# Dynamical Swirl Structures Powered by Microswimmers in Active Nematics

Partha Sarathi Mondal  Pawan Kumar Mishra  Tamás Vicsek  Shradha Mishra

###### Abstract

Active nematics, in their pure form, have demonstrated a plethora of dynamic
and steady-state behaviors, including large-scale dynamic structures,
collective flows, and intricate multi-spatial temporal dynamics. This
complexity further increases in the presence of external polar agents. We
investigate active nematics interspersed with polar microswimmers, akin to
active apolar cells infused with active impurities (microswimmers). Our
comprehensive numerical study reveals that varying the microswimmers’ motility
induces a novel spatiotemporal state in the active nematics backdrop. This
state is marked by macroscopic swirl-like structures and a reduction in the
overall order of the active nematics. Interestingly, this state emerges at
intermediate motility levels, where microswimmers form local clusters and
exhibit coherent motion. However, at higher motility levels, the swirls become
less coherent, and microswimmer clustering intensifies. We show that the
effect of the polar microswimmers on active nematics can be interpreted as a
spatiotemporally correlated colored noise on active nematics, which promotes
bend instability in active nematics, leading to the observed swirling
dynamics. Our findings indicate that the spatiotemporal states are highly
sensitive to the microswimmers’ motility, offering potential avenues for
pathogen identification based on known motility characteristics.

###### keywords:

Active Matter , Active Nematic , Polar Nematic Mixture , Spatiotemporally
Correlated Colored Noise, Topological Defects, Swirl Structures, Bending in
Active Nematics

[inst1]organization=Department of Physics, Indian Institute of Technology
(BHU) Varanasi,addressline=, city=Varanasi, postcode=00000, state=221005,
country=India

[inst2]organization=Department of Biological Physics, Eötvös
University,addressline=Pázmány Péter sétány 1A, city=Budapest,
postcode=H-1117, country=Hungary

A class of active systems has aligning motile impurities which we interpret as
microswimmers

Previous studies addressing inhomogeneities in active systems focused on
quenched or diffusive impurities.

The presence of microswimmers leads to dynamic swirl structures and
topological defects in the system.

Microswimmers influence active nematic in a way which is analogous to the
effect of a spatiotemporally correlated coloured noise.

Our findings offer potential avenues for pathogen identification in active
systems.

##  1 Introduction

Active matter systems are comprised of motile agents known as Self Propelled
Particles (SPPs) [1, 2, 3, 4, 5], which absorb energy from their environment
and convert it into mechanical motion. This distinctive interplay between
energy dissipation and self-propulsion sets active matter systems apart from
their equilibrium counterparts [5, 6, 7, 8, 9, 10, 11, 12]. These systems have
garnered significant interest due to their prevalence across various scales,
ranging from microscopic to macroscopic levels. Examples include cytoskeletal
filaments [13], molecular motors [14], bacterial colonies [15, 16, 17, 18],
schools of fish [19], herds of animals [20, 21], and flocks of birds [22]. The
classification of self-propelled particles into active polar or active
nematics categories, based on their symmetry, highlights substantial
differences in their dynamics. This classification aids in selecting the
appropriate category to address specific problems within these systems.
Numerous studies [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] have explored
the collective dynamics of SPPs in both pure polar and pure nematic systems.
However, the introduction of a foreign species into these systems adds a layer
of complexity to their behavior. When species with differing alignment
tendencies are mixed, the resulting system may exhibit features absent in pure
systems. Interactions both within and between species in the mixture dictate
how each species responds to the presence of the others, unveiling a host of
novel characteristics.
Additionally, if the foreign agents are nonmotile, then the system falls into
a distinct subclass, where the immobile species acts as a disorder [34, 35,
36, 37, 38, 39, 40, 41, 42, 43, 44]. The role of disorder of various origins
in active systems has recently gathered significant interest as it mirrors the
complexities encountered by SPPs in natural, heterogeneous environments,
resulting in noticeable alterations in their dynamics compared to homogeneous
environments. One key observation from these studies indicates that the
presence of disorder can induce rotational tendencies among the SPPs within
the systems. While most of these investigations have focused on quenched
disorder, it was demonstrated in reference [40, 45] that the influence of
diffusing obstacles on the system could vary depending on the dynamics of the
obstacles.
In this context, a new subclass known as Living Liquid Crystal (LLC) [46, 47,
48, 49, 50, 51] has emerged, where bacteria function as polar particles
immersed in a Liquid Crystal (LC). LLC systems demonstrate intriguing
characteristics, such as the formation of unique structures [47, 52, 53, 54,
55], accumulation of bacteria at defect cores [56], and enhanced dynamics of
+1/2 defects in LC [56], among others. In reference [57], the authors explored
the LLC system within the realm of active matter. The authors analyzed the
behavior of the background active nematics in the presence of polar particles
representing bacteria. By varying the bacterial density, they observed that
the global orientational order of the nematic exhibits a nonmonotonic
behavior, accompanied by complex patterns consisting of dynamic circular
patterns in the orientation field of the nematic for intermediate densities of
polar particles.
In this study, we investigate a mixture of active polar and nematic species
akin to that of reference [57]. We address an important question: How does the
activity of the polar particles affect the apolar particles? To this end, we
focus on the small densities of polar species in the mixture and adjust the
activity of the polar particles while keeping their density constant.
The primary observations of our study are as follows: (i) we observe a
nonmonotonic behavior in the global orientational ordering of the active
nematics upon tuning the activity of polar particles; (ii) the coupling
between apolar and polar particles induces collective bending of the nematic,
resulting in the formation of swirled and strongly bend structures in the
nematic. Furthermore, we propose that (iii) the influence of polar particles
on the active nematics can be interpreted as spatiotemporally correlated
colored noise acting on the active nematic.
The remainder of the paper is organized as follows: In Section 2, introduces
our model and details of the parameters utilized for numerical simulation.
Section 3 is dedicated to discussing the results of our study. Finally, in
Section 4, we summarize our findings and investigate potential applications
and future directions for further research.

##  2 Model

Our study examines a two-dimensional system featuring a mix of polar and
apolar active particles. Each particle (polar or apolar) in the system is
characterized by a position vector $\bm{r}_{i,p/ap}=(x_{i,p/ap},y_{i,p/ap})$
relative to an arbitrary origin and an orientation vector $\theta_{i,p/ap}$,
where, subscripts $p$, and $ap$ refer to polar and apolar particles,
respectively. The quantification of orientation for both types of particles is
elaborated in FIG.1. In the absence of perturbations, the polar particles
exhibit unidirectional motion along the direction of their head and are
referred to as polar swimmers, while the apolar particles display
bidirectional motion along their long axis, adhering to the symmetry of
$\theta\to\theta+\pi$. Self Propulsion velocities of polar and apolar
particles are denoted by $v_{p}$ and $v_{ap}$, respectively.
The particles interact through the alignment interaction, with the neighbours
within the interaction radius as introduced in reference [1]. The nature of
alignment solely depends on the intrinsic symmetry of the particles under
consideration. For polar particles, alignment is ferromagnetic, compelling
them to move along the mean direction of motion of their neighbors.
Conversely, apolar particles exhibit nematic alignment, randomly choosing
between parallel and antiparallel alignment with the mean direction of
orientation of their neighbors. Imperfections in alignment arise from errors
made by the particles in following their neighbors, introducing inherent
fluctuations into the system, such that the orientation of each particle
deviates by a small angle from the mean of its neighbours. Such noise is named
as scalar noise in previous studies [58, 59]. The specifics of particle
symmetry and interactions are elucidated in FIG.1. Further details outlining
the dynamical equations of both particle types are provided in A.
The system is simulated in a square box of size $L\times L$ subjected to
periodic boundary conditions to study bulk properties of the mixture and
minimize boundary effects. The density of apolar particles
($\rho_{ap}=\frac{N_{ap}}{L^{2}}$) is kept constant at 1.0, while the density
of polar particles ($\rho_{p}=\frac{N_{p}}{L^{2}}$) is maintained at a low
value, $\rho_{p}\in[0.02,0.03,0.04,0.05]$. In this setup, the self-propulsion
velocity of apolar particles ($v_{ap}$) is kept fixed at $0.25$, while we vary
the velocity of polar particles ($v_{p}$) from $0$ to $0.25$, encompassing the
activity ratio $\mu=\frac{v_{p}}{v_{ap}}$ within the interval $[0,1]$. At low
velocity, the polar particles have very slow dynamics. To overcome this, we
add a small diffusive part with magnitude $V_{R}=10^{-3}$ in the position
update of the polar particles.
Eq.(3-6) are solved numerically with step size $\Delta t=1$. One simulation
step is counted when the position and orientation of all the particles
(polar+apolar) are updated once. The results presented are based on
simulations using a $100\times 100$ system size unless otherwise specified.
For this system size, the system is simulated for a total of $1.7\times
10^{5}$ time steps, of which in the initial $1.3\times 10^{5}$ steps, the
system is allowed to reach the steady state and the remaining $0.4\times
10^{5}$ steps used for calculation of the observable. Additionally, some
results are showcased for a system size $130\times 130$, wherein the total
number of simulation steps is increased to $2.5\times 10^{5}$ for each
parameter set $[v_{p},\rho_{p}]$. The average over at least $15$ independent
realizations are performed to ensure better statistics. Other fixed parameters
in our model include $R_{p}=R_{ap}=1.0$ (interaction radii) and $\eta=0.2$
(strength of noise).

