# Odd Radio Circles as supernovae remnants in the intragroup medium

A. Omar
Aryabhatta Research Institute of observational-sciences, Manora Peak,
Nainital, 263001, India
E-mail<EMAIL_ADDRESS>(AO)

###### Abstract

A measurable fraction ($\sim 8$ per cent) of recently discovered arcmin-size
circular diffuse radio sources termed as Odd Radio Circles or ORCs can be
supernovae remnants in the intragroup medium, within the local group and its
immediate neighbour groups of galaxies. This estimate is based on the optical
detection rate of the intragroup supernovae events in the nearby ($z\sim
0.1-0.2$) galaxy groups. A rate of about 5400 intragroup supernovae per
million year is expected within the local and its immediate neighbour groups
of galaxies. For a radio detectability period of about $10^{4}$ years, on
average 1.3 intragroup medium supernovae remnants per 1000 square degree are
expected to be detected in the radio surveys with a sensitivity that led to
discovery of ORCs. The angular size, surface brightness and radio flux of the
supernova remnants up to a distance of $\sim 3$ Mpc in the intragroup medium
can be expected to be similar to the five known ORCs. The intragroup
supernovae remnants are not residing in the dense and cold interstellar medium
of the galaxies but evolving in low density ($10^{-4}-10^{-5}$ cm-3) warm
medium ($10^{5}-10^{6}$ K) in galactic halos or beyond, and may find their
progenitors in the diffuse stellar light associated with various tidal
streamers surrounding the Milky-Way and other nearby galaxies.

###### keywords:

galaxies: groups: intergalactic medium – supernovae: general – radio
continuum: general – supernovae remnants

##  1 Introduction

Some five Odd Radio Circles (ORCs) were recently discovered as a new
population of extremely faint (surface brightness a few tens of $\mu$Jy per
beam) diffuse radio sources in the celestial sky, all having circular ring-
like morphology with nearly identical angular diameters of $\sim 1$ arcmin
(Norris et al., 2021a; Norris, Crawford & Macgregor, 2021; Koribalski et al.,
2021). These sources were identified either in a deep pilot radio survey named
as Evolutionary Map of the Universe (EMU; Norris et al. 2021b), made using the
Australian Square Kilometre Array Pathfinder (ASKAP) or in the Giant Meter-
wave Radio Telescope (GMRT) archival data. The EMU survey attained surface
brightness sensitivity of $\sim 30$ $\mu$Jy per beam rms at an angular
resolution of nearly 12 arcsec near 1 GHz (0.944 GHz) frequency (Norris et
al., 2021b). Three of these ORCs were found in a search area of 270 deg2 of
the ASKAP data, one in 40 deg2 area around the galaxy NGC 253 in the ASKAP
data, and one in the archival observation data at 325 MHz in the cluster field
Abell 2142 from the GMRT. The EMU data cover fields around the Galactic
latitude of $-40$ degree. These radio objects are named as ORC-1 to ORC-5 with
their Galactic $(l,b)$ coordinates at (333.42, -39.0), (339.09, -39.5),
(339.08, -39.6), (44.36, 49.4) and (170.78, -86.6) degree respectively.

The morphological appearances of the five ORCs are strikingly similar to a
typical Galactic supernovae remnant (SNR). However, Norris et al. (2021a)
ruled out these as Galactic SNRs based on the high latitude positions of the
ORCs coming at odd with the statistical distribution of the known SNRs as a
function of the Galactic latitude. Although, a possibility for ORCs as a
previously unknown population of SNRs was not ruled out. All the five ORCs
have a similar angular diameter of nearly 1 arcmin in the sky. Optical
counterparts have been associated near the centers of ORC-1, ORC-4 and ORC-5
at photometric redshifts of 0.551, 0.457 and 0.270 respectively. ORC-2 and
ORC-3 could not be associated with any optical counterparts. The ORC-2 and
ORC-3 also form a close pair with an angular separation of nearly
$1.5^{\prime}$ in the sky. The flux densities near 1 GHz for four of these
ORCs are almost similar at $\sim 7$ mJy. The ORC-3 is the faintest with flux
density at $\sim 2$ mJy. The mean radio surface brightness of 4 bright ORCs is
nearly $10^{-21}$ W m-2 Hz-1 sr-1 and that of the faintest ORC is nearly
$3\times 10^{-22}$ W m-2 Hz-1 sr-1. Their radio spectral indices are negative,
indicating that the radio emission is of non-thermal origin.

The astrophysical origins of the ORCs are presently debated with a number of
possibilities. Norris et al. (2021a) discussed various possibilities to
associate ORCs with different types of known astrophysical sources after
excluding a possibility of Galactic SNRs. However, majority of other
possibilities are also ruled out on the basis of ORC’s distinct morphological
features such as edge-brightening, symmetry and a high degree of circularity,
which are not seen in other known similar diffuse radio sources such as lobes
of radio galaxies, face-on star-forming galaxies and cluster halos. Moreover,
ORCs are not physically associated with large face-on galaxies or rich galaxy
clusters. Norris et al. (2021b) and Koribalski et al. (2021) mentioned three
promising theoretical possibilities for the origins of some of the ORCs, viz.,
spherical shock resulting from a merger of two supermassive black-holes,
vortex rings in radio lobes viewed end-on in radio galaxies, and termination
shock of starburst winds from the galaxy. Presently, any single or all five
ORCs can not be firmly associated with a definite type of astrophysical object
or phenomena.

In this paper, we discuss a possibility that some ORCs may trace their origins
in SNRs in low density intragroup medium (IGrM). We will use single
abbreviation IGrM to denote both intragroup medium commonly associated with
the local group, and intergalactic medium or IGM commonly used for other
groups of galaxies. The motivation for exploring this possibility comes from
detections of a subtle population of stars in the IGrM in the form of various
known stellar streams in our local group and in other nearby groups of
galaxies, diffuse intracluster light in some clusters of galaxies, and
detections of a significant number of host-less supernovae.

##  2 ORCs as SNRs in low density hot medium

Based on a similar appearance as those of the Galactic SNRs, the ORCs are very
likely dynamically evolving objects, created via a catastrophic event in some
astrophysical objects. As the interest here is to explore a possibility of
some ORCs being SNR, and considering that the ORCs do not appear to be
residing in the Milky-Way or in any other large galaxy, a discussion of
evolution of SNRs in low-density hot medium, typical in IGrM environments will
be the most relevant here.

The radio brightness and morphology of a SNR depends on various factor such as
age, type (I or II) based on the properties of the progenitor star, and
density and temperature of the ambient medium in which it is evolving. Early
numerical studies related to surface brightness and size evolution of SNRs as
a function of ambient temperature and gas density were carried out by
Tomisaka, Habe & Ikeuchi (1980) and Higdon & Lingenfelter (1980). In their
minimalistic theoretical framework, the sizes of SNRs are expected to become
larger and surface brightness to become fainter in low-density warm medium
(e.g., $n\sim 10^{-4}$ cm-3, $T\sim 10^{5}$ K), compared to their counterparts
in dense and cold (e.g., $n\sim 10^{-1}$ cm-3, $T\sim 10^{2}$ K) interstellar
medium (ISM) in the Galactic plane. Evolutions of SNRs in hot medium with a
temperature up to $5\times 10^{7}$ K are described in Tang & Wang (2005).
Based on detailed hydrodynamical simulations, estimates for size and radio
surface brightness variations of SNRs with ambient density and time are
provided in more recent work of Pavlović et al. (2018). The main results from
all these studies can be summarized as follows – (i) Expansion deviates
quickly ($<10^{4}$ years) from the standard Sedov-Taylor solutions at ambient
temperatures above $10^{6}$ K. (ii) The sizes of SNRs near the end of the
Sedov-Taylor phase are expected to scale inversely with ambient density as a
power-law with an exponent of $\sim 1/3$, (iii) The SNRs expanding in a hot
medium are expected to have a lower radio surface brightness than those in a
warm medium. This behaviour is due to lower Mach number of the shock in a
hotter medium. (iv) The radio surface brightness decreases very fast in late
phases (i.e., Sedov-Taylor regime) of evolution with surface brightness to
diameter slopes in between $-4$ and $-6$. It means that while the diameter
will not increase very fast (i.e. expansion is slowed down), the surface
brightness decreases very fast in the late stages. These general results will
also be used in the next section to obtain a range of possible distances to
ORCs.

It is also worth to mention here that the shell component of SNRs in a low
ambient density will brighten at late stages and are likely to be detected in
radio surveys, compared to the plerionic component which will be bright in the
beginning for only a very short duration and will become extremely faint after
a few 100 years from the explosion (e.g., Bhattacharya 1990). The dynamical
lifetime of SNRs are expected to be $10^{4}-10^{5}$ years, after which SNRs
mix with the general ISM and cease to exist as a morphologically distinct
structure. It may be noted that the synchrotron lifetime of relativistic
electrons generated in a SNR are expected to be much higher (up to $\sim 100$
Myr) in low (a few $\mu$G) magnetic field strengths, however, these electrons
will diffuse out to much larger extents during this period and will contribute
to large-scale diffuse radio emission. This simple dynamical behaviour of SNRs
will have two consequences - firstly, small size SNRs representative of
younger age will be difficult to detect in a survey covering low volume in the
space and secondly, beyond a certain stage of expansion at later times, SNRs
will become too faint and large to be detected in a survey with a limited
sensitivity. Hence, statistically, if some ORCs are SNRs, then these are
detected at a particular stage of their evolution, when both surface
brightness and angular diameter become favourable for their detections in the
radio survey with a limited sensitivity. This stage will be neither too early
nor too late.

It is a well established fact that a large number of Galactic SNRs are missing
in the present radio surveys due to multiple reasons (see e.g., Bhattacharya
1990; Green 2004). The main reason is believed to be that many SNRs evolve in
low-density hot medium, created in the cavities blown by the progenitor
massive star thereby making such SNRs large and faint. The biases created due
to sensitivity limits in any radio surveys are also the reasons for missing
some SNRs. For instance, sensitivities of the radio interferometric surveys
such as ASKAP are limited in both flux and in detectable largest angular
diameter, which is decided by the shortest distance in the pairs of the
antennas in the radio array. Due to increase in sensitivity of the present
generation radio telescopes and with the advent of new detection techniques,
some very large and very faint SNRs have recently been detected. For instance,
a large size ($\sim 4.4$ degree diameter) SNR named ’Hoinga (G249.7+24.7)’
with very low radio surface brightness ($4\times 10^{-23}$ W m-2 Hz-1 sr-1 at
1.4 GHz) was detected at Galactic latitude of $\sim 25$ degree (Becker et al.,
2021; Fesen et al., 2021). Another SNR ’G354-33’ with angular size of
$11^{0}\times 14^{0}$ at Galactic latitude of $-33.5$ degree was detected in
radio (Fesen et al., 2021). Detections of such large SNRs are obviously more
favourable at higher Galactic latitudes due to reduced Galactic diffuse radio
background at high latitudes.

Since ORCs have low radio surface brightness and have been detected due to a
breakthrough in detection sensitivity with the ASKAP, it is worth to explore
their origins in low-density hot medium SNRs, which are believed to be missing
in previous generation radio surveys. The biggest challenge will be to
constrain the distances as discussed in the next section.

