\theta^{(1)}_{e/o}(t),\phi^{(1)}_{e/o}(t)$ fields decompose into two
$\pm\omega$ Fourier frequency components,

$\displaystyle\theta^{(1)}_{e/o}(t)$
$\displaystyle=\tilde{\theta}_{e/o}e^{i\omega
t}+\tilde{\theta}^{*}_{e/o}e^{-i\omega t},$ (S12)
$\displaystyle\phi^{(1)}_{e/o}(t)$ $\displaystyle=\tilde{\phi}_{e/o}e^{i\omega
t}+\tilde{\phi}^{*}_{e/o}e^{-i\omega t},$

where the coefficients of the $e^{\pm i\omega t}$ are complex conjugates of
each other, ensuring that $\theta^{(1)}_{e/o}(t),\phi^{(1)}_{e/o}(t)$ are
real. We now solve for the $\tilde{\theta}_{e/o},\tilde{\phi}_{e/o}$ Fourier
coefficients by comparing the coefficients of the $e^{i\omega t}$ terms in the
four coupled equations in Eq. S9 and S10, giving

$\begin{aligned}
&\tilde{\phi}^{(1)}_{e/o}=-\frac{B_{0}}{2}\Bigg{[}\frac{i(g_{e}+g_{o})}{i\alpha\delta_{4}\cos(4\theta_{0})-i\alpha\cos(2\theta_{0})(-2\delta_{2}-\delta_{4}+4(\Delta-1)J)+2\left(\alpha^{2}+1\right)\omega}\\\
&+\frac{i\omega(g_{e/o}-g_{o/e})}{2\left(\alpha^{2}+1\right)\omega^{2}-16(\Delta+1)J^{2}+(2\delta_{2}+\delta_{4})\cos(2\theta_{0})(4J+i\alpha\omega)+\delta_{4}\cos(4\theta_{0})(4J+i\alpha\omega)-4i\alpha(\Delta+3)J\omega}\Bigg{]},\\\
&\tilde{\theta}^{(1)}_{e/o}=\sin(\theta_{0})\Bigg{[}-\alpha\tilde{\phi}^{(1)}_{e/o}+\\\
&\frac{B_{0}}{2}\frac{4J(g_{e/o}-g_{o/e})}{2\left(\alpha^{2}+1\right)\omega^{2}-16(\Delta+1)J^{2}+(2\delta_{2}+\delta_{4})\cos(2\theta_{0})(4J+i\alpha\omega)+\delta_{4}\cos(4\theta_{0})(4J+i\alpha\omega)-4i\alpha(\Delta+3)J\omega}\Bigg{]}.\end{aligned}$

(S13)

_Check of Eq. S13 using the continuity equation —_ As mentioned in the main
text, there is a Noether current associated with the Goldstone mode,
$J^{\mu}=\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi_{e}}+\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi_{o}}$,
which satisfies the continuity equation $\partial_{\mu}J^{\mu}=0$, with
$\mu=\\{t,x,y,z\\}$, when $\alpha=0$. We have $J^{t}=m^{z}_{e}+m^{z}_{o}$, and
$J^{x,y,z}=0$, as we assumed the ferrimagnetic chain to be translationally
invariant. Thus, the continuity equation is just
$\dot{m}^{z}_{e}+\dot{m}^{z}_{o}=0$. This should be satisfied at all orders of
$\epsilon$. At first order in $\epsilon$, the equation reads
$\sin(\theta_{0})\left(\dot{\theta}^{(1)}_{e}+\dot{\theta}^{(1)}_{o}\right)=0$.
Setting $\alpha=0$ in Eq. S13, we see that
$\theta^{(1)}_{e/o}(t)\sim(g_{e/o}-g_{o/e})$ implies that this is indeed
satisfied.

Figure S5: Comparison of numerical and analytical results for the rotation
rate of the $xy$-order, $\varphi=\omega_{\text{rot}}t$, here shown as a
function of the frequency $\omega$ of the oscillating magnetic field,
$B_{z}(t)=B_{0}\cos(\omega t)$. For small $B_{0}$, a perfect agreement of
analytical and numerical results is obtained. Parameters:
$J=1,\Delta=0.8,\delta_{2}=-0.6,\delta_{4}=1,g_{1}=1,g_{2}=0.1,\alpha=0.1,B_{0}=0.1$,
$T=0$.

We now move on to the second order in $\epsilon$ response. Here, we are only
interested in obtaining $\omega_{\text{rot},e/o}$. To this end, we average Eq.
S10 over one period of oscillation $T=2\pi/\omega$ to remove all oscillating
terms and consider only the $\mathcal{O}(\epsilon^{2})$ terms, yielding

$\displaystyle\sin^{2}(\theta_{0})\omega_{\text{rot},e/o}=-2\sin(\theta_{0})\cos(\theta_{0})\left\langle\dot{\phi}^{(1)}_{e/o}\theta^{(1)}_{e/o}+\frac{1}{\alpha}\left(\theta^{(1)}_{e}+\theta^{(1)}_{o}\right)\left(\phi^{(1)}_{e/o}-\phi^{(1)}_{o/e}\right)\right\rangle_{T}.$
(S14)

Substituting in Eq. S13, after some algebra we can see that
$\omega_{\text{rot},e}=\omega_{\text{rot},o}=\omega_{\text{rot}}$, which can
be written in the more symmetric form

$\omega_{\text{rot}}=-\frac{\cos(\theta_{0})}{\sin(\theta_{0})}\left\langle\dot{\phi}^{(1)}_{e}\theta^{(1)}_{e}+\dot{\phi}^{(1)}_{o}\theta^{(1)}_{o}\right\rangle_{T}.$
(S15)

In the limit of small $m_{0}=\cos(\theta_{0})$, $\sin(\theta_{0})\approx 1$,
so $\omega_{\text{rot}}\propto m_{0}$. By construction, we also have
$\omega_{\text{rot}}\propto B^{2}_{0}$. A comparison of this analytical
finding to numerical data generated with Mumax3 is shown in Fig. S5. The
agreement is very good.

###  Appendix D Flying Domain Wall Velocity in the Presence of Finite Damping

Having studied the dynamics of the simple ferrimagnetic chain in App. C, we
are now prepared to tackle the challenge of calculating the dynamics of the
driven ferrimagnetic domain wall. Far away from the domain wall, the spins
rotate in the (anti)clockwise directions at the rate
$\pm|\omega_{\text{rot}}|$, depending on the sign of $m^{z}$. This creates a
dynamical frustration at the centre of the system, which the domain wall
resolves by acquiring a horizontal velocity $v$. In the main text, we consider
the limit of vanishing damping $\alpha$ and calculate $v$ by integrating the
conserved Noether current $\partial_{\mu}J^{\mu}=0$ over all space, and find
that $v\sim\sqrt{|\omega_{\text{rot}}|}$ for $\alpha\to 0$.

