# Enhancing (quasi-)long-range order in a two-dimensional driven crystal

R. Maire  Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,
91405 Orsay, France  A. Plati<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405
Orsay, France

###### Abstract

It has been recently shown that 2D systems can exhibit crystalline phases with
long-range translational order showcasing a striking violation of the
Hohenberg-Mermin-Wagner (HMW) theorem which is valid at equilibrium. This is
made possible by athermal driving mechanisms that inject energy into the
system without exciting long wavelength modes of the density field. However,
as thermal fluctuations are superimposed on the non-equilibrium driving, long-
range translational order is inevitably lost. In this paper, we discuss the
possibility of exploiting non-equilibrium effects to suppress arbitrarily
large density fluctuations even when a global thermal bath is coupled to the
system. We introduce a model of a harmonic crystal driven both by a global
thermal bath and by a momentum conserving noise, where the typical observables
related to density fluctuations and long-range translational order can be
analytically derived and put in relation. This model allows us to rationalize
the violation of the HMW theorem observed in previous studies through the
prediction of large-wavelength phonons which thermalize at a vanishing
effective temperature when the global bath is switched off. The conceptual
framework introduced through this theory is then applied to numerical
simulations of a hard-disk solid in contact with a thermal bath and driven
out-of-equilibrium by active collisions. Our numerical analysis demonstrates
how varying driving and dissipative parameters can lead to an arbitrary
enhancement of the quasi-long-range order in the system regardless of the
applied global noise amplitude. Finally, we outline a possible experimental
procedure to apply our results to a realistic granular system.

##  I  Introduction

The celebrated Hohenberg-Mermin-Wagner (HMW) theorem [1, 2, 3] is a
cornerstone of equilibrium statistical mechanics. It establishes the
impossibility of obtaining long-range order through continuous symmetry
breaking in 1D and 2D equilibrium spin systems with short-range interactions
at finite temperature. Similarly, 2D crystals with short-range interactions
exhibit translational quasi-long-range order with a correlation function
decaying as a power law due to strong phonons excitations at arbitrarily large
length scales [4]. However, their bond-orientational order is long-range [5,
6, 7, 8].

While the HMW theorem arises from equilibrium statistical physics, the
majority of systems encountered in nature operate out-of-equilibrium.
Consequently, the applicability of the theorem to such systems is not assured.
As an example, a notable breakdown is observed in the flocking behavior of
birds [9]. This process can be modeled by an active $XY$ model – the Vicsek
model [10], showcasing spin-spin long-range order in 2D [11, 12, 13, 14, 15],
in clear violation of the HMW theorem. Various studies have further explored
potential violations of this theorem in non-equilibrium spin-like systems due
to forcing at multiple temperatures [16, 17]; advection of the order parameter
through a shear flow [18, 19, 20, 21, 22]; or a colored noise [23, 24]. These
ingredients have been shown to be crucial in order to obtain spin-spin long-
range order. However, until recently, the prerequisites for translational
long-range order in 2D crystals remained unknown. Indeed, while exceptions
were found in nematic crystals [25] or chiral active matter [26], Mermin-
Wagner fluctuations are usually found to be enhanced in the presence of active
forces [27, 28, 29] and as a result, an increase of the lower critical
dimension is often observed due to giant number fluctuations [30, 31].

However, Galliano et al. presented compelling evidence of the breakdown of the
HMW theorem in 2D crystalline systems [32] formed in the active state of a
random organization model. The authors showed that, unlike in short-ranged
equilibrium systems, a simple non-equilibrium hyperuniform crystal exhibits
translational long-range order, as already hinted in Refs. 33, 34, due to the
absence of thermal fluctuations at large length scale.

Given these premises, it is important to point out that the recent
observations of crystalline phases with long-range translational order in two
dimensions only concern models where thermal-like fluctuations are absent by
construction [32, 35, 36, 37]. These studies leave open the following key
question: how is the translational order affected when thermal fluctuations
cannot be neglected?

To tackle this question, in this article, we will investigate the limits of
the HMW theorem and its potential breakdown in non-equilibrium systems through
a theoretical model of a harmonic crystal coupled to a local bath conserving
the center of mass (COM) and a global thermal-like bath. We will show that for
this system, the decay of the translational long-range order is controlled by
the temperature of the COM and not by the overall kinetic energy, as expected
at equilibrium [4]. This theory includes the case of true long-range
translational order as a particular limit and allows quantifying deviations
from it due to thermal effects.