##  3 Distances to ORCs

It becomes worth at this point to compare properties of the ORCs with those of
the Galactic SNRs and obtain some clues about their distances. Majority of the
Galactic radio SNRs have their 1 GHz radio luminosity in the range of
$10^{15}-10^{18}$ W Hz-1 and radio surface brightness in the range of
$10^{-22}-10^{-18}$ W m-2 Hz-1 sr-1 (Green, 2004). The average radio surface
brightness of five ORCs is in between $10^{-22}-10^{-21}$ W m-2 Hz-1 sr-1,
placing ORCs among the faintest bins of radio surface brightness of the
Galactic SNRs.

Considering ORCs as SNRs at late stages where their expansion is slowed down,
their angular sizes can provide another clue to their distances. The results
provided in Pavlović et al. (2018) for SNRs evolving in ambient medium
temperature of $10^{4}$ K and a range of densities between 2 to 0.005 cm-3 for
initial explosion energy in the range of 0.5 to 2.0$\times 10^{51}$ erg can be
adjusted for lower densities and higher temperatures in the IGrM, using the
general results described in the previous section. In ambient density
$10^{-3}$ cm-3, the 1 GHz radio surface brightness between $10^{-21}$ and
$10^{-22}$ W m-2 Hz-1 sr-1 will correspond to SNR diameter of $\sim 100$ pc.
In lower ambient density, e.g., in $10^{-4}$ cm-3 and $10^{-5}$ cm-3, the
sizes can be up to $\sim 200$ pc and $\sim 500$ pc respectively. It may be
noted that the effect of increased ambient temperature will be on the radio
surface brightness. The SNRs in hot (T$>10^{6}$ K) ambient medium (e.g., in
galaxy clusters or massive groups with diffuse X-ray emission) are expected to
become fainter than those in relatively warm (T$\sim 10^{5}-10^{6}$ K) ambient
medium, typically seen in low mass groups of galaxies. This effect can reduce
chances of detecting as well as identifying (from a background un-resolved
radio source population) SNRs in the cores of galaxy clusters and in groups
with diffuse X-ray emission.

Considering an upper limit to physical diameter as 500 pc corresponding to the
detected range of radio surface brightness of the ORCs, the maximum distance
to the ORCs in a low density ($10^{-5}$ cm-3) warm ($10^{5}$K) IGrM
environment can be about 1.5 Mpc for an average angular diameter of 1 arcmin.
If ORCs are evolving in higher density (e.g., $10^{-3}$ cm-3) ambient medium,
this distance can be revised down to 0.3 Mpc. The 1 GHz radio luminosity of
the Galactic radio SNRs are generally in the range of $10^{15}-10^{18}$ W Hz-1
(Green, 2004). Assuming that the radio luminosities of the ORCs are similar to
that of the general population of the Galactic SNRs, the ORCs with their
estimated flux density can be placed in a distance bracket within a range of
$\sim 0.1$ Mpc to $\sim 3$ Mpc. Therefore, the range of estimated distances of
the ORCs is nearly 0.1 - 3 Mpc and the two arguments, viz., using radio
brightness and size, and radio luminosity fetch similar results within the
uncertainties.

##  4 Discussions

###  4.1 On whereabouts of ORCs

From the arguments and estimates for the distances presented in the previous
section, the SNRs in IGrM appearing as ORCs can be best placed within the
extent of the local group of galaxies or some of its immediate neighbours (see
van den Bergh 1999 for properties of the nearby groups). The radius of the
local group is estimated as $\sim 1.2$ Mpc. More than 35 galaxies are
currently designated as members of the local group with a few additional
probable candidates and several other galaxies placed in the immediate
neighbour groups within a distance of $\sim 5$ Mpc. The locations of the
galaxies in the local group are not uniformly distributed in the sky. As no
bright galaxy from the local or its neighbour groups is detected at the
locations of the ORCs, some of the ORCs can be predicted to be residing in the
IGrM. The IGrM of the local group is not completely empty of stars. More than
40 stellar streams have been discovered at a range of distances surrounding
the Milky Way (Grillmair & Carlin, 2016). These streams originated during the
tidal interactions between galaxies in the group. In general, the environment
of a typical galaxy group is favourable to exert significant tidal
disturbances on galaxies, leading to removal of gas and stars from the parent
galaxies to much larger extents in the IGrM (e.g., Omar & Dwarakanath 2005).
Two largest stellar streams in the local group are associated with Sagittarius
dwarf galaxy and Large Magellanic Cloud (LMC) - Small Magellanic Cloud (SMC)
at a distance of nearly 60 kpc. The Sagittarius stream is seen almost over the
entire sky. The LMC-SMC stream is very wide and data from $Gaia$ suggest that
the stars are observed around LMC to an extent up to $20$ degree (Belokurov et
al., 2017).

The stellar streams are populated mostly by low-mass stars, which may undergo
type Ia supernovae (SNIa). Some ORCs may be SNRs in one of these stellar
streams. It is worth to note here that three ORCs are detected in a area
surrounding SMC and one near NGC 253 - a sculptor group galaxy at a distance
of $\sim 3$ Mpc. ORC-1, ORC-2 and ORC-3 are within 23-27 deg (about 30 kpc)
from the SMC. ORC-5 is at an angular separation of about 3 deg (about 170 kpc)
from NGC 253. The estimated separations of ORCs from the SMC or NGC 253 are
not large that the ORCs can not be associated with these galaxies. A number of
host-less (i.e., not associated with a galaxy) supernovae in the IGrM have
been detected in optical bands (Gal-Yam et al., 2003; Sand et al., 2008;
Dilday et al., 2010; Sand et al., 2011). McGee & Balogh (2010) provided a
statistics of the host-less SNIa detection rate in groups of galaxies based on
total supernovae detected in the Sloan digital sky supernovae survey (Friemann
et al., 2008) after assigning memberships of the detected supernovae across
1401 groups of galaxies in the redshift range $0.1-0.2$. A total of 22 SNIa
were found to be host-less in the galaxy groups, in the total survey-observing
period of nearly 9 months. The combined halo mass of all the groups was
estimated to be $5.4\times 10^{16}$ M⊙. It translates in to an average of 5400
IGrM or host-less supernovae per million year in a group mass of $10^{13}$ M⊙.

The total mass of the local group is estimated as $2.3\times 10^{12}$ M⊙ (van
den Bergh, 1999). Assuming that the masses of neighbour groups, namely Antlia-
Sextans group at 1.7 Mpc, the Sculptor group at 2.4 Mpc, the IC 342/Maffei
group at 3.2 Mpc and the M 81 group at 3.5 Mpc are of similar order, total
mass of the local group and its immediate neighbours can be taken as nearly
$10^{13}$ M⊙. Therefore, the SNIa rate in the IGrM associated with the local
group and its immediate neighbour groups is about 5400 supernovae per million
year. If the remnants of these supernovae are detectable up to $10^{4}$ years
in the radio bands, about 54 such SNRs may be detected in the sky with a
corresponding 1 GHz radio surface brightness between $10^{-22}-10^{-21}$ W m-2
Hz-1 sr-1. This estimate can be significantly uncertain as it depends on the
assumed density and temperature of the IGrM, which is poorly constrained.
However, this estimate is viable and can be considered as an upper limit as
long as the mean temperature of the IGrM does not exceed $10^{6}$ K (see
discussions in Mulchaey 2000 for IGrM temperature in the local group). If IGrM
temperatures are lower, SNRs detectability may increase significantly for the
older ($>10^{4}$ years) supernovae.

The number of progenitors or the intragroup SNIa rate may also differ from one
group to other as it can depend on the dynamical stage of the group and its
past evolutionary history in terms of tidal interactions and mergers.
Nevertheless, the average SNIa rate estimated from a large number (1401) of
low-redshift groups covered in the survey has a high statistical significance
and can be taken as a most probable average rate among the low redshift galaxy
groups. Therefore, a statistically significant possibility to detect some SNRs
in the IGrM of the local group and its immediate neighbour groups of galaxies
with ASKAP exists with a detection rate of about 1.3 SNR per 1000 square
degree. As the present detection rate of ORCs is much higher at about 1.6
every 100 square degree, not more than about 8 per cent of ORCs can be
associated with SNRs in the IGrM of local groups and its immediate neighbour
groups of galaxies. This detection rate is likely to change, as more ORCs will
be detected in future surveys covering a much larger area in the sky. The main
outcome of this analysis is that the expected detection rate of local group
SNRs in the IGrM is non-negligible yet much smaller than the present rate of
detections of ORCs. A similar analysis for the detectability of intergalactic
SNRs in the Virgo group of galaxies was previously presented in Maoz, Waxman &
Loeb (2005). They estimated about 10 SNRs per square degree with 1 GHz radio
flux of 0.1 mJy in Virgo galaxy cluster at an assumed distance of 10 Mpc and
for SNR detectability lasting for $10^{4}$ years.

###  4.2 On sizes and evolutionary stages of ORCs

Two most intriguing features of the ORCs are nearly identical angular size
($\sim 1$ arcmin) for all the sources and similar radio brightness for 4 out
of 5 sources. If ORCs are dynamically evolving objects similar to SNRs, a
distribution in size and flux density is expected in a survey. There could be
multiple reasons for not seeing this distribution. The ASKAP being an
interferometric survey loses sensitivity to detect extended sources beyond
about $\sim 1$ arcmin (Norris et al., 2021b). The convolved angular resolution
($\sim 20$ arcsec) of the ASKAP images may not be sufficient to distinguish
ORCs smaller than $\sim 1$ arcmin from the known population of other diffuse
radio sources. If the ORCs are residing at widely different redshifts ($z\sim
0.2-0.6$), the physical diameters of the distant ORCs will be larger than
those of the nearby ORCs, but both the cases having identical surface
brightness. If ORCs are dynamically evolving objects similar to SNRs, the
surface brightness is expected to reduce with increasing physical diameter.
Here, the similarity with a SNR is drawn in the context of dynamical behaviour
of an expanding shock and not in terms of input mechanical energy, which could
be much higher in case of super massive black hole merger scenario invoked to
explain ORCs in distant galaxies by Norris, Crawford & Macgregor (2021) and
Koribalski et al. (2021). On the other hand, if ORCs are nearly equidistant,
e.g., in some nearby structure or a group of galaxies covering large angular
extent in the sky, the variations in sizes and brightness are expected to be
less. Therefore, physical origins of the ORCs presently associated with
distant galaxies needs to be explained in a theoretical framework, which
allows a substantial variation of the key dynamical parameters such as input
mechanical energy from the initial explosion, and density and temperature of
the ambient medium. As ORCs seem to belong to different types of objects,
their similar appearances are likely to be related to dynamical evolution of
expanding shocks but with different energies in different astrophysical
processes.

The detection of two ORCs namely ORC-2 and ORC-3 at a very close separation in
the sky possibly also offers another clue about their evolutionary stages.
These two ORCs are separated by only about 1.5 arcmin in the sky. Considering
the rarity of ORCs, these two ORCs can be expected to be physically nearby in
space hence nearly equidistant and are not just a chance superposition in the
sky of two widely separated objects. As no optical counterparts have been
associated with these two ORCs, these may be genuine SNRs in the IGrM. These
two ORCs have nearly equal angular diameter but different flux, $\sim 7$ mJy
for ORC-2 and $\sim 2$ mJy for ORC-3. If these two ORCs are SNRs co-evolving
in an identical ambient environment and at a same distance from us, their
identical angular sizes may imply that these SNRs are seen at a late stage of
evolution where the expansion of the shock has already slowed down. The
difference in flux is then not due to a distance variation but due to their
age difference where the synchrotron losses makes the older object appear
fainter. The ORC-2 and ORC-3 can then be SNRs detected in the IGrM in a late
stage of evolution, when expansions of the SNRs have already slowed down. The
mystery still remains that how two such SNRs come so close in the space.