In the presence of finite damping, the Noether current is no longer conserved:
instead we have $\partial_{\mu}J^{\mu}=\mathcal{O}(\alpha)$. Nevertheless, the
velocity can still be calculated approximately. We present the full
calculation in this appendix, and show that a qualitatively new regime emerges
where $v\sim|\omega_{\text{rot}}|/\alpha$, when damping dominates driving,
$|\omega_{\text{rot}}|\lesssim\alpha^{2}$.

####  D.1 Derivation of the Continuum Description

Let us set up the calculation. The presence of the domain wall introduces a
spatial dependence (in addition to time dependence from the driving) to the
fields $\theta_{e/o},\phi_{e/o}$. To simplify the calculation, we make the
fields continuous functions of space and define a Hamiltonian density
$\mathcal{H}$, such that $H=\int\mathop{dx}\mathcal{H}$. This is a justified
approximation as long as the lattice spacing $a$ is much smaller than the
domain wall width $\xi_{0}$. We Taylor expand fields on the sites $x\pm a$
adjacent to site $x$ up to second order in the spatial derivatives,

$\displaystyle\theta_{e/o}(x\pm a,t)$ $\displaystyle=\theta_{e/o}(x,t)\pm
a\theta^{\prime}_{e/o}(x,t)+\frac{a^{2}}{2}\theta^{\prime\prime}_{e/o}(x,t)+\mathcal{O}(a^{3}),$
(S16) $\displaystyle\phi_{e/o}(x\pm a,t)$ $\displaystyle=\phi_{e/o}(x,t)\pm
a\phi^{\prime}_{e/o}(x,t)+\frac{a^{2}}{2}\phi^{\prime\prime}_{e/o}(x,t)+\mathcal{O}(a^{3}).$

We obtain the Heisenberg contribution to the Hamiltonian density from

$\displaystyle\mathcal{H}$ $\displaystyle=\frac{J}{2a}\sum_{\Delta x=\pm
a}\left(S^{x}_{e}(x)S^{x}_{o}(x+\Delta x)+S^{y}_{e}(x)S^{y}_{o}(x+\Delta
x)-\Delta S^{z}_{e}(x)S^{z}_{o}(x+\Delta x)\right)$ (S17)
$\displaystyle\qquad+\Big{(}S^{x}_{o}(x)S^{x}_{e}(x+\Delta
x)+S^{y}_{o}(x)S^{y}_{e}(x+\Delta x)-\Delta S^{z}_{o}(x)S^{z}_{e}(x+\Delta
x)\Big{)}.$

The continuum version of the Hamiltonian density therefore reads

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{J}{a}\Bigg{[}-2\left(\sin(\theta_{e})\sin(\theta_{o})\cos(\phi_{e}-\phi_{o})+\Delta\cos(\theta_{e})\cos(\theta_{o})\right)$
(S18)
$\displaystyle\qquad+\frac{1}{2}a^{2}\bigg{(}\sin(\phi_{e}-\phi_{o})\sin(\theta_{e})\sin(\theta_{o})(\phi_{e}^{\prime\prime}-\phi_{o}^{\prime\prime})$
$\displaystyle\qquad+\sin(\theta_{e})\sin(\theta_{o})\cos(\phi_{e}-\phi_{o})\left((\phi_{o}^{\prime})^{2}+(\phi_{e}^{\prime})^{2}\right)$
$\displaystyle\qquad+2\sin(\phi_{e}-\phi_{o})\left(\cos(\theta_{e})\sin(\theta_{o})\theta_{e}^{\prime}\phi_{e}^{\prime}-\cos(\theta_{o})\sin(\theta_{e})\theta_{o}^{\prime}\phi_{o}^{\prime}\right)$
$\displaystyle\qquad+\left(\sin(\theta_{e})\sin(\theta_{o})\cos(\phi_{e}-\phi_{o})+\Delta\cos(\theta_{e})\cos(\theta_{o})\right)\left((\theta_{e}^{\prime})^{2}+(\theta_{o}^{\prime})^{2}\right)$
$\displaystyle\qquad+\left(\Delta\cos(\theta_{o})\sin(\theta_{e})-\cos(\theta_{e})\sin(\theta_{o})\cos(\phi_{e}-\phi_{o})\right)\theta_{e}^{\prime\prime}$
$\displaystyle\qquad+\left(\Delta\cos(\theta_{e})\sin(\theta_{o})-\cos(\theta_{o})\sin(\theta_{e})\cos(\phi_{e}-\phi_{o})\right)\theta_{o}^{\prime\prime}\bigg{)}+\mathcal{O}(a^{3})\Bigg{]}$
$\displaystyle+\frac{1}{4a}\left(\left(\delta_{2}+\frac{\delta_{4}}{2}\right)\left(\cos(2\theta_{e})+\cos(2\theta_{o})\right)+\frac{\delta_{4}}{8}\left(\cos(4\theta_{e})+\cos(4\theta_{o})\right)\right).$

Note that the energy terms due to anisotropies and driving $B$-field remain
unchanged, as they are local.

####  D.2 Conserved Noether Current

Before stating and solving the continuous LLG equations resulting from Eq.
S18, we derive the Noether current associated with the Goldstone mode of the
system. The kinetic energy density of the spins is given by

$\mathcal{T}=\frac{1}{2a}\left(\boldsymbol{\omega}^{T}_{e}\mathcal{I}_{e}\boldsymbol{\omega}_{e}+\boldsymbol{\omega}_{o}^{T}\mathcal{I}_{o}\boldsymbol{\omega}_{o}\right),$
(S19)

where $\mathcal{I}$ is the inertia tensor and
$\boldsymbol{\omega}_{o/e}=(\omega^{x}_{o/e},\omega^{y}_{o/e},\omega^{z}_{o/e})^{T}$
is the angular velocity vector. Note that $\dot{\phi}_{o/e}=\omega^{z}_{o/e}$.
The spin angular momentum vector is related to the angular velocity via
$\mathbf{S}=\mathcal{I}\boldsymbol{\omega}$. We also have
$\mathbf{S}=\operatorname{sign}(\gamma)\frac{M_{0}a^{3}}{|\gamma|}\mathbf{m}$,
where $\gamma=-|e|g/2m_{e}$ is the gyromagnetic ratio for an electron with
charge $-|e|$, mass $m_{e}$ and $g$-factor $g$. As we set $\hbar=1$ in our
equations, we have $-\mathbf{m}=\mathcal{I}\boldsymbol{\omega}$. The
Lagrangian density is given by $\mathcal{L}=\mathcal{T}-\mathcal{V}$, where
$\mathcal{V}$ is the same as Eq. S18. Let us define the symmetric and
antisymmetric combinations

$\phi_{S/A}=\frac{1}{2}\left(\phi_{e}\pm\phi_{o}\right),\quad
m^{z}_{S/A}=\frac{1}{2}\left(m^{z}_{e}\pm m^{z}_{o}\right).$ (S20)