We next apply our theory to the time-continuous analogue [38] of a discrete
random organization model in which thermal fluctuations are taken into
account. It consists of a hard-disk solid in contact with a thermal bath and
driven out-of-equilibrium by active collisions. Through numerical simulations,
we show that, in the absence of thermal fluctuations, this system exhibits a
hyperuniform long-range ordered 2D crystal in agreement with the phenomenology
observed in [32, 35]. However, when thermal-like motion is taken into account,
perfect long-range order is lost. Despite this, we demonstrate how exploiting
the non-equilibrium properties of the model enables us to suppress arbitrarily
large density fluctuations. This makes it possible to enhance the quasi-long-
range translational order in a non-equilibrium crystal without the need to
neglect or fine-tune thermal fluctuations. Our model will also serve as a
coarse-grained representation of a confined quasi-2D vibrated granular system
[38] thus naturally providing a platform for investigating the enhancement of
quasi-long-range translational order in experimental systems.

The paper is organized as follows: In Sec. II we introduce the theoretical
model for a harmonic crystal and derive the analytical expression of the
typical observables related to density fluctuations and long-range
translational order. In Sec. III, we report the numerical results for the non-
equilibrium hard-disk solids. Finally, Sec. IV contains the conclusion and a
brief discussion about a possible experimental procedure to enhance quasi-
long-range order in a realistic granular system.

##  II  Theory

###  II.1 The model

We study a 2D crystal made of $N$ particles with masses $m$ arranged on a
periodic lattice of size $L\times L$ and lattice spacing $a$. Each particle on
the lattice site $\bm{n}$ interacts with its neighbors $\\{\bar{\bm{n}}\\}$
via a harmonic interaction $K$. Additionally, every particle is coupled to two
baths: a global one and a local one which conserves the momentum and COM. In
order to have a well-defined equilibrium limit, both of them separately
respect the fluctuation dissipation theorem (FDT) [39, 40, 41, 42]. From a
physical standpoint, the global bath can emerge from the coarse-graining of an
external energy source such as in the case of colloids diffusing in a fluid or
beads vibrating on a rough surface, while the local bath is more likely to
come from an internal source such as collisions between particles, as with the
noise current in fluctuating hydrodynamics [43, 44, 45, 46]. The displacement
$\bm{u}_{n}$ of each particle at lattice site $\bm{n}$ with respect to their
ideal lattice position is described by the following Langevin equation:

$m\ddot{\bm{u}}_{n}=-K\sum_{\\{\bar{n}\\}}({\bm{u}}_{n}-\bm{u}_{\bar{n}})+\bm{F}_{com}+\bm{F}_{loc}$
(1)

The first term on the right-hand side of the equation represents the harmonic
interaction between neighbors with spring constant $K$.

The second term $\bm{F}_{com}$ represents the coupling between the system and
a global bath, which does not conserve the position of the COM:

$\bm{F}_{com}=-\gamma_{com}\dot{\bm{u}}_{n}+\sqrt{2\gamma_{com}T_{com}}\bm{\xi}_{n},$
(2)

where $\gamma_{com}$ is a global damping and $T_{com}$ is the temperature of
the bath. The spatial components $\alpha$ of the white noise are Gaussian and
uncorrelated, with zero average:

$\langle\xi_{n}^{\alpha}(t)\xi_{m}^{\beta}(t^{\prime})\rangle=\delta(t-t^{\prime})\delta_{n,m}\delta^{\alpha,\beta}~{}~{}~{}~{}~{}~{}\langle\bm{\xi}_{n}(t)\rangle=0.$
(3)

Finally, the last term in Eq. (1) represents a second bath, at temperature
$T_{loc}$, conserving the momentum and the position of the COM. To fulfill
this requirement, we use a discretized version of Model B [47, 48, 49, 50, 51,
52]:

$\bm{F}_{loc}=-\gamma_{loc}\sum_{\\{\bar{n}\\}}({\bm{\dot{u}}}_{n}-\bm{\dot{u}}_{\bar{n}})+\sqrt{2\gamma_{loc}T_{loc}}(\bm{\nabla}\cdot\bm{\Xi}_{{n}}).$
(4)