##  5 Summary and concluding remarks

In this paper, radio surface brightness, luminosity, and angular size of the
ORCs were discussed. We attempted here to trace origins of some ORCs in the
SNRs in the IGrM. Our estimates made under a minimalistic framework, suggest
that some ORCs can be SNRs in the IGrM of the local group and in its immediate
neighbour groups of galaxies within a distance of about 3 Mpc. The estimated
host-less (presumably in IGrM) optical supernovae rate in other nearby groups
suggests that only about 1.3 supernovae per 1000 square degree may be
detected, assuming a detectability in radio for about $10^{4}$ years. The
detectability of old SNRs at an epoch may depend upon the temperature of the
IGrM in the sense that in hot IGrM (T$>10^{6}$ K), radio bright phase of the
SNRs will last for a shorter period than those in a relatively warmer IGrM.
This effect may favour larger number of SNR detections in the galaxy groups
than in the galaxy clusters. As the present detection rate of ORCs is much
higher at about 1.6 every 100 square degree, the ORCs as a class of objects
will be heterogeneous in nature and will very likely trace their origins in
multiple astrophysical objects or events. A similarity in physical appearances
of five ORCs indicates that although the ORCs may trace their origins in
multiple types of objects or events, the underlying phenomena responsible for
their morphologies is likely to be the expanding shock waves in IGrM. With the
present detection rate estimated from just five ORCs, not more than about 8
per cent of ORCs may be associated as SNRs in the IGrM of the local group and
its immediate neighbours. This value is likely to change, as more ORCs will be
detected in future.

Out of the five known ORCs, three ORCs have been associated with an optical
counterpart and two ORCs do not have optical counterparts. Presently, possible
explanations of the origins of ORCs do not seamlessly fit into a single
previously observed energetic phenomena. As some ORCs have an optical galaxy
and some appear host-less, finding origins of ORCs in multiple types of
physical objects and processes appears a clever choice. Given that the present
sample of ORCs is statistically very small and so any extrapolation of one
possible explanation to all such objects need not be carried out and at the
same time any possible or viable scenario should not be left out. Presently,
these important questions remain to be answered - (1) The nearly identical
sizes and radio flux of ORCs associated with an optical galaxy pose a
challenge to understand their origins in deep extra-galactic objects at widely
different redshifts. (2) Two ORCs with a very close separation in the sky. (3)
Missing smaller angular size ORCs in the ASKAP survey.

Finally, it is worth to mention here some important consequences of confirming
detections of SNRs in the IGrM. Domainko et al. (2004) and Zaritsky et al.
(2004) discussed importance of intracluster supernovae to match the observed
Fe abundance excess in the cool-core clusters. Supernovae are also considered
efficient sources of heating and turbulence over large volumes in low-density
hot medium typical of that in the intracluster medium (Valdarnini, 2003;
Domainko et al., 2004; Tang & Wang, 2005). Omar (2019) discussed a possibility
that radio emission from mini halos, another low radio surface brightness
structures on physical scales of a few hundred kpc, undoubtedly associated
with the brightest cluster galaxies in cool-core galaxy clusters, can be
explained in terms of relativistic electrons accelerated in the intracluster
supernovae events. A direct confirmed detection of intergalactic SNRs will
strengthen this possibility. The unprecedented sensitivity combined with the
anticipated sub-arcsec resolution of the upcoming Square Kilometer Array
telescope may enable us to identify SNRs in the IGrM of some nearby galaxy
groups and clusters.

## References

  * Becker et al. (2021) Becker, W., Hurley-Walker, N., Weinberger, Ch., Nicastro, L., Mayer, M. G. F., Merloni, A., Sanders, J., 2021, A& A, 648, A30
  * Belokurov et al. (2017) Belokurov, V., Erkal, D., Deason, A. J., Koposov, S. E., De Angeli, F., Evans, D. W., Fraternali, F., Mackey, D., 2017, MNRAS, 466, 4711
  * Bhattacharya (1990) Bhattacharya, D., 1990, JApA, 11, 125
  * Dilday et al. (2010) Dilday, B. et al., 2010, ApJ, 715, 1021
  * Domainko et al. (2004) Domainko, W., Gitti, M., Schindler, S., Kapferer, W., 2004, A&A, 425, L21
  * Fesen et al. (2021) Fesen, R. A. et al., 2021, ApJ, 920, 90
  * Friemann et al. (2008) Friemann, J. A. et al., 2008, AJ, 135, 338
  * Gal-Yam et al. (2003) Gal-Yam, A., Maoz, D., Guhathakurta, P., Filippenko, A. V., 2003, AJ, 125, 1087
  * Green (2004) Green, D., 2004, Bull. Astron. Soc. India, 32, 335
  * Grillmair & Carlin (2016) Grillmair, C. J., Carlin, J. L., 2016, Astroph. Space Sc. Library, 420, 87
  * Higdon & Lingenfelter (1980) Higdon, J., Lingenfelter, R. E., 1980, ApJ, 239, 867
  * Koribalski et al. (2021) Koribalski, B. S., Norris, R. P., Andernach, H., Rudnick, L., Shabala, S., Filipovic, M., Lenc, E., 2021, MNRAS, 505, L11
  * Maoz, Waxman & Loeb (2005) Maoz, D., Waxman, E., Loeb, A., 2005, ApJ, 632, 847
  * McGee & Balogh (2010) McGee, S. L., Balogh, M. L., 2010, MNRAS, 403, 79
  * Mulchaey (2000) Mulchaey, J. S., 2000, ARA&A, 38, 289
  * Norris et al. (2021a) Norris, R. P. et al., 2021a, PASA, 38, 3
  * Norris et al. (2021b) Norris R. P. et al., 2021b, PASA, 38, e046
  * Norris, Crawford & Macgregor (2021) Norris, R. P., Crawford, E., Macgregor P., 2021, Galaxies, 9, 83
  * Omar (2019) Omar, A., 2019, MNRAS, 484, L141
  * Omar & Dwarakanath (2005) Omar, A., Dwarakanath, K. S., 2005, JApA, 26, 71
  * Pavlović et al. (2018) Pavlović, M. Z., Urośevic, D., Arbutina, B., Orlando, S., Maxted, N., Filipovic, M. D., 2018, ApJ, 852, 84
  * Sand et al. (2008) Sand, D. J., Zaritsky, D., Herbert-Fort, S., Sivanandam, S., Clowe, D., 2008, AJ, 135, 1917
  * Sand et al. (2011) Sand, D. J. et al., 2011, ApJ, 729, 142
  * Tomisaka, Habe & Ikeuchi (1980) Tomisaka, K., Habe, A., Ikeuchi, S., 1980, Progress of Theoretical Physics, 64, 1587
  * Tang & Wang (2005) Tang, S., Wang, Q. D., 2005, ApJ, 628, 205
  * Valdarnini (2003) Valdarnini, R., 2003, MNRAS, 339, 1117
  * van den Bergh (1999) van den Bergh, S., 1999, ARA&A, 9, 273
  * Zaritsky et al. (2004) Zaritsky, D., Gonzalez, A. H., Zabludoff, A. I., 2004, ApJ, 613, L93


# Ordering kinetics in the active Ising model

Sayam Bandyopadhyay<EMAIL_ADDRESS>School of Mathematical &
Computational Sciences, Indian Association for the Cultivation of Science,
Kolkata – 700032, India.  Swarnajit Chatterjee  swarnajit.chatterjee@uni-
saarland.de Center for Biophysics & Department for Theoretical Physics,
Saarland University, 66123 Saarbrücken, Germany  Aditya Kumar Dutta  School of
Mathematical & Computational Sciences, Indian Association for the Cultivation
of Science, Kolkata – 700032, India.  Mintu Karmakar  School of Mathematical &
Computational Sciences, Indian Association for the Cultivation of Science,
Kolkata – 700032, India.  Heiko Rieger<EMAIL_ADDRESS>Center
for Biophysics & Department for Theoretical Physics, Saarland University,
66123 Saarbrücken, Germany  INM – Leibniz Institute for New Materials, Campus
D2 2, 66123 Saarbrücken, Germany  Raja Paul<EMAIL_ADDRESS>School of
Mathematical & Computational Sciences, Indian Association for the Cultivation
of Science, Kolkata – 700032, India.

###### Abstract

We undertake a numerical study of the ordering kinetics in the two-dimensional
($2d$) active Ising model (AIM), a discrete flocking model with a non-
conserved scalar order parameter. We find that for a quench into the liquid-
gas coexistence region and in the ordered liquid region, the characteristic
length scale of both the density and magnetization domains follows the
Lifshitz-Cahn-Allen (LCA) growth law: $R(t)\sim t^{1/2}$, consistent with the
growth law of passive systems with scalar order parameter and non-conserved
dynamics. The system morphology is analyzed with the two-point correlation
function and its Fourier transform, the structure factor, which conforms to
the well-known Porod’s law, a manifestation of the coarsening of compact
domains with smooth boundaries. We also find the domain growth exponent
unaffected by different noise strengths and self-propulsion velocities of the
active particles. However, transverse diffusion is found to play the most
significant role in the growth kinetics of the AIM. We extract the same growth
exponent by solving the hydrodynamic equations of the AIM.

##  I Introduction

Active matter systems involve the movement of large assemblies of individual
active particles that consume energy to self-propel and exhibit collective
behavior in a non-equilibrium steady state Ramaswamy (2010); Marchetti  _et
al._ (2013); Needleman and Dogic (2017); Gompper  _et al._ (2020). Collective
motion is ubiquitous in nature, observed in a wide array of different living
systems over a range of scales, from macroscopic fields like fleets of birds
Ballerini  _et al._ (2008) and schools of fish Becco  _et al._ (2006); Calovi
_et al._ (2014) to microscopic scales like hoards of bacteria Steager  _et
al._ (2008); Peruani  _et al._ (2012), cytoskeletal filaments and molecular
motors Schaller  _et al._ (2010); Sumino  _et al._ (2012); Sanchez  _et al._
(2012). It leads to the emergence of ordered motion of large clusters, called
flocks, with a typical size larger than an individual Ramaswamy (2010); Vicsek
and Zafeiris (2012); Shaebani  _et al._ (2020); De Magistris and Marenduzzo
(2015); Krishnan  _et al._ (2010). Over the past two decades, new models have
emerged to understand the various physical principles governing active matter
systems Shaebani  _et al._ (2020).

Vicsek and collaborators years ago introduced a minimal model Vicsek  _et al._
(1995) of active particles that move with a constant speed and orient via a
ferromagnetic interaction with a neighborhood similar to the XY model. In the
Vicsek model (VM) activity can stabilize the ordered phase even in two
dimensions which is not possible in the $2d$ XY model where the long-range
fluctuation destroys the ordered phase following Mermin-Wagner theorem Toner
and Tu (1995, 1998). Then, about a decade ago, Solon and Tailleur introduced
the active Ising model (AIM) Solon and Tailleur (2013, 2015) where the
continuous rotational symmetry of the VM is replaced by a discrete symmetry.
In the AIM, each particle assumes two possible states allowing the particle to
propel in a preferred direction which changes upon interaction with other
particles at the same lattice site. The AIM retains the essential part of the
VM physics and exhibits flocking behavior with three different phases at
steady state: disordered gas at high noise and low densities, polar liquid at
low noise and high densities, and a phase-separated liquid-gas coexistence
state at intermediate densities. However, a key difference between the VM and
the AIM arises in the steady-state behavior of the coexistence region. In this
region, AIM shows a macrophase separation associated with normal density
fluctuations whereas the VM is characterized by a microphase separation with
giant density fluctuations. The flocking transition in the AIM is a first-
order liquid-gas phase transition similar to the VM, however, for zero
activity, despite the dynamics being non-equilibrium, the AIM shows a second-
order phase transition belonging to the Ising universality class.

