Alignment destabilizes crystal order in active systems

Abstract

We combine numerical and analytical methods to study two-dimensional active crystals formed by permanently linked swimmers and with two distinct alignment interactions. The system admits a stationary phase with quasi-long-range translational order, as well as a moving phase with quasi-long-range active force director and velocity order. The translational order in the moving phase is significantly influenced by alignment interaction. For Vicsek-like alignment, the translational order is short ranged, whereas the bond-orientational order is quasi-long ranged, implying a moving hexatic phase. For elasticity-based alignment, the translational order is quasi-long ranged parallel to the motion and short ranged in the perpendicular direction, whereas the bond orientational order is long ranged. We also generalize these results to higher dimensions.

Full text

PHYSICAL REVIEW E 104, 064605 (2021)
Alignment destabilizes crystal order in active systems
Chen Huang ,1Leiming Chen,2and Xiangjun Xing 1,3,4,*
1Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240 China
2School of Materials Science and Physics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116 China
3Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240 China
4Shanghai Research Center for Quantum Sciences, Shanghai 201315 China
(Received 30 March 2021; revised 30 July 2021; accepted 4 November 2021; published 9 December 2021)
We combine numerical and analytical methods to study two-dimensional active crystals formed by perma-
nently linked swimmers and with two distinct alignment interactions. The system admits a stationary phase with
quasi-long-range translational order, as well as a moving phase with quasi-long-range active force director and
velocity order. The translational order in the moving phase is significantly influenced by alignment interaction.
For Vicsek-like alignment, the translational order is short ranged, whereas the bond-orientational order is
quasi-long ranged, implying a moving hexatic phase. For elasticity-based alignment, the translational order
is quasi-long ranged parallel to the motion and short ranged in the perpendicular direction, whereas the bond
orientational order is long ranged. We also generalize these results to higher dimensions.
DOI: 10.1103/PhysRevE.104.064605
I. INTRODUCTION
One of the most interesting and fundamental issues about
active systems [1–6] is the stability of orders. According to
the Mermin-Wagner theorem [7], two-dimensional (2D) equi-
librium systems with continuous symmetry and short-range
interaction cannot exhibit long-range order (LRO). However,
LRO was discovered in 2D polar active fluid both in sim-
ulation [8] and in hydrodynamic theory [9–12]. This LRO
is accompanied by super-diffusion and giant number fluctu-
ations [2,6,11,13], neither of which is seen in equilibrium
systems with short-range interactions. Many variants of Vic-
sek models with different particle polarity, alignments, and
exclusion [12–26] have been studied, and a variety of novel
phenomena have been discovered.
Dense active systems with repulsive interactions may also
exhibit translational orders. Solid phases as well as fluid-solid
phase separations have been repeatedly observed in active
colloidal systems both experimentally [27–29] and numeri-
cally [30–34]. In most of these works, there is no alignment
interaction and no visible collective motion. More recently,
Weber et al. [20] simulated a model of active crystal with
Vicsek-type alignment and discovered a stationary phase with
quasi-long-range (QLR) translational order, as well as a phase
of moving crystal domains separated by grain boundaries.1
Very recently, Maitra et al. [35] studied an active general-
ization of nematic elastomer [36] with spontaneous breaking
of rotational symmetry [37,38] and found QLR translational
orders in 2D. Their elastic energy contains a hidden rotational
symmetry (and its resulting Goldstone modes) involving both
*xxing@sjtu.edu.cn
1Note, however, that in this work the dynamical equations contain
no noise term.
shear deformation and orientational order, which are difficult
to realize experimentally.
Regardless of many previous studies, it is not clear whether
there exists a moving phase with certain translational order in
active systems with alignment interactions if the soft-mode in
Ref. [35] does not come into play. To address this interesting
question, here we combine analytic and numerical approaches
to study a model system of active crystal consisting of a 2D tri-
angular array of swimmers linked permanently by springs. We
introduce alignment interaction between neighboring swim-
mers that is either Vicsek-like (AD-I) or elasticity based
[39,40] (AD-II). In the strong noise–weak alignment regime,
we find a stationary phase with QLR translational order, which
was also seen in Refs. [20,41]. In the weak noise–strong align-
ment regime we find a moving phase with QLR active force
director and velocity order and with the nature of translational
order depending on the alignment. For AD-I, the moving
phase exhibits only short-range (SR) translational order and
QLR bond orientational order and hence should be identified
as a moving hexatic phase. For AD-II, the translational order
is QLR along the moving direction, and SR in the perpen-
dicular direction, whereas the bond-orientational order is LR.
We generalize the model to higher dimensions and show that
the active force director alignment in active systems tends to
destabilize crystal orders.
There are many experimental realizations of the models
studied in this work. Vicsek-like alignment, for example,
is believed to be relevant for living matters and flocking
behaviors [2,4–6]. It may also be realized effectively as a
consequence of collision [42] or interaction [43] between
active particles or in micro-robotics using remote-sensing
[44–46]. Both harmonic potential interaction and elasticity-
based alignment interactions may be realized by remote
sensing of the directions or positions between programed
robotics [44–46]. Finally, it is also conceivable to link
2470-0045/2021/104(6)/064605(15) 064605-1 ©2021 American Physical Society

HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 1. Our simulation model. Swimmers are connected by
springs and driven by active forces, shown as red arrows.
swimmers using polymers and to induce alignment interac-
tions using hydrodynamic effects or magnetic interactions.
The remainder of this work is organized as follows. In Sec. II
we present the simulation models and details. In Sec. III we
present numerical results on phase diagram and various corre-
lation functions both in the stationary phase and in the moving
phase. In Sec. IV we analytically study the models and show
that the results are fully consistent with the simulation results.
In Sec. Vwe draw concluding remarks. In the Appendices,
we present various details of simulations and analytic calcula-
tions.
II. MODEL AND SIMULATION DETAILS
As schematized in Fig. 1, our simulation model consists of
a triangular array of Nswimmers connected permanently by
harmonic springs. Each swimmer moves under the influences
of elastic force, friction and noise, as well as active force. The
position ri(t)oftheith swimmer in the laboratory frame obeys
the following overdamped Langevin equation:
γp˙
ri(t)=bpn(θi)+Fi(t)+γp2Dpξi(t),(1)
where γpis the friction coefficient and bpn(θi) is the active
force with fixed magnitude bpand director n(θi). The angle
θi(t) is defined with respect to the ˆ
xaxis, and is related to n(θi)
via n(θi)=(cos θi,sin θi). Throughout this section, we use
symbols with subscript pto denote parameters of the particle
model in order to distinguish them from the parameters of the
continuum model to be discussed in Sec. IV.
The second term in right-hand side of Eq. (1) is the elastic
force, given by
Fi=
jn.n.i
κ(|ri−rj|−a0)ri−rj
|ri−rj|,(2)
where the summation is over six nearest neighbors of swim-
mer i, while κand a0are respectively the elastic constant
and natural length of the springs. The last term of Eq. (1)is
the random force, with ξi(t) the unit-variance Gaussian white
noise. The translational diffusion coefficient Dpsatisfies the
Einstein relation Dp=T/γp, where Tis the temperature of
ambient fluid with the Boltzmann constant kB=1.
We consider two distinct alignment dynamics for the active
forces. The first alignment dynamics (AD-I) is Vicsek-like [8],
with each swimmer trying to align its director of active force
with its neighbors,2subject to an internal noise:
˙
θi(t)=gp(n(θi)×ni)·ˆ
z+2Dθ
pηi(t),(AD-I)
where gpis the alignment strength for (AD-I), ˆ
zis the unit nor-
mal to the plane, ni≡jn.n.in(θj)/6 is the average director
of all the six nearest neighbors, Dθ
pis the rotational diffusion
coefficient, and ηi(t) is a unit Gaussian white noise. Equation
(AD-I) describes the alignment of the active force of each
particle with those of its nearest neighbors. This alignment
mechanism by perceiving the average orientation of neighbors
has already been realized in swarm robots [44–46]. It may also
be realized in self-propelling Janus colloids that are able to
align directions [43].
The second alignment dynamics (AD-II) is elasticity based
[39,40,47], with each swimmer aligning its active force with
the local elastic force, so as to reduce the local elastic energy:
˙
θi(t)=cp(n(θi)×Fi)·ˆ
z+2Dθ
pηi(t),(AD-II)
where cpis the alignment strength. This mechanism can be
implemented, either by directly installing force sensors on ac-
tive particles or swarm robots or by indirect visual perception
of their relative positions [44–46].
A. Dimensionless forms and choice of parameters
Here we define various dimensionless parameters and vari-
ables and rewrite the dynamical equations in dimensionless
forms suitable for numerical computation. We use the lattice
constant a0as a unit of length and denote an arbitrary time unit
as τ0. The dimensionless parameters and variables are defined
as:
˜
r≡r/a0,
˜
bp≡bτ0/(γa0),˜κ≡κτ0/γ , ˜
Dp≡Dτ0/a2
0,
˜
Dθ
p≡Dθτ0,˜gp≡gpτ0,˜cp≡cγa0,(3)

