PHYSICAL REVIEW RESEARCH 3, 043104 (2021)

Algebraic correlations and anomalous fluctuations in ordered flocks of Janus particles fueled by an
AC electric field
Junichiro Iwasawa ,1,2,* Daiki Nishiguchi ,1,† and Masaki Sano

2,3,‡

1

Department of Physics, The University of Tokyo, Hongo 7-3-1, Tokyo, 113-0033, Japan
2
Universal Biology Institute, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Tokyo, 113-0033, Japan
3
Institute for Natural Sciences, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China(Received 30 August 2021; accepted 12 October 2021; published 10 November 2021)
We study the polar collective dynamics of Janus colloidal particles fueled by an AC electric field. Whenthe density is high enough, the polar interactions between the particles induce a polar orientationally orderedstate, which exhibits features reminiscent of the Vicsek model such as true long-range order and giant number
fluctuations. Independent measurements of the polarity and velocity at the single particle level allowed us
to investigate the single particle dynamics within the ordered state. We discovered theoretically unaddressed
statistical properties of the ordered state such as the asymmetric relation of polarity and velocity, enhanced
rotational diffusion stronger than in the disordered state, and an algebraic autocorrelation of the polarity. Our
experimental findings, at the crossroads of the Vicsek physics and the active Brownian particles physics, shedlight on the so-far-unexplored physics arising from the interplay between the polarity and the velocity.
DOI: 10.1103/PhysRevResearch.3.043104

I. INTRODUCTION

Active matter systems consume and dissipate energy at thelevel of their local units to generate systematic motion basedon their internal degrees of freedom (e.g., polarities) [1–4].
After decades of extensive research, it is now known that
active matter systems can exhibit a wide range of phenomena
such as phase separation [5–8], active chain formation [9,10],
active turbulence [11–14], etc. Among these topics, orientational order in active matter has gained considerable attention
since the introduction of the Vicsek model in 1995, wherepointwise polar particles move at constant speed toward theirpolarity and try to align their polarities with local neighbors
under the presence of noise [15]. Theoretical work concerning
the Vicsek-style models has been done including particle,
kinetic, and hydrodynamic levels [16–24]. In particular, thephenomenological hydrodynamic equations by Toner, Tu,
Ramawsamy, and their coworkers have predicted the existenceof true long-range order and giant number fluctuations (GNF)
for the orientationally ordered state [16–18,23,25]. In these
theoretical and numerical studies, each particle’s polarity is
assumed to be parallel to its velocity. In experimental systems,on the other hand, particles do not necessarily move toward
their polarities due to thermal and/or athermal noise and interparticle (hydrodynamic, excluded volume, or electrostatic)*jiwasawa@ubi.s.u-tokyo.ac.jp
nishiguchi@noneq.phys.s.u-tokyo.ac.jp‡
sano.masaki@sjtu.edu.cn†

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.
2643-1564/2021/3(4)/043104(10)043104-1

interactions (e.g., systems of elongated bacteria [26], vibrated
polar disks [27], motility assay [28], and active rolling colloids [29,30]).
One of the minimal models for investigating such active
systems whose particle velocities do not necessarily coincidewith their polarities is the active Brownian particles (ABP)model. In the ABP model, particles self-propel with a constantself-propulsive force under rotational/translational noise. The
particles interact only by volume exclusion and do not exhibit
any aligning interactions [31,32]. In spite of the absence ofalignment, intriguing collective phenomena such as motilityinduced phase separation have been reported for ABP [33].
However, it still remains unclear what lies between nonaligning ABP-like systems and the aligning Vicsek-style models.
Active colloidal systems similar to the ABP model have been
experimentally realized and studied by using, e.g., autocatalytic colloidal particles in hydrogen peroxide and rolling
colloids under a DC electric field. However, active particles
fueled by chemicals do not exhibit ordered states reminiscent of those in Vicsek-style models so far [6,7,34]. In thecase of rolling colloids, because the particles do not possess
intrinsic polarity and their polarities are only electronic andinvisible, in-depth study of the interplay between the polarityand velocity in the ordered state is not possible [30]. Thus,
our understanding of how polarity and velocity interact witheach other and how they behave in orientationally long-rangeordered colloidal systems with hydrodynamic interactions
remains elusive. For example, in the systems of elongatedbacteria and motility assay, although true long-range order has
been observed, particles could only be detected in a coarsegrained manner [26,28].
Janus particles [35], which are colloidal particles withtwo distinct hemispheres, offer a great platform for investigating active matter experimentally [6,8,9,32,36–39]. Among
Published by the American Physical Society