Figure 1: (color online) The cartoon illustration serves as a visual depiction
of our model, distinguishing polar particles through single-headed blue arrow
and apolar particles through double-headed red arrow. The cyan-colored arrows
on each apolar particle depict their instantaneous velocity direction.
Subplots (a) and (b) convey insights into the symmetries and characteristics
of these two particle types within our system. In subplot (a), the arrow head
signifies the self-propulsion direction of a polar particle, and its
orientation is measured by the angle that the arrow head makes with the
positive $X$-direction. In contrast, an apolar particle exhibits head-tail
symmetry, facilitating its movement in both directions along its axis with
equal probability. The orientation angle of an apolar particle is defined by
the angle its instantaneous velocity direction forms with the positive
$X$-direction as illustrated in subplot (b). This representation also
highlights the intrinsic $\theta\to\theta+\pi$ symmetry exhibited by apolar
particles. Subplots (c)-(f) elaborate on the interactions between different
particle types in our system. Instances before and after interactions are
marked by a black arrow symbolizing the arrow of time. Polar particles engage
in ferromagnetic interactions with neighbors, seeking to align parallel to the
instantaneous velocity direction of their neighbors. Subplots (c) and (e)
depict the interaction of a single polar particle ($i_{p}$) with a polar
($j_{p}$) and an apolar ($j_{ap}$) neighbour, respectively. In both subplots,
the dashed single-headed grey arrow on the right side of the black arrow
represents the orientation of particle $i_{p}$ before interaction. Conversely,
apolar particles participate in nematic interactions with neighbors, allowing
alignment either parallel or antiparallel to the instantaneous velocity
direction of each neighbor with equal probability. Subplots (d) and (f)
illustrate the alignment mechanism of an apolar particle ($i_{ap}$) with a
polar particle ($j_{p}$) and an apolar particle ($j_{ap}$), respectively. In
both subplots, the dashed double-headed grey arrow on the right side of the
black arrow represents the orientation of particle $i_{ap}$ before
interaction. Figure 2: (color online) The figure illustrates the variation of
$S_{ap}(\mu)$ with $\mu$. This non-monotonic behavior is consistent across
different values of $\rho_{p}=0.02,0.03,0.04,0.05$. (Inset) Three distinct
states are labeled as I, II, and III. Vertical dashed lines indicate the
boundaries between the different states for $\rho_{p}=0.04$. The errorbars
show the uncertainty in the value of $S_{ap}$. System size, $L=100$;

(a)

(b)

(c)

Figure 3: (color online) The top panel displays snapshots of the system in
three different States: (a) State-I: $\mu=0.004$, (b) State-II: $\mu=0.20$,
and (c) State-III: $\mu=0.68$. The OPDF of apolar, $f_{ap}(\theta_{ap})$ vs.
$\theta_{ap}$, and polar, $f_{p}(\theta_{p})$ vs. $\theta_{p}$ (corresponding
to the snapshots in the top panel) is shown below the snapshot (middle panel-
apolar particles, $\&$ bottom panel-polar particles) for each State. In
subplot (a), the green dashed line in the plot $f_{ap}(\theta_{ap})$ vs.
$\theta_{ap}$ marks the position of the maxima, referred to as
$\theta_{ap,m}$. In the $f_{p}(\theta_{p})$ vs. $\theta_{p}$ plot, two dashed
green lines show that the two peaks are positioned approximately at
$\theta_{ap,m}$ and $\theta_{ap,m}+\pi$. In subplot (c), the inset shows the
overlapping peaks of in the OPDF of polar and apolar particles, indicating
that the enhanced ordering in State-III originates from the coherent motion of
the polar clusters; $\rho_{p}=0.04$ and $L=100$.  Figure 4: (color online) The
figure illustrates : the behavior of the two-point correlation functions for
(a) Apolar Particles ($C_{ap}(r)$ versus $r$) and (b) Polar Particles
($C_{p}(r)$ versus $r$) in different States: State-I ($\mu=$ $0.004$ $\&$
$0.02$), State-II ($\mu=$ $0.20$ $\&$ $0.28$), and State-III ($\mu=$ $0.52$
$\&$ $0.68$); (c) System size dependence of Nematic Order Parameter, $S_{ap}$,
plotted in log-log scale. The data shown for different values of $\mu$ in
State-I ($\mu=$ $0.004$ $\&$ $0.02$), State-II ($\mu=$ $0.20$ $\&$ $0.28$),
and State-III ($\mu=$ $0.52$ $\&$ $0.68$) shows power-law variation. The
dotted lines show power-law fit: $S_{ap}\approx N_{ap}^{\chi}$. Rest of the
parameters are same as FIG.3.

##  3 Results

###  3.1 Nematic Order Parameter ($S_{ap}$)

To quantify the degree of order in the apolar particles, we use the nematic
scalar order parameter defined as:

$S_{ap}=\Bigg{<}2\sqrt{\bigg{(}\overline{\cos^{2}\theta}-\frac{1}{2}\bigg{)}^{2}+\bigg{(}\overline{\sin\theta\cos\theta}\bigg{)}^{2}}\Bigg{>}$

where the ‘bar’ denotes an average over all the particles, and $<...>$ denotes
an average over time and 15 different realizations. The uncertainty in the
order parameter value is represented as the errorbar which is estimated by the
standard deviation of $S_{ap}$ over different realizations. A value of
$S_{ap}\approx 1$ indicates the presence of global orientational order among
the apolar particles, while $S_{ap}\approx 0$ corresponds to a state with no
global orientational order. FIG.2 presents a plot of $S_{ap}$ as a function of
$\mu$ for four different densities of polar particles
($\rho_{p}=0.02,0.03,0.04$, and $0.05$). In FIG.2, three distinct regimes can
be easily identified based on the values of $S_{ap}$, referred to as different
States. We have labeled these three states in the inset of FIG.2.
First, we analyze the static properties of the three states using the
following observables:
1\. The probability density function of orientation (OPDF) of apolar
($f_{ap}(\theta_{ap})$ vs. $\theta_{ap}$) and polar particles
($f_{p}(\theta_{p})$ vs. $\theta_{p}$). OPDF $f(\theta)\Delta\theta$
represents the probability that a randomly selected particle will have
orientation in the range
$\bigg{[}\theta-\frac{\Delta\theta}{2},\theta+\frac{\Delta\theta}{2}\bigg{)}$.
The details of the calculation of OPDF of polar and apolar particles are
provided in B
2\. Spatial two-point correlation functions of the orientation of polar
($C_{p}(r)$) and apolar ($C_{ap}(r)$) particles. These correlations are
defined as:

$C_{p}(r)=\bigg{<}\cos(\theta_{p}(r_{0}+r)-\theta_{p}(r_{0}))\bigg{>}$ (1)

$C_{ap}(r)=\bigg{<}\cos(2\theta_{ap}(r_{0}+r)-2\theta_{ap}(r_{0}))\bigg{>}$
(2)

where $<....>$ implies an average over all the particles as well as time in
steady state and 15 independent realizations. We extract the correlation
lengths, $L_{p}$ and $L_{ap}$ for polar and apolar particles, respectively, as
the points (value of r) where the correlation function $C_{p}(r)$ and
$C_{ap}(r)$ crosses the $0.5$ value. These correlation lengths provide a
measure of the characteristic length scales over which the orientation of the
polar and apolar particles in the mixture are correlated.
The snapshots of the system configuration in each state and the corresponding
OPDF of apolar and polar particles are shown FIG.3(a-c). The plots of the
correlation function $C_{ap}(r)$ and $C_{p}(r)$, for apolar and polar
particles, respectively, in different states are depicted in FIG.4(a-b). In
State-I, for low $\mu$ values, the system demonstrates a strong nematic
ordering, characterized by a high $S_{ap}$ value and a single sharp peak in
the OPDF of apolar particles depicted in FIG.3(a). The persistent alignment of
apolar particles is further evident from the slow decay of $C_{ap}(r)$,
displaying a power-law behavior with a very small exponent for small $r$ and
saturating at large $r$ as can be seen in FIG.4(a). However, in the case of
polar particles, the effect of activity is overshadowed by the diffusion term,
causing random motion of isolated polar particles, resulting in the presence
of multiple small distinct peaks in OPDF of polar particles, as shown in
FIG.3(a). The swift decline of the correlation function, $C_{p}(r)$, further
underlines the minimal interaction between polar particles. However, the OPDF
of polar particles displays two relatively stronger peaks at $\theta_{ap,m}$
and $(\theta_{ap,m}+\pi)$, suggesting that due to alignment interactions,
individual polar particles orient themselves parallel to the local nematic
orientation direction. Hence, at lower values of $\mu$, the orientation of
polar particles is slaved by the global nematic orientation. Due to small
$\mu$, polar particles are almost immobile and act like obstacles for apolar
particles. Hence the dynamic nature of polar particles is suppressed for very
small $\mu$. This leads to the formation of stripe structure in apolar
particles, as shown in FIG.3(a), instead of macroscopic bands observed in
studies of pure active nematics [27, 57, 60, 61]. Similar stripe patterns are
found previously for the collection of polar self-propelled particles at high
velocity [59].
As $\mu$ increases, the influence of polar particle activity becomes more
pronounced, causing interactions and hence alignment among the polar
particles. This leads to the emergence of small clusters comprising only a few
polar particles. This developing spatial correlation among the polar particles
is evident from $C_{p}(r)$, while the multiple randomly positioned peaks in
the OPDF of polar particles indicate the random movement of these clusters, as
shown in FIG.3(b). In State-II, the random motion of these polar particle
clusters significantly impacts the nematic ordering of apolar particles.
Consequently, there is a considerable decrease in $S_{ap}$, highlighting the
breakdown of orientational order among the apolar particles, resulting in a
broadened OPDF of apolar particles as shown in FIG.3(b). The disordered nature
of the apolar particles is further reflected in the sharp decay of
$C_{ap}(r)$, illustrating the lack of spatial correlations among them. The
system snapshot in FIG.3(b) visually represents this disordered nature of
apolar particles. In this State, the system configuration, as shown in
FIG.3(b), underscores the dominant influence of polar-apolar interactions over
apolar-apolar interactions, significantly impacting the overall behavior of
the system.
Continuing to increase $\mu$, the polar particle clusters exhibit growth in
size and the emergence of coherent motion, evidenced by the sharp peak in the
OPDF of polar particles, (FIG.3(c)) and the slow decay of the two-point
correlation function, $C_{p}(r)$ (FIG.4(a)), eventually terminating at a non-
zero value. In this State, most of the polar particles are contained in one or
two coherently moving clusters, while a small fraction of polar particles form
tiny clusters moving randomly. This steady motion of polar clusters
contributes to increased nematic ordering in State-III, as evident from the
rise in $S_{ap}$ and the reappearing peak in OPDF of apolar particles shown in
FIG.3(c)). This increase in ordering ($S_{ap}$) is solely because of the
interaction of apolar particles with the coherently moving polar clusters. The
coinciding peaks in the OPDF of apolar and polar particles (inset of FIG.
3(c)) further validate this observation.
Further, to unravel the internal structure of the nematic ordering in
different states, we conducted a comprehensive study of its system size
dependence. Interestingly, our investigation revealed a power-law variation of
$S_{ap}$ with $N_{ap}$, expressed as $S_{ap}\approx N_{ap}^{-\chi}$ [60],
depicted in FIG.4(c). In State-I, the system exhibited a Quasi Long Range
Order, evident from $\chi<\dfrac{1}{16}$, which align perfectly the power-law
decay observed in $C_{ap}(r)$. Upon entering State-II, $\chi$ increases to
approximately $0.50$. This pronounced change in the value of $\chi$ signifies
a transition from an ordered to a disordered state for the orientation of the
apolar particles. In State-III, we found that $\chi$ approached $0.25$,
indicating a slightly enhanced ordering compared to State-II. Despite this,
the arrangement of apolar particles bears a closer resemblance to State-II
characteristics than to State-I.
Overall, the analysis in this section provides insight into the stationary
properties of the three states based on the behavior of the background
nematic, shedding light on the non-monotonic behavior of nematic ordering as
the system crosses from State-I to State-II and then to State-III. Animations
of the system in three different states are provided in supplementary movies
SM1-SM3 (F.1).
A comprehensive exploration of the dynamics of polar particles in different
states is presented in C.