Note that $\phi_{S}$ is the Goldstone mode, as $\mathcal{L}$ is independent of
it. Next, we calculate the associated Noether current density,
$J^{\mu}=\frac{\partial\mathcal{L}}{\partial\partial_{\mu}\phi_{S}}$. For
$J^{t}$ we have

$J^{t}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_{S}}=\frac{\partial\mathcal{T}}{\partial\dot{\phi}_{S}}=\frac{1}{a}\left(I_{e}(\dot{\phi}_{S}+\dot{\phi}_{A})+I_{o}(\dot{\phi}_{S}-\dot{\phi}_{A})\right)=-\frac{2m^{z}_{S}}{a}.$

For the spatial components $J^{x,y,z}$, we have $J^{y,z}=0$ and

$\displaystyle J^{x}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial\phi^{\prime}_{S}}=-\frac{\partial\mathcal{V}}{\partial\phi^{\prime}_{S}}$
$\displaystyle=-\frac{1}{a}Ja^{2}\left[2\sin(\theta_{e})\sin(\theta_{o})\cos(2\phi_{A})\phi^{\prime}_{S}+\sin(2\phi_{A})\left(\cos(\theta_{e})\sin(\theta_{o})\theta_{e}^{\prime}+\cos(\theta_{o})\sin(\theta_{e})\theta_{o}^{\prime}\right)\right].$

To leading order in $B_{0}$, we have $\phi_{A}=0$,
$\sin(\theta_{e/o}(x))=\sin(\theta_{0}(x))$. Thus, the leading order
contribution to $J^{x}$ reads

$J^{x}\simeq-\frac{1}{a}2Ja^{2}(1-(m^{z})^{2})\phi_{S}^{\prime},$ (S21)

where we replaced $\sin^{2}(\theta_{0})=(1-(m^{z})^{2})$. Substituting
$J^{t,x}$ into the continuity equation
$\partial_{t}J^{t}+\partial_{x}J^{x}=0$, we obtain

$\partial_{t}(m^{z}_{S})+Ja^{2}\partial_{x}\left((1-(m^{z})^{2})\phi_{S}^{\prime}\right)=0,$
(S22)

as stated in the main text. Note that the spin current in Eq. 5 is defined
with a relative minus compared to the Noether current in Eq. S21,
$j^{z}=-J^{x}$.

####  D.3 LLG Equations for the Continuum Model

Substituting Eq. S18 into Eq. S10, we obtain the four coupled EoMs for the
continuum model,

$\displaystyle\dot{\phi}_{e/o}\sin(\theta_{e/o})$
$\displaystyle-\alpha\dot{\theta}_{e/o}=-\sin(\theta_{e/o})\left(\delta_{2}\cos(\theta_{e/o})+\delta_{4}\cos^{3}(\theta_{e/o})\right)$
$\displaystyle+2J\left(\Delta\sin(\theta_{e/o})\cos(\theta_{o/e})-\cos(\theta_{e/o})\sin(\theta_{o/e})\cos(\phi_{e/o}-\phi_{o/e})\right)$
$\displaystyle-a^{2}J\Bigg{(}$
$\displaystyle\theta_{o/e}^{\prime\prime}(\Delta\sin(\theta_{e/o})\sin(\theta_{o/e})+\cos(\theta_{e/o})\cos(\theta_{o/e})\cos(\phi_{e/o}-\phi_{o/e}))$
$\displaystyle+(\theta_{o/e}^{\prime})^{2}(\Delta\sin(\theta_{e/o})\cos(\theta_{o/e})-\cos(\theta_{e/o})\sin(\theta_{o/e})\cos(\phi_{e/o}-\phi_{o/e}))$
$\displaystyle+2\theta_{o/e}^{\prime}\phi_{o/e}^{\prime}\cos(\theta_{e/o})\cos(\theta_{o/e})\sin(\phi_{e/o}-\phi_{o/e})$
$\displaystyle+\cos(\theta_{e/o})\sin(\theta_{o/e})\phi_{o/e}^{\prime\prime}(x)\sin(\phi_{e/o}-\phi_{o/e})$
$\displaystyle-\cos(\theta_{e/o})\sin(\theta_{o/e})(\phi_{o/e}^{\prime})^{2}\cos(\phi_{e/o}-\phi_{o/e})\Bigg{)}+\epsilon
g_{e/o}B_{1}(t)\sin(\theta_{e/o}),$ (S23)
$\displaystyle-\sin(\theta_{e/o})\dot{\theta}_{e/o}$
$\displaystyle-\alpha\dot{\phi}_{e/o}\sin^{2}(\theta_{e/o})=2J\sin(\theta_{e})\sin(\theta_{o})\sin(\phi_{e/o}-\phi_{o/e})$
$\displaystyle-
Ja^{2}\sin(\theta_{e/o})\Bigg{(}\sin(\phi_{e/o}-\phi_{o/e})\left(\sin(\theta_{o/e})\phi_{o/e}^{\prime
2}-\theta_{o/e}^{\prime\prime}\cos(\theta_{o/e})\right)$
$\displaystyle+2\theta_{o/e}^{\prime}\cos(\theta_{o/e})\phi_{o/e}^{\prime}\cos(\phi_{e/o}-\phi_{o/e})+\theta_{o/e}^{\prime
2}\sin(\theta_{o/e})\sin(\phi_{e/o}-\phi_{o/e})$
$\displaystyle+\sin(\theta_{o/e})\phi_{o/e}^{\prime\prime}\cos(\phi_{e/o}-\phi_{o/e})\Bigg{)}.$
(S24)

Despite looking impossible to solve at first glance, $v$ can be extracted from
Eq. S23 and S24 using the right ansatzes and approximations. For completeness,
we solve them first in the undriven case (static domain wall), before moving
on to the driven problem (moving domain wall).

####  D.4 Zeroth Order Solution: Static Domain Wall

In the absence of driving, Eq. S24 is trivially solved by setting
$\phi_{e/o}=\text{const.}$ everywhere. Eq. S23 is symmetric under the swap
$\theta_{e}\leftrightarrow\theta_{o}$, from which we conclude that
$\theta_{e}=\theta_{o}=\theta_{0}$, letting us rewrite it as

$\displaystyle 0$
$\displaystyle=\sin(\theta_{0})\cos(\theta_{0})\left(2J\left(\Delta-1\right)-\delta_{2}-\delta_{4}\cos^{2}(\theta_{0})\right)$
(S25)
$\displaystyle-a^{2}J\left(\theta_{0}^{\prime\prime}\left(\Delta\sin^{2}(\theta_{0})+\cos^{2}(\theta_{0})\right)+(\theta_{0}^{\prime})^{2}\sin(\theta_{0})\cos(\theta_{0})\left(\Delta-1\right)\right).$