IWASAWA, NISHIGUCHI, AND SANOPHYSICAL REVIEW RESEARCH 3, 043104 (2021)

these Janus particles, the ones fueled by an AC electricfield are especially suitable for analysis from the viewpointof statistical physics because they constantly convert energy into motion without running out of energy sources andthus, allow long enough measurements for statistical analysis [8,9,32,37,39,40], which is not the case for autocatalytic
particles due to the consumption of chemical fuels. Under an
AC electric field, Janus particles display two different mechanisms of swimming depending on the field frequency: induced
charge electrophoresis (ICEP) at low frequency [41–43];
self-dielectrophoresis (sDEP) at high frequency [44,45]. Importantly, the two swimming mechanisms induce motion
from the dielectric hemisphere to the metallic hemisphere[8,9,32,37,38,40,43] or vice versa [9,38,40,45], which allows
us to define and observe each particle’s polarity based on thelocation of its metallic hemisphere [8,9]. Not only the swimming mechanisms but also the interactions between particles
depend on the field frequency and ion concentration, whichlead to various types of collective behavior such as a turbulent
phase [37], a swarming state with vortices [38], flagella likemovement [9], and active clustering [8]. Thus, Janus particles
provide us an ideal venue for experimentally studying thecollective dynamics of ordered systems while keeping trackof polarity at the single particle level.
Here in this paper, we perform experiments on the quasitwo-dimensional system of active Janus particles fueled by an
AC electric field. We specifically use the AC field frequency
regime where Janus particles swim by sDEP and exhibit aflocking phase. Through the extraction of both the polarity andvelocity for each particle, we show that this flocking phaseof Janus particles can exhibit true long-range orientational
order. This ordered state of Janus particles also exhibited
statistical features such as GNF and algebraic correlations
for orientation fluctuations, which are reminiscent of the predictions of Toner and Tu [16,17,23]. We further investigate
the space-time correlations of density fluctuations leading toan estimate of dynamical and anisotropy exponents for theordered state of Janus particles. In addition, through single
particle measurements, we show that the particles counterintuitively exhibit larger diffusion in the ordered state than thatin the disordered state for short time scales. We suggest that
this anomalous enhanced diffusion stems from the coupling
between particle polarity and velocity through computationsof the cross-correlation function, which is a dismissed feature
in Vicsek-style models. Our work, through the observationof both velocities and polarities, unravels hitherto unexploredstatistical properties within orientationally ordered active colloidal systems.
II. EXPERIMENTAL SYSTEM

Our experimental system is composed of silica-based Janus
particles with 3.17 μm diameter [9]. To create distinct hemispheres, 35 nm of titanium followed by 15 nm of silica were
deposited to a monolayer of silica colloidal particles (Bangs
Laboratories, Inc., SS05N) using electron beam deposition
and thermal evaporation, respectively. The additional 15 nm
layer of silica was deposited to suppress oxidization and thesubsequent changes of electrical properties and color (fromblack to transparent) of the Ti surface. After deposition, theFIG. 1. Schematic image of the experimental system. (a) A
silica-based Janus particle with a metallic titanium hemisphere.
(b) The suspension of Janus particles were sandwiched by two
ITO coated electrodes. The particles sediment to the surface of the
ITO electrode, which leads to quasi-two-dimensional motion on thexy plane.