Although significant progress has been made to understand the steady state
properties of various active systems Toner  _et al._ (2005); Giomi  _et al._
(2013); Cates and Tailleur (2015); Solon  _et al._ (2015); Siebert  _et al._
(2018); Wysocki and Rieger (2020); Mangeat  _et al._ (2020); Chatterjee  _et
al._ (2020a); Kürsten and Ihle (2020); Fruchart  _et al._ (2021); Solon  _et
al._ (2022); Chatterjee  _et al._ (2022); Codina  _et al._ (2022); Chatterjee
_et al._ (2023); Karmakar  _et al._ (2023), there is much to explore in the
realm of ordering kinetics in active systems that relaxes to a non-equilibrium
steady state (NESS). Understanding the intrinsic non-equilibrium dynamics that
drive an active system towards its steady state is of fundamental as well as
practical relevance. Unlike active systems, ordering kinetics in non-
equilibrium passive systems have been studied over several decades Bray (2002,
1994); Puri (2009); Ahmad  _et al._ (2012); Shrivastav  _et al._ (2014); Kumar
_et al._ (2017); Chatterjee  _et al._ (2020b). Domain growth in passive
systems with non-conserved scalar order parameters follows the Lifshitz-Cahn-
Allen (LCA) growth law: $R(t)\sim t^{1/2}$ (Model A of order-parameter
kinetics) whereas passive systems with conserved order parameter follow a
Lifshitz-Slyozov-Wagner (LSW) growth law: $R(t)\sim t^{1/3}$ (Model B of
order-parameter kinetics), $R$ being average size of domains. Using tools that
quantify the kinetics of passive systems, several active systems have been
explored. These include Active Model B Wittkowski  _et al._ (2014); Pattanayak
_et al._ (2021a, b), active nematics Mishra  _et al._ (2014), self-propelled
particles in disordered medium Das  _et al._ (2018), Model B with
nonreciprocal activity Saha  _et al._ (2020), Kuramoto oscillators Rouzaire
and Levis (2022), active Brownian particles Dittrich  _et al._ (2023) and
motility-induced phase separated (MIPS) clusters Caporusso  _et al._ (2023).
Moreover, an interesting observation of multiple coarsening length scales was
made in the prototypical VM Katyal  _et al._ (2020) where velocities are found
to align over a faster-growing length scale compared to density. Another
intriguing result of an active system with a non-conserved vector order
parameter following the growth law of the non-conserved scalar order parameter
field has also been observed Dikshit and Mishra (2023). Since AIM is a minimal
flocking model with a rich phase behavior, studying the growth kinetics of
this model will allow us to interpret the origin of large flocks in terms of
microscopic interactions.

In this paper, we explore the phase ordering kinetics of the AIM with a non-
conserved scalar order parameter. Quenching the AIM inside the spinodal region
results in the formation of small positively or negatively magnetized clusters
which in the late stage of the coarsening merge to form a single, macroscopic
domain of one spin type Solon and Tailleur (2015). A few questions arise in
this context: (a) Does the domain morphology follow the same pattern and
growth law for quenches into the coexistence and in the ordered liquid region?
(b) Since the order parameter is non-conserved in AIM, how does the growth law
relate to the established growth law of similar passive systems? (c) Do the
density and magnetization align over the same length scale? Katyal  _et al._
(2020) (d) What is the impact of noise and particle activity on the domain
growth? and (e) What is the role of diffusion in the domain growth dynamics?
We address these issues by analyzing the ordering dynamics of the $2d$ AIM on
a square lattice via Monte Carlo simulations and by solving the AIM
hydrodynamic equations using the finite difference method.

This paper is organized as follows. In Sec. II we discuss the model and then
present the details of numerical simulations in Sec. III. In Sec.IV, we
present the growth law of the AIM from both numerical simulation and
hydrodynamic description. Finally, in Sec. V, we conclude this paper with a
summary and discussion of the results.

##  II Model

We consider $N$ particles on a two-dimensional square lattice $L\times L$ with
periodic boundary conditions. Thus the average particle density is
$\rho_{0}=N/L^{2}$. Each lattice site $i$ can accommodate an arbitrary number
of particles $n_{i}^{\sigma}$ with spin $\sigma=\pm 1$. Defining local density
$\rho_{i}=n_{i}^{+}+n_{i}^{-}$ and magnetization $m_{i}=n_{i}^{+}-n_{i}^{-}$
we note that $\rho_{i}$ has no upper bound, while $m_{i}$ is bounded by
$\rho_{i}$: $-\rho_{i}\leq m_{i}\leq\rho_{i}$. Each particle with a given spin
state $\sigma$ can either flip to $-\sigma$ or jump to a nearest-neighbor site
probabilistically.

The flipping rates are derived from a local ferromagnetic Potts Hamiltonian
Mangeat  _et al._ (2020); Chatterjee  _et al._ (2020a) defined as:

$H_{i}=-\frac{J}{2\rho_{i}}\sum_{k=1}^{\rho_{i}}\sum_{l\neq
k}\left(q\delta_{\sigma_{i}^{k},\sigma_{i}^{l}}-1\right)\,,$ (1)

where $J$ is the coupling between any two particles at site $i$ and the
Kronecker delta $\delta_{\sigma_{i}^{k},\sigma_{i}^{l}}$ survives only for
$\sigma_{i}^{k}=\sigma_{i}^{l}$. Eq. (1) with $q=2$ is the local Hamiltonian
defined for the AIM Solon and Tailleur (2015). Local interaction implies that
a particle can align with the average direction of all other particles at the
same site.

The spin-flip transition rates are derived from the Potts Hamiltonian $H_{\rm
APM}=\sum_{i}H_{i}$ according to the energy difference before and after the
spin flip. For $q=2$, a particle with spin $\sigma$ flips its state according
to the transition rate (Mangeat  _et al._ , 2020; Chatterjee  _et al._ ,
2020a):

$W_{\rm flip}(\sigma\rightarrow-\sigma)=\gamma\exp\left[-\frac{2\beta
J}{\rho_{i}}\left(\sigma m_{i}-1\right)\right]\,,$ (2)

where $\gamma$ is the rate of particle flipping when $\sigma m_{i}=1$. In this
paper, we choose $J=1$, and $\gamma=1$ without any loss of generality. The
parameter $\beta$, denoted as “inverse temperature”, $\beta=T^{-1}$, in
passive systems, controls the flip noise strength. Although the system under
consideration is athermal, we denote the parameter $T$ as “temperature” from
now on.

Moreover, each particle performs a biased diffusion on the lattice depending
on the spin state $\sigma$. For $q=2$ particles perform a one-dimensional
biased hopping along the $\pm x$-direction with the following hopping rate
Mangeat  _et al._ (2020); Chatterjee  _et al._ (2020a):

$W_{\rm hop}(\sigma,p)=D\left[1+\epsilon({2\delta_{\sigma,p}-1})\right]\,,$
(3)

where $p$ denotes the direction of bias-hopping ($p=+1$ along the $+x$
direction and $p=-1$ along the $-x$ direction). The presence of other
particles does not influence hopping rates and hence is independent of
particle density. The hopping rate $D=1$ is constant along the upward and
downward directions ($\pm y$-direction). The parameter $\epsilon\in[0,1]$
$(0\leq\epsilon\leq q-1)$ controls the asymmetry between the purely diffusive
limit $\epsilon=0$ and the purely ballistic limit $\epsilon=1$, while $D$
controls the overall hopping rate. On average, a particle drifts with speed
$v=2D\epsilon$ in the direction set by the sign of its spin state (where
lattice spacing is 1), while the total hopping rate $4D$, remains constant.

##  III Simulation details

A Monte Carlo (MC) simulation of the stochastic process defined above evolves
in unit Monte Carlo steps (MCS) $\Delta t$ resulting from a microscopic time
$\Delta t/N$. During $\Delta t/N$, a randomly chosen particle with spin
$\sigma$ flips with probability $W_{\rm flip}\Delta t$ or hops to one of the
neighboring sites with probability $W_{\rm hop}\Delta t$. Consequently,
$1-[4D+W_{\rm flip}]\Delta t$ is the probability that the particle does
nothing, and minimizing this we obtain $\Delta t=[4D+\exp(2\beta)]^{-1}$.

To study the morphology of the system during phase ordering kinetics, we use
the two-point equal-time $(t)$ correlation function of the local scalar order
parameters. The notion utilizes the spatial fluctuations in the density and
magnetization fields to estimate Dikshit and Mishra (2023):

$C_{\rho\rho}(r,t)=\frac{1}{L^{2}}\sum_{i=1}^{L^{2}}\langle\Delta\rho_{i,t}\Delta\rho_{i+r,t}\rangle$
(4) $C_{mm}(r,t)=\frac{1}{L^{2}}\sum_{i=1}^{L^{2}}\langle\Delta m_{i,t}\Delta
m_{i+r,t}\rangle\,,$ (5)

where $\langle\cdots\rangle$ denotes averaging over independent initial
realizations, $\Delta\rho_{i,t}=\rho_{i}-\rho_{0}$ and $\Delta
m_{i,t}=m_{i}-m_{0}$ are the local fluctuations in density and magnetization
from the mean, respectively. The above definition of $C_{\rho\rho}$
characterizes the morphology of the spatial structures and $C_{mm}$ evaluates
the correlations in the polar alignment among the evolving structures
separated by a distance $r$. Since we observe that $C_{\rho\rho}$ and $C_{mm}$
behave similarly in the AIM (see below), we focus here on $C_{\rho\rho}$,
which we denote as $C$ [$C_{\rho\rho}(r,t)\equiv C(r,t)$] from now on.
Following a temperature quench from a random initial configuration into the
ordered state, domains of both spin types appear and grow with time. Similar
morphology of the evolving domains with average domain size $R(t)$ would
correspond to a dynamical scaling relation Puri (2009); Bray (2002, 1994):

$C(r,t)=f\left(\frac{r}{R(t)}\right)$ (6)

where $f(x)$ is a time-independent scaling function. $R(t)$, estimated from
the decay of $C(r,t)$ generally show a power-law growth Puri (2009); Bray
(2002, 1994):

$R(t)\sim t^{\theta}\,,$ (7)

with $\theta$ as the growth exponent. Typically, the morphology of an ordering
system is studied by scattering experiments, which measure the structure
factor $S(k,t)$, defined by the Fourier transform of the correlation function
$C(r,t)$:

$S(k,t)=\int_{-\infty}^{\infty}C(r,t)e^{ikr}dr\,,$ (8)

and has a dynamical scaling form in $d$ dimensions:

$S(k,t)=R(t)^{d}g\left[kR(t)\right]\,.$ (9)

For scalar order parameters like the density field, the short-distance
(large-$k$) behavior of the structure factor scaling function is given by the
Porod’s law (for domains with smooth boundaries or scattering off sharp domain
interfaces) which corresponds to $g(k)\sim k^{-(d+1)}$. Next, we present
results for model parameters set by the average particle density $\rho_{0}$,
temperature $T=\beta^{-1}$, self-propulsion velocity $\epsilon$, and diffusion
constant $D$.

