††thanks: These authors contributed equally to the work††thanks: These authors
contributed equally to the work

# Polar order, shear banding, and clustering in confined active matter

Daniel Canavello  Rubens H. Damascena  Leonardo R. E. Cabral
<EMAIL_ADDRESS>Clécio C. de Souza Silva<EMAIL_ADDRESS>Departamento de Física, Centro de Ciências Exatas e da Natureza, Universidade
Federal de Pernambuco, Recife–PE, 50670-901, Brasil

###### Abstract

We investigate the collective behavior of sterically interacting self-
propelled particles confined in a harmonic potential. Our theoretical and
numerical study unveils the emergence of distinctive collective polar
organizations, revealing how different levels of interparticle torques and
noise influence the system. The observed phases include the shear-banded
vortex, where the system self organizes in two concentric bands rotating in
opposite directions around the potential center; the uniform vortex, where the
two bands merge into a close packed configurations rotating uniformly as a
quasi-rigid body; and the orbiting polar state, characterized by parallel
orientation vectors and the cluster revolving around the potential center,
without rotation, as a rigid body. Intriguingly, at lower filling fractions,
the vortex and polar phases merge into a single phase where the trapped
cluster breaks into smaller polarized clusters, each one orbiting the
potential center as a rigid body.

††preprint: APS/123-QED

##  I Introduction

Active matter encompasses a wide range of biological and synthetic entities,
including swimming bacteria, microalgae, living tissue, and artificial
swimmers, capable of converting energy absorbed from the environment into a
propelling force that drives them far from equilibrium [1, 2, 3, 4]. As a
result of their motility and mutual interactions, these systems exhibit a
wealth of self-organized collective phenomena, such as flocking, motility-
induced phase separation, dynamical clustering, and collective self-
optimization [5, 6, 7, 8, 9, 10, 11, 12]. Many of these phenomena stem from
the ability of self-propelled particles to reorient their propulsion direction
when interacting with each other. The reported mechanisms behind this polar
alignment extends from ad hoc aligning rules, such as the Vicsek interaction,
to simple steric forces and hydrodynamic interactions [6, 8, 13, 14, 15]. In
bulk, aligning interactions result in different kinds of orientational orders,
like ferromagnetic, nematic and vortex. These studies have significantly
contributed to our understanding of the underlying principles driving self-
organization and pattern formation in active matter.

In many situations of interest, active matter interact with confining walls,
obstacles, or potential energy landscapes, produced by optical and acoustic
tweezers, nonuniform hydrodynamic flows, and phoretic fields [16, 17, 18, 19,
20, 21]. Confinement plays a crucial role in shaping the behavior and
properties of active matter systems [22, 23, 24, 25, 26, 27] and can induce
novel collective phenomena that differ from their bulk counterparts, such as
the fluid pump phase predicted for confined particles interacting
hydrodynamically [28]. Moreover, confinement is an invaluable tool to
investigate non-equilibrium properties of active particles, specially those
that distinguish them from passive Brownian systems, as for instance noise-
induced escape problems that violate detailed balance [29, 30, 31]. However,
the impact of confinement on the collective dynamics and phase behavior of
self-propelled polar particles remains relatively unexplored.

The influence of confinement extends beyond forces to include torques that
realign the orientation of active particles. These aligning torques are often
overlooked in models of confined active matter, despite their significant
physical consequences. For instance, in the case of a single confined
particle, the interplay of motility and the confining force and torque gives
rise to a dynamical phase transition where the particle develops orbital
motion within the confining region [32, 33, 31]. This phenomenon has been
observed in diverse systems, from geometrically confined motile bacteria [34]
and microalgae [35] to a minirobot on a parabolic reflector [32]. Similarly,
active Janus colloids have been shown to experience an aligning torque induced
by a phoretic gradient [36]. Despite recent progress in understanding the
properties arising from aligning torques at the level of a single active
particle, their correlation with collective phenomena in confined active
matter remains unclear.

In this work, we investigate collective phenomena in a model system of
monodisperse self-propelled particles trapped by a harmonic potential and
interacting with each other via simple steric forces and torques. The
contribution of the steric torques and the torque induced by the confining
force is regulated by the angular mobility $\beta$. The higher $\beta$, the
more relevant is the role played by torques in the the overall dynamics. Fig.
1

Figure 1: Snapshots of positions (dots), orientations (colors) and velocities
(arrows) representative of the main phases observed in the system: (a)
radially polarized, (b) shear-banded vortex, (c) uniform vortex, (d) orbiting
“ferromagnetic” crystal, and (e) multi-cluster. In all cases, noise is turned
off and the particle diameter is $\sigma=0.1R_{\text{CI}}$, where
$R_{\text{CI}}$ is the radius of the critical isocline (blue dashes) of the
harmonic confinement. For (a) to (d) $N=350$ and the values of $\beta$ are,
respectively, $0.5,1.85$, $5.0$ and $5.0$, while for (e) $N=93$ and
$\beta=5.0$. The solid red lines depict the trajectory of the centroid of each
cluster. The dashed orange line in (b) and (c) denotes the theoretical
boundary between the shear bands (see text).

shows snapshots of positions, orientations (color coding) and velocities
(arrows) of the particles for different values of noise intensity $D$, angular
relaxation rate $\beta$, and particle size $\sigma$ representative of some of
the different phases observed in our study (see also the Videos in the
Electronic Supplementary Information, ESI). For small $\beta$ and at low
noise, the active particles are observed to arrange themselves as a radially
polarized fluid, with ring-shaped distribution (a). In increasing $\beta$, the
system rotates without revolving as either a shear-banded vortex (b), for
moderate filling fractions, or a uniform vortex (c), for large filling
fractions. At higher values of $\beta$, the system revolves, without rotation,
as a highly polarized (“ferromagnetic”) crystal (d). Finally, at low filling
fractions, the system self-organizes in multiple polarized clusters orbiting
the potential (e).

This paper is organized as follows. In Section II, we present the model and
derive analytical results for the orbiting polar and vortex phases in the
deterministic (quasi-)rigid-body limit. In Section III, we present detailed
numerical results on all observed phases and their dependence on the control
parameters. In particular, we validate and discuss the limitations of our
theoretical modelling on the deterministic vortex and orbiting polar phases.
We also discuss possible mechanisms for the observation of the shear-banding
transition, which is not captured by our model, and the effect of noise on all
observed phases. The conclusions are presented in Section IV.