By replacing $\cos(\theta_{0})=m_{0}\tilde{m}(\tilde{x})$ and
$2J(\Delta-1)-\delta_{2}=\delta_{4}m_{0}^{2}$, with the rescaled position
coordinate $\tilde{x}=m_{0}x$, we can write this as

$\displaystyle 0$
$\displaystyle=\left(\delta_{4}\tilde{m}(1-\tilde{m}^{2})+a^{2}J\Delta\tilde{m}^{\prime\prime}\right)m_{0}^{3}+\mathcal{O}\left(m_{0}^{5}\right).$
(S26)

For small $m_{0}$ (close to the phase transition [40]), we can ignore the
$m_{0}^{5}$ term. In this case, it is easy to check that the ansatz

$\tilde{m}(\tilde{x})=\tanh\left(\frac{\tilde{x}}{a}\sqrt{\frac{\delta_{4}}{2J\Delta}}\right)$
(S27)

solves the equation. Thus, switching back to the original coordinates we have

$\theta_{0}(x)=\arccos\left(m_{0}\tanh\left(x/\xi_{0}\right)\right)+\mathcal{O}((m_{0})^{5}),\qquad\xi_{0}=\frac{a}{m_{0}}\sqrt{\frac{2J\Delta}{\delta_{4}}}.$
(S28)

####  D.5 Driven Solution: Moving Domain Wall

We now switch on the driving field $B_{z}(t)$. We modify the ansatz Eq. S8 to
allow for a travelling solution,

$\displaystyle\theta_{e/o}(x,t)$
$\displaystyle=\theta_{0}(x-vt)+\Theta_{e/o}(x-vt)+\epsilon\theta^{(1)}_{e/o}(x-vt,t)+\epsilon^{2}\theta^{(2)}_{e/o}(x-vt)+\mathcal{O}(\epsilon^{2}),$
(S29) $\displaystyle\phi_{e/o}(x,t)$
$\displaystyle=\Phi_{e/o}(x-vt)+\epsilon\phi^{(1)}_{e/o}(x-vt,t)+\epsilon^{2}\tilde{\omega}t+\mathcal{O}(\epsilon^{2}).$

Here, $\theta_{0}$ is the same function as the one we solved for in the static
case, Eq. S28. $\Theta_{e/o}$ and $\Phi_{e/o}$ are new functions, whose exact
dependence on $B_{0}$ we are yet to find out, but which we know vanish when
$B_{0}=0$. The Goldstone mode activation principle still implies that all
$\phi_{e/o}$ may acquire a global rotational velocity, described by the
$\tilde{\omega}t$ term. Note that $\tilde{\omega}$ by itself is _not_ the same
as $\omega_{\text{rot}}$, as there will also be some rotation coming from the
$\Phi(x-vt)$ term. In addition, we still have the $\mathcal{O}(B_{0})$
oscillating terms $\theta_{e/o}^{(1)}(x-vt,t),\phi_{e/o}^{(1)}(x-vt,t)$, which
tend to the ferrimagnetic chain values Eq. S13 far away from the domain wall,
$|x|\gtrsim\xi_{0}$.

#####  D.5.1 Time-Averaged Equations of Motion

Next, we solve Eq. S23 and S24 in the presence of driving, $B_{0}>0$. By time
averaging the two equations over $T=2\pi/\omega$, we get rid of any
oscillating terms. We transform to a moving frame of reference where
$\tilde{x}=x-vt$, and immediately drop the tilde on the $\tilde{x}$ for
notational simplicity. Keeping only leading order terms linear in
$\Phi_{e/o},\Theta_{e/o}$, as well as the leading order
$\mathcal{O}(B_{0}^{2})$ terms resulting from products of the linear response
fields $\theta^{(1)}_{e/o},\phi^{(1)}_{e/o}$, we have

$\displaystyle(1-m^{2})\bigg{(}-v\Phi^{\prime}_{e/o}+\tilde{\omega}$
$\displaystyle+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}+f_{e/o}\bigg{)}+\alpha
v\left(-m^{\prime}+\partial_{x}\left(\sqrt{1-m^{2}}\Theta_{e/o}\right)\right)$
$\displaystyle=\sqrt{1-m^{2}}\Bigg{[}$
$\displaystyle\delta_{4}\left(2m^{2}(m_{0}^{2}-2m^{2})+3m^{2}-m_{0}^{2}\right)(\Theta_{e/o}+\theta^{(2)}_{e/o})$
$\displaystyle+a^{2}J\Bigg{(}-(\Theta^{\prime\prime}_{e/o}+\theta^{(2)^{\prime\prime}}_{e/o})\left(\Delta(1-m^{2})+m^{2}\right)$
$\displaystyle+2(\Delta-1)(\Theta^{\prime}_{e/o}+\theta^{(2)^{\prime}}_{e/o})m^{\prime}m$
$\displaystyle+(\Delta-1)(\Theta_{e/o}+\theta^{(2)}_{e/o})\left(2mm^{\prime\prime}+\frac{(m^{\prime})^{2}}{1-m^{2}}\right)\Bigg{)}\Bigg{]}+g_{e/o},$
(S30) $\displaystyle\left(\alpha
v+Ja^{2}\partial_{x}\right)\left((1-m^{2})\Phi^{\prime}_{e/o}\right)$
$\displaystyle-v\partial_{x}\left(m-\left(\sqrt{1-m^{2}}\Theta_{e/o}\right)\right)$
$\displaystyle=\alpha(1-m^{2})\left(\tilde{\omega}+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}+f_{e/o}\right),$
(S31)

with

$\displaystyle g_{e/o}(x)$
$\displaystyle=\left\langle\sin(\theta_{0}+\theta^{(1)}_{e/o})\text{RHS}[\text{\lx@cref{creftype~refnum}{eq:ThetaEoMModifiedContinuous}}]\right\rangle_{T}$
$\displaystyle(1-m(x)^{2})f_{e/o}(x)$
$\displaystyle=\alpha\left(-(1-m(x)^{2})\omega_{\text{rot}}\frac{m(x)}{|m_{0}|}+2m(x)\sqrt{1-m(x)^{2}}\langle\dot{\phi}^{(1)}_{e}(x)\theta^{(1)}_{e}(x)\rangle\right)$
$\displaystyle+\left\langle\text{RHS}[\text{\lx@cref{creftype~refnum}{eq:PhiEoMModifiedContinuous}}]_{\text{products
of first order terms}}\right\rangle_{T},$

Here, we used $m=\cos(\theta_{0}(x))$ and
$m_{0}=\cos(\theta_{0}(x\gg\xi_{0}))$. $g(x),f(x)$ are both odd functions of
$x$. Note also that we multiplied Eq. S23 with $\sin(\theta_{e/o})$ before
time averaging so that the resulting $(1-m^{2})$ prefactor of the
$\Phi^{\prime}$ term matches the one in Eq. S24. We define
$\theta^{(2)}_{e/o}(x)$ such that it balances $g_{e/o}(x)$ in Eq. S30, hence
$\theta^{(2)}_{e/o}(x)$ is also an odd function of $x$. Note also that
$f_{e/o}(x)$ is bounded to the region of the domain wall, i.e.
$f_{e/o}(|x|\gtrsim\xi_{0})=0$. This is because in the $|x|\gtrsim\xi_{0}$
regions, the $\omega_{\text{rot}}$ term perfectly cancels out all the time-
averaged products of first order fields, as explained in App. C.