particles were suspended into deionized water and washedby sonication to prevent particles from forming clusters. Thiswashing procedure was repeated three times after waiting for
30–60 min for particle sedimentation. The washed Janus particles were suspended in deionized water and then sandwichedby two 25-nm silica-deposited indium tin oxide (ITO) coatedelectrodes (Mitsuru Optical Co. Ltd.), separated by 130 μm
thick double-sided tape. Before applying the AC electric field,
we waited for a few minutes to ensure that the particles havesedimented to the surface of the bottom electrode, which leads
to a quasi-two-dimensional (quasi-2D) system of Janus particles (Fig. 1). Accordingly, the induced-charge electro-osmotic
flow caused by the electric field lets the particle’s polarity turnto the horizontal xy plane, leading to confined motion in thehorizontal 2D plane [47]. Note, because the particles have an
induced dipole distribution of charges, which is parallel to theelectric field, the particles will form clusters in the verticaldirection as in Ref. [48] if the electric field was applied before the particles were completely sedimented to the bottom
surface. It is also important to ensure that the Janus particles
have both high enough density and large enough radius so thatthey do not escape from the quasi-two-dimensional plane due
to vertical Brownian motion.
A sinusoidal AC electric field with f = 1 MHz frequency
and 16 Vpp voltage was applied for all experiments, where Vppdenotes the peak-to-peak voltage. In this frequency regime, it
is known that Janus particles tend to move toward its metallic
hemisphere, which contradicts with the ICEP theory [41–43].
This was first observed in Ref. [40], and now an explanation is given by the sDEP theory [44,45]. sDEP explains themetallic directed motion as a result of a surface force, arisingfrom localized nonuniform electric field gradients between theasymmetric particle and the surface of the electrode. Unlike
traditional dielectrophoresis where the motion is caused by theasymmetry of the external environment (e.g., electrode geometry), sDEP arises from the proximity of the asymmetric Janus
particle itself to the conducting electrode surface [44,45].
Note, the ordinary electrophoretic effects caused by the negative charge of the silica particles could be negligible in oursystem since we use an AC electric field. Therefore, ordinary
electrophoresis alone cannot compete with the ICEP/sDEP
mechanism. Experiments were performed under two different density conditions: the low-density regime where

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FIG. 2. Dynamics of Janus particles in the disordered and ordered states. (a), (b) Trajectories of the particles in the disordered state (3 s) andthe ordered state (2 s), respectively. The bright and dark colors indicate the early and late times of each trajectory, respectively. (c), (d) Velocity
distribution of the particles in the disordered (9909 particles) and ordered state (40205 particles), respectively [46]. The mean speeds for thedisordered and ordered state were v̄d = 14.6 μm/s and v̄o = 8.5 μm/s, respectively. (e) Binarized experimental snapshot of the Janus particles
in the ordered state. Each particle is colored based on the deviation of its polarity from the global order |θ − θ| (radian). The black arrowshows the direction of global order θ .

ρ  0.78 particles/1000 μm2 , and the high-density regime
where ρ  13 particles/1000 μm2 .
The observations of Janus particles were done using an
inverted microscope (Nikon ECLIPSE TE2000-U) with objective lens (Nikon Plan Fluor ELWD, 20×, NA = 0.45
and 40×, NA = 0.60 for the low- and high-density regime,
respectively). A halogen lamp was used to illuminate theparticles and a green filter was inserted in the light path before
the sample to increase the contrast of the obtained images.
Under this condition, the polarity of each Janus particle canbe distinguished using its dark gray hemisphere indicating
the titanium side, and the light gray hemisphere indicating
the silica-only side (for example see movies 2, 3 in the
Supplemental Material [46]). The dynamics were recorded
at 10 fps and 15 fps for the low- and high-density regime,
respectively, using a CMOS camera [Baumer LXG80, 3000 ×2400 pixels  430 μm × 340 μm (40×)]. This allowed us
to capture the large-scale statistics of the particles while detecting the polarities and velocities of each particle at thesame time.
Image analysis was performed using TRACKPY [49] andIMAGEJ [50]. Specifically, the polarities of the particles weredetected by taking the difference of the centroid and center of mass of image intensity for each particle using thebuilt-in algorithms in IMAGEJ [46]. The duration of the analyzed movies were 360 s and 212 s for the low- andhigh-density regime, respectively.
III. RESULTS
A. Binary polar interaction and following motion

To probe the interaction of the Janus particles in oursystem, we first measured the relative polarity distribution
P(θ ) for all Janus particle pairs in the low-density regime[Fig. 2(a)]. To examine the local interactions of the particles,
particle pairs within 6 μm and 30 μm distance were extracted,
respectively, and the relative polarity angle θ was measured
for each pair [Figs. 3(a) and 3(b)] [46]. The results show thatparticles within 6 μm range had a narrower distribution witha peak near θ = 0, which suggests the existence of polarinteraction within the Janus particles. Indeed, when observing
two Janus particles within a short distance, following motion
where one particle tends to follow the other particle could
often be observed [for example, see Fig. 3(c)].
The observed polar interaction and following motion canbe understood as a result of the balance of electrostatic and hydrodynamic interactions between the particles. Although theinteraction mechanism of Janus particles under an AC electric