##  II Model and analytical results

###  II.1 Equations of motion

We consider a 2D system of $N$ active Brownian particles (ABP), at positions
$\bm{r}_{i}=(x,y)$ and with orientation vectors
$\bm{n}_{i}=(\sin\theta,\cos\theta)$, confined in an isotropic harmonic
potential, $V(\bm{r}_{i})=\frac{1}{2}\kappa\bm{r}_{i}^{2}$. Their dynamics are
modeled by overdamped Langevin equations of motion,

$\displaystyle\dot{\bm{r}}_{i}$
$\displaystyle=v_{0}\bm{n}_{i}+\mu\bm{F}_{i}+\sqrt{2D_{t}}\bm{\xi}_{i}(t),$
(1) $\displaystyle\dot{\theta}_{i}$
$\displaystyle=\beta(\bm{n}_{i}\times{\bm{F}}_{i})\cdot\hat{\bm{z}}+\sqrt{2D}\zeta_{i}(t),$
(2)

where $v_{0}$ denotes the propulsion speed, which is the same for all
particles, $\hat{\bm{z}}$ is the unit vector perpendicular to the $xy$ plane,
$\mu$ ($\beta$) is the translational (angular) mobility, and
$\bm{F}_{i}=-\kappa\bm{r}_{i}-\nabla_{i}\sum_{j\neq
i}^{N}U(|\bm{r}_{i}-\bm{r}_{j}|)$ sum up the conservative forces acting on
particle $i$. The steric interactions are modeled by a truncated Lennard-Jones
(WCA) potential: $U(r)=4\epsilon[(\sigma/r)^{12}-(\sigma/r)^{6}]-\epsilon$,
for $r<2^{1/6}\sigma$, and $U(r)=0$, for $r>2^{1/6}\sigma$, where $\sigma$ is
the particle size and $\epsilon$ is the interaction energy scale. In this work
we fix $\epsilon=0.1v_{0}^{2}/(\mu^{2}\kappa)$. The translational,
$\bm{\xi}_{i}(t)$, and rotational, $\zeta_{i}(t)$, noise terms satisfy
$\langle\bm{\xi}_{i}(t)\cdot\bm{\xi}_{j}(t^{\prime})\rangle=2\delta_{ij}\delta(t-t^{\prime})$
and
$\langle\zeta_{i}(t)\zeta_{j}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime})$,
respectively, with $D_{t}$ and $D$ representing the respective diffusion
constants. Also, we focus mainly on the so-called active noise regime, which
is dominated by rotational diffusion and thus we set $D_{t}=0$ [37, 38, 39].
The effect of $D_{t}$ is discussed briefly in the Electronic Supplementary
Information (ESI).

The first term in the right-hand side of Eq. (2) represents the restoring
torque produced by the total force $\bm{F}_{i}$ acting on particle $i$. Since
from Eq. (1) $\bm{n}_{i}\times\mu\bm{F}_{i}=\bm{n}_{i}\times\dot{\bm{r}}_{i}$,
a non-zero restoring torque appears every time $\bm{n}_{i}$ and
$\dot{\bm{r}}_{i}$ misalign. This term adds considerable complexity to the
collective system dynamics as the terminal orientation of each particle
results from a compromise between the orientations of the neighboring
particles and that of the local confining force, which are all constantly
changing as the system evolves in time. The role of the restoring torque to
the overall dynamics depends crucially on the parameter $\beta$, which is a
measure of how fast the particle’s orientation relaxes towards the direction
of the local force field. For small $\beta$, the relaxation time grows making
it difficult for the particle to reach the preferred orientation. In this
case, the restoring torque plays a negligible role and one restores the
conventional ABP model.

###  II.2 Length scales and filling fraction

The deterministic, one-particle limit of equations (1) and (2) presents two
steady state solutions for radially symmetric confinement [32]: (i) The
climbing phase, where the particle climbs the potential well until reaching
the force balance condition $V^{\prime}(R_{\text{CI}})=v_{0}/\mu$; and (ii)
the orbiting phase, where the particle describes a circular orbit of radius
$R$ satisfying $R=v_{0}/\beta V^{\prime}(R)$ [31]. For the harmonic potential,
$V(r)=\frac{1}{2}\kappa r^{2}$, these length scales read:

$R_{\text{CI}}=\frac{v_{0}}{\mu\kappa},~{}R=\sqrt{\frac{v_{0}}{\kappa\beta}}$
(3)

The transition from climbing to orbiting takes place when $\beta$ becomes
larger than the critical value $\beta_{c}=\mu^{2}\kappa/v_{0}$.

The force balance condition establishes a critical isocline (CI) across which
the particle can never cross from inside out for any $\beta$. When many
particles are present, the force balance condition also involves interparticle
interactions. In this case, particles can cross the critical isocline assisted
by other ones. This is particularly the case when one exceeds the maximum
number of particles fitting the circle of radius $R_{\text{CI}}$. Therefore,
it is convenient to define the filling fraction as the ratio

$\displaystyle
f=N\frac{A_{\text{SPP}}}{A_{\text{CI}}}=\frac{N\sigma_{h}^{2}}{4R_{\text{CI}}^{2}}$
(4)

where $A_{\text{SP}}=\pi\sigma_{h}^{2}/4$ is the effective area occupied by
one particle, with $\sigma_{h}\simeq 2^{1/6}\sigma$ being the interparticle
distance in a head-on collision, and $A_{\text{CI}}=\pi R_{\text{CI}}^{2}$ is
the area enclosed by the critical isocline. We have observed that for
$f\gtrsim 1$ the particles typically organize themselves in a single cluster,
while for small $f$ the system can break into multiple clusters or in a
mixture of clusters and isolated particles.

From here on, we shall adopt a unit system where length is expressed in units
of $R_{\text{CI}}=v_{0}/(\mu\kappa)$, time in units of $t_{0}=1/(\mu\kappa)$,
and energy in units of $v_{0}^{2}/(\mu^{2}\kappa)$.

Figure 2:  Colormaps of the time and random noise averages of the (a) radial,
$\left\langle{\left\langle{P_{r}}\right\rangle}\right\rangle$, (b) azimuthal,
$\left\langle{\left\langle{P_{\phi}}\right\rangle}\right\rangle$, and (c)
modulus of the total,
$\left\langle{\left\langle{\left|{\bm{P}}\right|}\right\rangle}\right\rangle$,
polarizations, as well as (d) the hexatic order parameter,
$\left\langle{\left\langle{\psi^{G}_{6}}\right\rangle}\right\rangle$, as
function of ($\beta$, $D$) for $N=93$ and $\sigma=0.2$. The red, blue, and
white lines represent, respectively, the boundaries at which the ferromagnetic
(FM), uniform vortex (UV), and paramagnetic (PM) phases become unstable. The
blue dashed line separates the shear-banded vortex (SBV) and the radially
polarized (RP) phases.