##  IV Results

###  IV.1 Phase diagram and domain morphology

The steady-state behavior of the AIM is summarized in the temperature-density
($T-\rho_{0}$) [Fig. 1(a)] and velocity-density ($\epsilon-\rho_{0}$) [Fig.
1(b)] phase diagrams. The general structure of the phase diagrams consists of
a gas phase (G), a liquid phase (L), and a liquid-gas coexistence region (G+L)
separated by the gas and liquid binodals. Qualitatively similar results were
obtained earlier in Ref. Solon and Tailleur (2015). However, we obtain a
critical temperature $T_{c}\simeq 2$ (above which no phase separation occurs
irrespective of the density, critical density $\rho_{c}=\infty$) twice as
large as in Ref. Solon and Tailleur (2015). This can be understood if one
compares the flipping rate of Ref. Solon and Tailleur (2015), $W_{\rm
flip}(\sigma\rightarrow-\sigma)=\gamma\exp\left(-\sigma\beta\frac{m_{i}}{\rho_{i}}\right)$
with Eq. (2) for $J=1$ which denotes that the effective $\beta$ considered in
this paper is roughly twice as large as the $\beta$ of Ref. Solon and Tailleur
(2015). In the $\epsilon-\rho_{0}$ phase diagram, the two binodals converge at
a critical density $(\rho_{c}\simeq 2.9)$ for vanishing self-propulsion
($\epsilon=0$) which signifies a second-order phase transition belonging to
the Ising universality class Solon and Tailleur (2013).

We chose to quench the random initial systems in two different regimes of the
temperature-density phase diagram shown by the arrows in Fig. 1(a). The quench
occurs instantaneously from a very high-temperature regime
($T\rightarrow\infty$) shown by the blue stars to the liquid-gas coexistence
region or the polar liquid region, denoted by red stars, with the same final
temperature $T=0.9$ ($T<T_{c}$, corresponds to $\beta=1.1$) but different
densities.

To quantify the coarsening dynamics we performed simulations on a square
lattice of size $400^{2}$ with periodic boundary conditions applied on both
sides. Following quench at time $t=0$, the system is evolved up to $t=10^{5}$
using the MC algorithm described in Sec. III. All numerical data presented
here are averaged over at least 300 independent realizations.

Figure 1: (Color online) Phase diagrams of the AIM showing the quench
directions at fixed densities. (a) $(T,\rho_{0})$ phase diagram for self-
propulsion velocity $\epsilon=1$. (b) $(\epsilon,\rho_{0})$ phase diagram at
fixed temperature $T=0.9$. G and L denote the disordered gas and polar liquid
regions whereas G+L denotes the phase-coexistence region. The blue stars
indicate the initial high-temperature points in the phase diagram from where
the system is quenched (indicated by arrows) into the coexistence region
($\rho_{0}=3$) and to the liquid region ($\rho_{0}=10$) [red stars].
$\rho_{\rm gas}$ and $\rho_{\rm liquid}$ are the gas and liquid binodal,
respectively. Figure 2: (Color online) Time evolution snapshots of the local
magnetization field on a $400^{2}$ system after a quench from $T=\infty$ to
$T=0.9$ for $\epsilon=0$ (top panel) and $\epsilon=1$ (middle and lower
panel). The color bar denotes magnetization per site. (a–e) Curvature-driven
domain growth of the diffusive Ising model ($\rho_{0}=5$, $\epsilon=0$). (f–j)
Growth dominated by the dynamics of spinodal decomposition after the system is
quenched inside the spinodal region ($\rho_{0}=3$, $\epsilon=1$) and (k–o)
domain growth mediated by the merging of high-density clusters of $\sigma=\pm
1$ after the system is quenched deep inside the ordered liquid regime
($\rho_{0}=10$, $\epsilon=1$). Black arrows denote the direction of the
movement of the clusters and bands.

A typical simulation of coarsening dynamics starts with a homogeneous initial
configuration where particles with $\sigma=\pm 1$ are distributed randomly on
each lattice site. This can also be described as an equilibrium configuration
at infinite temperature because all configurations are equally likely. Then,
after a quench well below the critical temperature ($T_{c}\simeq 2$), the
homogeneous initial configuration starts evolving in time, and subsequent
dynamics are governed by the formation and growth of $\sigma=1$ and
$\sigma=-1$ rich domains. Such a time evolution of the local magnetization
field is shown in Fig. 2 for $\epsilon=0$ and $\epsilon=1$. In the latter
scenario, time evolution has been shown for a quench in both the coexistence
(middle row) and liquid regions (bottom row). The non-equilibrium steady
states (NESS) are characterized by a single band (at lower density) and a
polar liquid (at higher density).

The top panel, Fig. 2(a–e), depicts the ordering kinetics of a purely
diffusive $(\epsilon=0)$ polar liquid at $\rho_{0}=5$. The domain morphology
with increasing time exhibits a close resemblance with the passive Ising model
Majumder and Das (2011). In the Ising model, the driving force for the domain
growth is the curvature of the domain wall since the system surface energy can
only decrease through a reduction in the total net surface area. In the
$\epsilon=0$ limit, particles do not form high-density domains due to the
diffusive movement of particles. Therefore, density-wise, the system remains
homogeneous as we observe a steady growth of small, high-curvature to large,
low-curvature domains. This is unsurprising as the $2d$ AIM for $\epsilon=0$
belongs to the same universality class of the passive $2d$ Ising model Solon
and Tailleur (2015).

Next, we looked at the evolution of AIM with self-propulsion velocity
($\epsilon=1$) quenching the system inside the spinodal [Fig. 2(f–j)] and
homogeneous ordered regions [Fig. 2(k–o)]. The average densities representing
these two regions correspond to $\rho_{0}=3$ and 10, respectively. Inside the
spinodal region, the growth dynamics are driven by spinodal decomposition and
result in the formation of numerous small clusters of negative and positive
spins $(t=10^{2})$. The coarsening then stems from the merging of these
clusters $(t=10^{3}$ and $t=10^{4})$, until a single, macroscopic domain
emerges $(t=10^{5})$ in the steady state. A quench deep inside the homogeneous
ordered region also results in similar dynamics of cluster formation and
coalescence of those clusters into a single large liquid domain (in this
paper, for $\epsilon>0$, the word cluster is used interchangeably with
domain). With extreme self-propelled particles ($\epsilon=1$), although the
high-density clusters are strongly biased along the horizontal directions,
they can also grow along the transverse direction due to the constant
transverse diffusion $D$. When these clusters merge into a larger cluster, it
always tries to minimize the surface energy by decreasing the surface area.
Accordingly, the coarsening process leads to a single band [Fig. 2(j)] with
domain walls having the lowest curvature (the curvature of a straight line is
zero but a liquid band with a perfectly straight domain boundary can only
happen when there is no thermal fluctuation). The focus of this study is,
therefore, to analyze the coarsening dynamics shown in Fig. 2 which we will do
next.

###  IV.2 Dynamical scaling

Figure 3: (Color online) (a–c) Scaled correlation functions $C(r/R,t)$ versus
$r/R$, for the evolution of the $2d$ AIM shown in Fig. 2. Unscaled versions
are shown in the inset. “Liquid” and “coexistence” signify the region where
the system is quenched, $\epsilon$ denotes particle speed. (d) Log-log plot of
$1-C(r/R)$ versus $r/R$. The cusp exponent is estimated as $\alpha\sim 1$.
Parameters correspond to Fig. 2. Figure 4: (Color online) (a–c) Scaled
structure factors, $S(k,t)R(t)^{-2}$ versus $kR$, for the Fourier transform of
the correlation function data sets corresponding to the same values of time.
The line of slope $\simeq-3$ denotes the Porod’s law: $S(k)\sim k^{-(d+1)}$
for $d=2$.

In the theory of phase-ordering kinetics, the scaling hypothesis states that
if the system is characterized by a single length scale $R(t)$ [$R(t)$ is
equivalent to the average domain size], the domain morphology is statistically
the same at all times, apart from a scale factor. When all domain lengths are
measured in units of $R(t)$, the equal-time pair correlation function should
exhibit the dynamical-scaling property of Eq. (6). The equal-time correlation
function is a non-equilibrium quantity so as $R(t)$ which can be estimated
from the decay of the correlation function. To identify the self-similar
behavior of the evolving domains we plot the correlation function in Fig. 3.
The scaled $C(r/R,t)$ and unscaled $C(r,t)$ correlation functions are shown in
Fig. 3(a–c) for $\epsilon=0$ and $\epsilon=1$. $R(t)$ is determined from the
distance over which the correlation function decays to e.g., 0.2 of its
maximum value, that is, $C(r,t)=0.2C(0,t)$.

As the system coarsens, the correlation function decays slowly [insets of Fig.
3(a–c)], signifying the growth of the characteristic length scale $R(t)$. Upon
rescaling the spatial coordinates by this length scale, the correlation
function at different times collapses onto a single function $C(r/R,t)$ as
shown in Fig. 3(a–c), thus confirming a universal coarsening behavior with
time. Such scaling behavior implies that the structure is time-invariant and
consistent with a power-law growth of $R(t)$ with increasing $t$. This scaling
hypothesis is satisfied when the growing length is much smaller than the
system size to avoid the finite size effect. We further examine the AIM domain
morphology by approximating the small distance behavior of the scaled two-
point correlation function

$1-C(r)=\bar{C}(r)\sim r^{\alpha}$ (10)

in Fig. 3(d) which yields the cusp exponent $\alpha\sim 1$. This signifies the
existence of sharp domain interfaces [see Fig. 2(h) and Fig. 2(m)] and
translates into the power-law behavior of the scaled structure factor plotted
in Fig. 4 Shrivastav  _et al._ (2014). Fig. 3(d) also signifies that the
domain structure of the AIM for $\epsilon=1$ is statistically self-similar for
quenches into the coexistence and liquid region whereas different from the
domain morphology of the AIM for $\epsilon=0$, as evident from Fig. 2.

In Fig. 4, we plot the scaled structure factor, $S(k,t)R(t)^{-2}$ versus $kR$,
which is the Fourier transform of the correlation function. In Fourier space,
Eq. (10) translates into the following power-law behavior of the structure
factor: $S(k)\sim k^{-(d+\alpha)}$ and therefore, the large-$k$ behavior of
the structure factor tail generates a slope $\sim-3$ (in log-log plot) for
$d=2$ and $\alpha=1$ which denotes “Porod’s decay”: $S(k)\sim k^{-(d+1)}$,
associated with scattering from sharp interfaces Bray (2002, 1994). This
naturally originates from the long-range ordering in AIM leading to compact
high-density clusters with smooth boundaries. Domains with rough morphologies
having fractal interfaces do not follow Porod’s law and the large-$k$ tail of
the scaled structure factor yields a non-integer exponent Chatterjee  _et al._
(2018). A similar violation of the Porod law was also observed in the
coarsening of the VM due to the irregular morphology associated with the
cluster boundaries Dey  _et al._ (2012); Katyal  _et al._ (2020). $C(r,t)$
[and consequently, $S(k,t)$] exhibits two distinct power laws for small and
large $r/R$ limits [large and small $kR$ limits] in the VM Dey  _et al._
(2012) which we do not observe in the AIM. Furthermore, density fluctuations
might also play a role in determining whether a system will follow or violate
Porod’s law. In the VM, giant density fluctuations break large liquid domains
and restrict the formation of large compact domains (which eventually
manifests in the microphase separation of the coexistence region in the
steady-state) Solon  _et al._ (2015) and might be responsible for the non-
Porod behavior of the system Dey  _et al._ (2012). On the other hand, AIM
obeys the Porod behavior where density fluctuations are normal in the liquid
phase Solon and Tailleur (2015) and the steady-state manifests a bulk phase
separation.