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FIG. 3. Statistics of binary interactions of Janus particles.
(a) Schematic picture describing the relative angle of particle pairs.
(b) The relative polarity angle θ distribution of Janus particles ina low-density situation. R denotes the distance of two isolated Janus
particles. (c) Typical trajectories of two Janus particles when they are
in close distance.

field is still under discussion [9,38], our previous study experimentally validated the existence of polar interaction inducedby the effective electrostatic dipole-dipole interaction between
two particles both in the ICEP and the sDEP regimes [9]. Notethat although our study in the sDEP regime uses a similar
AC field frequency as in Ref. [9], the smaller ion concentration in this experiment than in our previous study√generates
a longer screening length (Debye length) λD ∝ 1/ c, wherec is the ion concentration [9]. The slip velocity us generated
in the electric double layer scales as us ∝ λD , follows fromthe Helmholtz-Smoluchowski formula [41]. This results in a
weak but relatively stronger pusher-type flow field than in ourprevious study [9], which prevents the particles from forming
active chains in the current setup. It should also be noted thatflow fields calculated based on pusher-type flow fields also
show following motion due to the reorientation of the following particle, which is consistent with our observations [51,52].
This synergy of electrostatic and hydrodynamic interactions
leads to polar alignment, allowing us to observe the ordered
dynamics of Janus particles as we see below.
B. True long-range order

Janus particles exhibited distinct global patterns when
we changed particle density. While Janus particles exhibited
uncorrelated disordered motion in the low-density regime(Figs. 2(a) and 2(c) supplemental movie 1), orientationallyordered coherent motion was observed in the high-density
regime [Figs. 2(b), 2(d), and 2(e), supplemental movies 3, 4]
due to the local polar interaction. This density dependency ofJanus particles is consistent with the previous experimental
results of Ref. [38]. Note, if we compare the Janus particle system with Vicsek-style models, the density and thenoise amplitude should be the basic factors, which control

the transition from the disordered state to the ordered state.
Although the effective noise amplitude, namely the Péclet
number, could be adjusted by changing the particle’s velocity
via the AC electric field voltage, we have not been able toinvestigate this in the current study. This is because the particles easily adhere to the surface of the ITO electrode at higher
voltages forbidding long time observation. Ideally, the effective noise amplitude could also be changed by using particles
with different diameters, but it would be nearly impossible tocontinuously change the noise amplitude in this case. It should
also be noted that the global orientation of the system varied
each time we conducted the experiment, and thus, is not deterministic. This is analogous to the setting of the Vicsek-stylemodels where the global orientation results from spontaneous
symmetry breaking of the continuous rotational symmetry of
the system.
To quantitatively assess the properties of orientational order in the low- and high-density regimes of the system, we
computed the polar order parameter,(1)C = eiθ S t ,for different sizes of rectangular (5:4) regions of interest(ROIs) taken in spatially uncorrelated locations (see Ref. [46]for details), where θ is the particle polarity and ·S and·t denotes the average within the same ROI and statistically independent time frames, respectively (Fig. 4, [46]).
In the low-density regime,√ the polar order parameter decays
as a power law C ∝ 1/ S [Fig. 4(a)]. This is consistent
with the case with N randomly oriented particles. In contrast, in the high-density regime, we can observe a decay
slower than a power law within S ∼ 105 μm2 . This asymptotic behavior characterizes the true long-range order, which
was previously reported in Vicsek-style models [4] and in
the bacterial experiment [26] where the orientational orderparameter converges to a positive finite value. Note, the decay slower than a power law in our Janus particle system
is observed for about 2.8 decades, which is larger than therange reported in the long-range ordered phase of bacteria[26] and even comparable to the range reported in numerical simulations (see Fig. 4(a) of Ref. [4]). To confirm theexistence of true long-range order, we performed nonlinear
fitting for the polar order parameter at S < 105 μm2 , using the
CURVE_ FIT function from the SCIPY package [53]. The data
was in good fit with an algebraic convergence to a finite value
C − C∞ ∼ S −γ /2 where C∞ = 0.74 and γ /2 = 0.53 [see theinset of Fig. 4(b)], which indicates true long-range order inthe system. It should be noted that recent simulations showed
γ = 0.64 for the Vicsek model [4]. At the same time, however,the decay of the order parameter deviates from this trend ofalgebraic convergence at S > 105 μm. This is caused by theexistence of counterflows of the particles (see supplemental
movies 3, 4), which is unavoidable due to the closed boundary conditions and the number conservation of the particles
unless we elaborate an effective periodic boundary condition
by, e.g., a racetracklike confinement [29]. However, in thecase of our electrically driven Janus particles, microstructuresfabricated by, e.g., photoresist distort the electric field, whichleads to unwanted complex 3D electroconvection and destroys
the two-dimensionality of the system. Therefore, we opted forusing a simple system by sandwiching as large a droplet of