###  II.3 Deterministic steady-state solutions of close-packed clusters

Here, we analyse the collective motion of the system when the particles move
as close-packed clusters in the absence of noise. In this situation, the
dimensionless version of Eqs. (1) and (2) become

$\displaystyle\dot{\bm{r}}_{k}+\bm{r}_{k}-\sum_{\begin{subarray}{c}l\neq
k\end{subarray}}^{N}{\bm{f}_{kl}}=\bm{n}_{k},$ (5)
$\displaystyle\dot{\theta}_{k}=\beta(\bm{n}_{k}\times\dot{\bm{r}}_{k})\cdot\hat{z},$
(6)

where $\bm{f}_{kl}=f(r_{kl}){\bm{r}_{kl}}/{r_{kl}}$ are arbitrary pair-wise
forces (in which the WCA model yields a particular case),
$\bm{r}_{k}=\left({r_{k}\cos\phi_{k},\,r_{k}\sin\phi_{k}}\right)$,
$\bm{n}_{k}=\left({\cos\theta_{k},\,\sin\theta_{k}}\right)$,
$\bm{r}_{kl}=\bm{r}_{k}-\bm{r}_{l}$, and $r_{kl}=|\bm{r}_{k}-\bm{r}_{l}|$.

By summing Eq. (5) for all the particles, the internal forces $\bm{f}_{kl}$
cancel out and we have an equation of motion for the system centroid
$\bm{R}=\sum_{k=1}^{N}\bm{r}_{k}/N$,

$\displaystyle\dot{\bm{R}}+\bm{R}=\dfrac{1}{N}\sum_{k=1}^{N}\bm{n}_{k}=\bm{P},$
(7)

where we have also defined the total polarization $\bm{P}$ of the cluster.

It is possible to show (see the ESI for details) that if all particles have
the same orientation, that is $\bm{n}_{k}=\bm{P}$ for all $k$, the cluster
moves as a rigid body. In this case, the dynamics of the polar cluster is
fully described by Eq. (7), supplemented by
$\dot{\theta}_{k}=\beta(\bm{P}\times\dot{\bm{R}})\cdot\hat{\bm{z}}$ for all
$k$, and thereby is equivalent to that of a single particle in a harmonic
confinement [33, 40]. For $\beta>1$, the steady state solution of this problem
is an orbital motion of the cluster, with centroid position $R$, angular
velocity $\Omega$, and orientation angle $\theta_{\rm c}$ given by,

$\displaystyle R$ $\displaystyle=\beta^{-1/2},$ (8) $\displaystyle\Omega$
$\displaystyle=\pm\sqrt{\beta-1},$ (9) $\displaystyle\theta_{\rm c}(t)$
$\displaystyle=\arccos\left({\beta^{-1/2}}\right)\pm\sqrt{\beta-1}t+\theta_{\rm
c}\left({0}\right).$ (10)

Another case of interest is the one in which we have a cluster of particles
rotating around the origin. In such vortex phase, $\bm{P}=0$, since
$\bm{R}=0$. A relation between the particles angular velocities
$\dot{\phi}_{k}$ and their radial positions $r_{k}$ can be obtained from the
vector product between $\bm{r}_{k}$ and Eq. (5). After summing all the
particles contributions, internal torques cancel, and we find

$\displaystyle\sum_{k}r_{k}^{2}\dot{\phi}_{k}=\sum_{k}r_{k}\sin{\chi}_{k},$
(11)

where we define the tilt angle $\chi_{k}=\theta_{k}-\phi_{k}$ with respect to
the radial position. As shown in the ESI, a strict rigid body rotation with
angular velocity $\omega$ would require $\dot{\theta}_{k}=\omega$, which is
admissible only for particles within the region $r>1/\beta$. Instead, we model
the system still assuming that all particles rotate with the same angular
velocity and keep their radial positions, but relaxing the $\dot{\chi}=0$
condition. Adopting this procedure, which we shall refer to as the quasi-
rigid-body (QRB) model, Eq. (6) yields (see ESI for more details)

$\displaystyle\dot{\chi}_{k}=\left({\beta r_{k}\cos\chi_{k}-1}\right)\omega.$
(12)

The solution of this equation reveals two distinct regimes of motion depending
on the particle distance to the origin. For $r_{k}>1/\beta$, $\chi_{k}$
depends monotonically on time,

$\displaystyle\chi_{k}=2\arctan\left[\left(\dfrac{C_{+}+C_{-}e^{-t/\tau_{\omega}}}{C_{+}-C_{-}e^{-t/\tau_{\omega}}}\right)\sqrt{\dfrac{\beta
r_{k}-1}{\beta r_{k}+1}}\right],$ (13)

where $C_{\pm}=\tan\left(\chi_{k}(0)/2\right)\pm\sqrt{\left(\beta
r_{k}-1\right)/\left(\beta r_{k}+1\right)}$, $\chi_{k}(0)$ is the initial
value of $\chi_{k}$ at $t=0$ and
$\tau_{\omega}=1/\omega\sqrt{\beta^{2}r_{k}^{2}-1}$. Notice that for long
times, i.e., $t\gg\tau_{\omega}$, the asymptotic value
$\chi_{k}=2\arctan\sqrt{\left(\beta r_{k}-1\right)/\left(\beta
r_{k}+1\right)}$ is achieved. This gives $\cos\chi_{k}=1/\beta r_{k}$, which
is the result expected by assuming $\omega=\dot{\theta}_{k}$. Meanwhile, for
$r_{k}<1/\beta$, $\chi_{k}$ has oscillatory response given by

$\displaystyle\chi_{k}=-2\arctan\left[\sqrt{\dfrac{1-\beta r_{k}}{1+\beta
r_{k}}}\tan\left(\nu t/2-C_{0}\right)\right],$ (14)

where $\nu=\sqrt{1-\beta^{2}r_{k}^{2}}\omega$ is the angular frequency of
$\chi_{k}$ and $C_{0}=\arctan\left[\sqrt{\left(1-\beta
s_{k}\right)/\left(1+\beta
s_{k}\right)}\tan\left(\chi_{k}(0)/2\right)\right]$.