###  IV.3 Growth law

Figure 5: (Color online) (a) $R(t)\sim t^{1/2}$ for a quench into the
coexistence and liquid regimes. The late time saturation of $R(t)$ signifies
that the system has reached the corresponding NESS. As the density is higher
for the liquid quench, the saturation appears late. Parameters: $\rho_{0}=3$
(coexistence), $\rho=10$ (liquid), $L=400$, $\beta=1.1$ and $\epsilon=1$. (b)
$R(t)$ versus $t$ for different thermal noises ($T=\beta^{-1}$, coexistence
regime). Parameters: $L=400$, $\rho_{0}=3$ and $\epsilon=1$. (c) Role of self-
propulsion on the characteristic length $R(t)$ as a function of $t$. For a
larger $\epsilon$, the system reaches the NESS faster while for $\epsilon=0$,
the corresponding dynamics is slow. Parameters: $L=400$, $\beta=1.1$, and
$\rho_{0}=5$. The growth law $R(t)\sim t^{1/2}$ is unaffected by thermal
fluctuations and particle activity.

In passive systems with non-conserved scalar order parameters, the late-stage
domain growth is governed by the diffusive Lifshitz-Cahn-Allen (LCA) growth
law $R(t)\sim t^{1/2}$ Bray (1994). For non-conserved systems described by
scalar fields such as the Ising model, the growth process is driven by the
diffusion of the domain walls (the simplest form of topological defect) caused
by the local changes in the order parameter. Diffusion also influences
coarsening in non-conserved systems with vector fields, such as the $2d$ XY
model, where domain evolution occurs when point topological defects, such as
vortices and anti-vortices, diffuse, interact, and annihilate Yurke  _et al._
(1993). Both systems exhibit a diffusive growth exponent $\theta=\frac{1}{2}$
(in the XY model, $R(t)\sim(t/\ln t)^{1/2}$, the logarithmic correction is due
to the free vortices Yurke  _et al._ (1993)). Therefore, a 0.5 growth exponent
signifies a coarsening process dominated by the diffusion of defects. For
example, if the time evolution of a non-conserved scalar order parameter such
as the magnetization $m$ follows the diffusive equation
$\dot{m}=D\nabla^{2}m$, then the length scale will exhibit a $\sqrt{t}$
dependence.

Therefore, the immediate question one can ask is whether the LCA growth law
also governs the late-stage coarsening dynamics of an active system with non-
conserved scalar order parameters such as the AIM and whether the coarsening
process is diffusion-dominated. To characterize this, we plot the length scale
data with time in Fig. 5 for various control parameters and quench regimes. We
find that the late-stage growth kinetics of the domain in the coexistence and
liquid regimes exhibit a $R(t)\sim t^{1/2}$ growth law [Fig. 5(a)]. We extract
the same growth law for different temperatures [Fig. 5(b)], and self-
propulsion velocities [Fig. 5(c)]. Domains identified by the correlation of
density and magnetization fields also show similar growth behavior (see
Appendix A). An interesting feature of the domain growth kinetics is that the
domain size for a quench into the liquid regime is larger than the
corresponding domain size in the coexistence regime [see Fig. 2 at $t=10^{3}$
and Fig.5(a)], although the domain morphologies for both these quench regimes
are self-similar. The larger domains in the liquid regime are likely to be
arising from higher density. Also, notice that a larger $\beta$ in Fig. 5(b)
signifies a reduced thermal noise that slows down the local ordering of spins.
Thus smaller $\beta$ allows the formation of larger domains. Nevertheless,
different thermal fluctuations exhibit the same growth as thermal noise is
asymptotically irrelevant for ordering in systems that are free from disorder
Puri (2009).

Fig. 5(c) shows the increasing length scale with time for different
$\epsilon$. For $\epsilon=0$, the system is purely diffusive, and domain
growth proceeds via the coarsening of connected domains (curvature-driven
growth facilitated by diffusing domain wall) [see Fig. 2(c–d)]. The domain
morphology looks very similar to the evolution of an Ising ferromagnet
quenched below the critical temperature. Thus, a $\sim t^{1/2}$ growth law for
$\epsilon=0$ similar to the pure Ising model is physically plausible. But,
when $\epsilon>0$, the domains of each spin no longer remain connected and
form high-density clusters that self-propel along the horizontal direction. As
time progresses, these clusters spread in the transverse direction due to
constant diffusion ($D=1$) and merge with other clusters. Therefore, the
domain growth for $\epsilon>0$ is again a diffusive phenomenon, and
consequently, the growth kinetics exhibits a $R(t)\sim t^{1/2}$ growth law
similar to $\epsilon=0$. We have identified this novel mechanism of diffusion-
driven domain growth in the AIM after a thorough investigation of the altered
diffusion coefficient in Sec. IV.4 and Appendix B.

It should also be noted that while approaching the NESS via coarsening, the
high-density AIM domains of individual spins, upon merging, try to minimize
the surface energy by decreasing the surface area similar to the dynamics of
the passive non-conserved scalar order parameter. Therefore, although the
system is active, the critical mechanisms (diffusion-dominated growth and
minimization of surface energy during coarsening) of domain coarsening in AIM
are similar to the $2d$ Ising model, and thus, it is not surprising that we
extract a $R(t)\sim t^{1/2}$ growth law for both the passive and active
models. For a more precise quantification of the asymptotic growth law, we
determine the effective growth exponent, defined as:

$Z_{\rm eff}=\frac{d[\ln R(t)]}{d[\ln t]}\,.$ (11)

In Fig. 6(a), we plot $Z_{\rm eff}$ versus $t$ for different $\epsilon$
corresponding to the data in Fig.5(c). All data show extended flat regimes at
late times (after $\sim t>10^{2}$). The effective exponent turns out to be in
the regime $0.45<Z_{\rm eff}<0.52$, denoted by the dashed lines. We perform a
similar study for the quench into the coexistence and liquid regimes. Fig.
6(b) shows $Z_{\rm eff}$ vs. $t$ once again confirms $Z_{\rm eff}\sim 0.5$,
irrespective of the quench regimes.

Figure 6: (Color online) Effective growth exponent $Z_{\rm eff}$ versus time
$t$ for (a) different self-propulsion ($\epsilon=0,0.5$, and $1$, $\beta=1.1$
and $\rho_{0}=3$) and (b) quenching into different NESS ($\rho_{0}=3$ for
coexistence and $\rho_{0}=10$ for liquid, $\beta=1.1$ and $\epsilon=1$). The
dashed lines are a guide to the eyes.

###  IV.4 Role of transverse diffusion

Figure 7: (Color online) Time evolution of the local magnetization field for
$D_{\perp}=0$ [top panel, (a–c)] and $D_{\perp}=1$ [bottom panel, (d–f)] after
quenching the system from a disordered gaseous phase to a high-density region.
Parameters: $L=400,\beta=1.1$, $\rho_{0}=10$, and $\epsilon=1$.

In the AIM, particles self-propel only in the horizontal direction with
average velocity $2D\epsilon$. The directional hopping of the particles is a
function of the spin type ($+\sigma$ or $-\sigma$). To test the hypothesis
that the late time domain growth in the AIM is also a diffusion-driven
process, we decompose the diffusion into two components, $D_{\perp}$ (along
$\pm y$ direction) and $D_{\parallel}$ (along $\pm x$ direction). We vary
$D_{\perp}$, keeping $D_{\parallel}=D=1$ henceforth, and show the domain
evolution in Fig.7.

The time evolution of the domains for $D_{\perp}=0$ is shown in Fig. 7(a–c).
Starting from a disordered configuration, the time evolution of the system
progresses via the formation of $1d$ rings with domains of alternating
polarity along the horizontal direction ($x-$direction). These domains are one
lattice unit wide along the transverse direction ($y-$direction) and are
independent of the neighboring rings [Fig. 7(b)]. As particles can not diffuse
along the transverse direction, these domains can only grow horizontally
without merging with the neighboring rings. At late times, we observe narrow
horizontal stripes of alternating magnetization [Fig. 7(c)]. Therefore, AIM
with $D_{\perp}=0$ does not manifest the observed domain morphology
representative of the growth law $\sim t^{1/2}$. $D_{\perp}=0$ also signifies
$L_{y}$ numbers of one-dimensional periodic rings on which the AIM is defined.
Such $1d$ AIM has been found to display flocking of a single dense ordered
aggregate at intermediate temperatures (this flock undergoes stochastic
reversals of its magnetization with time) but an aster phase consisting of
sharp peaks of positive and negative magnetizations in a jammed state at lower
temperatures Benvegnen  _et al._ (2022). The one lattice unit-wide (along $y$)
domains in Fig. 7(c) manifest the characteristics of the flocking state of
$1d$ AIM for the given parameters (see Appendix B for details) but the system
does not exhibit flocking as a whole since the $L_{y}$ number of $1d$ rings do
not interact with each other for $D_{\perp}=0$. Altering the transverse
diffusion $D_{\perp}$ to a nonzero value, particle diffusion occurs along the
transverse $\pm y$ direction. Therefore, the random initial state coarsens to
form high-density clusters which coarsen further to give rise to a large
flocking domain [Fig. 7(d-f)].

Figure 8: (Color online) $R(t)$ versus $t$ (on a log-log scale) showing a 0.5
growth exponent for different transverse diffusion $D_{\perp}=0.3$ and
$D_{\perp}=1$. Inset: $R(t)$ versus $t$ in the absence of any transverse
diffusion ($D_{\perp}=0$) showing no growth of domains with time. Parameters:
$L=400$, $\beta=1.1$, $\epsilon=1$, and $\rho_{0}=8$.

To quantify the role of the magnitude of transverse diffusion coefficient
($D_{\perp}$), we plot $R(t)$ versus $t$ in Fig. 8 for two different values of
$D_{\perp}$. The plot shows that the system exhibits the power-law growth of
$R(t)\sim t^{1/2}$ for different values of $D_{\perp}$. Interestingly, even a
small $D_{\perp}=0.3$ is sufficient to drive the proper domain growth. A
larger $D_{\perp}$ only increases the average domain size. As discussed
earlier in Fig. 7(b–c), a vanishing $D_{\perp}$ can not initiate a domain
growth as $R(t)$ remains constant with $t$ (see inset of Fig. 8). Therefore,
transverse diffusion plays the most crucial role in the AIM growth kinetics.

###  IV.5 Growth law from the hydrodynamic description of the AIM

In this section, we want to investigate whether the continuous description of
the AIM also manifests the same time dependency $(\sim t^{1/2})$ of the
coarsening length scale as observed in our numerical analysis. We consider
refined mean-field equations similar to Ref. Solon and Tailleur (2015) for the
spatiotemporal evolutions of the density $(\rho)$ and magnetization $(m)$
fields:

$\displaystyle\dot{\rho}=D\nabla^{2}\rho-v\partial_{x}m\,,$ (12)
$\displaystyle\dot{m}=D\nabla^{2}m-v\partial_{x}\rho+2\left(2\beta-1-\frac{r}{\rho}\right)m-\alpha\frac{m^{3}}{\rho^{2}}\,,$
(13)

where $v=2D\epsilon$, $\alpha=4\beta^{2}(1-2\beta/3)$, and
$r=3\alpha\alpha_{m}/2$ is a positive function of $\beta$ Solon and Tailleur
(2015). In constructing Eq. (13), we slightly modify the flipping rate
equation of Eq. (2) to read $W_{\rm flip}=\exp\left(-2\beta\sigma
m/\rho\right)$. For the mean-field equations, $r=0$, which can not capture the
AIM physics correctly as the system always exhibits homogeneous profiles (gas
and liquid), and the inhomogeneous phase-separated profiles are never observed
Solon and Tailleur (2015). In Eqs. (12) and (13), local fluctuations are taken
into account, which are generally neglected in the mean-field approximations
Solon and Tailleur (2015).