###  3.2 Dynamic Swirl State

(a)

(b)

Figure 5: (color online) Visuals of Circular Defects in State-II and State-III
of the Mixture. Snapshots (a) and (b) showcases the system configuration for
$\mu=$ $0.20$ (State-II) $\&$ $0.68$(State-III) respectively. The inset
provides a zoomed view of the region inside the rectangular box. All other
parameters are same as in FIG.3.

As previously discussed, the increase in $\mu$ disrupts the strong nematic
ordering among apolar particles observed in State-I due to the influence of
polar particles. Consequently, both State-II and State-III, exhibit system
configurations resembling a disordered state concerning the orientation of
apolar particles. However, it is essential to note that these seemingly
disordered States, State-II and State-III, are significantly different from
the typical disorder states at large noise strengths characterised by the
random orientation of particles. These states are highly dynamic, exhibiting
robustness and complexity that are not commonly found in disordered states by
high noise levels.
In State-II and State-III, we observe the formation of $\pm\dfrac{1}{2}$
topological defects corresponding to apolar particles, a characteristic
feature of nematically interacting systems. The characteristics of these
defects are very different in active nematics in contrast to the corresponding
equilibrium system [62, 63, 33]. Further details on the properties of these
defects in active nematics are provided in F.2. Additionally, intriguing
higher-order swirl-like structures emerge within the nematic system. These
swirl structures consist of a high concentration of apolar particles along the
perimeter or edge, with apolar particle expulsion occurring at the core, as
depicted in FIG.5. The swirls undergo periodic cycles of formation,
dissociation, and reformation, contributing to the highly dynamic and complex
nature of the apolar system in State-II and State-III. Similar structures were
reported in [57], where the authors studied a system by varying the density of
polar particles at fixed activity in a setup similar to ours. Further, the
configuration of the apolar particles in the State-II, shows that the system
consists of regions of strong order separated by regions of strong bend
distortions and swirls. The system confiuration is similar to the ‘Mixed
Phase’ observed in ref. [64], which examines the behavior of the polar flock
subjected to a time delay in the particle-particle interaction.
We refer to the state exhibiting the ongoing periodic formation and
dissociation of swirl structures as the “Dynamic Swirl State”. It is similar
to the active turbulence phenomena observed in active nematics. In active
nematics, the active turbulence phase emerges as a function of the activity of
nematic particles, characterized by the spontaneous formation and annihilation
of $\pm\dfrac{1}{2}$ defects [65, 66, 67, 68]. Similarly, by increasing the
activity of polar particles in the active nematics system, we observe the
continuous formation and dissociation of swirl-like structures. In this
section, we focus on examining the stability of these swirl structures,
intending to gain a deeper understanding of their behavior, dynamics, and
their overall impact on the system.
Swirl structures are regions exhibiting strong bend distortions within the
active nematics system. The presence of these structures signifies the local
rotational tendencies in the orientation field of apolar particles, resulting
in a non-zero curl of the local nematic order parameter of apolar particles
denoted by the vector $\bm{J}=(J_{x},J_{y})$, where, $J_{x}$ and $J_{y}$ are
the two components of the local order parameter. We define an observable
termed Swirling Parameter ($SP$), which is calculated as the width of the PDF
of $\bm{\nabla}\times\bm{J}$. The $SP$ provides a quantitative measure of the
strength of these swirl structures. The details on the calculation of the $SP$
is provided in the D.
When plotted against $\mu$, $SP$ shows a non-monotonic behavior exhibiting a
peak in State-II, as shown in FIG.6. The low value of $SP$ in State-I is due
to the presence of strong nematic ordering and the minimal rotational
tendencies of the apolar particles only at the strip edges. In contrast, in
State-II and State-III, the presence of swirl structures results in a much
larger value of $SP$. Further, the interaction between the swirl structures
and the polar particle clusters plays a crucial role in determining the fate
of the system in State-II and III. In State-II, the polar clusters consist of
only a few particles, making them highly susceptible to changes on interacting
with the swirled structures. FIG.S2 in F.3 provides a visual description of
this interaction: upon interaction with a swirl, a tiny polar cluster
dissociates, while a slightly larger cluster merely undergoes a change in its
orientation. This influence of the swirl structures on the dynamics of the
polar clusters indirectly sets a positive feedback loop that enhances the
stability of the swirl structures in this state. However, in State-III, the
polar clusters grow much larger in size, bestowing them with the capability to
distort and ultimately disrupt these swirl structures. Consequently, the
stability of the swirl structures in State-III diminishes. One such
interaction between a large polar cluster and a swirl structure in State-III
is displayed through a series of snapshots in FIG.S3 in F.4. A detailed
discussion on the stability of the swirl structures in different states is
provided in F.5.

Figure 6: (color online) The diagram depicts the Swirling Parameter ($SP$)
variations across different states for various $\mu$ values, alongside the
behavior of $S_{ap}$ for the original model in (a), and the model with colored
noise in (b). In panel (b), the filled symbol (with solid line) and empty
symbols (with dashed line) show the results for the original model and active
nematics with colored noise, respectively. The vertical black dashed lines
denote the boundary between different states, and the vertical green dashed
line highlights the correspondence between the peak of $SP$ and the minima of
$S_{ap}$, indicating the increased strength of swirled structures in State-II.
The rest of the parameters are the same as in FIG.3.

Furthermore, we observe the entrapping of isolated polar particles within void
regions at the core of the swirl structures, visually represented through a
series of snapshots in FIG.S2 in F.3 and FIG.S3 in F.4. These particles seem
to lack the necessary drive to escape and continue to undergo frequent
reorientations. This behavior highlights the profound impact of the swirl
structures on the individual particles.

###  3.3 Mechanism of the emergence of the Swirl structures

In this section, our primary focus is to discern the underlying mechanisms
leading to the formation of swirls in active nematics under the influence of
polar microswimmers. Our investigation employs a two-step approach. Firstly,
we aim to unveil how the presence of polar particles stimulates the apolar
system to exhibit intriguing behaviors across different States. Subsequently,
we delve into understanding how this leads to the observed formation of
swirled structures.

####  3.3.1 Polar particles act like spatially-temporally correlated colored
noise for apolar system

Our observations thus far emphasize the crucial role played by the
interactions between polar and apolar particles in determining the system’s
dynamics across different states. Here, we delve deeper into this interaction
by proposing that polar particles serve as a form of spatiotemporally
correlated colored noise for the apolar particles. The rationale supporting
this interpretation is detailed in the E.1. Before proceeding with our
discussion to evaluate the validity of this interpretation, it is essential to
explore the concept of colored noise within the context of the different
states.
In State-I, where polar particles exhibit limited motion and primarily
reorient within specific regions of space, their influence can be analogized
to a quenched random field distributed throughout the space i.e. a colored
noise with minimal spatiotemporal correlation. In State-II, polar particles
form multiple small clusters moving randomly, hence showing persistent motion
over a larger length and time scale compared to State-I. Thus, in this State,
the correlation length and time of the colored noise are significantly larger.
In State-III, most of the polar particles are contained inside one or two
large polar clusters moving coherently through the system, and actively
contribute to enhancing the nematic ordering of apolar particles through
alignment interactions. Hence, in this State, the effect of polar particles
can be considered as a moving field with spatial and temporal correlation.
However, this proves to be tough, if not unattainable, to accurately simulate
with colored noise. Consequently, our comparative analysis between the two
setups remains constrained to State-I and State-II exclusively.
In our model, since we are trying to mimic the effect of polar particles with
colored noise, we feed the data of correlation length ($L_{p}$) and
correlation time ($\tau_{p}$) of polar particles for different $\mu$ into the
colored noise (correlation length $L_{c}=L_{p}$, and correlation time
$\tau_{c}=\tau_{p}$ as shown in Fig.C2). As discussed in the E.1, the presence
of polar particles in the system influences the interaction among the apolar
particles in the system. Consequently, the colored noise in the active
nematics is introduced to perturb the alignment of apolar particles with their
neighbors within the interaction radius based on their distance. The details
of the equation of motion of apolar particles with colored noise are provided
in E.2.
Our observations with this setup are as follows: For $L_{c}$ and $\tau_{c}$
values corresponding to State-I, we observe strong nematic ordering of the
apolar particles, and we observe the formation of stripes like structures of
the apolar particles similar to those observed in the State-I in our original
model. As $L_{c}$ and $\tau_{c}$ increase, the nematic ordering of the system
diminishes, as shown in FIG.6(b). Finally, the system transitions to a
disordered state for $L_{c}$ and $\tau_{c}$ values corresponding to State-II.
In this case, we observe the formation of swirled structures akin to State-II
in our original setup. However, these structures are smaller in size and
transient in nature i.e. they persist in the system for very short period of
time. The snapshots of the apolar particle system with colored noise are shown
in E.3
Further, to characterize these structures, we calculate the $SP$ (for this
spatiotemporal colored noise model), which exhibits a peak at values
corresponding to State-II, as shown in FIG.6(b). However, the magnitude of
$SP$ in this model is small, indicating the weak nature of these structures.
This suggests that, although using a colored noise with fixed $L_{c}$ and
$\tau_{c}$ is appropriate to reproduce some aspects of the system’s behavioral
transition from State-I to State-II with increasing $\mu$, it is not entirely
accurate. The motion of polar particles (and clusters) depends significantly
on the status of nematic particles in their neighborhood. Hence, rather than
adopting a mean value of $L_{c}$ and $\tau_{c}$ throughout the system
depending on $\mu$, $L_{c}$ and $\tau_{c}$ should be functions of the local
nematic ordering i.e. $L_{c}=L_{c}(\mu,Q)$ and $\tau_{c}=\tau_{c}(\mu,Q)$,
where, $Q$ is the nematic tensor order parameter defined in the neighborhood
of each particle. However, determining the functional dependence of $L_{c}$
and $\tau_{c}$ on the local nematic ordering, $Q$, remains an intricate task,
adding a layer of complexity to our understanding.
Next, we explain how the polar particles change the mechanics of well formed
nematics as we increase their velocity.