Fi≡Fiτ0/(γa0),ηi≡η√τ0,˜
ξi≡ξ√τ0.
The dimensionless form of Eq. (1)is
˙
˜
ri(˜
t)=˜
bpn(θi)+
Fi(˜
t)+2˜
Dp˜
ξi(˜
t).(4)
The dimensionless forms of the alignment dynamics are as
follows:
˙
θi(˜
t)=˜gp(n(θi)×ni)·ˆ
z+2˜
Dθ
p˜ηi,(AD-

I)
˙
θi(˜
t)=˜cp(n(θi)×
Fi)·ˆ
z+2˜
Dθ
p˜ηi.(AD-
II)
We use a rhombic cell with periodic boundary condition
(cf. Fig. 1) and numerically integrate the dynamical equations,
Eq. (4), (AD-

I), and (AD-
II), using the first-order Euler-
Maruyama scheme [48]. The discretized equations are shown
in Appendix A, with the simulation time step ˜
t=10−3.
2Note that in the original Vicsek model [8], swimmers control their
velocities instead of active forces. In the overdamped regime, this
difference is inessential.
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ALIGNMENT DESTABILIZES CRYSTAL ORDER IN … PHYSICAL REVIEW E 104, 064605 (2021)
We choose the parameters ˜
bp=2, ˜κ=100 for AD-I and
˜κ=20 for AD-II. The dimensionless translational diffusion
coefficient is chosen to be ˜
Dp=0.01. The value of ˜gp,˜cp, and
˜
Dθ
pare specified in the following sections, respectively. Nu-
merical results indicate that, with these choices of parameters,
the system is in the small deformation regime.
III. NUMERICAL RESULTS
A. Phase diagram
We define the order parameter P=|1
Nkn(θk)|t,which
is the steady-state time average of the magnitude of the
system-averaged active force director. Summing Eq. (1) over
all swimmers, we see that the elastic forces vanish because of
Newton’s third law, and hence we obtain bp|1
Nkn(θk)|=
γp|1
Nk˙
ri(t)−1
N2Dpkξi(t)|. For infinite system size,
the sum over noises converges to zero according to the law
of large numbers, and we see that the order parameter is
strictly proportional to the absolute value of average velocity
of all swimmers, further averaged over time. Hence for infinite
system size, the order parameter can be used to distinguish
the stationary phase (P=0) from the moving phase (P= 0).
For finite system size, the proportionality between Pand the
average velocity of the system is only approximate. Nonethe-
less, we can obtain a good estimate of the phase boundary by
simulating systems with sizes 32 ×32 and 64 ×64.
As shown in Fig. 2(a), for weak alignment–strong active
noise (dark blue in upper left) there is a stationary phase where
Pis approximately zero,3whereas for strong alignment–weak
active noise (bright yellow in lower right) there is a collec-
tively moving phase where Pis finite. As shown in Fig. 2(b),
fitting of Pas a function of alignment strength for a relatively
larger system size 64 ×64 suggests that these two phases are
separated by a line of second-order phase transitions.
B. Definitions of correlation functions
In this subsection we define various correlation functions
in an active crystal.
We first define the displacement vector field ˜
ui(in di-
mensionless units) as the deviation from the perfect lattice
location of each particle i, either in the laboratory frame for
the stationary phase or in the comoving frame for the moving
phase. Then we numerically Fourier transform the displace-
ment vector field ˜
uinto the reciprocal k-space vector field
ˆ
u(k)=(ˆux(k),ˆuy(k)), where k=(kx,ky) is the dimension-
less wave vector reciprocal to the dimensionless distances. We
compute averages of the norm squared |ˆux(k)|2,|ˆuy(k)|2,
which will be hereafter referred to as the u-correlations (in
momentum space). The technical details of numerical Fourier
transformation on the triangular lattice are presented in Ap-
pendix B.
The real-space translational correlation function gq(˜r)for
a lattice characterizes the correlation of translational order
with Bragg vector q=(qx,qy) at two regions separated by
3It is never strictly zero because the system is finite and there are
always instantaneous fluctuations at each time.
FIG. 2. (a) Phase diagram of AD-I (left) and AD-II (right).
Vertical axis: Dimensionless noise strength ˜
Dθ
p; horizontal axis: di-
mensionless alignment ˜gp(AD-I) or ˜cp(AD-II). The color represents
the order parameter of average alignment P, defined in the main
text. (b) Average alignment Pof AD-I (left) and AD-II (right) as a
function of alignment interaction strength, from a cut indicated by the
black line segments in (a), suggests second-order phase transitions.
The fitting is applied on the curve of system size 64 ×64. (c) Log-
log plot of the ucorrelations |ˆux|2and |ˆuy|2of the stationary
phase along the ˆ
kxand ˆ
kyaxis, with parameters ˜
bp=2, ˜
Dp=0.01,
˜
Dθ
p=0.3, (˜κ, ˜gp)=(100,0.1) in AD-I, ( ˜κ, ˜cp)=(20,0.1) in AD-
II, and ( ˜κ, ˜gp,˜cp)=(100,0,0) in AD-0. Bottom left inset: A typical
configuration of active forces in the stationary phase.
a dimensionless distance ˜r≡r/a0[49]:
gq(˜r)=j=kζ(˜r−|˜
rj−˜
rk|)Re(eiq·(˜
rj−˜
rk))
2π˜r˜rρN,(5)
where ˜
rj=(˜xj,˜yj) is the dimensionless Cartesian coordinates
of the jth particle and ζ=1 if the distance |˜
rj−˜
rk|is in the
interval ˜r∼˜r+˜r; otherwise, ζ=0. The number density
ρ=(√3/2)−1is the inverse of the area of a unit cell in the
triangular lattice of unit length. The dimensionless reciprocal
vector qis evaluated as the Bragg peaks of the structure factor:
Q(qx,qy)=1
Nρ(qx,qy)ρ(−qx,−qy),(6)
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HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 3. The perfect triangular lattice in real space (left) and its
corresponding Bragg peaks in the reciprocal space (right). Only
parts of the lattices near the origin are displayed. In dimensionless
length scales, the real-space lattice has unit lattice constant, while
the reciprocal lattice has lattice spacing 2π/(√3/2). The symbols
q0
x,q0
y,andq0
tare defined in Sec. III D.
where
ρ(qx,qy)=
N