In order to know the radial and angular components of $\bm{n}_{k}$, namely
$\sin\chi_{k}$ and $\sin\chi_{k}$ respectively, we computed their time
averages from the above time dependencies of $\chi_{k}$. For $r>1/\beta$ the
time averages for $t\gg t_{\omega}$ are given by the asymptotic behavior of
$\chi_{k}$. On the other hand, for $r<1/\beta$, we calculate the time
averages, $\left\langle{\sin\chi_{k}}\right\rangle$ and
$\left\langle{\cos\chi_{k}}\right\rangle$ over one period of oscillation,
$T=2\pi/\nu$. Therefore,

$\displaystyle\left\langle{\sin\chi_{k}}\right\rangle$
$\displaystyle=\left\\{\begin{array}[]{l
l}\sqrt{1-\dfrac{1}{\beta^{2}r_{k}^{2}}},&\text{ for }r_{k}>1/\beta,\\\
0,&\text{ for }r_{k}<1/\beta,\end{array}\right.,$ (17)
$\displaystyle\left\langle{\cos\chi_{k}}\right\rangle$
$\displaystyle=\left\\{\begin{array}[]{l l}\dfrac{1}{\beta r_{k}},&\text{ for
}r_{k}>1/\beta,\\\ \dfrac{1-\sqrt{1-\beta^{2}r_{k}^{2}}}{\beta r_{k}},&\text{
for }r_{k}<1/\beta.\end{array}\right.$ (20)

Now, we return to Eq. (11) and compute its time average, substituting the
appropriate dependencies of $\sin\chi_{k}$ given by Eq. (17). Assuming
$\left\langle{\dot{\phi}}\right\rangle\approx\left\langle{\omega}\right\rangle$
we find the time-average angular velocity of the rotating cluster

$\displaystyle\left\langle{\omega}\right\rangle=\dfrac{\displaystyle\sum_{k}{}^{\prime}\,\sqrt{r_{k}^{2}-\dfrac{1}{\beta^{2}}}}{\displaystyle\sum_{k}r_{k}^{2}},$
(21)

where $\displaystyle\sum_{k}{}^{\prime}$ means sum over only the particles
outside of $r=1/\beta$, while $\displaystyle\sum_{k}$ is the sum over all the
particles. Moreover, substituting Eq. (20) in Eq. (6) enable us to obtain an
approximate time average of $\dot{\theta}_{k}$, i.e.,

$\displaystyle\\!\\!\\!\left\langle{\dot{\theta}_{k}}\right\rangle=\left\\{\begin{array}[]{rr}\left\langle{\omega}\right\rangle,&\text{
for }r_{k}>\beta^{-1},\\\
\left({1-\sqrt{1-\beta^{2}r_{k}^{2}}}\right)\left\langle{\omega}\right\rangle,&\text{
for }r_{k}<\beta^{-1}.\end{array}\right.$ (24)

It is important to mention that these results for the vortex phase implies
nonzero $\dot{\bm{r}}_{k}$ in Eq. (5), particularly for $r<1/\beta$. These
lead to nonzero radial distances variations $\delta r_{k}$ of the particles
positions. Therefore, our model for the vortex phase is expected to hold for
$\delta r_{k}\ll r_{k}$.

##  III Numerical results

To investigate in further detail the different dynamical phases and phase
transitions exhibited by this system, we numerically integrated Eqs. (1) and
(2), using a second-order stochastic Runge-Kutta algorithm, exploring a wide
range of values of angular mobility $\beta$, rotational noise intensity $D$,
and filling fraction $f$. In Fig. 2, we present the dynamical phase diagram in
the $\beta D$ parameter plane for a system of $N=93$ self-propelled particles
of size $\sigma=0.2$, which corresponds to an intermediate-valued filling
fraction $f=1.17$, in the active noise ($D_{t}=0$) regime. The heatmaps
correspond to time and random noise averages,
$\langle\\!\langle\,\cdots\rangle\\!\rangle$,111 Unless stated otherwise, time
averages were computed over $100t_{0}$ after a $300t_{0}$ waiting time while
noise averages were performed over 50 independent realizations of the
stochastic forces. of the total, radial, and azimuthal polarizations, given
respectively by

$\displaystyle\bm{P}=\sum_{i=1}^{N}\frac{\bm{n}_{i}}{N},~{}{P}_{r}=\sum_{i=1}^{N}\frac{\bm{n}_{i}\cdot\hat{\bm{r}}_{i}}{N},~{}{P}_{\phi}=\sum_{i=1}^{N}\frac{\bm{n}_{i}\cdot\hat{\bm{\phi}}_{i}}{N},$
(25)

where $\hat{\bm{r}}_{i}=\bm{r}_{i}/|\bm{r}_{i}|$ and
$\hat{\bm{\phi}}_{i}=\hat{\bm{z}}\times\hat{\bm{r}}_{i}$, and the hexatic
order parameter, defined as

$\psi_{6}^{G}=\left|\frac{1}{N}\sum_{i=1}^{N}\left[\frac{1}{{\cal
N}_{i}}\sum_{j=1}^{{\cal N}_{i}}e^{i6\theta_{ij}}\right]\right|,$ (26)

where $\mathcal{N}_{i}$ is the number of nearest neighbors of particle $i$ and
$\theta_{ij}$ is the angle of the link between $i$ and neighbor $j$ with
respect to the $x$ axis.

These quantities allow for identifying in the parameter space the distinct
phases represented by the configurations shown in Fig. 1 (a)-(d). The radially
polarized (RP) state is characterized by large (close to one)
$\langle\\!\langle P_{r}\rangle\\!\rangle$ and vanishingly small
$\langle\\!\langle P_{\phi}\rangle\\!\rangle$ and
$\langle\\!\langle|\bm{P}|\rangle\\!\rangle$. The vortex phases present two
distinct behaviors: for $\beta\gtrsim 1$, $\langle\\!\langle
P_{r}\rangle\\!\rangle$ is still large, but $\langle\\!\langle
P_{\phi}\rangle\\!\rangle$ is non-zero. The small azimuthal alignment of the
particle orientations is just enough to make the system rotate in either
clockwise or anticlockwise direction as a shear-banded vortex (SBV); for
$\beta\gg 1$ or large values of $f$, the system rotates uniformly and
typically has $\langle\\!\langle P_{\phi}\rangle\\!\rangle>0.5$ and
$\langle\\!\langle P_{r}\rangle\\!\rangle<0.5$, which reflects the global
azimuthal alignment of the particle orientations. This uniform vortex (UV)
phase is also characterized by large hexactic order as a result of the close-
packed arrangement of the particles. The orbiting ferromagnetic (FM) phase is
characterized by a high degree of orientational and translational order as
revealed by the large values of $\langle\\!\langle|\bm{P}|\rangle\\!\rangle$
and $\langle\\!\langle|\psi_{6}^{G}|\rangle\\!\rangle$. It dominates a large
part of the parameter space and coexist with the vortex phases for small $D$
and intermediate $\beta$ values. Finally, in the small $\beta$ and large $D$
region all polar order parameters vanish, which characterize the unpolarized
or paramagnetic (PM) state. In the following subsections, we discuss in more
detail the properties of each polar phase and the effect of changing $\beta$,
$D$, number of particles, and particle size.