Figure 7: (color online) The figure depicts the configuration of the system
during State-II. Green dashed arcs delineate regions with pronounced bend
distortions, while a green dashed circle indicates the presence of a swirl
structure within the apolar particles. Black dashed squares emphasize domains
characterized by robust nematic ordering. These domains are entirely isolated,
and the ordering direction within each domain is completely independent. The
figure highlights the significant correlation between the positions and
orientations of the domains and the locations of swirls and pronounced bend
structures. The parameters are the same as in FIG.3.

####  3.3.2 Polar Particles promote collective bending of apolar particles

As previously discussed, the influence of polar particles on apolar particles
disturbs the alignment between the apolar particles and can be mimicked by
introducing spatiotemporally correlated colored noise in the apolar-apolar
interaction. Experimental findings [65] on the system comprising microtubules
and kinesin molecular motors, exhibiting behaviors akin to active nematics,
reported the facilitation of collective in-plane bending of cells by weakening
cell-cell interactions. In a recent study [33], it is reported that the
elasticity of a liquid crystal system consisting of a dense suspension of
semiflexible biopolymers can be manipulated over a relatively wide range of
elastic moduli.
Corroborating with these observations, we propose an explanation for the
mechanism underlying the formation of swirled structures in our system: as
$\mu$ increases, polar particles starts to form clusters. As these clusters
traverse the ordered nematic, they induce bend-type distortions in the nematic
ordering. Simultaneously, inherent $\pm\dfrac{1}{2}$ defects in active
nematics consist of bend-type distortions. When another cluster of polar
particles moves through regions with bend distortion or near a defect
structure, the distortion is amplified, squeezing apolar particles together
and creating void spaces in the neighborhood of the bend. Ultimately, the
collective action of multiple polar clusters leads to the formation of strong
bend distortions which look like macroscopic swirl like structures in the
system.
To substantiate this argument, we conducted simulations starting with strong
nematically ordered initial condition, identical to State-I, and initial
condition of polar identical to State-II and III. The observations show that
the polar clusters drive the formation of swirls consisting of strong bend
distortions in the nematic, aligning with our argument. A zoomed in animation
of the system, showcasing the formation of a swirl induced by the polar
clusters, is provided in the supplementary movie SM4 (F.1 ).
Furthermore, we observed a distinct pattern among the apolar particles: the
presence of strongly ordered domains. These domains are entirely disconnected
from one another, with each domain’s orientation being independent of the
others, resulting in a lack of global ordering in the system. In FIG.7, we
have highlighted some of these domains, demonstrating their correlation with
regions exhibiting strong bend distortions and swirl structures.
To characterize these domains, we calculate the following interaction
energies:

$E_{ap,ap}=-{{\sum}}_{i,ap}{{\sum}}_{|\bm{r}_{i,ap}-\bm{r}_{j,ap}|<R_{0,ap}}cos(2\theta_{i,ap}-2\theta_{j,ap})$

and

$E_{ap,p}=-{{\sum}}_{i,ap}{{\sum}}_{|\bm{r}_{i,ap}-\bm{r}_{j,p}|<R_{0,ap}}cos(2\theta_{i,ap}-2\theta_{j,p})$

where, $E_{ap,ap}$ and $E_{ap,p}$ accounts for the apolar-apolar and apolar-
polar interaction, respectively. FIG.8(a) shows the variation of $E_{ap,ap}$
vs. $\mu$ in State-II and State-III, exhibiting a minimum in deep State-II,
signifying the presence of strongly ordered domains of apolar particles in the
system. The variation of $E_{ap,p}$ vs. $\mu$ is illustrated in FIG.8(a). It
exhibits a maximum in State-II (approximately 0), indicating a misalignment of
apolar particles with polar particles in their vicinity. Additionally, the
Probability Density Function (PDF) of $\phi=(\theta_{i,ap}-\theta_{j,p})$,
$f(\phi)$, displays a higher order peaks at a finite $\phi=\pm\pi/4$ and
$\pm\pi/2$, depicted in FIG.8(b). This accounts for the observed peak in the
plot of $f(\phi)$ $vs.$ $\phi$ and explains the origin of bend distortions
induced in the well-ordered nematic by polar clusters. The polar particle
clusters in State-II approach the strongly ordered nematic at an angle to the
ordering direction. Consequently, a passing polar cluster induces distortion
in the strongly ordered nematic, resulting in bend distortion.

Figure 8: (color online) The figure depicts (a) the variation of $E_{ap,ap}$
and $E_{ap,p}$ vs. $\mu$ across different States; (b) the PDF of
$\phi=(\theta_{i,ap}-\theta_{j,p})$, $f(\phi)$, in State-II, giving a
quantitative measure of relative orientation of the apolar particles with the
polar particles inside the interaction radius. The parameters are the same as
in FIG.3.

###  3.4 Number Fluctuation

From an observational point of view, strong Number Fluctuation is one of the
novel features distinguishing active systems from their equilibrium
counterparts. For 2D active nematics, it is predicted numerically to vary as:
$\Delta N\sim N^{\alpha}$ with $\alpha\sim 1$, both theoretically [60, 69] as
well as experimentally[70] compared to $\Delta N\sim\sqrt{N}$ in equilibrium
systems, hence called Giant Number Fluctuation (GNF). Later, the study in
ref.[71] showed that the presence of an external field makes the orientational
fluctuations in active nematics finite resulting in the suppression number
fluctuations in active nematics. In our study, we calculated the number
fluctuation of apolar particles by measuring the number of particles in a
subsystem of size $l\times l$ such that $l<\frac{L}{2}$ to take care of PBC,
and then $\Delta N$ is calculated as Root Mean Square (RMS) fluctuation of N.
We observed a variation of $\Delta N$ $vs.$ $N$ as : $\Delta N\sim N^{\alpha}$
as shown in FIG.9.
In State-I of our mixture, $\alpha$ takes value close to $\approx 0.50$,
suggesting a uniform arrangement of apolar particles throughout the system.
This suppression of GNF becomes apparent from the snapshots of the system
shown in FIG.3(a), which shows strip like high density structures in apolar
system in contradiction to well known phenomena of band formation in active
nematic. This phenomenon can be attributed to apolar-polar interactions.
Although on a global scale the polar particles are disordered, the OPDF of
polar and apolar particles shows that there is a strong correspondence between
them. On a micro scale, the apolar-polar interaction suggests that the
presence of polar particles acts like a local orienting field to the apolar
particles and this field changes direction randomly because of frequent
reorientation of polar particles. This leads to suppression of GNF.
However, as we go from State-I to State-II, the formation of randomly moving
polar clusters and the emerging circular vortices in apolar particles cause
enhancement of number fluctuation in the system compared to State-I and the
value of $\alpha$ is close to $\approx 0.75$. Upon entering State-III, the
coherent motion of the polar clusters leads to suppression the number
fluctuation in this state compared to State-II and $\alpha$ takes the value
$\approx 0.70$.
The Dynamic Swirl State we observe here is not only a dynamic state in nematic
ordering, but also has large density fluctuation, which is very non-trivial as
in previous studies GNF is observed in ordered nematic.

Figure 9: (color online) The figure depicts the behavior of number
fluctuation, $\Delta N$ in different States: State-I ($\mu=$ $0.004$ $\&$
$0.02$), State-I ($\mu=$ $0.20$ $\&$ $0.28$) and State-I ($\mu=$ $0.52$ $\&$
$0.68$). The plot is shown on a log-log scale. The bottom and top dashed lines
in ‘cyan’ color indicate lines of slope $\frac{1}{2}$ and $1$, which
correspond to the equilibrium system and the pure active nematics system,
respectively. System size, $L=130$. The rest of the parameters are the same as
in FIG.3.