j=1
exp [i(qx˜xj+qy˜yj)].(7)
The Bragg peaks of a perfect triangular lattice also form a
perfect triangular lattice in the reciprocal space, see Fig. 3.
However,as demonstrated in Ref. [49], because of sample-to-
sample fluctuations of crystal structure, the Bragg peak qof
gq(˜r) must be identified carefully as the peak value of each
sample state, i.e., the true Bragg peaks are near but not exactly
equal to the Bragg peaks of a perfect hexagonal lattice.
The correlation function of the active force director
is defined as n(0) ·n(˜r)c≡(n(0) −n)·(n(˜r)−n)=
n(0) ·n(˜r)−n2where the system-average director n≡
N−1N
i=1n(θi), and
n(0) ·n(˜r)=j=kζ(˜r−|˜
rj−˜
rk|)n(θj)·n(θk)
2π˜r˜rρN.(8)
Similarly, the velocity correlation function is defined
as ˆv(0) ·ˆv(˜r)c≡ˆv(0) ·ˆv(˜r)−ˆv2where the system-
average unit velocity ˆv≡N−1N
i=1ˆvi, and
ˆv(0) ·ˆv(˜r)=j=kζ(r−|˜
rj−˜
rk|)ˆvj·ˆvk
2π˜r˜rρN,(9)
and the normalized velocity is ˆvj≡˜
vj/|˜
vj|. The velocity ˜
vjof
each particle is evaluated by subtracting the coordinates of the
two configurations in the laboratory frame separated by 100
simulation steps, which corresponds to a dimensionless time
interval 0.1. This choice is made such that during this time
interval, the random forces in Eq. (4) are mostly averaged out,
yet the lattice moves very little.
The local bond orientational parameter of the particle at
˜
rjis defined as ψ6(˜
rj)=n−1n
m=1exp(i6θj
m) where the sum
runs over its nVoronoi neighbors at position ˜
rm, and θj
mis the
angle of the bond (˜
rm−˜
rj) relative to any fixed axis,say the ˆ
x
axis. The bond-orientational correlation function g6(˜r) is then
defined as [49]:
g6(˜r)=Reψ6(˜
rj)ψ∗
6(˜
rk)
≡j=kζ(˜r−|˜
rj−˜
rk|)Re(ψ6(˜
rj)ψ∗
6(˜
rk))
2π˜r˜rρN.(10)
To study all the above orders in both the stationary and the
moving phase in the following sections, we simulate a larger
system with size 256 ×256. The total number of simulation
steps is 2 ×106and simulation samples are collected every
2000 steps.
C. Correlation function in the stationary phase
In this section we study the u-correlation function of the
stationary phase. We choose the alignment strength ˜gp=0.1
for AD-I and ˜cp=0.1 for AD-II and the magnitude of the
angular noise ˜
Dθ
p=0.3 for both. The simulation starts from
a perfect triangular lattice where all the active forces orient
randomly.
In Fig. 2(c) we plot |ˆux(k)|2and |ˆuy(k)|2in kspace
along the ˆ
kxand ˆ
kyaxis in log-log scale in both AD-I and
AD-II. For comparison we also plot the corresponding result
for a model without any alignment (AD-0), which corresponds
to ˜gp=˜cp=0 in Eqs. (AD-I) and (AD-II).
For a passive isotropic crystal at equilibrium, we have bp=
0, and the ucorrelations along the ˆ
kxand ˆ
kyaxes [50]:
|ˆux(k,0)|2= γD
3λk2,|ˆux(0,k)|2= γD
λk2,
|ˆuy(k,0)|2=γD
λk2,|ˆuy(0,k)|2= γD
3λk2,
(11)
which satisfy the following equalities:
|ˆux(k/√3,0)|2=|ˆux(0,k)|2
=|ˆuy(k,0)|2=|ˆuy(0,k/√3)|2.(12)
This prompts us to plot four functions: |ˆux(k/√3,0)|2,
|ˆuy(0,k/√3)|2,|ˆux(0,k)|2, and |ˆuy(k,0)|2for active
crystal in the same plot and compare them with the result
of equilibrium crystals. As shown Fig. 2(c), all four curves
collapse onto each other. The master curve exhibits k−2in the
long length scales, as well as k−4scaling in the intermediate
length scale (0.1k1). The latter signifies anomalously
large structure fluctuations in the real-space length scales
6a060a0that are caused by the fluctuations of active
forces, where kin reciprocal space satisfies /a0=2π/k.At
longer length scales or smaller k,theucorrelations crossover
to k−2scaling, which indicates a QLR translational order and
is consistent with the simulation result obtained in Ref. [20].
The transition from k−4to k−2scaling is also consistent with
very recent works on active solids by Caprini et al. [41,51,52]
in the absence of explicit alignment, where the u-correlation
function under overdamped Langevin dynamics [41] has the
form |ˆ
u(k)|2∝(k2+f(Dθ)k4)−1with f(Dθ) antipropor-
tional to Dθ. Hence we conclude that in the stationary phase,
activity only affect the fine structure of the lattice but does not
alter the QLR translational order at long length scales.
D. Correlation functions in the moving phase
We are most interested in the orders of the collective mov-
ing phase. For this phase, we choose the alignment strength
˜gp=10 for AD-I and ˜cp=1 for AD-II and the magnitude of
the angular noise ˜
Dθ
p=0.05 for both such that the fluctuation
of angle is small enough.
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ALIGNMENT DESTABILIZES CRYSTAL ORDER IN … PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 4. Bragg peak of Q(qx,qy) which are numerically evaluated (qx,qy,andqt) and that of a perfect hexagonal lattice (q0
x,q0
y,and
q0
t), and the corresponding translational correlation function in AD-I with [(a)–(c)] qx=(12.5172,−0.0292) and q0
x=(4π,0); [(d)–(f)]
qy=(0.0196,7.2905) and q0
y=(0,2π/(√3/2)); [(g)–(i)] qt=(6.2998,3.6363) and q0
t=2π(1,1/√3). The second column is in log-log
scale, while the third column is in log-linear scale.
The simulation starts again from a perfect lattice with all
the active directors orienting in the positive ˆ
xaxis, i.e., θi=0
for all i∈{1,...,N}. If we directly simulate Eqs. (AD-I)
and (AD-II) for a finite system, the average velocity would
evolve slowly but randomly, which complicates the computa-
tion of the translational correlation function gq(˜r) as defined
in Eq. (5). This is because gq(˜r) depends both on the Bragg
peak qand on the average velocity, and we need a large
number of data points with both of them fixed. To avoid this
problem, right after each sample has been collected (per 2000
time steps), we rotate the average velocity back to the ˆ
xaxis
and then continue the simulation.
Under the constraint of collective motion along the ˆ
x
axis, we evaluate gq(˜r) at three Bragg peaks, i.e., the first
Bragg peak qynear that of a perfect triangular lattice q0
y=
(0,2π/(√3/2)) along the ˆ
yaxis (perpendicular to the col-
lective moving direction), qtnear q0
t=2π(1,1/√3) along
a tilted direction, and the second Bragg peak qxnear q0
x=
(4π,0) along the ˆ
xaxis (parallel to the collective moving di-
rection), as shown in Fig. 3. Simulation data are collected after
the system has reached a steady state of collective moving
along the ˆ
xaxis, with P=0.9915 in AD-I and P=0.9830
in AD-II.
In principle we could also choose to fix the average veloc-
ity along an arbitrary direction instead of the positive ˆ
xaxis.
The numerical computation will then become more compli-
cated, since it would be difficult to specify the corresponding
Bragg peaks that are either parallel or perpendicular to this
arbitrary collective moving direction. From a theoretical point