Fig. 2 also features estimates of the stability boundaries of the phases,
shown as continuous white, red, and blue lines. They were obtained by
initializing the system in a given phase and sweeping $\beta$ and $D$ until
the main polar order parameter characterizing that phase falls below a
threshold value (we choose 0.5 as the threshold for all polar order
parameters, but see below for details). Beyond this point, the phase is
considered to be extinct and is replaced by a new phase, either in a
continuous or abrupt way. This allows a reasonably accurate identification of
the stability limit of the phases for the cases of abrupt transition, such as
the UV to FM and FM to UV transitions, and a rough estimate for smooth
transitions, such as all transitions involving the paramagnetic state. The
consistency of these criteria was validated by videos of the system evolution
at points nearby the estimated boundaries. The transition between the radially
polarized and the shear-banded vortex state requires a different criterion. In
that case, since rotation commences as soon as a small azimuthal polarization
appears, we use the more stringent condition $\langle\\!\langle
P_{\phi}\rangle\\!\rangle>0.1$ (dashed blue line) as an estimate of the break
down of the RP phase in favour of the SBV phase. The SBV phase has typically a
larger value of $\langle\\!\langle P_{r}\rangle\\!\rangle$ as compared to the
UV phase, so the $\langle\\!\langle P_{\phi}\rangle\\!\rangle=0.5$ line can be
taken as a rough estimate for the SBV to UV transition.

We have also analysed the impact of non-negligible translational noise on the
observed phases and their respective boundaries. For that, we considered the
case where the random noise is dominated by thermal fluctuations, for which
$D_{t}=\sigma^{2}D/3$ [37, 38]. While no new phases emerged in this case,
notable changes were observed in some phase boundaries compared to the
$D_{t}=0$ scenario depicted in Fig.2. The impact is particularly pronounced
for large values of $\beta$ and $D$, leading to significant alterations in the
FM-PM phase boundary. Conversely, transitions between the RP phase and the
vortex phases as well as the RP-PM phase boundary, occurring at small $D$ and
relatively small $\beta$, remain essentially unchanged (see Supplemental
Material, ESI).

###  III.1 Radially polarized phase

The radially polarized phase is reminiscent of the climbing phase, where non-
interacting active particles under radial confinement are predicted to
accumulate at a certain distance away from the potential center [16, 41, 19,
42, 43, 44]. This distance coincides with the radius of the critical isocline
for small $D$, as discussed in Sec. II.2. Here, because of the steric
interparticle interactions, this accumulation manifests itself in different
ways. For $D=0$ and $\beta\lesssim 1$, the particles exhibit a close-packed
configuration in a ring-shaped crystal [see Fig. 1-(a) for a representative
configuration], resulting in high values of the hexatic order parameter. At
non-zero but small values of $D$, the hexatic order decays considerably and
the system exhibits a fluidlike behavior, while still keeping the radial polar
order (see Supplementary Video 1, ESI). Upon increasing $D$ further at small
$\beta$, the system loses the radial polar order and enters the unpolarized
phase. On the other hand, increasing $\beta$ at $D=0$ makes the system to
transition to the vortex phase and subsequently, at a higher value of $\beta$,
to the orbiting polarized state as explained below.

###  III.2 The ferromagnetic and uniform vortex phases: global properties and
phase coexistence

In the orbiting ferromagnetic (FM) phase [Fig. 1-(d)], the particles form a
single close-packed cluster that revolves around the potential center but
never rotates around its own centroid. Additionally, all orientation vectors
are parallel to each other and rotate at the same rate as the angular speed of
the centroid (see Supplementary Video 2, ESI). These attributes suggest the FM
phase can be well described by our rigid-body model presented in Sec. II.3,
which reduces the problem to that of a single particle with constant orbital
radius and angular speed given by Eqs. (8) and (9), respectively. In Fig
3-(a),

Figure 3:  Comparison between the analytical predictions and numerical results
for a cluster with $N=93$ and $\sigma=0.2$ in (a) the orbiting polar and (b)
the vortex phases. The solid lines represent the numerical results of the mean
angular frequency (a) and (b) and centroid radial distance (inset) for
different values of $D$. The dashed lines correspond to the analytical results
[Eqs. (9), (8), and (21)]

we confront these analytical results with measurements of the mean angular
speed and radial position of the centroid of a system of $N=93$ particles of
diameter $\sigma=0.2$ in the orbiting polar phase. The simulations were
performed as a function of $\beta$ for different $D$. The numerical results
are shown to be in excellent agreement with the theory, even in the presence
of moderate noise. Indeed, as already anticipated in the diagrams of Fig. 2,
the orbiting polar phase is highly resilient to noise, being stable even at
high values of $D$, specially for high angular mobility.

The uniform vortex (UV) phase is also characterized by a close-packed
arrangement of the particles, but now the system rotates around its own
centroid which remains essentially fixed at the potential minimum. In Fig.
3-(b), we show the average angular speed of the particles along $100t_{0}$
time window as a function of $\beta$ for different values of $D$ and compare
with the analytical formula Eq. (21). For $D=0$, the simulation results are in
excellent agreement with the model. However, the agreement becomes
considerably poor for $D>0.1$. Notice that $D=0.5$ is well within the
stability range of the UV phase, which becomes unstable only at $D\gtrsim 4$
for $10<\beta<30$. Still, the mean angular speed of the system at this noise
level is considerably smaller than the $D=0$ limit, suggesting that the UV
phase is typically much more sensitive to fluctuations than the FM phase.

The FM and UV phases have a large intersection where both phases coexist, that
is, depending on the initial conditions, the system self-organizes in either
FM or UV phases, with the FM (UV) being more probable for higher (lower)
values of $\beta$, as indicated by the color gradients in Fig. 2 (b) and (c).
As a result, the transition from UV to FM and back exhibits pronounced
hysteretic behavior as illustrated in Fig. 4.