The conservation of $\sum_{n}\dot{\bm{u}}_{n}$ is ensured by a discrete
Laplacian of $\dot{\bm{u}}$ for the damping $\gamma_{loc}$, taken to be the
average over first neighbors and a divergence acting on a rank 2 random tensor
$\bm{\Xi}$. This type of damping naturally arises in granular gases where
collisions tend to align particles [53], in active matter with effective
alignment [54, 46] or in Dissipative Particle Dynamics [55, 56, 57]. More
generally, with the corresponding equilibrium noise, it plays the role of a
discrete hydrodynamic viscosity. Note that if every particle moves in the same
direction, neither damping nor noise is applied, in line with the idea that
this local noise might arise from collisions between particles and acts
locally. Our model constitutes a simplified lattice equivalent to a
fluctuating hydrodynamics description of a solid [58, 59, 60, 61, 62, 63, 64].
The variance and the average of the rank 2 tensor $\bm{\Xi}_{n}$ are fully
determined by our assumption that the FDT holds separately for both noises
[42, 65]:

$\langle\bm{\Xi}_{n}^{\alpha,\mu}(t)\bm{\Xi}_{m}^{\beta,\nu}(t^{\prime})\rangle=\delta(t^{\prime}-t)\delta^{\alpha,\beta}\delta^{\mu,\nu}\delta_{n,m},$
(5)

We further discuss the peculiarity of the noise as well as the expression of
the discrete Laplacian and divergence in Appendix A following Ref. 66.

For simplicity, in the following, we chose to work in Fourier space by
performing a semi-discrete Fourier transform (see Appendix B) on Eq. (1) [67,
68]:

$\begin{split}\left(-mw^{2}-iw(\gamma_{com}+\gamma_{loc}\omega_{k}^{2})+K\omega_{k}^{2}\right)\tilde{\bm{u}}_{k}=&\sqrt{2\gamma_{com}T_{com}}\tilde{\bm{\xi}}_{k}\\\
+&\sqrt{2\gamma_{loc}\omega_{k}^{2}T_{loc}}\tilde{\bm{\eta}}_{k},\end{split}$
(6)

with $\omega_{k}$ the dispersion relation of the lattice divided by the
natural frequency of the lattice $\sqrt{K/m}$ or equivalently, the eigenvalue
of the discrete Laplacian. For instance, in a square lattice,
$\omega_{k}^{2}=2(2-\cos(k_{x}a)-\cos(k_{y}a))$. Both noises have unit
variance and zero average, from a Fourier transform we obtain:

$\begin{split}\langle\tilde{\xi}_{k}^{\alpha}(w)\tilde{\xi}_{q}^{\beta}(w^{\prime})\rangle=\delta(w+w^{\prime})\delta_{q,-k}\delta^{\alpha,\beta}~{}~{}~{}~{}~{}\langle\tilde{\xi}_{k}^{\alpha}(w)\rangle=0\\\
\langle\tilde{\eta}_{k}^{\alpha}(w)\tilde{\eta}_{q}^{\beta}(w^{\prime})\rangle=\delta(w+w^{\prime})\delta_{q,-k}\delta^{\alpha,\beta}~{}~{}~{}~{}~{}\langle\tilde{\eta}_{k}^{\alpha}(w)\rangle=0\end{split}$
(7)

The noises are simple to generate in the reciprocal space compared to the ones
in real space. Moreover, we clearly see that the local bath does not act on
the COM since it vanishes at $\bm{k}=0$ implying that, as wanted, the motion
of the COM is completely governed by the global bath.

Due to the harmonicity of the crystal, Eq. (6) is non-interacting in $k$-space
and every mode reaches an independent equilibrium-like steady state at an
effective $k$-dependent temperature:

$\tilde{T}_{k}=\dfrac{\gamma_{com}T_{com}+\gamma_{loc}\omega_{k}^{2}T_{loc}}{\gamma_{com}+\gamma_{loc}\omega_{k}^{2}}.$
(8)

Since the dispersion relation must vanishes at low $k$:
$\omega_{k}^{2}\sim(a\bm{k})^{2}$ [68] we obtain the two limiting cases:

$\begin{split}\tilde{T}_{k\to 0}&=T_{com}\\\
\tilde{T}_{k\to\infty}&=T_{loc}\quad\textrm{if}\quad\gamma_{loc}\gg\gamma_{com}.\end{split}$
(9)

That is, the large length scale temperature is controlled by $T_{com}$ since
the energy created on small length scales by the local bath is damped by
$\gamma_{com}$ on every scale. While the small length scale temperature is
controlled by both the global and local bath. In the following, we will assume
that the system is more weakly coupled to the local bath compared to the
global bath: $\gamma_{loc}\gg\gamma_{com}$ which leads to
$T_{k\to\infty}=T_{loc}$.