of view, we do not expect the operation of constraining its
motion along the ˆ
xaxis to alter the system’s long length-
scale physical properties, because the long-scale continuum
elasticity theory of the triangular lattice is isotropic. See the
next section for more details.
We first study the translational correlation function gq(˜r)
for a typical configuration of the moving phase. We plot in
Figs. 4and 5, for AD-I and AD-II, respectively, the struc-
ture factor Q(qx,qy) (the first column) and the corresponding
translational correlation function (the second column in log-
log scale and the third in log-linear scale) at the Bragg peaks
qx,qy, and qt. In AD-I, the translational correlation function
decays faster than power law along all the three wave vec-
tors, indicating short-range translational orders. In AD-II, the
system also exhibits short-range order along qyand qt, while
it has a power-law decay along the ˆ
xaxis indicating a QLR
translational order.
In Figs. 6(a) and 6(b) we plot the correlation functions
of the active force director n(0) ·n(˜r)cof 10 independent
samples (indicated by different colors). We find a power-law
decay indicating QLRO of the active force directors in both
AD-I and AD-II. Similarly, the velocity correlation functions
ˆv(0) ·ˆv(˜r)cas shown in Figs. 6(c) and 6(d) also exhibit
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HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 5. Bragg peak of Q(qx,qy) which are numerically evaluated (qx,qy,andqt) and that of a perfect hexagonal lattice (q0
x,q0
y,and
q0
t), and the corresponding translational correlation function in AD-II with [(a)–(c)] qx=(12.5667,−0.0015) and q0
x=(4π,0); [(d)–(f)]
qy=(0.0166,7.2272) and q0
y=(0,2π/(√3/2)); [(g)–(i)] qt=(6.2847,3.6213) and q0
t=2π(1,1/√3). The second column is in log-log
scale, while the third column is in log-linear scale.
power-law decay, which again indicates QLRO of the velocity
field in both cases. Furthermore, a linear fitting to the average
of the 10 independent sample curves all give a power-law
exponent very close to unity.
In Figs. 6(e) and 6(f), we show that the bond-orientational
correlation function g6(˜r) has a power-law decay with a
nonzero but small exponent −0.0284 in AD-I, while it con-
verges to a finite limit at large distances for AD-II. Hence the
bond orientational order is quasi-long ranged for AD-I and
long ranged for AD-II.
In summary, the moving phase of AD-I exhibits QLRO in
active force director, velocity, and bond orientation, but only
has SR translational order. Hence it should be categorized as
amoving hexatic phase. By contrast, the moving phase of
AD-II exhibits QLRO in active force director and velocity and
LRO in bond orientation, yet the translation order is quasi-
long ranged along the moving direction and short ranged in
the other directions. This resembles the active smectic phase
[53,54], even though there is no visible layer structures in
our system. These numerical results indicate that there is
no enhancement of velocity order due to translational order.
On the contrary, alignment interactions tend to destabilize
translational order in active crystals and that the destabilizing
effect is stronger for Vicsek-like alignment (AD-I) than for
elasticity-based alignment (AD-II).
IV. ANALYTIC RESULTS
To obtain a more thorough understanding of the moving
phase of active crystals, we study a continuum theory of active
solid which is inspired by our simulation model. Although a
systematic procedure of coarse-graining [55–57] may be ap-
plied on the simulation model to obtain the continuum theory,
we shall be content with heuristic and informal derivations.
In the present case, the derivation of the continuum equa-
tions constitutes of three steps: (1) We replace the variables
ri,θi,n(θi), and ξi(t) and associated parameters, which are
defined on lattice points, by continuous fields R(r), θ(r),
n(θ(r)), and ξ(r,t) and parameters; (2) we replace the elastic
force Eq. (2) acting on a swimmer by its counterpart from
continuum elasticity theory; and, finally, (3) we Taylor expand
the dynamical equations up to first order in θand second order
in derivatives.
Since we are interested in the moving phase, it is conve-
nient to parametrize the instantaneous positions of swimmers
in the laboratory frame as
R(r,t)≡r+u(r,t)+v0t,(13)
where ris the Lagrangian coordinate, v0is the velocity of
collective motion, and u(r,t) is the displacement field defined
relative to the undeformed uniformly moving state.
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ALIGNMENT DESTABILIZES CRYSTAL ORDER IN … PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 6. [(a) and (b)] Correlation functions of the active force
director n, [(c) and (d)] correlation functions of velocity ˆv,and
[(e) and (f)] the bond-orientational correlation functions g6(˜r)of10
independent samples at different simulation steps as indicated by
different colors. Data are collected after the system reaches a steady
state with P≈0.9915 in AD-I and P≈0.9830 in AD-II. The linear
fitting at large ˜ris applied to the average of the 10 sample curves. Left
column: AD-I; right column: AD-II. The parameters are specified in
the main text.
We assume that the system is moving along the ˆ
xaxis and
that the fluctuations of the director field away from the ˆ
xaxis
are small, such that n(r,t)≈ˆ
x+θ(r,t)ˆ
ywith θ(r,t)=0
and |θ(r,t)|1. We substitute Eq. (13) back into Eq. (1) and
find:
γv0ˆ
x+γ˙
u(r,t)
≈bˆ
x+bθ(r,t)ˆ
y+F(r,t)+γ√2Dξ(r,t),(14)
where γ,b, and Dare the continuum counterpart of the pa-
rameters γp,bp, and Dpin the particle model and ξ(r,t)isa
normalized two-dimensional Gaussian white noise:
ξi(r,t)=0,(15)
ξi(r,t)ξj(r,t)=δijδ(r−r)δ(t−t).(16)
From this we immediately see that the collective moving
velocity is v0=b/γ , and hence Eq. (14) becomes a stochastic
partial-differential equation (SPDE):
γ˙
u(r,t)=bθ(r,t)ˆ
y+F(r,t)+γ√2Dξ(r,t),(17)
A nonvanishing θ(r,t) implies rotation of the active force
away from the ˆ
xaxis, which, according to Eq. (17), leads to a
nonvanishing active force along the ˆ
yaxis. This is, of course,
obviously correct.
TheelasticforceinEqs.(14) and (17) are described by
linear isotropic elasticity theory [50,58,59]:
F=μ∇2u+(λ+μ)∇(∇·u),(18a)
where λand μare two independent Lam´ecoefficients charac-
terizing the solid elasticity. This is the continuum counterpart
of Fiin Eq. (2). In components we have
Fx=(λ+2μ)∂2
xux+μ∂2
yux+(λ+μ)∂x∂yuy,
Fy=(λ+2μ)∂2
yuy+μ∂2
xuy+(λ+μ)∂x∂yux,(18b)
where λ=μ=κ√3/4 for two-dimensional triangular lattice
[60] with κthe elastic constant shown in Eq. (2). Note that the
elasticity theory shown in Eq. (18a) is isotropic, and hence
the direction of collective motion can be arbitrarily chosen.
The choice of the direction of collective moving along ˆ