Figure 4:  (a) Time and noise averages of the polar order parameters,
$\langle\\!\langle|\bm{P}|\rangle\\!\rangle$ and
$\langle\\!\langle|P_{\phi}|\rangle\\!\rangle$, calculated for fixed
$\beta=25$ and sweeping $D$ from zero up to 10 (solid lines) and back to zero
(dashes). The shadows correspond to standard deviations from the mean. (b)
Same as (a), but fixing $D=2.5$ and sweeping $\beta$ from 0 to 60 and back.
The simulations were performed for $N=93$ and $\sigma=0.2$ as in Fig. 2. The
insets depict representative snapshots of the system at the points indicated
by the arrow. The color scheme is the same as in Fig. 1.

In panel (a) the system is initialized in the vortex phase at $\beta=25$ and
$D=0$. Upon increasing $D$ slowly up to 10, the vortex phase decays into the
orbiting polar phase, as indicated by the sharp drop in the azimuthal polar
order parameter accompanied by an increase in the total polarization. Then,
the FM phase continuously crosses over to the PM phase. Now, decreasing $D$
from 10, the transition from PM to FM follows approximately the same path,
thus emphasizing the continuous character of this transition. In contrast, by
decreasing $D$ even further, the FM phase never decays back to the vortex
phase, following a different path from the upward branch. Similar hysteretic
phenomena is also observed by sweeping $\beta$ up and down at fixed $D=2.5$,
as shown in panel (b). In this case, the system is initialized in the
unpolarized state which is the only one available for $(\beta,D)=(0,2.5)$. In
increasing $\beta$, it crosses over reversibly to the uniform vortex state and
at $\beta\simeq 40$ decays to the FM phase. Reversing $\beta$ from 60 down to
0, the FM phase decays back to UV phase only at $\beta\simeq 13$, thus
revealing a pronounced hysteresis.

Fluctuations can often induce spontaneous transitions between coexisting
states. To check the possibility of similar noise-induced switching between
distinct phases in our system, we selected a few $(\beta,D)$ points and, for
each of them, calculated the azimuthal and total polarization order parameters
as a function of time along a $10^{4}t_{0}$ time window, starting from one of
the coexisting phases. In Fig. 5

Figure 5:  Time series of the total polarization modulus (red) and the
azimuthal polarization (blue) featuring: (a) and (b) a transition between UV
and FM phases for $\beta=25$ and $D=2.85$ [(b) shows a zoomed-in view of the
transition]; (c) multiple switchings between clockwise and anticlockwise
vortex phases for $\beta=9.2$ and $D=2.5$.

we illustrate two kinds of noise-induced switching between coexisting
dynamical states observed in this study. Panel (a) shows a typical phase
switching from the uniform vortex to the ferromagnetic phase at a point within
the UV-FM coexisting zone, $(\beta,D)=(25,2.85)$. After initializing the
system at the uniform vortex state, a sudden drop of the azimuthal order
parameter and concomitant rise of the total polarization take place at
$t\simeq 2700t_{0}$. Conversely, by initializing the system in the FM state at
the same point of the parameter space, no transition is observed within the
simulated time window. This suggests that at this point the FM state is more
stable than the UV state. In panel (c), we show the time evolution of the
system for a point in the phase diagram where only the UV phase is stable. The
data reveal multiple switchings of the sign of the azimuthal order parameter,
thus reflecting transitions between clockwise and counterclockwise UV
rotation. Unlike the UV-FM spontaneous transitions, here the inversions of the
UV rotation to one direction or the other occur at similar rates, as a result
of the symmetry of both states.

###  III.3 Anatomy of the uniform vortex phase

As mentioned above, the uniform vortex phase is characterized by uniform
rotation of the particles, that is, all $\dot{\phi}_{k}$ are essentially the
same. In contrast, the polarization vectors of the particle showcase a highly
non-uniform radial distribution. To illustrate and better understand these
intriguing local properties, we consider a smaller particle size,
$\sigma=0.1$. This allows for investigating clusters with considerably larger
number of particles, which in turn renders better local statistics while
keeping the filling fraction close to one. In Fig. 6

Figure 6: Time-averaged radial distribution of $P_{R}$, $P_{\phi}$,
$\dot{\theta}$ and $\dot{\phi}$ calculated from the simulations (filled areas)
and predicted by the model (dashed lines). The red and blue dashed lines are
given by Eqs. (17) and (20), while the purple and green lines were obtained
from Eqs. (21) and (24). The situation in (a) and (b) ($N=500$ and
$\beta=1.85$) corresponds to the UV state, which is fully captured by the
model. For (c) and (d) ($N=350$ and $\beta=1.85$), the system is in the SBV
state, in which case the model fails for $r<\beta^{-1}$ (see text).

(a) and (b) we show the time-averaged radial distributions of $P_{r}$,
$P_{\phi}$, $\dot{\phi}$, and $\dot{\theta}$ for $N=500$ particles and
$(\beta,D)=(1.85,0)$. For these parameters, the particles form a close-packed
cluster that rotates uniformly, that is $\dot{\phi}(r)\simeq\text{constant}$.
Nevertheless, all other measured quantities present strikingly different
behavior in two different regions. For $r>1/\beta$, these quantities follow
closely the theoretical prediction for a perfect rigid-body vortex:
$P_{r}=\cos\chi=1/\beta r$ and $P_{\phi}=\sin\chi=\sqrt{1-1/\beta^{2}r^{2}}$.
For $r<1/\beta$, the rigid-body condition $\dot{\theta}=\dot{\phi}$ breaks
down. However, since $\dot{\phi}$ is still uniform, the approximations that
lead to Eq. (12) still apply. Indeed, the profiles of $P_{r}$, $P_{\phi}$, and
$\dot{\theta}$ in this region are well described by Eqs. (17), (20), and (24),
respectively, in the full range of $r$.

To gain deeper insights into the efficacy of the quasi-rigid-body model
presented in Sec. II.3, we closely examine the time-dependence of the radial
position and the tilt angle $\chi=\theta-\phi$ of four particles, two of them
with average radial positions $\left\langle{r}\right\rangle>1/\beta$ and the
other two with $\left\langle{r}\right\rangle<1/\beta$, as shown in Fig 7.