Figure 9: (Color online) Time evolution snapshots of the magnetization field
$m$ obtained by solving the AIM hydrodynamic equations Eq. (12) and Eq. (13)
after a quench from the homogeneous gaseous phase to an ordered liquid phase.
Parameters: $L=400$, $\beta=0.75$, and $\rho_{0}=5$.

Eqs. (12) and (13) immediately show the importance of the transverse
diffusion. If we start with a $x$-independent initial condition, for instance,
a horizontal thin stripe of width $w$, where: $\rho(y)=\rho_{0}/w$,
$m(y)=m_{0}>0$ is the stationary solution of Eq. (13) for $0<y<w$, and
$\rho(y)=\delta\ll\rho_{0}/w$ and $m_{0}=0$ otherwise. Then the solution stays
$x$-independent at all later times: $\rho(x,y,t)=\tilde{\rho}(y,t)$ and
$m(x,y,t)=\tilde{m}(y,t)$ where the equations for $\tilde{\rho}$ and
$\tilde{m}$ are:

$\displaystyle\dot{\tilde{\rho}}=D\partial_{yy}\tilde{\rho}\,,$ (14)
$\displaystyle\dot{\tilde{m}}=D\partial_{yy}\tilde{m}+2\left(2\beta-1-\frac{r}{\tilde{\rho}}\right)\tilde{m}-\alpha\frac{\tilde{m}^{3}}{\tilde{\rho}^{2}}\,.$
(15)

Eq. (14) shows that the thin stripe expands diffusively in the $y$-direction
(transverse direction) yielding a $\sqrt{t}$ dependence of the stripe width.
For general initial conditions, we use explicit Euler FTCS (Forward Time
Centered Space) Press (2007) differencing scheme to numerically integrate Eqs.
(12) and (13). We solve these two coupled partial differential equations on a
square domain of size $L\times L$ with periodic boundary conditions applied in
both directions. In our simulation, $L=400$ and the maximum simulation time is
$t_{\rm sim}=5\times 10^{6}$. To maintain the numerical stability criteria, we
set $\Delta x=1$ as the discretization in space and $\Delta t=10^{-3}$ as the
discretization in time. These discretization parameters satisfy the Courant-
Friedrichs-Lewy (CFL) stability condition. In our numerical implementation, we
fix $D=r=v=1$ and the initial system is prepared as a high-noise homogeneous
gas phase with $\rho=\rho_{0}$ and $m=0$ by adding a zero-mean scalar Gaussian
white noise to Eq. (13) Solon  _et al._ (2015). We then calculate the spatial
dependence of the density and magnetization correlation using Eqs. (4) and (5)
for 25 independent realizations. Finally, $R(t)$ is determined where the
ensemble-averaged correlation functions decay to 0.2 of its maximum value.

Figure 10: (Color online) $R(t)$ versus $t$ (on a log-log scale) exhibiting a
growth exponent $\theta=1/2$ by solving the AIM hydrodynamic equations Eq.
(12) and Eq. (13) using the FTCS scheme. Parameters: $L=400$, $\beta=0.75$,
and $\rho_{0}=5$.

In Fig. 9, we plot the time evolution of the magnetization field $m$ by
solving Eqs. (12) and (13) via the finite difference FTCS scheme. The
formation of self-propelling clusters with smooth interfaces and their growth
with time resembles the dynamics shown in Fig. 2 for the time-evolution of the
microscopic model [Eqs. (1–3)]. We also extract the LCA growth law $R(t)\sim
t^{1/2}$ (as shown in Fig. 10 on a logarithmic scale) by solving the
hydrodynamic equations after quenching the system from a disordered gaseous
phase to an ordered liquid phase. The length scale in Fig. 10 is obtained from
the equal-time spatial correlation of the density fields although the length
scale obtained from the correlation of the magnetization fields also exhibits
the same growth law. Therefore, the self-propulsion terms in the hydrodynamic
equations (12) and (13) of the AIM do not affect the asymptotic growth law of
the non-conserved Model A.

##  V Summary and Discussion

We conclude this paper with a summary and discussion of our results. We study
the ordering kinetics of the active Ising model (AIM), a flocking model with a
non-conserved scalar order parameter, after it is quenched from a disordered
high-temperature gaseous phase to the phase-coexistence region and the polar-
ordered liquid phase. We observe the formation of connected domains of
negative and positive spins similar to the ferromagnetic Ising model in the
zero activity diffusive limit of the AIM. But for self-propelled particles,
AIM manifests an extensive number of disconnected small clusters of the
negative and positive spins which eventually merge to a single, macroscopic
liquid domain Solon and Tailleur (2015). The domain evolution morphology is
characterized by the equal-time two-point correlation function and its Fourier
transform, the structure factor. The scaling of the correlation function
exhibits good data collapse, signifying a self-similar nature of the domain
growth and the large-$k$ behavior of the scaled structure factor tail shows
the Porod’s decay which signifies the smooth spatial structure of the AIM
domains. The growth law we extract for the AIM follows the Lifshitz-Cahn-Allen
(LCA) growth law Bray (2002, 1994); Puri (2009) $R(t)\sim t^{1/2}$ of the non-
conserved scalar order parameter and is unaffected by the system control
parameters such as temperature, density, and particle velocity. Unlike the VM
Katyal  _et al._ (2020), in AIM, the density domain aligns over the same
length scale as the orientation and we do not observe any activity-induced
correction to the growth law of non-conserved scalar order parameter discussed
in the context of the active polar fluid Dikshit and Mishra (2023). We further
investigate the role of diffusion on the AIM growth kinetics. We observe that
due to the horizontal biased hopping (along $\pm x$-direction) of the AIM
clusters, particle diffusion along the vertical $\pm y$-direction is the
predominant mechanism through which coarsening happens in the AIM. This
establishes diffusion as the principal growth mechanism rather than activity,
leading to a $R(t)\sim t^{1/2}$ growth law for the AIM. We further solve the
AIM hydrodynamic equations via a finite difference scheme and the extracted
coarsening length scale validates the growth exponent $\theta\simeq 1/2$
observed in the microscopic simulation.

An interesting extension of this study would be to explore the phase ordering
kinetics in flocking models in the presence of disorder as experimental
systems always contain both quenched and mobile impurities. The similarity in
the growth law of a passive system and its active counterpart is an
interesting result and therefore, further studies on the coarsening dynamics
of active systems, such as the active Potts model or active clock model where
the growth law of the corresponding passive models are well-known Grest  _et
al._ (1988); Chatterjee  _et al._ (2018, 2020b), are required to confirm (or
contradict) this theoretical observation.

##  VI Acknowledgments

MK acknowledges financial support through a research fellowship from CSIR,
Govt. of India (Award Number: 09/080(1106)/2019-EMR-I). RP, SB, AD, and MK
thank the Indian Association for the Cultivation of Science (IACS) for its
computational facility and resources. SC and HR are financially supported by
the German Research Foundation (DFG) within the Collaborative Research Center
SFB 1027-A3 and INST 256/539-1. SC acknowledges many helpful discussions with
Dr Matthieu Mangeat.

##  Appendix A Comparison of the growth law for the density and magnetization
field

Figure 11: (Color online) $R(t)$ versus $t$ for density and magnetization
field yields the same growth exponent 0.5 for a quench in the coexistence
regime, $\rho_{0}=3$. (Inset) A similar growth law is extracted for a quench
in the liquid regime, $\rho_{0}=10$. Parameters: $L=400$, $\beta=1.1$ and
$\epsilon=1$.

The ordering kinetics of the AIM discussed in this paper are studied mainly by
extracting the length scale $R(t)$ from the equal-time two-point density
correlation function defined in Eq. (4). However, one can also extract $R(t)$
from the magnetization correlation function defined in Eq. (5). Now, it was
argued in the context of the coarsening dynamics of the VM that, unlike
generic coarsening systems which typically exhibit a single dominant length
scale, the Vicsek model exhibits distinct coarsening length scales for the
density and velocity correlations Katyal  _et al._ (2020). In VM, despite the
density and velocity fields being fully coupled, the velocity length scale
grows much faster compared to the density length scale because velocity order
extends over longer distances than density clusters due to the irregular
fractal morphology of the density clusters. In AIM, however, besides being the
density and magnetization fields fully coupled, the clusters are also
regularly shaped with smooth boundaries, and therefore, the temporal behavior
of the two length scales are found similar as shown in Fig. 11.

Fig. 11 shows, for the coarsening of the AIM, that the two length scales
exhibit the same growth law, $R(t)\sim t^{1/2}$ (although the domain size for
the density field is marginally larger than the corresponding magnetization
field) for a quench into the coexistence region and into the polar ordered
liquid regime (see inset of Fig. 11). Therefore, we can conclude that the
$R(t)\sim t^{1/2}$ growth law is reasonably universal in the AIM as it neither
depends on the quenching regime nor the local order parameter (be it density
or magnetization).

##  Appendix B 2d AIM with $D_{\perp}=0$

Figure 12: (Color online) (a) $m(y)$ versus $t$ for $L_{y}\in[200:210]$
starting from a random disordered configuration ($m\sim 0$). The color bar
denotes the averaged magnetization along the $x$-axis. (b) Late time averaged
magnetization profile as a function of $y$. Parameters: $L=400$,
$\rho_{0}=10$, $\beta=1.1$, and $\epsilon=0.8$.

The role of transverse diffusion in domain growth of $2d$ AIM is immense. Here
we present a more detailed picture of the $D_{\perp}=0$ scenario. In Fig.
12(a) we plot $m(y)=\frac{1}{L_{x}}\sum^{L_{x}}_{i=1}m(i,y)$ versus time $t$
for $L_{y}\in[200:210]$. As the density is very large $(\rho_{0}=10)$, we see
highly magnetized one lattice unit-wide domains that can not merge to create a
larger domain as time progresses because we inhibit the diffusive hopping in
the transverse direction. These $1d$ domains are single dense ordered
aggregates that stochastically reverse their magnetization Benvegnen  _et al._
(2022). The magnetization profile in Fig. 12(b) is averaged over $x$ and shows
sharp peaks of positive and negative magnetizations, spread over either one
site or a few sites. The density profile of such an arrangement shows a
homogeneous profile around the average density which is similar to the density
profile of a large liquid domain but the alternating magnetization profile
signifies that there is no domain growth at the asymptotic limit in the $2d$
AIM for $D_{\perp}=0$.