xaxis
does not lose generality for the continuum model.
A. Analytic results of AD-I
The derivation of the continuum dynamic equation for θ
can be similarly carried out. For AD-I, the details and the
result are explained in Appendix C. Combining with Eq. (17),
we obtain for AD-I the following set of linearized SPDEs:
γ˙ux=Fx+γ√2Dξx
γ˙uy=bθ+Fy+γ√2Dξy
˙
θ=gθ +√2Dθη
.(19)
where η=η(r,t) is a normalized Gaussian white noise with
η(r,t)=0,η(r,t)η(r,t)=δ(r−r)δ(t−t).
Following the method discussed in Appendix D,weimple-
ment a Fourier transform (FT) for the spatial coordinates of
Eqs. (19) and obtain a Lyapunov equation:
⎛
⎝
˙
ˆux
˙
ˆuy
˙
ˆ
θ
⎞
⎠+⎛
⎝
ˆux
ˆuy
ˆ
θ
⎞
⎠=⎛
⎝
√2Dξx
√2Dξy
√2Dθη
⎞
⎠,(20)
where the matrix is given by
=1
γ⎡
⎣
λ(3k2
x+k2
y)2λkxky0
2λkxkyλ(3k2
y+k2
x)−b
00γg(k2
x+k2
y)⎤
⎦.(21)
We further define the diagonal noise matrix B=
diag(D,D,Dθ) and the correlation matrix:
M=⎛
⎝|ˆux|2ˆuxˆu∗
yˆuxˆ
θ∗
ˆuyˆu∗
x|ˆuy|2ˆuyˆ
θ∗
ˆ
θˆu∗
x
ˆ
θˆu∗
y|
ˆ
θx|2
⎞
⎠.(22)
The steady-state correlation functions for u(k,t) and θ(k,t)
can then be obtained by solving Eq. (D6):
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HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
|ˆux(k)|2=Dk2
x+3k2
yγ
3k4λ+b2Dθk2
xk2
y(gγ+4λ)
3k10λg(g2γ2+4gγλ+3λ2)≡W1(α)
k6+W2(α)
k2,(23a)
|ˆuy(k)|2=D3k2
x+k2
yγ
3k4λ+
b2Dθg3k4
x+3k2
xk2
y+k4
yγ+3k2
x+k2
y2λ
3k10λg(g2γ2+4gγλ+3λ2)≡W3(α)
k6+W4(α)
k2,(23b)
|ˆ
θ(k)|2= Dθ
gk2.(23c)
where k=|k|=(k2
x+k2
y)−1/2and αis the polar angle of k. The functions Wi(α) are defined as
W1(α)=b2Dθ(gγ+4λ)sin
22α
12gλ(g2γ2+4gγλ+3λ2),W2(α)=γD(2 −cos 2α)
3λ,(24a)
W3(α)=b2Dθ[gγ(15 +8 cos 2α+cos 4α)+λ(36 +32 cos 2α+4 cos 4α)]
24gλ(g2γ2+4gγλ+3λ2),(24b)
W4(α)=γD(2 +cos 2α)
3λ.(24c)
Various functions appearing in Wi(α) are plotted in Fig. 7.It
can be seen there that W2(α), W3(α), and W4(α) are all strictly
positive, whereas W1(α) is positive except along the ˆ
kxand
ˆ
kyaxes, where α=0,π/2,π,3π/2, respectively. Hence only
along these axes does |ˆux(k)|2scale as k−2instead of k−6
along other directions.
The real space fluctuations of displacement fields and di-
rector field can be obtained by implementing inverse Fourier
transform to Eqs. (23a), (23b), and (23c) over the kspace.
Integrating Eqs. (23a) and (23b) we see that ux(r,t)2and
uy(r,t)2diverge in power law with system size, indicating
SR translation orders which agree with our numerical obser-
vations in Fig. 4. On the other hand, integrating Eq. (23c)we
see that θ(r,t)2diverges logarithmically with system size,
and hence the active force exhibits QLRO. Since the velocity
is massively coupled to the active force, it should also exhibit
QLRO, as demonstrated by our numerical simulation.
B. Analytic results of AD-II
For AD-II, the derivation of the dynamic equation for θ,
Eq. (AD-II), is trivial: We only need to replace various vari-
ables and elastic force by their continuum counterparts, and
further expand the equation to first order in θ. The details
and the result are also explained in Appendix C. The resulting
dynamical equations are
γ˙ux=Fx+γ√2Dξx,
γ˙uy=bθ+Fy+γ√2Dξy,
˙
θ=cF
y+√2Dθη.
(25)
Again using the method in Appendix D, we implement the
Fourier transform to the spatial coordinates and find:
⎛
⎝
˙
ˆux
˙
ˆuy
˙
ˆ
θ
⎞
⎠+⎛
⎝
ˆux
ˆuy
ˆ
θ
⎞
⎠=⎛
⎝
√2Dξx
√2Dξy
√2Dθη
⎞
⎠,(26)
where the matrix Band Mare the same as those in (AD-I) and
is given by
=1
γ⎛
⎝
λ(3k2
x+k2
y)2λkxky0
2λkxkyλ(3k2
y+k2
x)−b
2γλckxkyγλc(3k2
y+k2
x)0
⎞
⎠.(27)
The steady-state correlation functions for u(k,t) and
θ(k,t) are then obtained in a similar way:
|ˆux(k)|2=b16Dθk2
xk2
y+c2Dk4
x+12k2
xk2
y+27k4
yγ2+12cDk4k2
x+3k2
yγλ
3ck4λbck2
x+9k2
yγ+12k4λ(28a)
=W5(α)
k2+O(1), as k→0,(28b)
|ˆuy(k)|2=3b2cDθk2γ+b3k2
x+k2
y4Dθ3k2
x+k2
y+c2Dk2
x+9k2
yγ2λ+12cDk43k2
x+k2
yγλ
2
3ck4λ2bck2
x+9k2
yγ+12k4λ(28c)
=W6(α)
k4+O1
k2,ask→0,(28d)
|ˆ
θ(k)|2=b2c2Dθk2
x+3k2
yγ2+bck2γ13Dθk2+c2Dk2
x+9k2
yγ2λ+12k6(Dθ+c2γ2D)λ2
bck2λbck2
x+9k2
yγ+12k4λ(28e)
=W7(α)
k2+O(1), as k→0.(28f)
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ALIGNMENT DESTABILIZES CRYSTAL ORDER IN … PHYSICAL REVIEW E 104, 064605 (2021)
where W5(α), W6(α), and W7(α) are defined as
W5(α)=c2γ2D(12 +2 cos 4α−13 cos 2α)+2Dθ(1 −cos 4α)
3c2γλ(5 −4 cos 2α),(29a)
W6(α)=bDθ
cλ2(5 −4 cos 2α),W7(α)=γDθ(2 −cos 2α)
λ(5 −4 cos 2α).(29b)
The functions appearing in W5(α), W6(α), and W7(α)are
also plotted in Fig. 7. It is obvious that W5(α), W6(α), and
W7(α) are all strictly positive. Implementing inverse Fourier
transform to Eqs. (28b) and (28d), we see that translational
order for AD-II is QLR along the ˆ
xaxis and SR along the ˆ
y
axis, consistent with the numerical results displayed in Fig. 5.
Since Eq. (28f) scales the same as Eq. (23c), we see that the
active force director order (as well as the velocity order) is
also quasi-long ranged.
To make further comparison between the numerical and an-
alytic results, we draw in Fig. 8the contour plots of correlation
functions |ˆux(k)|2,|ˆuy(k)|2, and |ˆ
θ(k)|2in the collective
moving phase and find qualitative agreements between numer-
ical and analytic results. The numerical results are shown in
the first and third rows, whereas the analytical results are in
the second and fourth rows. For dimensionless equations and
parameters of the continuum model, see Appendix E.
We also plot in Fig. 9the cut of |ˆ
θ(k)|2along the positive
ˆ
kxaxis and ˆ
kyaxis in kspace, observing a k−2decay in AD-I
and a k−2decay followed by a constant in AD-II. These scal-
ing properties match the analytical results along ˆ
kxand ˆ
kyof
Eqn. (23c) for AD-I with the form |ˆ
θ(kx,0)|2=Dθ/(gk2
x),
|ˆ
θ(ky,0)|2=Dθ/(gk2
y), and (28e) for AD-II with the form:
|ˆ
θ(kx,0)|2=γDθ
λk2
x+Dθ+c2γ2DT
bc ,(30)
|ˆ
θ(0,ky)|2=γDθ
3λk2
y+Dθ+c2γ2DT
bc .(31)
In the limit b→0, the ucorrelations in both AD-I and
AD-II, i.e., Eqs. (23a), (23b), (28a), and (28c), reduce exactly
FIG. 7. Various functions appearing in Wi(α). It can be eas-
ily seen that all the terms are positive except sin22αat α=
0,π/2,π,3π/2. Since W1(α) is proportional to sin22α,italsovan-
ishes in these places.
to those of passive crystals [50]:
|ˆux(k)|2=γD(2 −cos 2α)
3λk2,
|ˆuy(k)|2=γD(2 +cos 2α)
3λk2,
(32)
which further reduce to Eqs. (11) along ˆ
kxand ˆ
kyaxes.
C. Generalization to higher dimensions
To achieve more thorough understanding of our models
of active solids in the moving phase, we shall generalize the
analytical theory to arbitrary spacial dimensions d>2. For
a pedagogical discussion of d-dimensional elasticity theory
of isotropic solid, see Ref. [50]. The elastic force of a d-
dimensional isotropic solid can by obtained by generalization
of Eqs. (18). In fact, Eq. (18a) remains valid if we understand
∇and uas d-dimensional gradient operator and displacement
field. In components we have
Fi=μ
d