#####  D.5.2 Solving Eq. S30 and S31 in the Absence of Damping

For future reference, we present first how to solve Eq. S30 and S31 in the
absence of damping, $\alpha=0$. The equations read

$\displaystyle(1-m^{2})\Bigg{(}-v(\Phi^{\alpha=0}_{e/o})^{\prime}+\tilde{\omega}$
$\displaystyle+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}+f_{e/o}\Bigg{)}$
$\displaystyle=\sqrt{1-m^{2}}\Bigg{[}$
$\displaystyle\delta_{4}\left(2m^{2}(m_{0}^{2}-2m^{2})+3m^{2}-m_{0}^{2}\right)\Theta^{\alpha=0}_{e/o}$
$\displaystyle+a^{2}J\Bigg{(}-(\Theta^{\alpha=0}_{e/o})^{\prime\prime}\left(\Delta(1-m^{2})+m^{2}\right)$
$\displaystyle+2(\Delta-1)(\Theta^{\alpha=0}_{e/o})^{\prime}m^{\prime}m$
$\displaystyle+(\Delta-1)\Theta^{\alpha=0}_{e/o}\left(2mm^{\prime\prime}+\frac{(m^{\prime})^{2}}{1-m^{2}}\right)\Bigg{)}\Bigg{]},$
(S32) $\displaystyle
Ja^{2}\partial_{x}\left((1-m^{2})(\Phi^{\alpha=0}_{e/o})^{\prime}\right)$
$\displaystyle-v\partial_{x}\left(m-\left(\sqrt{1-m^{2}}\Theta^{\alpha=0}_{e/o}\right)\right)=0.$
(S33)

Eq. S33 can be directly integrated over space to give

$(1-m^{2})(\Phi^{\alpha=0}_{e/o})^{\prime}=\frac{1}{v}(1-m_{0}^{2})\tilde{\omega}+\frac{v}{Ja^{2}}\left(m-\sqrt{1-m^{2}}\Theta^{\alpha=0}_{e/o}\right),$
(S34)

where $\frac{1}{v}(1-m_{0}^{2})\tilde{\omega}$ was added as constant of
integration. How did we know to add this particular constant? Far enough away
from the domain wall, $|x|\gtrsim\xi_{0}$, we enter back into the
ferrimagnetic chain regime, where $\Theta^{\alpha=0}_{e/o}(x)$ and its
derivatives vanish. Consequently, the RHS of Eq. S32 vanishes, as does
$f_{e/o}$, and the equation simplifies to

$(1-m_{0}^{2})\left(-v(\Phi^{\alpha=0}_{e/o})^{\prime}+\tilde{\omega}+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}\right)=0.$

Substituting in Eq. S34, we see that we have two kinds of terms: $m$-dependent
and $m$-independent. Balancing the $m$-dependent terms, we have
$\frac{v^{2}m}{Ja^{2}}=(1-m_{0}^{2})|\omega_{\text{rot}}|\frac{m}{|m_{0}|}$,
giving us our velocity formula at zero damping,
$v^{2}=Ja^{2}(1-m_{0}^{2})\frac{|\omega_{\text{rot}}|}{|m_{0}|}$. Meanwhile,
the remaining $m$-independent terms give us the constant of integration
$\frac{1}{v}(1-m_{0}^{2})\tilde{\omega}$.

For future reference, we also state the uncoupled EoM for
$\Theta^{\alpha=0}_{e/o}$, obtained by substituting Eq. S34 into Eq. S32,

$\displaystyle(m_{0}^{2}-m^{2})\Bigg{(}\tilde{\omega}$
$\displaystyle+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}\Bigg{)}+(1-m^{2})f_{e/o}$
$\displaystyle=\sqrt{1-m^{2}}\Bigg{[}\Theta^{\alpha=0}_{e/o}\bigg{(}\delta_{4}\left(2m^{2}(m_{0}^{2}-2m^{2})+3m^{2}-m_{0}^{2}\right)-\frac{v^{2}}{Ja^{2}}\bigg{)}$
$\displaystyle+a^{2}J\Bigg{(}-(\Theta^{\alpha=0}_{e/o})^{\prime\prime}\left(\Delta(1-m^{2})+m^{2}\right)+2(\Delta-1)(\Theta^{\alpha=0}_{e/o})^{\prime}m^{\prime}m$
$\displaystyle+(\Delta-1)\Theta^{\alpha=0}_{e/o}\left(2mm^{\prime\prime}+\frac{(m^{\prime})^{2}}{1-m^{2}}\right)\Bigg{)}\Bigg{]}.$
(S35)

The source term on the LHS of Eq. S35 vanishes for $|x|\gtrsim\xi_{0}$,
consequently so does $\Theta^{\alpha=0}_{e/o}(|x|\gtrsim\xi_{0})$ and its
derivatives.

#####  D.5.3 Solving Eq. S30 and S31 in the Presence of Damping: Far-Field
Asymptotics

Let us consider Eq. S30 and S31 in the far-field limit, $|x|\gg\xi_{0}$. In
this limit, $m\to\pm|m_{0}|$ and $m^{\prime},m^{\prime\prime}\to 0$, and the
equations simplify to

$\displaystyle(1-m_{0}^{2})\Bigg{(}-v\Phi^{\prime}_{e/o}+\tilde{\omega}$
$\displaystyle+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}\Bigg{)}+\alpha
v\sqrt{1-m_{0}^{2}}\Theta_{e/o}^{\prime}$
$\displaystyle=\sqrt{1-m_{0}^{2}}\Bigg{[}2\delta_{4}m_{0}^{2}(1-m_{0}^{2})\Theta_{e/o}-a^{2}J\Theta^{\prime\prime}_{e/o}\left(\Delta(1-m_{0}^{2})+m_{0}^{2}\right)\Bigg{]},$
(S36) $\displaystyle\left(\alpha v+Ja^{2}\partial_{x}\right)$
$\displaystyle\left((1-m_{0}^{2})\Phi^{\prime}_{e/o}\right)+v\sqrt{1-m_{0}^{2}}\Theta_{e/o}^{\prime}$
$\displaystyle=\alpha(1-m_{0}^{2})\left(\tilde{\omega}+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}\right),$
(S37)

We can solve these by making the following ansatz

$(1-m_{0}^{2})\Phi_{e/o}^{\prime}=\frac{1}{v}(1-m_{0}^{2})\left(\tilde{\omega}+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}\right)+\Psi_{f}e^{\kappa
x},\qquad\Theta_{e/o}=\Theta_{f}e^{\kappa x},$ (S38)

which reduces the coupled equations Eq. S37 and S36 to a $2\times 2$ matrix
equation,