Figure 7: Time evolution of (a) radial position and (b, c) tilt angle
$\chi=\theta-\phi$ of 4 different particles in a system with $N=500$ and
$\beta=1.85$. The dashed lines in (b) and (c) correspond to the theoretical
prediction Eqs. (13) and (14). For particles with mean radial position above
$\beta^{-1}$, $\chi$ fluctuates around a constant value as the particle orbits
the potential well, whereas for particles in the inside region $\chi$ performs
full revolutions due to the orientation $\theta$ and radial position $\phi$ no
longer having the same frequency.

All four particles oscillate radially with similar amplitudes $\sim\sigma/2$.
Notably, the oscillations at different mean radial positions are highly
correlated, indicating that a collective mode of the cluster is excited. This
subtle wobbling of the inner region relative to the outer region, as revealed
in Video 4 (ESI), underpins the observed highly correlated radial
fluctuations. That is to say, the cluster can no longer be treated as a rigid
system. Nevertheless, the behavior of the tilt angle is found to align with
the quasi-rigid-body model. Remarkably, despite the uniformity of the angular
velocity, it unveils radically different polarization dynamics in both
regions: for particles with $\left\langle{r}\right\rangle>1/\beta$, $\chi(t)$
only fluctuates slightly around a constant angle, which is in excellent
agreement with the asymptotic limit of Eq. (13); for
$\left\langle{r}\right\rangle<1/\beta$, $\chi$ undergoes full rotations
periodically, in close agreement with Eq. (14). Deviations from the model can
be attributed to the fluctuations in the radial position seen in (a).

Another striking feature of the UV phase can be observed in Fig. 8 (c) and
(d),

Figure 8:  Instantaneous configurations for different numbers of particles and
$\beta=1.85$ (see labels in the top right corner of the figures). The dashed
blue and orange circles represent the isocline ($r=1$) and the critical radius
($r=1/\beta$) respectively. The color pattern within the particles depicts
their orientation, following the same scheme as described in Fig 1.

where we show snapshots of positions and polar angles $\theta$ of particles
for $N=500$ and $N=800$ clusters in the UV phase. In both cases, the particles
orientations in the region $r<1/\beta$ are completely random. This can also be
understood from Eq. (14). Although particles within the same radial distance
have the same oscillation frequency, their phase depends on the value of
$\chi$ held by the particle at the moment the system reached its close packed
form, $\chi_{k}(0)$ in Eq. (14). Therefore, the color pattern seen in the
region $r<1/\beta$ encodes a memory of how the UV phase was formed in the
past. Indeed, the transition between the different phases typically passes
through a disordered state, resulting in the observed random phase
distribution.

Altogether, these results suggest that the UV phase rotates not as a strict
rigid body but rather as a deformable solid. Such deformations allow the
system to violate the $\dot{\chi}=0$ condition in the region $r<1/\beta$ while
still keeping its close-packed structure. Interestingly, a strict rigid-body
vortex phase is not possible when particles occupy the $r<1/\beta$ region.
Therefore, elastic deformations seem to be essential for the stabilization of
the uniform vortex phase.

###  III.4 Shear-banding transition

When decreasing beta $\beta$ at fixed $D$ and $f$, the close-packed UV phase
becomes unstable with respect to the expansion of the cluster. This can be
visualized in Fig. 9 (a),

Figure 9: SBV to UV transition of a cluster with $N=93$ and $\sigma=0.2$. (a)
Time-averaged radial position of the outermost ($R_{\text{ext}}$, red line)
and the innermost ($R_{\text{int}}$, green line) particle in the cluster as a
function of $\beta$. The black dashed line marks the critical radius
$(1/\beta)$ separating regions of different polar behavior (see text). (b)
Time-averaged angular speed of the particles belonging to the regions
$r>1/\beta$ (black line) and $r<1/\beta$ (blue line). At $\beta=3$ (grey
dashed line), the cluster is nearly fully compacted, and the shear banding
phenomenon begins to fade.

where we plot the time-averaged radial positions of the outermost
($R_{\text{ext}}$) and the innermost ($R_{\text{int}}$) particles of the
cluster as a function of $\beta$ for $D=0$, $N=93$, and $\sigma=0.2$. For
$\beta\gtrsim 3$, $R_{\text{int}}<\sigma$ and both $R_{\text{int}}$ and
$R_{\text{ext}}$ do not change further with $\beta$. This is a result of the
cluster being in a fully compact state. However, below this threshold, the
cluster expands into a ring with $R_{\text{ext}}>R_{\text{int}}>\sigma$.
Additionally, the uniformity of the angular speed of the particles, which
characterizes the UV phase, breaks down. This is evidenced in Fig. 9-(b),
where we show the time-averaged angular speed of particles belonging to two
distinct regions: $r>1/\beta$ and $r<1/\beta$. For $\beta\gtrsim 3$, particles
in both regions have the same angular speed, as expected for the UV phase. In
contrast, for $\beta\lesssim 3$, particles occupying the $r<1/\beta$ region
have, in average, considerably lower $\omega$, that is, the system is in the
SBV phase, with two distinct concentric bands rotating at different angular
speeds.

The concomitant breakdown of cluster compaction and uniformity of the angular
speeds, marking the transition from the UV to the SBV phase, can be achieved
not only by changing $\beta$, but also by changing the filling fraction. Fig.
8 presents snapshots of clusters with different filling fractions but the same
$\beta$ and $D$, (1.85 and 0, respectively). These figures exemplify that, by
increasing the filling fraction at a fixed $\beta$, the empty region is
progressively populated causing $R_{\text{int}}$ to decrease to the point
where the cluster becomes compact again. At this point, angular velocities
become uniform all over the cluster as required for UV rotation.

The time-averaged radial distributions of $P_{r}$, $P_{\phi}$, $\dot{\phi}$,
and $\dot{\theta}$ for $N=350$ particles and $(\beta,D)=(1.85,0)$ is shown in
Fig. 6 (c) and (d). The $r>1/\beta$ region is still well described by the
quasi-rigid body model, as the particles arrange in an almost crystalline
structure in this region. In contrast, the inner band presents highly non-
uniform profiles of both, $\dot{\phi}$, and $\dot{\theta}$, thus indicating a
fluidlike behavior. Surprisingly, some particles even rotate in the opposite
direction of the outer band flow. Another remarkable feature of the inner band
is that the particles in this region have in average a considerably high
radial polarization, as if they were in a kind of climbing state.