##  4 Discussion

To summarize, in this study, we explore the characteristics of a mixture of
apolar rods and polar swimmers on a two dimensional substrate by keeping the
density of both species fixed ($\rho_{ap}$ $\&$ $\rho_{p}$) and varying the
activity ratio, $\mu$, within the low noise regime. On varying $\mu$, the
system is found to exist in three distinct states based on the strength of the
nematic ordering of apolar particles. The dynamics of the polar particles are
influenced by two competing factors: self-propulsion and diffusion. At low
value of activity ratio $\mu$, in State-I, diffusion dominates, causing
diffusive motion of isolated polar particles, whereas the apolar particles
display strong nematic ordering. As we increase $\mu$, the activity of the
polar particles starts playing a dominant role, causing alignment interaction
between the particles, leading to the formation of small clusters. These
clusters undergo random motion, disrupting the nematic ordering of apolar
particles and causing the polar particles’ mean-squared displacement (MSD) to
become super-diffusive in State-II. As $\mu$ further increases, these clusters
grow larger in State-III. In this state, we observe the formation of one or
two large polar clusters, which move coherently through the system enhancing
the nematic ordering of apolar particles. However, there are a few small polar
clusters that move randomly and prevent complete alignment.
In State-II and III, which are disordered from the perspective of global
nematic ordering of apolar particles, we observe the formation of swirled
structures, accompanied by half-integer defects. The fate of polar particles
in this state is strongly influenced by these swirl structures and vice versa.
To quantify the swirled structures, we introduce a parameter which we call the
Swirling Parameter ($SP$). The $SP$ proves valuable in characterizing the
swirling pattern of apolar particles and provides insights into the strength
and stability of the swirls in different parameter regimes of the system. The
non-monotonic behavior of $SP$ and the observed peak in State-II highlight the
intricate interplay between the polar clusters and the swirled structures,
offering valuable insights into the underlying mechanisms driving the system
dynamics. The dynamic swirl state appears when polar particles assemble in
many small clusters. The structures become weak when individual polar
particles are moving randomly, as well as when they form macroscopic clusters.
Additionally, we observe the trapping of isolated polar particles at the core
of swirl structures.
These swirled structures undergo a continuous cycle of formation,
dissociation, and subsequent reformation over time, creating a dynamical
state, which we term the “Dynamic Swirl state”. We observe that the “Dynamic
Swirl state” bears a resemblance to the active turbulence observed in active
nematics [65, 66, 67, 68]. Active turbulence, marked by the spontaneous
creation and annihilation of half-integer defects, emerges as a function of
the activity of apolar particles in pure active nematics systems. In our
study, the transition to the Dynamic Swirl state arises by increasing the
activity of polar particles, leading to the persistent formation and
dissolution of swirl structures in apolar particles. The presence of polar
particles breaks the nematic symmetry of the system and leads to the dynamic
state with continuous formation and dissociation of swirl structures. However,
it is essential to emphasize that these two phenomena originate from entirely
different underlying mechanisms, and the comparison drawn here serves as a
simplified analogy rather than an in-depth equivalence. Further, active
turbulence is observed in dense active nematic, whereas we observe these
dynamic swirl structures in low density active nematics.
Furthermore, our attempt to unravel the underlying mechanism of the formation
of the swirled structures uncovers a subtle intricacy: the influence of polar
particles on apolar particles can be mimicked through colored noise, wherein
the correlation length and time dynamically depend on the local nematic
ordering and activity of the apolar particles. This opens up a new way of
understanding the complex pattern in the mixture. Such colored noise
highlights the subtleties inherent in capturing the dynamic interplay between
apolar and the polar particles.
A distinctive feature of our model is the nonreciprocal interaction between
the polar and apolar particles in the system: polar particles align
ferromagnetically with apolar particles, while apolar particles align
nematically with polar particles. In a recent study [72], it is reported that
non-reciprocity in the interaction between the particles can have a
substantial effect on the dynamics of the interacting many body systems.
Therefore, it would be interesting to investigate how non-reciprocity impacts
the behavior of the two types of particles in our model and how this effect
varies with the activity levels of polar and apolar particles.
Following the results of this study, several other questions remain open. An
important characteristics of liquid crystal systems is the presence of
topological defects of charge $\pm\frac{1}{2}$. However, our microscopic
approach in this study does not allow us to explore the effect of the polar
particles on the dynamics of the defects (see F.2 ). It requires an in depth
analysis of the kinetic of the system by constructing a coarse grained version
of the model which is a work in progress. Further, our model can also be
extended to include chirality in polar particles, as most of the microswimmers
in nature are chiral. Our preliminary observations (see F.6), reveals a trade-
off between the chirality ($\omega$) and activity of polar particles
($v_{p}$), and within a specific range of parameter values this stabilizes the
swirl structures. However, a detailed exploration of the parameter space
($v_{p},\omega$) is necessary to develop an understanding of the effect of
chirality which is beyond the scope of this study.
At the experimental level, our study assumes significant importance, as it
opens up an unexplored direction with promising possibilities. Through
simulations, we introduce an active background nematic, a departure from the
traditional LLC system. Yet, the essence of LLC remains evident as the
activity of polar particles gives rise to novel orientational patterns, which
would otherwise elude observation. The dependence of stability and scale of
these intricate patterns on the magnitude of polar particle activity provides
us with a powerful tool for detecting bacteria with precision. Additionally,
we unveil the emergence of dynamic swirl structures, offering a potential
means to trap bacteria at their core, a practical application worth exploring,
and considerable potential for bacterial manipulation.

##  5 Acknowledgement

The authors thank Venceslas Ngounou for useful discussions. P.S.M, P.K.M and
S.M. thank Jacques Prost for useful discussions and Sriram Ramaswamy for
critical reading of the manuscript and providing useful comments. S.M. thank
Pranay Bimal Sampat for the useful discussions at the start of the project.
P.S.M., P.K.M. and S.M., thanks PARAM Shivay for computational facility under
the National Supercomputing Mission, Government of India at the Indian
Institute of Technology, Varanasi and the computational facility at I.I.T.
(BHU) Varanasi. P.S.M. and P.K.M. thank UGC for research fellowship. S.M.
thanks DST, SERB (INDIA), Project No.: CRG/2021/006945, MTR/2021/000438 for
financial support. T.V. acknowledges funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 955576.

##  Appendix A Dynamical Equations of apolar and polar particles

Update rules for apolar particle:- Equations for position and orientation
update of apolar particles are given by,

$\bm{r}_{i,ap}(t+1)=\bm{r}_{i,ap}(t)+\bm{v}_{i,ap}(t)\Delta t,$ (3)
$\theta_{i,ap}(t+1)=\dfrac{1}{2}\bigg{<}\theta(t)\bigg{>}_{R_{0,ap}}+\zeta_{i,ap}(t)$
(4)

In Equation (3), $v_{i}(t)$ is the instantaneous velocity of the $i^{th}$
apolar particle at time $t$. In Equation (4), first term represents the
alignment term and second term is a delta correlated white noise with zero
mean and uniform distribution in the range
$\bigg{[}-\frac{\eta}{2},+\frac{\eta}{2}\bigg{]}$. The alignment term, which
is the mean orientation of particles inside the interaction radius of apolar
particle, is evaluated as

$\bigg{<}\theta(t)\bigg{>}_{R_{0,ap}}=\arg\bigg{(}{{\sum}}_{|\bm{r}_{i,ap}-\bm{r}_{j,ap}|<R_{0,ap}}e^{2i\theta_{j,ap}(t)}+{{\sum}}_{|\bm{r}_{i,ap}-\bm{r}_{j,p}|<R_{0,ap}}e^{2i\theta_{j,p}(t)}\bigg{)}$

In the above expression, first and second term represents the interaction of
$i^{th}$ apolar particle with apolar and polar neighbors respectively. The
instantaneous velocity vector of the $i^{th}$ apolar particle at time $(t+1)$
is given by,

$\bm{v}_{i,ap}(t+1)=R_{i,ap}(t+1).v_{ap}\bigg{(}\cos(\theta_{i,ap}(t+1),\sin(\theta_{i,ap}(t+1)\bigg{)}$

The term $R_{i,ap}$ is chosen randomly between $\pm 1$ with equal probability
to incorporate it’s bidirectional motion along the long axis of apolar
particles.

Update rules for polar particle:- The position and orientation update equation
of polar particles are given by,

$\bm{r}_{i,p}(t+1)=\bm{r}_{i,p}(t)+\bm{v}_{i,p}(t)\Delta t+\sqrt{2V_{R}\Delta
t}\bm{\xi_{i}}$ (5)
$\theta_{i,p}(t+1)=\bigg{<}\theta(t)\bigg{>}_{R_{0,p}}+\zeta_{i,p}(t)$ (6)

In equation (5) third term is an additional term added to standard self-
propulsion dynamics of polar particle’s position update. It accounts for
diffusion,
$\bm{\xi_{i}}\equiv\bigg{(}\cos(\phi_{i}(t)),\sin(\phi_{i}(t))\bigg{)}$ is a
random unit vector with $\phi_{i}(t)$ is randomly chosen from the range
$[0,2\pi]$ and $V_{R}$ is the diffusion constant.
In Equation (6), the second term is a delta-correlated white noise with zero
mean, chosen uniformly from the range
$\bigg{[}-\frac{\eta}{2},+\frac{\eta}{2}\bigg{]}$. The first term represents
alignment and captures the average of instantaneous motion direction of all
other particles within the interaction radius of the $i^{th}$ polar particle.
It is evaluated as,

$\bigg{<}\theta(t)\bigg{>}_{R_{0,p}}=\arg\bigg{(}{{\sum}}_{|\bm{r}_{i,p}-\bm{r}_{j,ap}|<R_{0,p}}\bm{v}_{j,ap}(t)+{{\sum}}_{|\bm{r}_{i,p}-\bm{r}_{j,p}|<R_{0,p}}\bm{v}_{j,p}(t)\bigg{)}$

where the first and second terms correspond to the interactions of the
$i^{th}$ polar particle with apolar and polar particles in its neighborhood,
respectively. The instantaneous velocity vector of the $i^{th}$ polar particle
at time $(t+1)$ is,

$\bm{v}_{i,p}(t+1)=v_{p}\bigg{(}\cos(\theta_{i,p}(t+1)),\sin(\theta_{i,p}(t+1))\bigg{)}$

##  Appendix B Probability Density Function of Orientation(OPDF) of apolar and
polar particles

Consider a system consisting of N particles and their orientation at a given
instant of time $t$ is denoted by $\\{\theta_{i},i=1,N\\}$ where
$-\theta_{1}\leq\theta_{i}\leq+\theta_{1}$. In our model, $(\theta_{1},N)$ =
$(\pi,N_{p})$ and $(\pi/2,N_{ap})$ for the polar and apolar particles,
respectively. The steps to calculate the probability density function of
orientation (OPDF) of the system of particles are described as follows:

  * (i)

Divide the total range of $\theta\in[-\theta_{1},+\theta_{1}]$ in $N_{bin}$
number of bins of width $\Delta\theta$ and calculate the number of particles
in each bin.