j=1
∂2
jui+(λ+μ)
d

j=1
∂i∂juj.(33)
More specifically, for the 3D case, we have
Fx=μ∂2
xux+∂2
yux+∂2
zux
+(λ+μ)(∂2
xux+∂x∂yuy+∂x∂zuz),
Fy=μ∂2
xuy+∂2
yuy+∂2
zuy
+(λ+μ)(∂y∂xux+∂2
yuy+∂y∂zuz),
Fz=μ∂2
xuz+∂2
yuz+∂2
zuz
+(λ+μ)(∂z∂xux+∂z∂yuy+∂2
zuz).(34)
As before we shall choose the positive ˆ
xaxis to be the
direction of collective motion. It is then convenient to decom-
pose uand ∇into u=(ux,u⊥) and ∇=(∂x,∇⊥), where u⊥
and ∇⊥denote respectively the (d−1)-dimensional gradient
operator and displacement field in the perpendicular subspace.
The director field can be decomposed as n=(1,δn⊥), where
again we have assumed that |δn⊥|1. The elastic force can
be similarly decomposed, F=(Fx,F⊥), and each component
is given by
Fx=μ∇2ux+(λ+μ)∂x(∇·u),(35a)
F⊥=μ∇2u⊥+(λ+μ)∇⊥(∇·u).(35b)
The overdamped dynamics of uin the moving phase is then
given by
γ˙
u(r,t)=bn⊥+F(r,t)+γ√2Dξ(r,t).(36)
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HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 8. Contour plot of the logarithm of the ucorrelations |ˆux|2,|ˆuy|2,and|ˆ
θ|2in kspace in the collective moving phase for (a) AD-I
and (b) AD-II. The first and third rows are results from simulations, whereas the second and fourth rows are results from analytic models.
where ξ(r,t)=ξxˆ
x+ξ⊥is a d-dimensional Gaussian white
noise. The dynamics of n⊥can be analogously obtained:
˙
n⊥(r,t)=dn⊥+√2Dθη⊥(r,t),(AD-I)
˙
n⊥(r,t)=c(n×F)×n+√2Dθη⊥(r,t)
=cF⊥+√2Dθη⊥(r,t).(AD-II)
FIG. 9. The cut of |ˆ
θ|2along the positive ˆ
kxaxis and ˆ
kyaxis
in kspace. In AD-I, we observe a k−2decay along both directions,
while in AD-II both of the two k−2scaling curves are followed by a
constant term.
We can substitute Eq. (35) into Eq. (36) together with Eqs.
(AD-I) and AD-II) and obtain the d-dimensional dynamical
equations for AD-I and AD-II. For AD-I we have
γ˙ux=Fx+γ√2Dξx
γ˙
u⊥=bn⊥+F⊥+γ√2Dξ⊥(37)
˙
n⊥=dn⊥+√2Dθη⊥.
For AD-II we have
γ˙ux=Fx+γ√2Dξx
γ˙
u⊥=bn⊥+F⊥+γ√2Dξ⊥(38)
˙
n⊥=cF⊥+√2Dθη⊥.
It can be easily seen that Eqs. (37) and (38) have exactly
the same structure as Eqs. (19) and (25). The correlation
functions of ufield and n⊥field can be calculated using the
same method, and the results have exactly the same structure
as Eqs. (23) and (28), as long as we replace θby n⊥.
We can now deduce the stability of director order and
translational order by evaluating the real space fluctuations of
the director field and the displacement field. For both AD-I
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ALIGNMENT DESTABILIZES CRYSTAL ORDER IN … PHYSICAL REVIEW E 104, 064605 (2021)
TABLE I. Stability of the translational order, bond-orientational
order, active force director order, and velocity order of active crystals
in 2D and 3D. In the stationary phase, bond-orientational order is
always long ranged, whereas active force director is always short
ranged.
Stationary Moving
phase phase
uψ6uu⊥n&ˆvψ6
2D AD-I QLRO LRO SRO SRO QLRO QLRO
AD-II QLRO LRO QLRO SRO QLRO LRO
3D AD-I LRO LRO SRO SRO LRO LRO
AD-II LRO LRO LRO SRO LRO LRO
and AD-II, correlation of n⊥scales as k−2for small k.This
allows us to calculate n⊥fluctuations in real space:
n⊥(r)2=ddkn⊥(k)2∼2π/a0
2π/L
ddk
k2,(39)
where 2π/Lis the low momentum cutoff set by the system
size Land 2π/a0is the high momentum cutoff set by lattice
spacing. This integral diverges with system size for d2,
which means that the lower critical dimension for director is
dv
c=2. Similarly, Eqs. (23) show that the ucorrelation for
AD-I is k−6, except on four discrete directions [the narrow
regions where W1(α) are vanishingly small do not affect the
estimate of the divergence below]. The real space fluctuations
of displacement field u⊥,uxare then
u(r)2=ddku(k)2∼2π/a0
2π/L
ddk
k6,(40)
which diverges with Lfor d6. Hence the lower critical
dimension for translational order for AD-I is dt
c=6. Finally,
Eqs. (28) show that correlation function of u⊥(displacement
in transverse direction) scales as k−4, whereas that of ux
(displacement in parallel direction) scales as k−2. A similar
calculation then shows that the lower critical dimension of
translational order is dt⊥
c=4 in perpendicular direction and
dt
c=2 in the parallel direction. The nature of various orders
for the 2D and 3D cases are summarized in Table I.The
nature of bond-orientational order, however, cannot be easily
determined from our analytic theory. Nonetheless, given our
2D results, we deduce that the bond orientational order is long
ranged above two dimensions both for AD-I and AD-II.
D. Validity of the linearization of θ
We have assumed that |θ|1 in the theoretical analysis
as well as in the simulation, under the choice of collective
moving along the ˆ
xaxis. In our simulation, the fluctuation
of θis indeed very weak, being of the order of 1/10 radian.
Hence the linearization of θis well justified.
For system with infinite system size, the real space fluctu-
ations of θand uindeed diverge both for AD-I and for AD-II,
as is demonstrated by our continuum theory. This implies
that we need to study possible topological defects of active
director, just as in classical equilibrium XY model. Is there a
defects-unbinding transition similar to the Kosterlitz-Thouless
(KT) transition [61] in equilibrium 2D XY model? Does solid
elasticity play an essential role and change the physics of KT
transition? These questions are certainly interesting but can
only be addressed by future studies.
V. CONCLUSION
In this work, we have provided a complete characterization
of various orders in two different models of active solids.
These results may be readily checked by future experiments of
active systems. It would also be interesting to study whether
the phases we discovered are related to those solidlike moving