$\displaystyle\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\begin{pmatrix}\Psi_{f}\\\
\Theta_{f}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}0\\\ 0\end{pmatrix},$
(S39)

with

$\displaystyle A$ $\displaystyle=-v\sqrt{1-m_{0}^{2}},$ $\displaystyle B$
$\displaystyle=\bigg{(}\alpha
v\kappa-2\delta_{4}m_{0}^{2}(1-m_{0}^{2})+a^{2}J\kappa^{2}\left(\Delta(1-m_{0}^{2})+m_{0}^{2}\right)\bigg{)},$
$\displaystyle C$ $\displaystyle=\left(\alpha
v+Ja^{2}\kappa\right)\sqrt{1-m_{0}^{2}},$ $\displaystyle D$
$\displaystyle=\kappa v.$

Non-trivial solutions of Eq. S39 occur only when the determinant of the matrix
is zero, which gives us a cubic equation in $\kappa$. We are interested in the
limit where $v\ll 1$. In this limit, two of the roots tend to a finite value
of the same order of magnitude as the inverse length of the domain wall width,
$1/\xi_{0}$, i.e., they decay very quickly away from the centre of the domain
wall and are therefore irrelevant in the far field. On the other hand, the
third root, given by $\kappa=-\frac{\alpha v}{Ja^{2}}+\mathcal{O}(\alpha
v^{3})\ll 1/\xi_{0}$, is much smaller than the width of the domain wall, and
therefore persists a large distance beyond it. This is the root giving the
non-trivial far-field behaviour. We now wish to obtain a relation between
$\Psi_{f}$ and $\Theta_{f}$. Using that $\kappa,\kappa^{2}$ are much smaller
than $J,\delta_{2},\delta_{4},m_{0}$, we set them to zero in the first line of
Eq. S39, giving

$v\Psi_{f}=-2\delta_{4}m_{0}^{2}\sqrt{1-m_{0}^{2}}\Theta_{f}.$ (S40)

Thus we have $\Theta_{f}\sim v\Psi_{f}$. We will use this result in App.
D.5.4.

#####  D.5.4 Solving Eq. S30 and S31 in the Presence of Damping: Near-Field
Approximate Solution

We are finally ready to approximately solve Eq. S30 and S31 in the near field
in the presence of damping. Combining the insights gained in App. D.5.2 and
D.5.3, we use the following ansatzes for
$\Phi^{\prime}_{e/o}(x),\Theta_{e/o}(x)$

$\displaystyle(1-m^{2})\Phi^{\prime}_{e/o}$
$\displaystyle=\frac{1}{v}(1-m_{0}^{2})\left(\tilde{\omega}+|\omega_{\text{rot}}|\frac{m}{|m_{0}|}\right)-\frac{v}{Ja^{2}}\sqrt{1-m^{2}}\Theta^{\alpha=0}_{e/o}+e^{\kappa
x}\psi_{e/o}(x),$ (S41) $\displaystyle\Theta_{e/o}$
$\displaystyle=\Theta^{\alpha=0}_{e/o}+e^{\kappa x}\vartheta_{e/o}(x).$

where

$\psi_{e/o}(|x|\gtrsim\xi_{0})=\begin{cases}\Psi_{f},&\kappa x<0,\\\ 0,&\kappa
x>0.\end{cases},\quad\vartheta_{e/o}(|x|\gtrsim\xi_{0})=\begin{cases}\Theta_{f},&\kappa
x<0,\\\ 0,&\kappa x>0.\end{cases}$ (S42)

Ansatz Eq. S41 replicates the correct behaviour Eq. S38 in the far field
regions. Inside the domain wall, $-\xi_{0}\lesssim x\lesssim\xi_{0}$,
$\Theta^{\alpha=0}_{e/o}$ balances the source terms which do not explicitly
depend on $\alpha$, according to Eq. S35. The job of balancing the additional
$\alpha$-dependent source term $-\alpha vm^{\prime}$ on the LHS of Eq. S30
falls to $e^{\kappa x}\psi_{e/o}(x),e^{\kappa x}\vartheta_{e/o}(x)$. Inside
the domain wall, $\psi_{e/o}(x)$ and $\vartheta_{e/o}(x)$ are functions which
interpolate between $0$ and $\Psi_{f}$, $\Theta_{f}$, respectively. As a
considerable simplification, we approximate this interpolation to be linear in
$x$,

$\psi_{e/o}(x)=\frac{\Psi_{f}}{2}\left(1+\operatorname{sign}(v)\frac{x}{\xi_{0}}\right),\quad\vartheta_{e/o}(x)=\frac{\Theta_{f}}{2}\left(1+\operatorname{sign}(v)\frac{x}{\xi_{0}}\right).$
(S43)

Substituting Eq. S43 into Eq. S30 and setting $x=0$, we find

$-\frac{v\Psi_{f}}{2}+\alpha
v\left(-\frac{m_{0}}{\xi_{0}}+\frac{\Theta_{f}}{2}\left(\kappa+\frac{\operatorname{sign}(v)}{\xi_{0}}\right)\right)=\frac{\Theta_{f}}{2}m_{0}^{2}\left(-\delta_{4}+a^{2}J(\Delta-1)\frac{1}{\xi_{0}^{2}}\right).$

Using that $\Theta_{f}\sim v\Psi_{f}$, we can neglect the term proportional to
$\alpha v\Theta_{f}$ on the LHS, as it is suppressed by a factor $\alpha v\ll
1$ compared to the other terms containing a factor $\Psi_{f}$. Eliminating
$\Theta_{f}$ using the relation Eq. S40, we solve for $\Psi_{f}$,

$\Psi_{f}=\frac{2\alpha
m_{0}}{\xi_{0}}\left[\frac{a^{2}J(\Delta-1)\frac{1}{\xi_{0}^{2}}-\delta_{4}}{2\delta_{4}\sqrt{1-m_{0}^{2}}}-1\right]^{-1}.$
(S44)

Thus, $\Psi_{f}$ is _independent_ of $v$. $\Psi_{f}$ is however proportional
to $\alpha$, which means that the exponential tail $\Psi_{f}e^{\kappa x}$
vanishes in the limit of zero damping, $\alpha=0$, as expected.