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FIG. 4. The polar order parameter for the ordered and disordered states. (a) Log-log plot of the polar order parameter C vs S, the area of
the ROI. The blue and red points show the order parameter for the ordered and disordered state, respectively. (b) The same data as in (a) for
the ordered state in a magnified range (log-log). (c) The same data as in (b) with the asymptotic value C∞ = 0.74 subtracted (log-log). The
solid black line shows the nonlinear fit: C = C∞ + kS −γ /2 with C∞ = 0.74, k = 3.1, and γ /2 = 0.53. The data for ROIs with S > 105 μm2
have been excluded from the nonlinear fitting due to the existence of counterflows in the system. Error bars in (a), (b), (c): standard error.

the suspension of Janus particles as possible between the twoelectrodes. Overall, the finite size scaling analysis assures that
the Janus particles exhibit polar collective motion with truelong-range order.
C. Interplay of polarity and velocity

To investigate the detailed dynamics of single particle trajectories, we computed the polarity autocorrelation functionf p (τ ) for the disordered and ordered states, respectively. Here,f p (τ ) = n p (t ) · n p (t + τ )t ,(2)

where n p (t ) denotes the polarity for the particle at time t, and·t takes the average for particle trajectories longer than 24 s
[Figs. 5(a) and 5(b)]. We can first observe that the polarity
autocorrelation decays exponentially in the disordered state.
At small time scales, this decay was slightly faster than thetheoretical estimation for an active Brownian particle f p (τ ) =exp (−2Dr τ ) [31,36], where 1/Dr ∼ 24.2 s is the rotational
diffusion constant [Fig. 5(a)] from the Einstein-Stokes relation, which is due to the polarity detection errors [32]. We

thus focus on the time scale τ = 0–12 s, where the decay of
the autocorrelation exhibited a slope of 21.15 ± 0.08 s, close
to that of the theoretical estimation.
In the ordered state, on the other hand, f p (τ ) exhibited aslow power-law-like decay in the long time scale (τ > 1 s,[46]). Since particle polarities in the ordered state tend to alignto the global order, we suspected that this power-law behaviorof f p (τ ) is related to the fluctuations of the global order. To
assess this hypothesis, we compared the connected polarity
autocorrelation function in the ordered state with that of the
global order (Fig. 6). In Fig. 5(c), we show the connected
polarity (velocity) autocorrelations,f p,C (τ ) = n p (t ) · n p (t + τ )t ,(3)

fv,C (τ ) = nv (t ) · nv (t + τ )t ,(4)

where n p,v (t ) = n p,v (t ) − n p,v (t ), and n p,v (t ) is the
direction of global order defined by polarity or velocity, respectively. A power-law-like decay with a slope close to−0.7 for f p,C (τ ) could be observed for both the polarityand velocity autocorrelations. Interestingly, the connected

FIG. 5. Autocorrelation functions. (a) Polarity autocorrelation function in the ordered (red) and disordered state (blue) on a semilog scale.
τ denotes the time delay. The dashed line shows exp (−2Dr τ ) where Dr  1/24.2 s−1 is the rotational diffusion constant for a 3.17 μm
diameter particle derived from the Einstein-Stokes relation. (b) The same data as in (a) in a magnified view, which is shown as a gray box in(a) (semilog). Error bars show the standard error. (c) The connected autocorrelation functions for polarity: f p,C (τ ) and velocity: fv,C (τ ) in theordered state. The blue dashed line with a slope: −0.7 is a guide for the eye.
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FIG. 6. Fluctuations of the mean polarity in the ordered region.
(a) Time series of the fluctuations of global order θ − θordered
where ·ordered takes the average within the highly ordered region(S < 105 μm). (b) Autocorrelation of the polarity fluctuations fromglobal order fglobal,C (τ ) = ñ p (t ) · ñ p (t + τ ), where ñ p (t ) =n p − n p t,ordered ordered . The blue dashed line with a slope: −0.85
is a guide for the eye.