structures observed in Vicsek-type models with short-range
interactions [21,22].
ACKNOWLEDGMENTS
X.X. acknowledge support from NSFC via Grant No.
11674217, as well as additional support from a Shanghai
Talent Program. This research is also supported by Shanghai
Municipal Science and Technology Major Project (Grant No.
2019SHZDZX01).
APPENDIX A: NUMERICAL INTEGRATOR
In this Appendix, we present the discretized dynamical
equations for ˜
rand θ, obtained by applying the first-order
Euler-Maruyama scheme [48]toEqs.(4) and (AD-

I) or (AD-

II) in the main text. The equation for ˜
ris
˙
˜
ri(˜
t+˜
t)=˙
˜
ri(˜
t)+˜
t[˜
bpn(θi(˜
t)) +
Fi(˜
t)] +V2˜
Dp˜
t.
(A1)
The equations of θfor AD-I or AD-II are respectively:
˙
θi(˜
t+˜
t)=˙
θi(˜
t)+˜
t[˜
dp(n(θi(˜
t))
×ni)·ˆ
z]+W2˜
Dθ
p˜
t,(A2)
˙
θi(˜
t+˜
t)=˙
θi(˜
t)+˜
t[˜cp(n(θi(˜
t)) ×
Fi)·ˆ
z]
+W2˜
Dθ
p˜
t.(A3)
where Vand Ware normalized two- and one-dimensional
Gaussian variables, respectively. The time step is chosen as
˜
t=10−3.
APPENDIX B: FOURIER TRANSFORM
ON TRIANGULAR LATTICE
For brevity and clarity, we omit the tilde symbol for the
dimensionless variables in this section, keeping in mind that
the quantities all refer to their dimensionless counterparts
when we apply the technique in our simulations.
The simulations use a triangular lattice with a rhomblike
boundary, where the lattice points are shown in Fig. 10 as the
solid black points. We wish to implement the FT to the data
(e.g., the displacement ux,uy, and the angle θ) sampled on
this triangular lattice into the reciprocal kspace. Although it
is easy to implement FT on a square lattice by using the fast
Fourier transform (FFT) techniques, no direct method exists
for FT on a triangular lattice. However, the triangular lattice
064605-11

HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
FIG. 10. A lattice with lattice constant a0=1andL=8. The black points denote (a) sample points in real space and (b) the corresponding
k-space points. The red triangles are the standard grid in each space, respectively. The black dashed lines in (a) denote the boundary of the
simulation box.
can be deemed as originating from a linear transform of the
square lattice. Since FT is also a linear transform, we expect
a combination of the two linear transforms may serve our
purpose. Below is a derivation of this process.
Suppose we sample a complex function f:Rn→Con
a set of n-dimensional points ∈Rnin real space. Let
:Rn→Rnbe a linear isomorphic map under which n-
dimensional integers are mapped to n-dimensional real values
such that Zn→. We may obtain the FT of fin terms of the
FT of ( f◦), which is a function on Zn. The FT of ( f◦)
is
F(f◦)(s)≡f◦(x)e−2πis,xdx
=1
|det |f(y)e−2πi−Ts,ydy
=1
|det |
F(f)−Ts.(B1)
Here a substitution y=xwas made, and it was used that
s,
−1y=−Ts,y.Note that the conventions with re-
spect to the sign of the exponent and factors of 2πmay differ.
We now discuss the meaning of the above formula. The
function f◦is the value of the sample points of farranged
in an n-dimensional array lying on a standard grid (equal-
axis orthogonal grid). Its Fourier transform gives an array of
the same shape, but the interpretation is different: The value
at index sin reality is the value at −Ts∈Rnin kspace,
differing by a multiplier |det |.
In our simulation’s setup, the space dimension n=2 and
∈R2is the set of the triangular lattice points in real space.
The function fwhose FT we want to compute can be the
displacement field ux,uyor the orientation field θ. The linear
map y=xwith =(11
2
0√3
2
) transforms a square regu-
lar lattice with lattice constant a0into the desired triangular
lattice with the same lattice constant. The transverse of its
inverse is −T=(10
−1
√3
2
√3). The function f◦takes the
value of the sample points of fon , but its arguments lie on
the corresponding square lattice points. We apply FFT on this
real-space square lattice (standard grid), obtain its FT values
onak-space square lattice (standard grid) indexed by s, and
assign the FT values to its true coordinate in kspace by left
multiplying −Tto s. Suppose the standard grid in real space
has a linear size L×Land each dimension is divided into
equally spaced npoints with distance of unity a0=1; then
there are n×ntotal sample points in both real and kspace.
In Fig. 10, a lattice of L=8 and n=8 is shown. The
lattice constant of the standard grid in real space is a0=1
and that in kspace is k=2π/L=π/4. The k-space vector
of the standard grid with the minimal magnitude s1=(0,2π
L)
transforms to the true k-space coordinate, g1=(0,2π
L
2
√3),
while s2=(2π
L,0) transforms to g2=(2π
L,−1
√3
2π
L) with both
|gi|=2π
L
2
√3.
APPENDIX C: CONTINUUM DESCRIPTION OF θ
ALIGNMENT DYNAMICS
In this Appendix, we heuristically derive a continuum de-
scription of θalignment dynamics for both AD-I and AD-II.
Similarly to the discussion at the beginning of Sec. IV for
u, the derivation of the continuum equations for θdynamics
in AD-I and AD-II constitutes the following steps: (1) We
replace the variables θi,n(θi), and ηi(t) and associated pa-
rameters gp,cp, and Dθ
p, which are defined on lattice points,
by continuous fields θ(r), n(θ(r)), and η(r,t) and parameters
g,c, and Dθand (2) we Taylor expand the alignment terms up
to first order in θand second order in derivatives.
We first derive the continuous form of the alignment term
n(θi)×ni)·ˆ
zin Eq. (AD-I) of Sec. II into its counterpart
θ in Eq. (19) of Sec. IV A. Assuming that the director
n(θi) changes slowly in space and can be described by a
continuous unit vector field n(x,y), such that for dx,dy of the
order of lattice spacing a0, the vector field near the position
(x,y) can be well approximated by its Taylor expansion to the
064605-12