#####  D.5.5 Evaluation of $v$

To calculate $v$, we proceed as before by integrating Eq. S31 with the
ansatzes Eq. S41 and S43 over space. It is sufficient to do this between
$-\xi_{0}\lesssim x\lesssim\xi_{0}$, as all terms in the equation vanish
outside this range. We obtain

$\displaystyle\alpha
v\Psi_{f}\frac{1}{\kappa}\left(\left(1-\frac{\operatorname{sign}(v)}{\kappa\xi_{0}}\right)\sinh(\kappa\xi_{0})+\operatorname{sign}(v)\cosh(\kappa\xi_{0})\right)$
$\displaystyle+\left(Ja^{2}\Psi_{f}+v\sqrt{1-m_{0}^{2}}\Theta_{f}\right)\left(\sinh(\kappa\xi_{0})+\operatorname{sign}(v)\cosh(\kappa\xi_{0})\right)=2m_{0}\left[v-\frac{Ja^{2}}{v|m_{0}|}(1-m_{0}^{2})|\omega_{\text{rot}}|\right].$

Next, we use $\kappa\xi_{0}\ll 1$ to approximate $\cosh(\kappa x)\approx
1,\sinh(\kappa\xi_{0})\approx 0$, which gives

$\operatorname{sign}(v)\left[\left(\alpha
v\xi_{0}+Ja^{2}\right)\Psi_{f}+v\sqrt{1-m_{0}^{2}}\Theta_{f}\right]=2m_{0}\left[v-\frac{Ja^{2}}{v|m_{0}|}(1-m_{0}^{2})|\omega_{\text{rot}}|\right].$

Once again, we suppress the terms $\alpha
v\xi_{0}\Psi_{f},v\sqrt{1-m_{0}^{2}}\Theta_{f}$, and keep only the dominant
$Ja^{2}\Psi_{f}$ term. This simplifies the equation to

$v^{2}-\frac{Ja^{2}}{2m_{0}}\operatorname{sign}(v)\Psi_{f}v-\frac{Ja^{2}}{|m_{0}|}(1-m_{0}^{2})|\omega_{\text{rot}}|=0,$

where we multiplied everything by $v$. As $\Psi_{f}$ is independent of $v$,
this is a quadratic equation in $v$. However, care needs to be exercised in
solving it, due the sign change of the linear in $v$ term for negative vs
positive $v$. For the parameter range we are interested in,
$\frac{Ja^{2}}{2m_{0}}\Psi_{f}$ is always negative. This means that the
minimum of the parabola, located at
$\operatorname{sign}(v)\frac{Ja^{2}}{4m_{0}}\Psi_{f}$, is never reached. Thus,
the regions $v>0$ and $v<0$ only contribute _one_ root each. These two roots
are given by

$v=\pm\left(\sqrt{\left(\frac{\alpha\rho_{s}}{\xi_{0}\eta}\right)^{2}+\frac{\rho_{s}|\omega_{\text{rot}}|}{|m_{0}|}}-\frac{\alpha\rho_{s}}{\xi_{0}\eta}\right),$
(S45)

with

$\eta=2(1-m_{0}^{2})\left[1-\frac{\left(a^{2}J(\Delta-1)\frac{1}{\xi_{0}^{2}}-\delta_{4}\right)}{2\delta_{4}\sqrt{1-m_{0}^{2}}}\right]=2(1-m_{0}^{2})+\sqrt{1-m_{0}^{2}}\left(1+\frac{m_{0}^{2}(1-\Delta)}{2\Delta}\right).$
(S46)

$\eta$ interpolates smoothly between $\eta=3$ for a small FM magnetisation,
$m_{0}\to 0$, and $\eta=0$ for $m_{0}\to 1$, respectively. Fig. 2 of the main
text shows that our approximate formula for $v$, while not exact, describes
with a good accuracy the numerically observed velocity with an error of only a
few percent for magnetic fields $B_{0}\lesssim 0.2\,J$. As our analysis is
only valid for small $B_{0}$, it cannot describe the breakdown of the moving
domain wall solution for larger $B_{0}$.

###  Appendix E Further Numerical Results

In this section, we display and discuss some extra numerical data to support
the conclusions reached in the main text.

####  E.1 Short-Ranged Correlations of the $xy$-Order

Figure S6: Decay of correlations of the AFM $xy$-order parameter,
$C_{\text{AFM}}(x)=\sum_{j}(-1)^{i}\langle S^{x}_{j}S^{x}_{j+x/a}\rangle$ for
integer even $x/a$. Error bars denote the standard deviation of the mean.
Parameters as in Fig. S2. In the equilibrium system, $xy$-correlations decay
exponentially on a length scale of about $67$ lattice sites (solid line), in
the temperature range investigated here. In the driven system, the correlation
length is considerably smaller, only 20-30 sites.

The discussion in the main text focuses mainly on the giant correlation length
of the FM magnetisation, which is enhanced by several orders of magnitude in
the driven system. In contrast, the correlation functions associated with the
AFM $xy$-order decay faster in the driven system than in the equilibrium case,
as shown in Fig. S6. For the chosen parameters, the $xy$-order decays on a
length scale of only 20 or 30 sites, even in a regime where $m^{z}$-domains
have sizes exceeding $10^{5}$ sites. The reason is that, embedded in these
large $m^{z}$ domains, there are smaller domains with opposite $m^{z}$
orientation. Thus, $xy$-spins rotate in opposite directions which reduced
$xy$-correlations.

Remarkably, despite this very short correlation length of the $xy$-order, our
analytical formulas for the domain wall velocities, which ignored noise and
start from a system with long-ranged $xy$-order, correctly predict the
velocity of the domain walls.

####  E.2 Dynamics of Ordering

Here, we give more information on how long-ranged $m^{z}$ order develops in
the absence of thermal noise after a quench from the AFM phase. This physics
is also discussed in Fig. 4 of the main text, which shows that at first the
correlation length grows very slowly, then speeds up and grows linearly in
time for long times. The initial slow growth is explained by the fact that the
domain walls need a few rotation periods $T_{\text{rot}}$ to reach their final
velocity. The corresponding velocity distributions of domain walls are are
shown in Fig. S7 for three different times. While for short times, $t\sim
T_{\text{rot}}$, the velocity distribution is centered around zero, the
velocity distribution becomes peaked at $\pm v$ in the long-time limit. An
analytical theory for $v$ is provided in Sec. D.

Figure S7:  Distribution of domain wall velocities after a quench from an AFM
ordered state into a ferrimagnetic state for three different times after the
quench. For long times, the velocity distribution is strongly peaked at the
velocity of a single domain wall, $\pm v$. Same parameters as Fig. S2 at $T=0$
for a system of 40,000 spins.

####  E.3 Quench from the Ferrimagnetic Phase and Convergence of Results

Our driven system shows order on very long length scales. Therefore the
equilibration time needed to reach a steady state also becomes very long. To
investigate whether the finite $T$ correlation functions discussed in the main
text reflect steady-state properties, we compare data arising from two initial
states in Fig. S8. The first one, a perfectly ordered AFM state with a small
randomness in the initial value of $S_{i}^{z}=\pm 0.1$, initially produces a
very large number of short domains. This is compared to a system which starts
from a perfectly ordered ferrimagnetic state, consisting of a single domain
which includes all 250,000 spins of our simulation. In the first case, the
motion and collisions of the domain walls increases the correlation length
until a steady state is reached in the noisy system. In the second case, the
noise has to induce more and more defects to reach the same steady state for
$t\to\infty$. Fig. S8 shows that for the higher temperatures the system does
reach the same steady state within the simulated time. This is, however, not
the case for the lowest simulated temperature.