The equilibrium-like nature of our equation stems from the assumption that
both baths are delta correlated and that each mode is independent on the
others in $k$ space. This would not be the case in a realistic system with
anharmonic terms for instance. Nonetheless, energy injection would still be
localized at the scale of collisions – albeit with an energy flux between
modes [69] – and large wavelength thermal fluctuations would remain damped by
$\gamma_{com}$. Therefore, the rough phenomenology introduced in this section
is expected to hold as well, in these more complex cases.

###  II.2 Hyperuniformity

As already noted in Ref. 32, a key requirement to obtain a long-range ordered
2D crystal is hyperuniformity i.e. the suppression of density fluctuations at
arbitrarily large length scales. Within the context of the proposed
theoretical model, we derive here the analytical expression of the structure
factor $S(\bm{k})$ whose vanishing low-$\bm{k}$ limit indicates the presence
of hyperuniformity in a system [33].

We start by computing the static displacement function (SDF)
$C_{uu}(\bm{k})\equiv C_{uu}(\bm{k},t=0)$:

$\begin{split}C_{uu}(\bm{k})&=\dfrac{1}{2\pi}\int\mathrm{d}w\langle\tilde{\bm{u}}_{k}(w)\cdot\tilde{\bm{u}}_{-k}(-w)\rangle\\\
&=\dfrac{2}{K}\dfrac{\gamma_{com}T_{com}+\gamma_{loc}\omega_{k}^{2}T_{loc}}{\omega^{2}_{k}(\gamma_{com}+\gamma_{loc}\omega_{k}^{2})}=\dfrac{2}{K\omega_{k}^{2}}\tilde{T}_{k}.\end{split}$
(10)

where Eq. (6) and (7) have been used to calculate averages over the noise. The
above identity represents the equipartition of elastic energy at an effective
temperature $\tilde{T}_{k}$ which is given by Eq. (8). The standard
equipartition theorem for phonons $K\omega_{k}^{2}C_{uu}(\bm{k})/2=T_{loc}$ or
$K\omega_{k}^{2}C_{uu}(\bm{k})/2=T_{com}$ is recovered in the three
equilibrium limits of the model, that is either when
$\gamma_{com}=0,\gamma_{loc}=0$ or $T_{com}=T_{loc}$. In the large lengths
limit where $\omega_{k}^{2}\sim(a\bm{k})^{2}$, the SDF reads:

$\begin{split}C_{uu}(\bm{k})=&\dfrac{2T_{com}}{K(a\bm{k})^{2}}+\dfrac{2\gamma_{loc}(T_{loc}-T_{com})}{K\gamma_{com}}\\\
&+\mathcal{O}\left((T_{loc}-T_{com})(a\bm{k})^{2}\right).\end{split}$ (11)

and diverges as $1/(a\bm{k})^{2}$ at equilibrium or more generally when
$T_{com}\neq 0$. However, when $T_{com}=0$ and $\gamma_{com}\neq 0$, the
phonons created locally are damped over large distances and long wavelength
phonons are completely suppressed since their elastic energy
$K\omega_{k}^{2}C_{uu}(\bm{k})/2$ goes to 0. The same type of result can be
found for the static velocity factor
$mC_{\dot{u}\dot{u}}(\bm{k})/2=\tilde{T}_{k}$ showing a depletion of kinetic
energy on large scales.

At this point, it is illuminating to note that the long-range structure factor
$S(\bm{k})$ and the SDF are linked by the relation [34, 33]:

$\begin{split}S(\bm{k})&=\left\langle\dfrac{1}{N}\left|\sum_{\bm{n}}e^{i\bm{k}\cdot(a\bm{n}+\bm{u}_{n})}\right|^{2}\right\rangle\\\
&=\left\langle\left|\bm{k}\cdot\tilde{\bm{u}}_{k}\right|^{2}\right\rangle+\mathcal{O}\left(\left|\bm{k}\cdot\tilde{\bm{u}}_{k}\right|^{4}\right)\\\
&=|\bm{k}|^{2}C_{uu}(|\bm{k}|)+\mathcal{O}\left(\left|\bm{k}\cdot\tilde{\bm{u}}_{k}\right|^{4}\right)\\\
\end{split}$ (12)