  * (ii)

Let, $n_{i}$ is the number of particles in the $i^{th}$ bin and $\theta^{i}$
is the center of the $i^{th}$ bin. The probability that a randomly chosen
particle will have an orientation in the range
$(\theta^{i}-\frac{\Delta\theta}{2},\theta^{i}+\frac{\Delta\theta}{2})$ is
$P(\theta^{i})=\frac{n_{i}}{N}$.

  * (iii)

The OPDF, $f(\theta)$ is defined as the probability per unit length of the
interval
$(\theta^{i}-\frac{\Delta\theta}{2},\theta^{i}+\frac{\Delta\theta}{2})$ i.e.
$f(\theta^{i})=\frac{P(\theta^{i})}{\Delta\theta}$.

  * (iv)

The normalization condition is given by,
${\int}_{-\theta_{1}}^{\theta_{1}}f(\theta)d\theta={\sum}_{i=1}^{N_{bin}}f(\theta^{i})\Delta\theta=\Delta\theta{\sum}_{i=1}^{N_{bin}}f(\theta^{i})=1$

In Fig.3 of the main text, presents the OPDF for apolar and polar particles,
depicted in the middle and bottom panels, respectively. For each case, the
OPDF is derived from a single time snapshot, shown in the top panel. The
justification for utilizing just one snapshot to calculate the OPDF is as
follows: In State-I, the polar particles reorient randomly, and in State-II,
small polar clusters move without a fixed direction. Consequently, over time,
while the location of peaks in the OPDF shifts, the general features remain
constant. Averaging over time would thus flatten the OPDF, resulting in loss
of detail and obscuring insights into the system’s behavior. Similarly, for
apolar particles, the dynamic nature of swirl structures, through their
formation, dissociation, and reformation, in State-II and III suggests that
time-averaging would erase these distinctive features.
Therefore, we present the OPDF of both polar and apolar particles at a single
time instant, corresponding to the snapshot displayed in the top panel.

##  Appendix C Characteristics of Polar Particles

In this section, we employ several tools to gain insights into the dynamics of
polar particles and its impact on the nematic in different states of the
mixture:

  1. 1.

The order parameter for Polar particles,

$<\phi_{p}(\mu)>=\bigg{<}\frac{1}{N_{p}v_{p}}\bigg{|}{\sum}_{i=1}^{N_{p}}\bm{v}_{i,p}(t)\bigg{|}\bigg{>}$
(7)

where $N_{p}$ is the total number of polar particles and $<...>$ implies
average over time in steady state as well as over independent realizations.
$<\phi_{p}(\mu)>$ quantifies the degree of ordering of polar particles within
the system. A value close to 1 indicates a highly ordered state of the polar
particles, while a value close to zero suggests a disordered spatial
arrangement of polar particles.

  2. 2.

The Mean Square Displacement (MSD) of polar particles, represented as

$\Delta_{p}(\mu,t)=\bigg{<}\dfrac{1}{N_{p}}{\sum}_{i=1}^{N_{p}}|\bm{r}_{i,p}(t_{0}+t)-\bm{r}_{i,p}(t_{0})|^{2}\bigg{>}$
(8)

where, $<....>$ carries the same meaning as in Eq.(9). MSD, displayed in
FIG.C1(a), provides insights into the nature of trajectories of polar
particles. It exhibits power law behavior as $\Delta_{p}(\mu,t)\sim t^{\beta}$
where the exponent $\beta$ is indicative of the type of motion exhibited by
the polar particles: $\beta=1$ and $\beta=2$ correspond to diffusive and
ballistic motion, respectively, while $1<\beta<2$ indicates super diffusion.

  3. 3.

Velocity Auto Correlation Function (VACF) of polar particles,

$\gamma_{p}(\mu,t)=\bigg{<}\frac{1}{v_{0}^{2}N_{p}}{\sum}_{i=1}^{N_{p}}\bm{v}_{i,p}(t_{0}+t)\cdot\bm{v}_{i,p}(t)\bigg{>}$
(9)

where $<....>$ implies average over reference times, $t_{0}$, in the steady
state as well as independent realizations. VACF, demonstrated in FIG.C1(b),
examines the correlation between the velocities of polar particles at
different time intervals during it’s motion and it reveals the timescale over
which a polar particle retains its orientation. We fit $\gamma_{p}(\mu,t)$
with exponential, $exp\bigg{[}{-\dfrac{t}{\tau_{p}(\mu)}}\bigg{]}$ and
calculated the correlation time $\tau_{p}(\mu)$.

Figure C1: (color online) The figure illustrates the behavior of the (a) Mean
Square Displacement (MSD), $\Delta_{p}(\mu,t)$, and (b) the Velocity Auto
Correlation Function (VACF), $\gamma_{p}(\mu,t)$, of polar particles in
different states of the system: State-I ($\mu=0.004$ $\&$ $0.02$), State-II
($\mu=0.20$ $\&$ $0.28$), and State-III ($\mu=0.52$ $\&$ $0.68$). The MSD plot
is shown on log-log scale and the VACF plot is shown on the log-y scale. In
sublot (a), the two dashed lines in cyan represent the lines of slope
$\beta=1$ and $2$ corresponding to diffusive and ballistic motion,
respectively. The rest of the parameters are the same as in FIG.3 Figure C2:
(color online) This figure illustrates behavior of various observables for
polar particles across different States. Two vertical dashed black lines serve
as a visual guide, indicating the boundaries between these distinct States. To
ensure clarity and meaningful comparison, different quantities, namely
$<\phi(\mu)>$, $f(\mu)$, $\tau_{p}(\mu)$, $L_{p}$, and $\beta$, are
appropriately scaled on the Y-axis, as indicated by the accompanying legends.
The parameters used in this plot are the same as in FIG.3.

  4. 4.

The fraction of the total number of polar particles, on average, that reside
within the interaction radius of each polar particle can be quantified using
the following expression:

$f(\mu)=\frac{<n>}{N_{p}},$ (10)

where $<n>$ denotes the average number of polar particles found within the
interaction radius of the each polar particle. This quantity provides a
measure of the average size of the polar particle clusters present in the
system.

We show the variation of $<\phi_{p}(\mu)>$, $f(\mu)$, $\tau_{p}(\mu)$,
$L_{p}(\mu)$ and $\beta$ with activity ratio, $\mu$ in FIG.C2.

In State-I, the small magnitude of $\mu$ results in the diffusion term
dominating over the activity term. As a consequence, the motion of polar
particles closely resembles that of random walkers in two dimensions. This
behavior is evident from the rapid decay of $\gamma_{p}(\mu,t)$, as shown in
FIG.C1(b), with an extremely small correlation time $\tau_{p}(\mu)$,
indicating the absence of temporal correlations in their motion. Furthermore,
the low value of $f(\mu)$ suggests that, on average, only few polar particles
resides within the interaction radius of another polar particle. The
combination of a small correlation length $L_{p}$, a short correlation time
$\tau_{p}(\mu)$ (FIG.C2), and the low value of $f(\mu)$ supports the notion
that polar particles do not form clusters but rather engage in random walk
independent of others within their species. The diffusive nature of this
behavior is further substantiated by the exponent $\beta\approx 1$ observed in
the Mean Square Displacement (MSD), $\Delta_{p}(\mu,t)$, as shown in
FIG.C1(a). Additionally, the presence of small random peaks in $P(\theta_{p})$
vs. $\theta_{p}$ and the visual snapshots of the system configuration, as
shown in Figure 3(a), reinforce the diffusive characteristics observed in this
state (see the supplementary movie SM1 in F.1).
In State-II, increase in the magnitude of $\mu$ leads to the emergence of
spatial (FIG.4(b)) and temporal (FIG.C1(b)) correlations in the motion of
polar particles. This is reflected in the growth of the correlation length
$L_{p}$ and correlation time $\tau_{p}$, indicating the formation of clusters
among the polar particles, as shown in FIG.C2. The fraction $f(\mu)$
increases, indicating a higher average number of polar particles within the
interaction radius of each polar particle (C2). Consequently, the order
parameter $<\phi_{p}(\mu)>$ increases, signifying a higher degree of ordering
among the polar particles. The animation of the system (supplementary movie
SM2 in F.1), in State-II clearly shows the dynamic nature of these clusters,
characterized by their formation and dissolution. When a particle is part of a
cluster, it exhibits persistent motion aligned with the overall direction of
the cluster. However, upon leaving the cluster, the particle undergoes
diffusion until it encounters and aligns with another cluster. As a result,
the overall motion of polar particles displays superdiffusive behavior in the
MSD, FIG.C1(a), with an exponent $\beta>1$, FIG.C2. Snapshots of the system,
as shown in FIG.3(b), configuration highlight the uncorrelated motion of the
clusters and the disruption of strong orientational order among the apolar
particles.
In State-III, the fraction $f(\mu)$ significantly increases, indicating the
formation of larger polar particle clusters, as shown in FIG.3(c). In this
State, we observe one or two large cluster of polar particles moving
coherently and few small clusters moving randomly (see supplementary movie SM3
in F.1). This is supported by the presence of a large correlation length
$L_{p}$, suggesting that the majority of particles are bound within these
clusters. The motion of polar particles in this state becomes persistent,
exhibiting a ballistic Mean Square Displacement (MSD) with an exponent
$\beta\approx 2$. Consequently, polar particles move consistently in the
direction of their head, influencing the apolar particles coming in their
path. Occasionaly a cluster of polar particles splits in two to three
clusters, when it encounters a dense nematically ordered region. In this
manner the dynamic nature of this state is maintained. The polar order
parameter $<\phi_{p}(\mu)>$ assumes a much higher value, approximately 0.50,
indicating the existence of correlations within the motion of the polar
clusters. In State-III, the dynamics of the system are predominantly governed
by these polar clusters, which in turn induce alignment among the apolar
particles in their wake, leading to enhanced ordering among the apolar
particles compared to State-II. However, the presence of small random-moving
clusters prevents the overall global alignment of the nematic.
Apolar particles disturb the polar ordering :- Further to investigate the
influence of apolar particles on the dynamics of polar particles, we conducted
simulations of a system of polar swimmers while keeping other parameters
constant. The observations revealed higher values of the order parameter
$<\phi_{p}(\mu)>$, with the difference increasing as $\mu$ increases. This
finding suggests that the presence of apolar particles in the system
suppresses the ordering among the polar particles in all three States,
consequently affecting the nematic ordering of the apolar particles.