Figure S8:  Comparison of the FM correlation function $C(x)=\langle
S^{z}_{i}S^{z}_{i+x/a}\rangle$ measured a time $t\approx 520{,}000/J\approx
850\,T_{\text{rot}}$ after the quench, for four different temperatures, for a
system with 250,000 spins. The solid lines show data from a quench from an
initial AFM state (also shown in Fig. 6 of the main text). We compare this to
data obtained by starting from a perfectly ordered ferrimagnetic state, shown
as points. The plot shows that for the three highest temperatures, the
correlation function is independent of the initial state within error bars.
For the lowest $T$ (blue) this is not the case: the simulated time is shorter
than the time required to reach a steady state. Also, the system size is
smaller than the correlation length in this case. Parameters: As in Fig. S2.
For both initial conditions, averaging was performed over four simulations and
error bars show standard deviations of the mean.

An important characteristic of the correlation functions shown in Fig. S8 is
an extremely slow decay of $C(x)$, roughly following

$\displaystyle C(x)\approx
C(0)\left(1-\ln\left[\frac{x}{a}\right]/\ln\left[\frac{\xi_{\text{max}}}{a}\right]\right).$
(S47)

The slow decay with $x$ arises because there is a very broad distribution of
domain sizes. Furthermore, the system is characterised by domains within
domains. A consequence of this fact is that the half-width $\xi_{1/2}$ of
$C(x)$, defined by $C(\xi_{1/2})=\frac{1}{2}C(0)$, can not be used as a useful
measure of domain sizes. Consider, for example, Fig. 5 of the main text. For
the two temperatures shown, $\xi_{1/2}\approx 615$ for $T=0.0334E_{\text{DW}}$
and $\xi_{1/2}\approx 35$ for $T=0.349E_{\text{DW}}$ as can be read off from
Fig. S8. However, a short inspection of Fig. 5 of the main text reveals that
the true correlations are orders of magnitude larger. To obtain a more
reasonable correlation length and error bars for the inset of Fig. 6 of the
main text, we proceed in the following way. We first use a fit to Eq. (S47) to
determine $\xi_{\text{max}}$. For the fit, we use values of $x/a$ smaller than
$8{,}000$, $20{,}000$, $1{,}000$ and $300$ for $T=0.334,0.342,0.349$ and
$0.357\,E_{\text{DW}}$. The small cutoff value for the lowest temperature
takes into account that the system has not equilibrated at long distances, see
Fig S8. The crudeness of the fit is reflected in generous error bars, which we
choose as follows. A lower estimate $\xi^{<}_{\text{max}}$ is obtained from
$C(\xi^{<}_{\text{max}})=0.1\,C(0)$. We estimate $\xi^{>}_{\text{max}}$ from
the point where $C(x)\approx 0$ within the error bars. As the lowest $T$ (blue
curve) is not equilibrated, we can only estimate
$\xi^{>}_{\text{max}}=\infty$. With this procedure, we obtain the estimates
shown in Table S1.

$T/E_{\text{DW}}$ | $\xi^{<}_{\text{max}}/a$ | $\xi_{\text{max}}/a$ | $\xi^{>}_{\text{max}}/a$
---|---|---|---
0.334 | 45,200 | 475,000 | $\infty$
0.342 | 10,400 | 18,100 | 108,000
0.349 | 800 | 1,800 | 13,600
0.357 | 400 | 700 | 5,900
Table S1: Estimates for the correlation length used for the plot in the inset
of Fig. 6 of the main text.

####  E.4 Correlations in space-time

Even in the presence of thermal noise, the spatial and temporal correlations
of the FM magnetisation

$\displaystyle C(x,t)=\left\langle
S^{z}_{i}(t_{0})S^{z}_{i+x/a}(t_{0}+t)\right\rangle$ (S48)

(averaged over $i$ and a few $t_{0}$ after the system has equilibrated) are
dominated by the fact that domain walls move actively with the velocity $\pm
v$. This is clearly visible in Fig. S9, which shows the correlation function
in the steady state. The correlation function shows pronounced peaks at $x=\pm
vt$.

Remarkably, the correlation function looks practically identical when the
roles of $x$ and $vt$ are exchanged. This is is shown in Fig. S10a): the $t=0$
and $x=0$ correlation functions are identical within error bars. Furthermore,
the width of the peak at $x=\pm vt$ also scales linearly in $t$ for long
times, see Fig. S10b).

The correlation length along the line $x=\pm vt$ is also much longer than in
the time or space direction only, see Fig. S10a). To compare these
correlations, it is useful to plot the correlations as function of
$\lambda=\sqrt{x^{2}+(vt)^{2}}$, see Fig. S10a). When one moves a distance
$\lambda$ along a line parallel to $x=vt$, one crosses no right-moving domains
but only left-moving domains (with a factor $\sqrt{2}$ increased density).
However, these geometric factors alone cannot explain the increased
correlations along $x=\pm vt$, which are roughly a factor $5$ longer for the
chosen parameters.

Figure S9: Steady-state correlation function $C(x,t)=\left\langle
S^{z}_{i}(t_{0})S^{z}_{i+x/a}(t_{0}+t)\right\rangle$ of a driven ferrimagnet
as a function of space and time in the presence of thermal noise with
$T=0.349\,E_{\text{DW}}$. Other parameters as in Fig. S8. The data has been
averaged over $t_{0}$ and 5 independent runs. Various cuts through this
function are shown in Fig. S10.

a)

b)

Figure S10: Cuts of the steady-state correlation function $C(x,t)$ shown in
Fig. S9, in the presence of thermal fluctuations ($T=0.349\,E_{\text{DW}}$).
Panel a) shows that the equal time correlation function $C(\lambda,t=0)$ is
within errors indistinguishable from the local correlation function $C(x=0,t)$
for $t=\lambda/v$. Here, the average velocity of domain walls, $v\approx
0.135\,Ja$, is about $10\,\%$ faster than the velocity of a domain wall in the
noiseless system. A much slower decay of correlations is obtained along the
diagonal of Fig. S9, i.e., for $C(\lambda/\sqrt{2},\lambda v/\sqrt{2})$ (green
curve, panel a), see also inset which shows the same data with a log scale for
$\lambda$), $\lambda=\sqrt{x^{2}+(vt)^{2}}$. For all three curves, $\lambda$
measures the distance from the origin, with $\lambda^{2}=x^{2}+(vt)^{2}$.
Panel b) shows $C(x,t)/C(vt,t)$ as function of $x/(vt)$. The plot shows that
for long times, the width of the peak at $x=vt$ also scales linearly with $t$.