The spontaneous segregation of the system in two rotating bands can be
understood as follows. Particles at the inner edge of the ring are pushed away
from the cluster towards the center of the potential by the potential itself
as well as by the steric forces induced by the other particles. These
frustrated particles are constantly trying to fit into the flock. Some of them
eventually succeed, thanks to sliding events causing a rearrangement of the
system, which involves particles from both bands (see Supplementary Video 4,
ESI). In this case, the lucky particle is replaced by another one. Those that
don’t succeed are ejected back and the whole process repeats. In order to
reach the flock, the stray particles must first climb the confining potential.
At this point, they can do that with a slight trend towards either clockwise
or counterclockwise rotation. Particles with a clockwise trend have a better
chance to get absorbed by the flock if the flock also rotates clockwise and
vice-versa, while particles moving opposite to the main flow will remain at
the inner band. This process naturally results in an average opposite motion
of particles belonging to the inner band, thus explaining the sign change
observed in the radial distribution of the angular velocity.

The shear banding transition also occurs when one starts from $\beta<1$ at the
climbing phase and increases $\beta$ above a threshold value, $\beta_{c}\sim
1$, at fixed small $D$. Just above threshold, the symmetry is spontaneously
broken and the system rotates on average either clockwise or anticlockwise.
For moderate filling fractions, $f\sim 1$, and just above $\beta_{c}$, the
ring-shaped cluster yields and self-organizes in a shear banded vortex. For
large $f$, the radially polarized phase transitions directly to the uniform
vortex phase.

###  III.5 Multi-cluster phase

We have observed that, for small filling fractions, $f<1$, and moderate values
of $\beta$, the system splits into multiple clusters, as illustrated in Fig.
1-(e). The resulting configurations and dynamics possess characteristics of
both orbital and vortex orders: all particles within a cluster have the same
alignment $\bm{n}_{i}$ and thereby can be treated as a single giant active
particle, orbiting around the confinement center following Eqs. (8), (9) and
(10); at the same time, the set of several such giant particles can be seen as
being in a vortex state, as their polarizations acquire the same azimuthal
component, thus leading to the observed coherent rotation of the clusters (see
Supplementary Video 5, ESI).

Figure 10: Density plots of the average number of clusters as a function of
particle size $\sigma$ and total number of particles $N$ for $D=0$ and
different values of $\beta$, indicated in upper right corners. Higher values
of $\beta$ lead to tighter systems with only one cluster, even for smaller
$N$. Dashes represent the $f=1$ line as defined by Eq (4).

To investigate the multi-cluster phase in more detail, we calculate the number
of close-packed clusters averaged over 100 independent initial conditions,
$\langle N_{c}\rangle$, exploring the $\sigma N$ parameter space in the range
$0.1<\sigma<0.3$ and $10<N<100$. The simulations are performed at $D=0$ for
three different values of $\beta$ and the results are depicted in Fig. 10.
Notice that for $f\gtrsim 1$, the system typically arrange into a single
cluster, while for $f<1$ the particles can either form one or several clusters
or none for very disperse systems, depending in an intricate way on $\sigma$,
$N$, and $\beta$. At some points, different initial conditions produce
different numbers of clusters leading to non-integer $\langle N_{c}\rangle$.
Notice that higher values of $\beta$ typically induces the formation of a
single cluster. For $\beta\gtrsim 10$, the single cluster phase dominates the
parameter space. This can be understood as a result of the orbit radius of the
clusters, given by $1/\sqrt{\beta}$, becoming so small that different clusters
can no longer orbit the potential well without colliding with each other,
thereby causing them to merge into a single cluster, which evolves dynamically
either in the FM or vortex state.

##  IV Conclusions

In summary, our analytical and numerical investigations have unveiled a
remarkable variety of emergent dynamical phases of harmonically confined self-
propelled particles interacting via steric forces and torques. The coupling
between torques and the orientational dynamics of the particles, controlled by
the angular mobility $\beta$, revealed to be a crucial ingredient for the
richness of the phase diagram. For small $\beta$, the particles self-organizes
into a static radially polarized state, reminiscent of the climbing phase of
single confined self-propelled particles. As one increases $\beta$ past a
critical value, the spontaneous symmetry is broken and the system initiates a
rotational motion in either the clockwise or counterclockwise direction. The
resulting state of the system can be either of several distinct dynamical
phases: Orbiting ferromagnetic state, where the cluster orbits the potential
center with fully aligned polarizations as in a magnet; Uniform vortex, where
the net azimuthal components of the particles orientations add up to rotate
the whole system uniformly concentric to the confining center; Shear-banding
vortex, where the system segregates in two concentric bands rotating with
different angular speeds; and the multicluster phase, where the system divides
into several smaller polarized clusters orbiting the potential center. The
specific phase exhibited is contingent on several parameters, including
filling fraction, noise intensity, and $\beta$, as well as the historical
context of parameter changes.

Digging deep into the properties of these phases reveals a few surprising
features. For instance, in the uniform vortex phase, the angular speed of the
particles is uniformly distributed, but the dynamics of their polarization
vectors are highly nonuniform. This might initially appear paradoxical since
rigid body rotation typically requires simultaneous evolution of both angular
positions and polarization vectors at the same rate. However, our
investigation shows that the uniform vortex phase behaves not as a strict
rigid body but rather as a deformable solid. This unique characteristic allows
for the decoupling of translational and polar dynamics, resolving the apparent
contradiction. Moreover, an intriguing observation emerges in the shear-banded
vortex phase, where particles in the inner band move in the opposite direction
to those in the outer band. This counterflow results from a filtering effect,
where particles from the inner band are more likely to be absorbed by the
outer band if they move in the same direction, leading to an overpopulation of
particles rotating in the opposite direction in the inner band region.

An intriguing avenue for further research on this system involves a deeper
investigation into the deformable properties of the vortex phase, specially
considering the relevance of deformable active systems for smart soft
materials and their potential application in understanding biological
functions [45]. Specifically, focusing on identifying possible selection
mechanisms of collective modes [46, 47] and exploring whether external
excitation of such modes could induce a transition from a vortex to the
ferromagnetic state could unveil novel insights. Another crucial aspect
deserving dedicated attention is the role of defects within the observed
vortex states. In our simulations, these defects in the polarization space
appear stable over the entire simulated time window, raising questions about
their translocation and potential annihilation. A detailed study addressing
how such defects could be manipulated or eliminated would contribute
significantly to our understanding of the stability and dynamics of vortex
states in confined active matter systems.

###### Acknowledgements.

This work was financed in part by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior - Brasil (CAPES), Finance Code 001. CCSS is funded by Conselho
Nacional de Desenvolvimento Científico e Tecnológico - Brasil (CNPq), Grant
No. 312240/2021-0.

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