autocorrelation of the global order fglobal,C (τ ) [46] also
showed a power-law-like decay with a slope of −0.85, close tothat of f p,C (τ ) [Fig. 6(b)]. Although there is, to the best of ourknowledge, no theory predicting the slope for these power-law
decays in global order and polarity (velocity) autocorrelations,these observations support our hypothesis that the power-law
behavior of the polarity autocorrelations is a consequence ofthe fluctuations in global order. However, it should be notedthat in a finite system, the direction of global order diffusessuch that the autocorrelation function decays exponentially[24]. Therefore, we should expect that the power-law behaviorof fglobal,C (τ ) that we observe is a transient to an exponential
decay. Indeed, when we focus on longer time scales (τ >10 s), we can observe a decay faster than a power law for
fglobal,C (τ ) [46]. Although the limited length scale of ourexperimental system hinders the long time behavior of single
particle trajectories, we expect that the connected polarity
autocorrelation function f p,C (τ ) will also show a crossover toan exponential decay, reflecting the fluctuations of the global
order.
The polarity autocorrelation function for particles in
the ordered state also exhibited peculiar behavior at shorttime scales (τ ∼ 1 s). Figure 5(b) shows a magnified viewof the behavior of f p (τ ), and it can be observed that f p (τ )of the ordered state decays faster than that of the disorderedstate at short time scales. Because this time scale is similar
to the time scale of the mean-free time (distance to the nearest
neighbor ∼7 μm)/(average velocity v̄o) ∼0.8 s, we speculated
that this enhanced diffusion was caused by the frequent andcontinuous interactions among closely distanced particles in
the ordered state. To confirm this speculation, we computed
the polarity-velocity cross-correlation function,g pv (τ ) = n p (t ) · nv (t + τ )t ,(5)

where nv (t ) = v(t )/|v(t )| is the unit velocity vector at time t(Fig. 7). The assessment of g pv (τ ) revealed the asymmetric relation of polarity and normalized velocity in the ordered state,
where polarity tended to correlate the most with the velocityof 0.2 s in the past (τ ∼ −0.2). In other words, the polarity inthe ordered state tended to follow the direction of the velocityof the particle with a finite time delay. It should be noted that,
although not statistically significant, g pv (τ ) in the disordered

FIG. 7. Polarity-velocity cross-correlation functions. (a) The
cross-correlation function g pv (τ ) for Janus particles in the ordered(blue) and disordered state (red). (b) The same data as in (a) in amagnified view for the gray box shown in (a). Error bars: standard
error. The difference in error bar size is caused by the sparsity in thedisordered state.

state showed a weak, if any, asymmetry around τ = 0 as well.
These results imply that the asymmetry in g pv (τ ) is amplified
in higher-density situations. We therefore suspect that thisasymmetry is caused by electrostatic and/or hydrodynamic
torques induced by collisions between individual particles.
Overall, our results suggest that the particle polarities tend tofollow the particle’s velocity, yet local interactions between
nearby particles disturb their alignment causing a time delay
especially at the high-density state. These effects might explain the enhanced diffusion observed in the ordered state.
D. Giant number fluctuations in the ordered state

Giant number fluctuations (GNF), where the variance ofthe number of particles N 2 grows faster than its mean N(N 2 ∝ N2α with 2α > 1) in long-range ordered phases,
are a hallmark of the Toner-Tu phase [16,17,23]. Since its
theoretical prediction by Toner, Tu, Ramaswamy, and theircoworkers [16,17,25], the presence of GNF has been extensively studied in numerical systems [18,21,24,54]. However,in addition to experimental difficulty in realizing collective
motion with true long-range order, experimental detection of
GNF requires delicate treatment due to many possible pitfalls
such as difficulty in disentangling fluctuations arising from,e.g., phase coexistence or boundary effects as critically discussed in Ref. [26]. Therefore, GNF as the hallmark of the
Toner-Tu-Ramaswamy phase had remained undetected until
recent experiments on a long-range ordered nematic phase ofelongated bacteria [26] and then on the polar ordered phase ofcolloidal rollers [30]. Here in our Janus particles system, as
we have confirmed the true long-range order, it is natural toquestion whether the ordered state shows anomalous number
fluctuations.
To quantify number fluctuations, the mean N andstandard deviations N of the number of particles for different sizes of ROIs were calculated for statistically independent
frames (Fig. 8). For the disordered state, normal number
fluctuations with N ∝ N0.5 were observed, which is consistent with the central limit theorem. In contrast, anomalous
number fluctuations were observed in the ordered state whereN ∝ Nα with α > 0.5. In fact, the estimated exponent was
α = 0.790(5) with 95% confidence level, which is very close
to the theoretical prediction α = 0.80 [16,17] and the recent