## References

  * Ramaswamy (2010) S. Ramaswamy, “The mechanics and statistics of active matter,” Annual Review of Condensed Matter Physics 1, 323–345 (2010).
  * Marchetti  _et al._ (2013) M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, “Hydrodynamics of soft active matter,” Reviews of Modern Physics 85, 1143 (2013).
  * Needleman and Dogic (2017) D. Needleman and Z. Dogic, “Active matter at the interface between materials science and cell biology,” Nature Reviews Materials 2, 1 (2017).
  * Gompper  _et al._ (2020) G. Gompper, R. G Winkler, T. Speck, A. Solon, C. Nardini, F. Peruani, H. Löwen, R. Golestanian, U. B. Kaupp, L. Alvarez, _et al._ , “The 2020 motile active matter roadmap,” Journal of Physics: Condensed Matter 32, 193001 (2020).
  * Ballerini  _et al._ (2008) M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale, and V. Zdravkovic, “Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study,” Proceedings of the National Academy of Sciences 105, 1232 (2008).
  * Becco  _et al._ (2006) Ch. Becco, N. Vandewalle, J. Delcourt, and P. Poncin, “Experimental evidences of a structural and dynamical transition in fish school,” Physica A: Statistical Mechanics and its Applications 367, 487–493 (2006).
  * Calovi  _et al._ (2014) D. S. Calovi, U. Lopez, S. Ngo, C. Sire, H. Chaté, and G. Theraulaz, “Swarming, schooling, milling: phase diagram of a data-driven fish school model,” New Journal of Physics 16, 015026 (2014).
  * Steager  _et al._ (2008) E. B. Steager, C. B. Kim, and M. J. Kim, “Dynamics of pattern formation in bacterial swarms,” Physics of Fluids 20, 073601 (2008).
  * Peruani  _et al._ (2012) F. Peruani, J. Starruß, V. Jakovljevic, L. Søgaard-Andersen, A. Deutsch, and M. Bär, “Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria,” Physical Review Letters 108 (2012).
  * Schaller  _et al._ (2010) V. Schaller, C. Weber, C. Semmrich, E. Frey, and A. R. Bausch, “Polar patterns of driven filaments,” Nature 467, 73–77 (2010).
  * Sumino  _et al._ (2012) Y. Sumino, K. H. Nagai, Y. Shitaka, D. Tanaka, K. Yoshikawa, H. Chaté, and K. Oiwa, “Large-scale vortex lattice emerging from collectively moving microtubules,” Nature 483, 448–452 (2012).
  * Sanchez  _et al._ (2012) T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, “Spontaneous motion in hierarchically assembled active matter,” Nature 491, 431–434 (2012).
  * Vicsek and Zafeiris (2012) T. Vicsek and A. Zafeiris, “Collective motion,” Physics Reports 517, 71–140 (2012).
  * Shaebani  _et al._ (2020) M. R. Shaebani, A. Wysocki, R. G. Winkler, G. Gompper, and H. Rieger, “Computational models for active matter,” Nature Reviews Physics 2, 181 (2020).
  * De Magistris and Marenduzzo (2015) G. De Magistris and D. Marenduzzo, “An introduction to the physics of active matter,” Physica A: Statistical Mechanics and its Applications 418, 65–77 (2015).
  * Krishnan  _et al._ (2010) J. M. Krishnan, A. P. Deshpande, and P. B. S. Kumar,  _Rheology of complex fluids_ (Springer, 2010).
  * Vicsek  _et al._ (1995) T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel type of phase transition in a system of self-driven particles,” Physical Review Letters 75, 1226 (1995).
  * Toner and Tu (1995) J. Toner and Y. Tu, “Long-range order in a two-dimensional dynamical $\mathrm{XY}$ model: How birds fly together,” Physical Review Letters 75, 4326 (1995).
  * Toner and Tu (1998) J. Toner and Y. Tu, “Flocks, herds, and schools: A quantitative theory of flocking,” Physical Review E 58, 4828 (1998).
  * Solon and Tailleur (2013) A. P. Solon and J. Tailleur, “Revisiting the flocking transition using active spins,” Physical Review Letters 111, 078101 (2013).
  * Solon and Tailleur (2015) A. P. Solon and J. Tailleur, “Flocking with discrete symmetry: The two-dimensional active ising model,” Physical Review E 92, 042119 (2015).
  * Toner  _et al._ (2005) J. Toner, Y. Tu, and S. Ramaswamy, “Hydrodynamics and phases of flocks,” Annals of Physics 318, 170–244 (2005), special Issue.
  * Giomi  _et al._ (2013) L. Giomi, M. J. Bowick, X. Ma, and M. C. Marchetti, “Defect annihilation and proliferation in active nematics,” Physical Review Letters 110, 228101 (2013).
  * Cates and Tailleur (2015) M. E. Cates and J. Tailleur, “Motility-induced phase separation,” Annual Review of Condensed Matter Physics 6, 219–244 (2015).
  * Solon  _et al._ (2015) A. P. Solon, H. Chaté, and J. Tailleur, “From phase to microphase separation in flocking models: The essential role of nonequilibrium fluctuations,” Physical Review Letters 114, 068101 (2015).
  * Siebert  _et al._ (2018) J. T. Siebert, F. Dittrich, F. Schmid, K. Binder, T. Speck, and P. Virnau, “Critical behavior of active brownian particles,” Physical Review E 98, 030601 (2018).
  * Wysocki and Rieger (2020) A. Wysocki and H. Rieger, “Capillary action in scalar active matter,” Physical Review Letters 124, 048001 (2020).
  * Mangeat  _et al._ (2020) M. Mangeat, S. Chatterjee, R. Paul, and H. Rieger, “Flocking with a $q$-fold discrete symmetry: Band-to-lane transition in the active potts model,” Physical Review E 102, 042601 (2020).
  * Chatterjee  _et al._ (2020a) S. Chatterjee, M. Mangeat, R. Paul, and H. Rieger, “Flocking and reorientation transition in the 4-state active potts model,” Europhysics Letters 130, 66001 (2020a).
  * Kürsten and Ihle (2020) R. Kürsten and T. Ihle, “Dry active matter exhibits a self-organized cross sea phase,” Physical Review Letters 125, 188003 (2020).
  * Fruchart  _et al._ (2021) M. Fruchart, R. Hanai, P. B. Littlewood, and V. Vitelli, “Non-reciprocal phase transitions,” Nature 592, 363 (2021).
  * Solon  _et al._ (2022) A. Solon, H. Chaté, J. Toner, and J. Tailleur, “Susceptibility of polar flocks to spatial anisotropy,” Physical Review Letters 128, 208004 (2022).
  * Chatterjee  _et al._ (2022) S. Chatterjee, M. Mangeat, and H. Rieger, “Polar flocks with discretized directions: the active clock model approaching the vicsek model,” Europhysics Letters 138, 41001 (2022).
  * Codina  _et al._ (2022) J. Codina, B. Mahault, H. Chaté, J. Dobnikar, I. Pagonabarraga, and X. Shi, “Small obstacle in a large polar flock,” Physical Review Letters 128, 218001 (2022).
  * Chatterjee  _et al._ (2023) S. Chatterjee, M. Mangeat, C. U. Woo, H. Rieger, and J. D. Noh, “Flocking of two unfriendly species: The two-species vicsek model,” Physical Review E 107, 024607 (2023).
  * Karmakar  _et al._ (2023) M. Karmakar, S. Chatterjee, M. Mangeat, H. Rieger, and R. Paul, “Jamming and flocking in the restricted active potts model,” Physical Review E 108, 014604 (2023).
  * Bray (2002) A. J. Bray, “Theory of phase-ordering kinetics,” Advances in Physics 51, 481–587 (2002).
  * Bray (1994) A. J. Bray, “Theory of phase-ordering kinetics,” Advances in Physics 43, 357 (1994).
  * Puri (2009) S. Puri, “Kinetics of phase transitions,” (CRC press, 2009) p. 13.
  * Ahmad  _et al._ (2012) S. Ahmad, F. Corberi, S. K. Das, E. Lippiello, S. Puri, and M. Zannetti, “Aging and crossovers in phase-separating fluid mixtures,” Physical Review E 86, 061129 (2012).
  * Shrivastav  _et al._ (2014) G. P. Shrivastav, M. Kumar, V. Banerjee, and S. Puri, “Ground-state morphologies in the random-field ising model: Scaling properties and non-porod behavior,” Physical Review E 90, 032140 (2014).
  * Kumar  _et al._ (2017) M. Kumar, S. Chatterjee, R. Paul, and S. Puri, “Ordering kinetics in the random-bond XY model,” Physical Review E 96, 042127 (2017).
  * Chatterjee  _et al._ (2020b) S. Chatterjee, S. Sutradhar, S. Puri, and R. Paul, “Ordering kinetics in a $q$-state random-bond clock model: Role of vortices and interfaces,” Physical Review E 101, 032128 (2020b).
  * Wittkowski  _et al._ (2014) R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, “Scalar $\varphi^{4}$ field theory for active-particle phase separation,” Nature Communications 5, 4351 (2014).
  * Pattanayak  _et al._ (2021a) S. Pattanayak, S. Mishra, and S. Puri, “Ordering kinetics in the active model B,” Physical Review E 104, 014606 (2021a).
  * Pattanayak  _et al._ (2021b) S. Pattanayak, S. Mishra, and S. Puri, “Domain growth in the active model b: Critical and off-critical composition,” Soft Materials 19, 286 (2021b).
  * Mishra  _et al._ (2014) S. Mishra, S. Puri, and S. Ramaswamy, “Aspects of the density field in an active nematic,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, 20130364 (2014).
  * Das  _et al._ (2018) R. Das, S. Mishra, and S. Puri, “Ordering dynamics of self-propelled particles in an inhomogeneous medium,” Europhysics Letters 121, 37002 (2018).
  * Saha  _et al._ (2020) S. Saha, J. Agudo-Canalejo, and R. Golestanian, “Scalar active mixtures: The nonreciprocal cahn-hilliard model,” Physical Review X 10, 041009 (2020).
  * Rouzaire and Levis (2022) Y. Rouzaire and D. Levis, “Dynamics of topological defects in the noisy kuramoto model in two dimensions,” Frontiers in Physics 10, 976515 (2022).
  * Dittrich  _et al._ (2023) F. Dittrich, J. Midya, P. Virnau, and S. K. Das, “Growth and aging in a few phase-separating active matter systems,” Physical Review E 108, 024609 (2023).
  * Caporusso  _et al._ (2023) C. B. Caporusso, L. F. Cugliandolo, P. Digregorio, G. Gonnella, D. Levis, and A. Suma, “Dynamics of motility-induced clusters: coarsening beyond ostwald ripening,” Physical Review Letters 131, 068201 (2023).
  * Katyal  _et al._ (2020) N. Katyal, S. Dey, D. Das, and S. Puri, “Coarsening dynamics in the vicsek model of active matter,” The European Physical Journal E 43, 10 (2020).
  * Dikshit and Mishra (2023) S. Dikshit and S. Mishra, “Ordering kinetics in active polar fluid,” Europhysics Letters 143, 17001 (2023).
  * Majumder and Das (2011) S. Majumder and S. K. Das, “Diffusive domain coarsening: Early time dynamics and finite-size effects,” Physical Review E 84, 021110 (2011).
  * Chatterjee  _et al._ (2018) S. Chatterjee, S. Puri, and R. Paul, “Ordering kinetics in the $q$-state clock model: Scaling properties and growth laws,” Physical Review E 98, 032109 (2018).
  * Dey  _et al._ (2012) S. Dey, D. Das, and R. Rajesh, “Spatial structures and giant number fluctuations in models of active matter,” Physical Review Letters 108, 238001 (2012).
  * Yurke  _et al._ (1993) B. Yurke, A. N. Pargellis, T. Kovacs, and D. A. Huse, “Coarsening dynamics of the XY model,” Physical Review E 47, 1525 (1993).
  * Benvegnen  _et al._ (2022) B. Benvegnen, H. Chaté, P. L. Krapivsky, J. Tailleur, and A. Solon, “Flocking in one dimension: Asters and reversals,” Physical Review E 106, 054608 (2022).
  * Press (2007) W. H. Press,  _Numerical recipes: The art of scientific computing_ (Cambridge university press, 2007).
  * Grest  _et al._ (1988) G. S. Grest, M. P. Anderson, and D. J. Srolovitz, “Domain-growth kinetics for the Q-state potts model in two and three dimensions,” Physical Review B 38, 4752 (1988).