##  Appendix D Calculation of Swirling Parameter (SP)

We first discretize the system into an underlying lattice. We consider a small
$b\times b$ box with the lattice point at its center, such that $b<<L$(system
size). For our calculation, we used $b=4$. Inside each box we calculate the
components of the local ordering due to apolar particles,
$\bm{J}=(J_{x},J_{y})$ as :

$J_{x}=\bigg{<}\dfrac{1}{n}{\sum}_{i,ap=1}^{n}\cos(2\theta_{i,ap})\bigg{>}$,
$J_{y}=\bigg{<}\dfrac{1}{n}{\sum}_{i,ap=1}^{n}\sin(2\theta_{i,ap})\bigg{>}$

where, $n$ is the number of apolar particles inside a box whose $x$ and $y$
coordinate lies in the range
$x\in\bigg{[}x_{c}-\dfrac{b}{2},x_{c}+\dfrac{b}{2}\bigg{]}$ and
$y\in\bigg{[}y_{c}-\dfrac{b}{2},y_{c}+\dfrac{b}{2}\bigg{]}$ with
$(x_{c},y_{c})$ being the center of the box. The swirling nature of the
orientational field of the apolar particles can be estimated by calculating
the curl of $\bm{J}$ as,
$\bm{R}=(\bm{\nabla}\times\bm{J})_{z}=\bigg{(}\dfrac{\partial J_{x}}{\partial
y}-\dfrac{\partial J_{y}}{\partial x}\bigg{)}$. The vector nature of $\bm{R}$
will give positive and negative values in two dimensions.
The partial derivatives $\dfrac{\partial J_{y}}{\partial x}$,$\dfrac{\partial
J_{x}}{\partial y}$ are calculated using central difference method as
described below,

$\dfrac{\partial J_{y}(i,j)}{\partial
x}=\dfrac{J_{y}(i+1,j)-J_{y}(i-1,j)}{2b}$

$\dfrac{\partial J_{x}(i,j)}{\partial
y}=\dfrac{J_{x}(i,j+1)-J_{x}(i,j-1)}{2b}$

Next, we calculate the probability density function (PDF) of $\bm{R}$ and
average it over 15 independent realizations. For all values of $\mu$, the PDF
is symmetric about $\bm{R}=0$. That represents the equal probability of
clockwise and anti-clockwise pattern of circular structure.
The PDF of $\bm{R}$, $f(\bm{R})$, in three different states is shown in
FIG.E1(a). The detailed steps of calculation of $f(\bm{R})$ is same as that of
OPDF provided B. It can be seen that the width of the PDF changes non-
monotonically as we go from State-I to State-II and then to State-III. The
width of the PDF gives information about the quantitative measure of the swirl
structures present in the system. To characterize it, first, we fit the PDF
with the Lorentzian distribution, as shown in FIG.E1(b) for one set of
parameters, given by

$F(x)=A\dfrac{\lambda}{(x-x_{0})^{2}+\lambda^{2}}$

where, $x_{0}$ is the median and $\lambda$ is the scale parameter. The scale
parameter $\lambda$ carries the information about the width of $f(\lambda)$
i.e. larger $\lambda$ means larger width of the $f(\lambda)$. The value of
$\lambda$ obtained from this fit is our order parameter, which we termed
Swirling Parameter $SP$.

Figure E1: (Color Online) This figure illustrates (a) The PDF of $R$, $f(R)$,
for different $\mu$ values in three different States: State-I ($\mu=$ $0.004$
$\&$ $0.02$), State-II ($\mu=$ $0.20$ $\&$ $0.28$), and State-III ($\mu=$
$0.52$ $\&$ $0.68$). $f(R)$ is averaged over 10 independent realizations; (b)
Fitting of PDF $f(R)$ vs. $R$ with Lorentzian distribution. Parameters are the
same as FIG.3 in the main text.

##  Appendix E Modelling apolar particles with spatiotemporally correlated
colored noise

###  E.1 Interpretation of the effect of polar particles on apolar particles
as colored noise

Here we delve into the interplay between polar and apolar particles within a
system, particularly emphasizing the influence of polar particles on the
apolar particles. Within our theoretical framework, each particle, based on
its interaction radius, engages with surrounding particles, with the
interaction’s nature being contingent upon the particle’s symmetry—polar or
apolar. This context raises a pertinent question: Can an apolar particle,
devoid of any polar particles within its interaction radius, still experience
the effects of polar particles? The answer is, yes. This is due to cascading
of information inherent to all interacting active matter systems. This
phenomenon in the context of our model happens in such a way that an apolar
particle can be affected by polar particles even in their absence within the
interaction radius. This is achieved through a domino effect, where
information is transferred from one particle to another via nearest-neighbor
interactions. Hence the alignment among apolar particles is invariably
disturbed by the presence of polar particles in the system. This comes in
addition to the noise already present in the system. This disturbance is not
static; it changes with the motion of polar particles clusters, leading to a
dynamic and correlated perturbation in both space and time.
There are several studies examining the effect of quenched obstacles
(disorder) on active matter systems [34, 35, 36, 37, 38, 39, 40, 41]. In the
quenched case, the obstacles are fixed in space and time. Therefore, the
spatiotemporal dependence of the effect of the obstacle on the particles is
due to the motion of the particles. In contrast, in our case, the obstacles
(polar particles) themselves are motile, introducing additional spatiotemporal
dependence. Thus, the effect of the obstacles (polar particles) on the apolar
particles strongly depends on how these polar particles or clusters move. This
intrinsic distinction sets our model apart from other active matter systems
with quenched obstacles.
Further, recent studies [41, 40] have revealed that disorder in active matter
systems can induce collective rotational motions. In our model, the
introduction of polar particles into the system leads to the emergence of
swirl-like patterns among the apolar particles, visible within a specific
ranges of polar particle activity. The apparent similarity of the these
observations prompts us to raise the question: Can the influence of polar
particles on apolar particles be effectively emulated by applying a
spatiotemporally correlated colored field to the latter? Given that in our
foundational model, the primary mode of interaction is alignment among
particles, addressing disorder as a field might prove to be a more apt
approach than considering physical disorder directly.

###  E.2 Equations of motion of active nematic with colored noise

Here, we propose a theoretical framework for studying a collection of apolar
particles subjected to spatiotemporally correlated colored noise. The
mathematical model governing the dynamics of these particles, focusing on
their position and orientation evolution, is as follows.
The posiotion update of the $i^{th}$ particle at subsequent time steps is
defined by:

$\bm{r}_{i}(t+\Delta t)=\bm{r}_{i}(t)+\bm{v}_{i}(t)\Delta t$ (11)

where, $\bm{v}_{i}(t)$ is the instantaneous velocity of the $i^{th}$ particle
at time $t$.
The orientation update mechanism for each particle involves alignment with
nearby apolar particles within the interaction radius, modulated by the
colored noise. The orintation update rule of each particle is the following:

$\theta_{i,ap}(t+1)=\dfrac{1}{2}\bigg{<}\theta(t)\bigg{>}_{R_{0}}+\zeta_{i,ap}(t)$
(12)

Where, the $1^{st}$ term on the right hand side captures the aforementioned
effects and the second term represents a delta correlated Gaussian white noise
with zero mean and uniformly distributed in the range
$[-\frac{\eta}{2},\frac{\eta}{2}]$. The $1^{st}$ term is evaluated as follows:

$\bigg{<}\theta_{i,ap}(t)\bigg{>}_{R_{0},ap}=\arg\bigg{(}{{\sum}}_{|\bm{r}_{i,ap}-\bm{r}_{j,ap}|<R_{0,ap}}e^{i[2\theta_{j,ap}(t)+\xi(|\bm{r}_{i,ap}-\bm{r}_{j,ap}|,t)]}\bigg{)}$

In the above expression $R_{0}$ is the interaction radius and $\xi(r,t)$
represents the colored noise, with its spatial and temporal correlation
described as:
$\bigg{<}\xi(r,t)\xi(r=0,t=0)\bigg{>}=Ae^{-\frac{r}{L_{c}}}e^{-\frac{t}{\tau_{c}}}$,
where $A$ is the strength of the noise and , $L_{c}$ and $\tau_{c}$ are
respectively the correlation length and correlation time of the noise,
respectively.

###  E.3 Comparision of the structure of different states in the original
model and active nematics with colored noise

Colored Noise Original Model

(a)

(b)

(c)

(d)

Figure E2: (color online) The figure depicts a visual comparison between the
structures observed in the State-I and State-II in both the original model and
the model of active nematics with a colored noise as discussed in the main
text. In the first row, snapshots of State-I in the two models reveal a
similar structure with the formation of stripe-like patterns in both the
models. The second row features snapshots of State-II, we observe similarities
such as the emergence of swirl and strong bend structures in the model with a
colored field, akin to the original model. Some of these structures are
highlighted in both the snapshots. Despite these shared features, distinctive
visual differences are apparent between the two models. These differences
offer valuable insights, as expounded upon in Sec.III(C) of the main text. The
remaining parameters remain consistent with those detailed in FIG.3 of the
main text.