Where we used the fact that in our model the longitudinal dispersion relation
is equal to the transversal ones. Otherwise, the last relation only holds for
longitudinal polarization of the displacement [34]. In an equilibrium system,
the value of the structure factor at $\bm{k}=0$ is a constant proportional to
the isothermal compressibility and the temperature [70, 71]. However in our
case, from Eq. (11), it follows that:

$\begin{split}S(\bm{k})=&\dfrac{2T_{com}}{Ka^{2}}+\dfrac{2\gamma_{loc}(T_{loc}-T_{com})}{\gamma_{com}}\bm{k}^{2}\\\
&+\mathcal{O}\left((T_{loc}-T_{com})a^{2}\bm{k}^{4}\right).\end{split}$ (13)

When $T_{com}=0$ and $\gamma_{com}\neq 0$, since the SDF is finite at low $k$,
the structure factor behaves as $S(\bm{k})\sim\bm{k}^{2}$, unveiling the
hyperuniformity of our non-equilibrium system when the COM position is
conserved by the noise at finite global damping. This aligns with results
found in Refs. 26, 50, 72, 73, 74, 32, 75 and confirms that our model includes
a suitable limit to study the emergence of long-range order in dimension lower
than three.

###  II.3 (Quasi-)long-range order

To quantitatively describe the emergence of long-range order in our model, we
consider the crystalline translational correlation function, defined as [76,
77]:

$g_{\bm{G}}(|\bm{n}-\bm{m}|)=\left\langle
e^{i\bm{G}\cdot(\bm{u}_{n}-\bm{u}_{m})}\right\rangle=e^{-\langle(\bm{G}\cdot(\bm{u}_{n}-\bm{u}_{m}))^{2}\rangle/2},$
(14)

where $\bm{G}$ is one of the inner Bragg-peak vectors of the crystal and the
last equality follows from the gaussianity of the stationary probability
distribution function. In a long-range ordered crystal, the correlation
function decays to a constant value while in a quasi-long-range 2D equilibrium
crystal it exhibits an algebraic decay [4, 7, 77, 78, 79, 80]. From Eq. (14),
it is clear that the asymptotic behavior of the displacement correlation
function $\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle$ governs the translational
long-range behavior of the system. Specifically, if the displacement
correlation function reaches a constant or decays at large distances, true
long-range order is expected, otherwise, only quasi-long-range order or short-
range order is found. In the infinite system size limit and at large distances
compared to the lattice spacing it can be approximated as (see Appendix C) :

$\begin{split}\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle&\simeq\dfrac{a^{2}}{\pi}\int_{\frac{1}{a|\bm{n}-\bm{m}|}}^{\frac{\pi}{a}}\mathrm{d}kkC_{uu}(|\bm{k}|)\\\
&\simeq\left\\{\begin{array}[]{ll}\dfrac{2T_{com}}{\pi
K}\log(\pi|\bm{n}-\bm{m}|)\quad|\bm{n}-\bm{m}|\gg\delta^{-1}\\\ \\\
\dfrac{2T_{loc}}{\pi
K}\log(\pi|\bm{n}-\bm{m}|))\quad|\bm{n}-\bm{m}|\ll\delta^{-1}.\end{array}\right.\end{split}$
(15)

where $\omega_{k}^{2}=(a\bm{k})^{2}$ was assumed and
$\delta=\sqrt{\gamma_{com}/\gamma_{loc}}$ is the adimensionalized natural
inverse length scale of the system. At $T_{com}=T_{loc}$ we recover the
equilibrium logarithmic increase with distance, proportional to the
temperature [4]. In a non-equilibrium setting with $T_{com}\neq T_{loc}$, the
divergence is still present, but with $T_{com}$ as the prefactor at large
scales and $T_{loc}$ at small scales (still larger than the lattice spacing).
The numerical ”exact” computation of the integral for a hexagonal lattice and
its comparison with our approximate formula are shown in Fig. 1. As expected,
the approximations work well at large distances. At short distances, although
the global scaling remains the same, strong oscillations appear on top of the
logarithmic scaling since the phonons start to feel the discreteness of the
lattice. For large $\delta$, the transition between the two regimes happens
far away from the lattice spacing allowing for a large range of values over
which both scalings are clear and oscillations small.