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FIG. 8. Scaling of number fluctuations. (a) Number fluctuationsin the ordered (blue) and disordered (red) state. (b) Same data as in(a) with the number fluctuations normalized by N0.5 . The cyan andmagenta dashed lines show N ∝ N0.8 and N ∝ N0.5 , respectively, as a guide to the eye. The gray dashed line indicates the scaleof counterflows up to which the true long-range order persists.

numerical estimate α = 0.84 for the Vicsek model [24]. It is
worth noting that the existence of nontrivial GNF has to be
discussed under the presence of true long-range orientational
order. Indeed, there are several experimental studies that report GNF for systems that are not in the fully ordered state(for details see the Supplemental Material of Ref. [26]). Here,
we have set all ROIs to be placed at the center of the whole
image so that the ROIs correspond to those that were used forthe computation of the order parameter. Accordingly, the graydashed line in Fig. 8 indicates the length scale S ∼ 105 μm2
at which the order parameter in the ordered phase starts todeviate from algebraic convergence to a finite value (i.e., truelong-range order). Therefore, the observation of anomalous
fluctuations below this line imply that ordered Janus particles
do exhibit GNF in the context of the Toner-Tu phase.
E. Algebraic scars of long-range order

When a continuous symmetry is spontaneously broken,
the entire ordered phase should also be characterized by analgebraic decay of its connected correlation functions [55].
In the case of the Vicsek model, a power-law decay for theorientation fluctuation correlation function exemplifies the
spontaneous symmetry breaking of the rotational symmetry of
the system, and its exponents have been predicted by the hydrodynamic theory of Toner and Tu [16,17,23]. Interestingly,
a recent numerical study of the Vicsek model suggested thatthe ordered state could exhibit GNF exponents close to theconjectured values even though there existed a discrepancy
in the exponents of the correlation functions [24]. Thus, inaddition to number fluctuations, the fluctuation correlation
functions are important measures to probe the statistical properties of orientational order in the system.
To address the questions of whether the orientational fluctuations of Janus particles exhibit scale-free correlations, andif so, whether they follow exponents of the Toner-Tu conjecture, we measured the orientation fluctuation correlation
function,CC (R) ≡ δn p⊥ (t, r)δn p⊥ (t, r + R)r t ,(6)

where δn p⊥ (t, r) = n p (t, r) − n p  is the deviation of a single
particle polarity n p from the global order n p , and ·r takes

FIG. 9. Orientation fluctuation correlation function in the ordered state. (a) Color map of the correlation function CC (R).
The global mean director is set aligned with the vertical axis (R ).
The region of |R| < 6 μm were excluded from calculations due tothe small statistics. (b) Log-log plot of CC (R) both in the transverse(R⊥ , purple) and longitudinal (R , green) direction. The purple andgreen solid lines are a guide for the eye, which show a slope of−0.52. Note, the exponents from the Toner-Tu conjecture predict−2/5, R−2/3 [16,17,23] and recent
an anisotropic decay of CC (R) ∝ R⊥
−0.62
numerical simulations suggested CC (R) ∝ R⊥
, R−0.65 [24].