ALIGNMENT DESTABILIZES CRYSTAL ORDER IN … PHYSICAL REVIEW E 104, 064605 (2021)
second order:
n(x+dx,y+dy)≈n(x,y)+(dx∂x+dy∂y)n(x,y)
+1
2(dx∂x+dy∂y)2n(x,y).(C1)
For a given lattice point, its six nearest neighbors are given by
(dx,dy)=(±a0,0),(±a0/2,±√3a0/2).(C2)
The average active director of the six neighbors, after combin-
ing like terms, is
ni≡
jn.n.i
n(θj)/6≈n(θi)+a2
0
4∂2
y+∂2
yn(θi)
=n(θi)+a2
0
4n(θi).(C3)
Substituting Eq. (C3) into the term (n(θi)×ni)·ˆ
zin Eq.
(AD-I), we obtain:
(n(θi)×ni)·ˆ
z=cos θisin θi+a2
0
4sin θi
−sin θicos θi+a2
0
4cos θi
=a2
0
4θ. (C4)
This builds up the correspondence between (n(θi)×ni)·ˆ
z
in Eq. (AD-I) of the particle model and its counterpart θ in
Eq. (19) of the continuum model. Omitting the coefficients in
Eq. (C4), the θdynamics of AD-I in the continuum model is
˙
θ=gθ +√2Dθη, (C5)
as given in Eq. (19).
The treatment for the θdynamics of AD-II is similar:
Decompose the elastic force as F=(Fx,Fy) and then put it
into the continuous version of Eq. (AD-II) together with the
assumption n(r,t)≈ˆ
x+θ(r,t)ˆ
yand keep only the leading
term, and the alignment term (n(θ)×F)·ˆzbecomes Fy. Then
we have the θdynamics in the continuum model of AD-II:
˙
θ=cF
y+√2Dθη, (C6)
as given in Eq. (25).
APPENDIX D: CALCULATING CORRELATION
FUNCTIONS
Here we discuss a general method for finding steady-state
correlation of general multidimensional Langevin equations:
˙
x(k,t)+x(k,t)=ζ(k,t),(D1)
where ζi(k,t)ζT
j(k,t)=ζi(k,t)ζj(−k,t)=2Bijδ(k−
k)δ(t−t). The matrices Band are assumed to be real, and
x(k,t) and ζ(k,t) are column vectors. Defining y=etx,itis
easy to prove dy/dt =etζ, and we have the formal solution:
x(k,t)=t
−∞
e−(t−τ)ζ(k,τ)dτ. (D2)
The correlation matrix is
x(k,t)xT(k,t)=t
−∞ t
−∞
e−(t−τ)2Bδ(k−k)δ(τ−τ)e−T(t−τ)dτdτ
=min(t,t)
−∞
e−(t−τ)2Bδ(k−k)e−T(t−τ)dτ. (D3)
Defining the autocorrelation matrix M(k,t)≡Mas
M=x(k,t)xT(k,t)dk=t
−∞
e−(t−τ)2Be−T(t−τ)dτ, (D4)
we have
M+MT=t
−∞
e−(t−τ)2(B+BT)e−T(t−τ)dτ=2e−tt
−∞
d
dτeτBeTτdτe−t=2B.(D5)
Hence we have
M+MT=2B,(D6)
which can be used to find the autocorrelation matrix Min
terms of and B. Equation (D6) is known as the Lyapunov
equation [62].
APPENDIX E: DIMENSIONLESS FORMS OF THE
CONTINUUM MODEL
Here we derive the dimensionless forms of the continuum
model. Using the space and time units a0and τ0, we define the
reduced quantities:
˜
b≡bτ0
γa0
,
F≡Fτ0
γa0
,˜μ
˜
λ≡τ0
γa2
0μ
λ,
˜
D≡Dτ0
a2
0
,˜
Dθ≡Dθτ0,˜
u≡u/a0,
˜
d≡dτ0
a2
0
,˜c≡cγa0,˜η
˜
ξ≡√τ0η
ξ.(E1)
Equations (19) and (25) can then be rewritten into the follow-
ing dimensionless forms:
˙
˜
u(˜
r,˜
t)=˜
bθˆ
y+
F(˜
r,˜
t)+2˜
Dξ(˜
r,˜
t),(E2)
064605-13

HUANG, CHEN, AND XING PHYSICAL REVIEW E 104, 064605 (2021)
˙
θ(˜
r,˜
t)=˜
d˜
θ +2˜
Dθ˜η(˜
r,˜
t),(AD-˜
I)
˙
θ(˜
r,˜
t)=˜c(n×
F)·ˆ
z+2˜
Dθ˜η(˜
r,˜
t),(AD- ˜
II)
where ˜
≡a2
0. Since the Lam´ecoefficients of a triangular
lattice are equivalent and can be related to the elastic constant
via ˜
λ=˜μ=˜κ√3/4[60], we also apply this relation in the
continuous model in order to compare the two models. For
a qualitative comparison with the particle model in Fig. 8,
the other parameters are chosen as ˜gp=˜g=10 and ˜κ=100
for AD-I, ˜cp=˜c=1 and ˜κ=20 for AD-II, and ˜
bp=˜
b=2,
˜
Dp=˜
D=0.01, and ˜
Dθ
p=˜
Dθ=0.05. These parameters are
put into Eqs. (23) and (28) to compute the correlation func-
tions.
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