Figure 1: Numerical and approximate values of
$\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle$ at
$\gamma_{com}/\gamma_{loc}=10^{-3}$ given by Eqs. (15) and (36). For the
numerically solved integral, we use the dispersion relation of a hexagonal
lattice.

From Eq. (14) and (15) we obtain that the translational correlation function
exhibits a double power law behavior:

$g_{\bm{G}}(|\bm{n}-\bm{m}|)\sim\left\\{\begin{array}[]{ll}|\bm{n}-\bm{m}|^{-|\bm{G}|^{2}T_{com}/(2\pi
K)}\quad|\bm{n}-\bm{m}|\gg\delta^{-1}\\\ \\\
|\bm{n}-\bm{m}|^{-|\bm{G}|^{2}T_{loc}/(2\pi
K)}\quad|\bm{n}-\bm{m}|\ll\delta^{-1},\end{array}\right.$ (16)

where the angle average of the dot product appearing in Eq. (14) was taken
into account by assuming an isotropic crystal. At non zero $T_{com}$, the
large-scale decay of the correlation function is algebraic indicating quasi-
long-range order [81]. Nonetheless, in contrast to the equilibrium picture,
the average kinetic energy of the system can be arbitrarily high, the only
crucial factor for the translational quasi-long-range behavior of the system
at large scale, is the temperature of the COM or equivalently, the temperature
of the phonons existing on these scales. We also note that, as expected when
the COM is conserved at finite global damping ($T_{com}=0$ and
$\gamma_{com}\neq 0$), the system displays genuine long-range order as
observed in Ref. 32 and the HMW theorem is broken.

We finally point out that the connection between hyperuniformity and the
emergence of long-range order is made explicit by Eqs. (15) and (12) which
show the relationship between the large distance behavior of the displacement
correlation function and the low-$k$ limit of the structure factor. Indeed,
for large distances, we find in 2D:

$\begin{split}\langle(\bm{u}_{n}-\bm{u}_{0})^{2}\rangle&\propto\int_{\frac{1}{a|\bm{n}|}}^{\frac{\pi}{a}}\mathrm{d}k\,kC_{uu}(|\bm{k}|)\\\
&\propto\int_{\frac{1}{a|\bm{n}|}}^{\frac{\pi}{a}}\mathrm{d}k\,\frac{S(|\bm{k}|)}{k}\\\
&\propto\left\\{\begin{array}[]{ll}S_{0}\log(|\bm{n}|)&\mbox{if
}S(k)=S_{0}+\mathcal{O}(k)\\\ S_{1}&\mbox{if
}S(k)=S_{1}k^{\beta}+\mathcal{O}(k^{\beta+1})\end{array}\right.\end{split}$
(17)

with $\beta>0$.

###  II.4 Breakdown of the HMW theorem

The breakdown of the HMW theorem when $T_{com}=0$ ultimately stems from the
fact that large wavelength phonons are thermalized with a bath at 0 effective
temperature; a temperature for which, even at equilibrium, long-range order is
expected by the HMW theorem. In this sense, the theorem is broken in a way
expected from its equilibrium definition while simultaneously being strongly
broken since the large-scale SDF at $\bm{k}\to 0$ is constant and does not
diverge at all. This can be traced back to the strong hyperuniformity
$S(\bm{k})\sim\bm{k}^{2}$ of our system. We remark that our rationalization of
the emergence of stable long-range order based on large wavelength phonons
thermalizing at a vanishing effective temperature can be applied also for the
non-equilibrium crystal obtained with the random organization model [32].

It is important to note that an exponent 2 for the small $k$ structure factor
is not a necessary condition to break the HMW theorem. Indeed, as understood
from Eq. (17), any hyperuniformity in 2D would sufficiently reduce the
divergence of the SDF to allow long-range order. This mechanism parallels
bird-flocking dynamics, where alignment among neighbors reduces the infrared
divergence of the correlation functions to a scaling of $1/k^{a}$ with $a<2$
[11, 13], allowing for long-range order.