an average for all |R| distanced particle pairs (see Fig. 9).
Here, the global order n p  was calculated by averaging thepolarity n p both spatially and temporally. Figure 9(b) shows
the decay of CC (R) in the transverse and longitudinal directions to global order. Although there is an anisotropy in
the absolute value of the correlation function, it decays algebraically in both directions with the same exponent close to−0.52. This algebraic decay signifies the spontaneous symmetry breaking of the rotational symmetry of the system andconsequent scale-free features. However, the exponents of the
decay differ from the Toner-Tu conjecture, which predicts an2χ
anisotropic decay of CC (R) ∝ R⊥
, R2χ/ξ , where χ = −0.2
and ξ = 0.6 are the roughness and anisotropy exponent, respectively [16,17,23]. This discrepancy might stem from thedifference in microscopic properties including steric and hydrodynamic interactions as we will discuss in the following
section. Interestingly, recent experimental [30] and numerical[24] studies for polar orientational order have both reported
weak, possibly vanishing, anisotropy for the orientational
fluctuation correlation function. In addition, our analysis on
the space-time correlations of density fluctuations also suggests weakly anisotropic sound modes propagating in ourJanus colloidal flocks [46]. This leads to the estimate ofthe dynamical exponent z ∼ 1.2 while the anisotropy exponent ξ ∼ 1, suggesting a nondiffusive but weakly anisotropic
dependence that qualitatively agrees with recent numericalsimulations [24]. It should also be noted that the roughness
exponent χ also takes similar but distinct exponents for theory, simulation, and experiments where χ = −0.20 for the
Toner-Tu conjecture [16,17,23], χ ∼ −0.31 for numericalsimulations [24], χ ∼ −0.38 for active Quincke rollers [30],and χ ∼ −0.26 for our system of Janus particles. These differences in the exponents might indicate that some of thenonlinearities that were neglected in the original calculations[16,17] are relevant asymptotically, as suggested in Ref. [24].
Our measurements of CC (R) and the sound modes, along withrecent numerical [24] and experimental [30] studies, may pave
the way for further theoretical work on collective systems.

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IWASAWA, NISHIGUCHI, AND SANOPHYSICAL REVIEW RESEARCH 3, 043104 (2021)

IV. DISCUSSION

We studied the collective motion of active Janus particles fueled by an AC electric field. At high density, theJanus particles exhibited polar orientational order, which allowed us to investigate and compare the statistical propertiesof the system with that of the Toner-Tu phase. We haveconfirmed GNF within the polar orientational ordered statewith an exponent similar to but different from the theoretical predictions of Toner and Tu. In addition, an algebraic
decay was observed for the orientation fluctuation correlation function, which is also associated with the spontaneous
symmetry breaking of the rotational symmetry of the system.
The observed exponents showed a discrepancy with the theoretical predictions and recent numerical simulations [24] of
the Vicsek model, suggesting a different mechanism for the
emergence of long-range order. As pointed out in Ref. [26]and then both experimentally and numerically verified inRefs. [28,56], weak volume exclusion that allows crossing
over other particles is considered a key to make collective
motion of elongated objects similar to the Vicsek-style interactions and thus achieve long-range order at least for theotherwise turbulent or clustering 2D nematic systems such asbacteria and gliding microtubules [26,28]. In the case of polar
collective motion, the weak exclusion may not be a requirement for the long-range order because the particles cannot
overlap each other in our experimental setup as in granularexperiments [27,57]. What realizes long-range order in polarsystems could be the following motion exemplified in Fig. 3,
as discussed previously in, e.g., mutant Dictyostelium cells[58], and/or the torque aligning the polarity to the velocity,
as studied in vibrated polar disks [59].
Our results might suggest that, although GNF could be observed, algebraic decay as in the Toner-Tu phases are difficult
to observe in experimental systems as well as in numericalsimulations. The exponents of the algebraic decay for theorientation fluctuation correlation function differed from thatobserved for polar flocks of Quincke colloidal rollers [30].
Although Janus particles and Quincke rollers are both considered as polar particles with polar interaction, the polarity
acquisition mechanisms are completely different. Whereas
the polarity of Janus particles is a priori defined by its
metallic hemisphere, Quincke rollers acquire polarity by the
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particle scale. It could be interesting to investigate whether
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colloidal particles, which opens a new avenue on further theoretical and experimental works for the field of active matter.

ACKNOWLEDGMENTS

We thank Benoît Mahault and Hugues Chaté for insightful
discussions and valuable comments on an early version of themanuscript. We also thank Tetsuya Hiraiwa and Kazumasa A.
Takeuchi for fruitful discussions. This work was supported by
KAKENHI (Grant No. JP25103004). J.I. was supported by
a Grant-in-Aid for the Japan Society for Promotion of Science Fellows (Grant No. JP18J21942). D.N. was supported by
JSPS KAKENHI Grants No. JP19K23422, No. JP19H05800,
and No. JP20K14426.

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