##  Appendix F Supplementary Material

###  F.1 Description of the Supplementary Movies

The supplementary movies (SM1-SM3) provide a visual representation of the
dynamics of polar and apolar particles in different states, as discussed in
Sec.III(A). The movies are captured within the time steps $1.3\times 10^{5}$
to $1.5\times 10^{5}$. The specific details of each movie are as follows:

  * •

SM1: This movie shows the dynamics of the particles in the system in State-I
for the activity ratio, $\mu=0.004$. Rest of the parameters are same as in
FIG.3 in the main text. (SM1.mp4)
Link: https://youtu.be/5u_nVrtrET0

  * •

SM2: This movie illustrates the dynamics of the particles in State-II with the
activity ratio, $\mu=0.20$. Rest of the parameters are same as in FIG.3 in the
main text. (SM2.mp4)
Link: https://youtu.be/iyld_KGpv-Y

  * •

SM3: This movie depicts the dynamics of the particles in the system in State-
III with the activity ratio, $\mu=0.52$. Rest of the parameters are same as in
FIG.3 in the main text. (SM3.mp4)
Link: https://youtu.be/EsXY8qbFYaI

Additionally, the supplementary movie SM4 provides a visual description of the
formation of a swirl structure in active nematics through their interaction
with randomly moving polar clusters, as discussed in Sec.III(C2). The
simulation begins with the initial condition of the active nematics identical
to the steady state configuration in State-I (see FIG.3(a) in the main text),
and the initial condition of polar particles similar to the steady state
configuration in State-II (see FIG.3(b) in the main text), for a system size
of $L=100$. The animation is zoomed in to a small part of the whole system for
convenience. (SM4.mp4)
Link: https://youtu.be/HYGKscAr1Ag

###  F.2 Topological defects in active nematics

Topological defects with charges of $\pm\frac{1}{2}$ are distinguishing
features in nematic systems [73, 74, 75]. In equilibrium settings, such
defects diffuse through the system, eventually annihilating in pairs of
opposite charges as propelling the system towards an equilibrium state [76,
77, 78, 79]. The emergence of these defect pairs is associated with an energy
cost, preventing their spontaneous formation in equilibrium systems.
Therefore, as the system evolves, the quantity of defects decreases over time.
The time scale of achieving a defect-free equilibrium state extends with the
size of the system, leading to what is known as Quasi Long Range ordering.
However in active nematic systems, owing to the local input of energy,
spontaneous proliferation of defects are possible and the $\pm\frac{1}{2}$
defects persist even in the steady state. The asymmetrical shape, coupled with
activity-induced stress, results in $+\frac{1}{2}$ defects exhibiting self-
propelled particle-like motion within active nematic systems[63].
In our study, we identify the presence of $\pm\frac{1}{2}$ defects in the
steady state during State-II and State-III. These defects are visually
identifiable, as demonstrated in FIG.S1. However, conducting a quantitative
analysis of these defects presented challenges. Given that the particles in
our model are point-like without physical shape, and we operate at a
relatively low density ($\rho_{ap}\approx 1$), there are numerous void spaces
within the system. This complexity impedes the effective application of
algorithms for locating the defects and tracking it over time [80, 81, 82,
83]. Consequently, our exploration does not delve deeply into the quantitative
dynamics of $\pm\frac{1}{2}$ defects in the system.

(a)  (b)  (c)

Figure S1: (color online) The figure depicts the $\pm\frac{1}{2}$ defects in
active nematic systems. The structure of $+\frac{1}{2}$ and $-\frac{1}{2}$
defects are described in (a); In (b) and (c) some $+\frac{1}{2}$ and
$-\frac{1}{2}$ defects are marked with black circulr arc and black ’Y’
symbols.

###  F.3 Interaction of Polar clusters with Swirl Structures in State-II

(a)  (b)  (c)  (d)

Figure S2: (color online) This figure captures an interaction between Polar
cluster and swirl structures in State-II of the mixture. Two distinct polar
particle clusters are identified as 1 and 2, and their direction of motion is
illustrated by black arrows. The boundary of the circular Vortex structure is
indicated by a dashed green circle. The series of snapshots (a), (b), (c), and
(d) depict the progressive evolution of the system over time. Upon interaction
with the circular defect, the smaller cluster (1) undergoes dissociation,
while the larger cluster (2) merely gets reoriented. Moreover, throughout all
the snapshots, a dashed black circle highlights the presence of an isolated
polar particle, which becomes entrapped at the center of the vortex and
exhibits random motion. The parameters used in this simulation are the same as
those of FIG.3 in the main text.

###  F.4 Interaction of Polar clusters with Swirl Structures in State-III

(a)  (b)  (c)  (d)

Figure S3: (color online) This figure illustrates an interaction between a
Polar cluster and a strong swirl structure in State-III of the mixture. The
motion of a Larger Polar particle cluster is tracked and black arrow shows its
direction of motion at a particular instant. The boundary of the circular
defect is highlighted by a dashed green circle. Snapshots (a)-(d) exhibit the
dynamic evolution of the system over time. Remarkably, in State-III, the
larger polar cluster remains largely unaffected by the vortex structure. It
successfully breaks through the vortex, restoring nematic ordering along its
wake, showcasing the dominance of polar particle clusters in State-III of the
mixture. Throughout all the snapshots, a dotted circle is utilized to mark an
isolated polar particle, which becomes entrapped at the center of the vortex.
The remaining parameters are consistent with those shown in FIG.3 in the main
text.

###  F.5 Swirl structures are intrinsic in the system

Here, we discuss the intrinsic nature of the swirl structure with the help of
the time series of SP, $\lambda$, and its probability density function (PDF),
$f(\lambda)$ vs. $\lambda$. The details of the calculation of $f(\lambda)$ is
similar to that of OPDF described in the Appendix.B in the main text.
The plots for the same are shown in FIG.S4. In State-I, the nematic is
strongly ordered, and the swirly character of the nematic orientation field is
minimal except at the edge of the stripes. This is represented by the a small
mean value of $\lambda$. Further, the small fluctuations in $\lambda$ about
the mean, which leads to a sharp $f(\lambda)$, implies that this state
persists in the steady state of the system in State-I. In State-II, the mean
value of $\lambda$ is much larger, which implies the presence of strong bend
and swirl structures in the orientation field of apolar particles. The large
fluctuations of $\lambda$ about the mean, leading to a broader $f(\lambda)$,
can be interpreted as the dynamic nature of the swirl structures in the
system, i.e., in the steady state, the swirl structures undergo formation,
dissociation, and reformation. Moving on to State-III, the mean value of
$\lambda$ decreases whereas the range of fluctuations around the mean is much
larger. This implies the weak and transient nature of swirl structures of
apolar particles in comparison to State-II. FIG.S4 clearly shows that the mean
value of $\lambda$ in State-II is approximately $2-2.5$ times larger than in
State-I. Compared to State-III, the value of $\lambda$ comes out to be
$1.2-1.3$ times larger in State-II, and the fluctuations are much smaller. The
argument presented here clearly shows that the swirl structures are much
stronger and persist much longer in State-II compared to the other two States.

Figure S4: (Color online) The figure illustrates the characteristics of the
SP, $\lambda$, in the steady state of the system in the three states. The (a)
panel displays the the time series of SP, $\lambda$, in steady state, and the
(b) panel shows the PDF of $\lambda$. In both (a) and (b) panels different
colors are used for different $\mu$ values in three different states : State-I
($\mu=$ $0.004$ $\&$ $0.02$), State-II ($\mu=$ $0.20$ $\&$ $0.28$), and State-
III ($\mu=$ $0.52$ $\&$ $0.68$). The parameters are same as FIG.3 in main
text.

###  F.6 Adding chirality to the polar particles

Our study shows that the motile nature of the polar particle promotes the
formation of dynamic swirl structures in active nematics. Till now, we have
examined the effect of achiral polar particles on the active nematics.
However, in nature, most of the natural microswimmers are chiral [84, 85, 86,
87]. Hence, to understand the effect of the chirality of microswimmers on the
system, we conducted additional simulations with chiral polar particles. The
modified orientation update equation of the polar particles in the presence of
chirality is the following:

$\theta_{i,p}(t+1)=\bigg{<}\theta(t)\bigg{>}_{R_{0,p}}+\omega+\zeta_{i,p}(t)$
(13)

where the terms on the right-hand side, apart from the newly introduced second
term, remain consistent with our original model. This second term represents
the chirality effect, with $\omega$ denoting the chirality coefficient that
can be both positive and negative, representing right and left handed
chirality, respectively.
The introduction of chirality adds a rotational bias to the movement of polar
particles, introducing a new rotational length scale proportional to
$\frac{v_{p}}{\omega}$. This creates a complex interplay between the activity
of particles and chirality, potentially leading to a range of exciting
features within the system [88, 89, 90]. Our preliminary analysis indicates
that for certain ranges of activity and chirality, there is a noticeable
enhancement in the stability of swirled structures. In FIG.S5, we illustrate
through a sequence of snapshots how a cluster of polar particles moves in a
circular trajectory within the swirl structures, significantly extending the
persistence of these structures.
The comprehensive examination of the full spectrum of effects on the dynamics
of our system due to chirality remains an area for future investigation.

(a)  (b)  (c)  (d)  (e)  (f)

Figure S5: (color online) This figure highlights the impact of chirality on
the stabilization of swirl structures, illustrated through snapshots (a)-(f)
taken at successive time intervals. These images depict a polar cluster
(marked by black dashed circle and the direction of motion of the cluster is
shown with black arrow) as it travels in a counter clockwise circular motion
within the swirl structures, significantly enhancing their persistence for
$\mu=0.68$ and $\omega=0.01$. Rest of the parameters are same as in FIG.3 in
the main text.

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