From a mathematical point of view this breakdown of the HMW theorem in the
case $T_{com}=0$ at finite $\gamma_{com}$ is manifested through the
regularization of the infrared divergence of the integral used to compute the
displacement correlation function (Eq. (35)) via $\gamma_{com}$:

$\begin{split}\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle\simeq\dfrac{2a^{2}}{K\pi}\int_{\frac{1}{a|\bm{n}-\bm{m}|}}^{\frac{\pi}{a}}\mathrm{d}kk&\left(\dfrac{1}{\omega_{k}^{2}}\dfrac{T_{com}}{1+(\omega_{k}/\delta)^{2}}\right.\\\
&\quad\left.+\dfrac{T_{loc}}{\delta^{2}+\omega_{k}^{2}}\right).\end{split}$
(18)

When $T_{com}=0$, the first term vanishes. In the second term, the phonons
exhibit a behavior akin to particles with an effective ”mass”
$\sqrt{K}\delta=\sqrt{K\gamma_{com}/\gamma_{loc}}$ and an adimensionalized
dispersion relation $\Omega_{k}=\sqrt{\delta^{2}+\omega_{k}^{2}}$. Indeed, an
equilibrium system with Hamiltonian [68]:

$\mathcal{H}/K=\dfrac{a^{2}}{2}\left(2\operatorname{Tr}\left[\bm{\mathsf{u}}\cdot\bm{\mathsf{u}}^{T}\right]-\operatorname{Tr}\left[\bm{\mathsf{u}}\right]^{2}\right)+\dfrac{\delta^{2}}{2}\bm{u}^{2},$
(19)

with $\bm{\mathsf{u}}$ the infinitesimal strain tensor:
$\bm{\mathsf{u}}=(\bm{\nabla}\bm{u}+(\bm{\nabla}\bm{u})^{T})2$, would produce
a similar equilibrium displacement correlation. This effective mass pins down
the particles on their ideal lattice position allowing long-range order. Of
course, when $T_{com}\neq 0$, the first term of Eq. (18) becomes significant
at small $k$ and induces the logarithmic divergence at large scales discussed
above.

It is interesting to note that hyperuniformity of equilibrium crystals is
solely achieved through long-range or external interactions [33], in contrast
with our system and that of others, where hyperuniformity is allowed through
the breakdown of the FDT and correlated noise [24, 82, 26, 83, 84]. This is
reminiscent of long-range correlations induced by bulk conservation law and
dissipation in self-organized criticality [85, 86, 44, 53, 87, 88, 89, 90, 91,
92]. See Ref. 93 for a recent review. Moreover, COM conservation is not the
only route to hyperuniformity and systems with multiplicative [94, 82] or time
correlated noise can have suppressed long-range fluctuations [35, 24].
Additionally, recent work on particular non-equilibrium collective excitations
in active solids, known as entropons [67, 95], highlight that on a general
basis, active forces only add an additional term to the SDF, linked to entropy
production but do not affect its phononic part. Thus, active forces alone
cannot tame the $1/k^{2}$ divergence of the phononic part of the equilibrium
SDF and in general, increase the density fluctuations [29, 27, 28]. Indeed,
active short-ranged systems exhibiting hyperuniformity usually lose this
property in the presence of a global thermal equilibrium-like noise [96, 37,
97, 36]. This supports the idea that non-equilibrium-like colored or
multiplicative noises are crucial for hyperuniformity and translational long-
range order in 2D crystals featuring short-ranged interactions. Further
discussions about entropons in our system can be found in Appendix D. Note
however, that hyperuniformity in passive or active scalar field theories has
been found even in the presence of a thermal noise for model B type equations
[98, 99, 100, 101].

Of course, the analysis performed so far predicts the violation of the HMW
theorem only in the singular limit $T_{com}=0$, $\gamma_{com}\neq 0$. However,
the results obtained in Sec. II.3 suggest how to approach this physical
condition when $T_{com}\neq 0$. Indeed, the key quantity that controls the
strength of the quasi-long-range order (and its possible limit to true long-
range) is the exponent of the algebraic decay of the translational correlation
function for $|\bm{n}-\bm{m}|\gg\delta^{-1}$ in Eq. (16) which is proportional
to the ratio $T_{com}/K$. This means that, for a finite $T_{com}$, one can
still obtain an arbitrary slow decay by increasing the elastic constant $K$.
This is clearly a quite limiting and trivial way to enhance the quasi-long-
range order of the system since it consists of a direct increase of the
particle interaction strength and would also work at equilibrium. However, in
the next section, we will see how the phenomenology of a realistic model of a
non-equilibrium crystal can be mapped into the one of our model with an
effective $K$ which depends on the non-equilibrium properties of the realistic
system (i.e. dissipation and driving parameters). This will be the key point
of our strategy to enhance quasi-long-range through non-equilibrium effects.

##  III  Hard-disk crystal driven by active collisions

###  III.1  A coarse grained model

We explore a practical application of our theoretical framework to investigate
the breakdown of the HMW theorem and the dependence of (quasi-)long-range
order on the noise in realistic non-equilibrium systems. To conduct this
investigation, we use the model introduced in Ref. 38 with an additional
global noise. This can be thought as an underdamped continuous-time analogue
of a random organization model [102] describing sheared suspensions [103]. In
the absence of global noise, these models present an absorbing phase
transition, are hyperuniform [33] at the transition point [73, 82] and, if the
COM is conserved, in the active state [72, 75, 26]. As will be further
discussed in the conclusion, this model has been used to describe the
experimental realization of vibrated quasi-2d granular systems [104, 38, 105].

We study this system by performing hybrid event-driven/time-stepped molecular
dynamics simulations [106] (see Appendix E) of $N$ active hard disks of
diameter $\sigma$ and mass $m$ in a 2D square of size $L$ with periodic
boundary conditions. We define an arbitrary unit of time $\hat{\tau}$. The
(granular) temperature $T$ of the system is defined as its global kinetic
energy:

$T=\dfrac{1}{2}m\sum_{i=1}^{N}\bm{v}_{i}^{2},$ (20)

with $\bm{v}_{i}$ the velocity of the particle $i$. The disks experience a
global white bath during their free flight:

$\dfrac{\mathrm{d}\bm{v}}{\mathrm{d}t}=-\gamma_{com}\bm{v}+\sqrt{2\gamma_{com}T_{com}}\bm{\eta}(t),$
(21)

with

$\langle{\eta}_{i}^{\alpha}(t){\eta}_{j}^{\beta}(t^{\prime})\rangle=\delta(t-t^{\prime})\delta_{i,j}\delta^{\alpha,\beta}~{}~{}~{}~{}~{}\langle{\eta}_{i}^{\alpha}(t)\rangle=0.$
(22)

Upon collision, two disks undergo a momentum-conserving active collision [104,
107, 108]:

$\begin{split}\bm{v}_{i}^{\prime}&=\bm{v}_{i}+\dfrac{1+\alpha}{2}(\bm{v}_{ij}\cdot\hat{\bm{\sigma}}_{ij})\hat{\bm{\sigma}}_{ij}+\Delta\hat{\bm{\sigma}}_{ij}\\\
\bm{v}_{j}^{\prime}&=\bm{v}_{j}-\dfrac{1+\alpha}{2}(\bm{v}_{ij}\cdot\hat{\bm{\sigma}}_{ij})\hat{\bm{\sigma}}_{ij}-\Delta\hat{\bm{\sigma}}_{ij},\end{split}$
(23)

where $0\leq\alpha\leq 1$ is the coefficient of restitution, $\Delta>0$ is a
velocity injection term, $\bm{v}_{i}^{\prime}$ the post-collision velocity of
particle $i$, $\bm{v}_{i}$ its pre-collision velocity, and
$\hat{\bm{\sigma}}_{ij}$ and $\bm{v}_{ij}$ are respectively the unit vector
joining particles $i$ and $j$ and the relative velocity between them. In the
limit $\Delta=0$, the usual collision rule of dissipative hard disks is
recovered.

As evidenced by the energy change during a collision

$E^{\prime}-E=m\Delta^{2}+m\alpha\Delta|\bm{v}_{ij}\cdot\hat{\bm{\sigma}}_{ij}|-m\dfrac{1-\alpha^{2}}{4}|\bm{v}_{ij}\cdot\hat{\bm{\sigma}}_{ij}|^{2},$
(24)

the parameter $\Delta$ controls the intensity of the energy injection at
collision. These collisions will thus play the role of the momentum-conserving
bath introduced in Sec. II at temperature $T_{loc}$ and associated effective
damping $\gamma_{loc}\omega_{k}^{2}$.

Hence, to recapitulate the differences and similarities between the harmonic
crystal presented in the theory and the granular hard disks of interest here:
both models feel an external global Gaussian bath at temperature $T_{com}$ and
damping $\gamma_{com}$, however, the active collisions in the granular system
are modeled as an effective local bath at temperature $T_{loc}$ and damping
$\gamma_{com}\omega_{k}^{2}$ in the theory. From a physical standpoint, the
effective $T_{loc}$ in the granular system is technically a function of every
parameter of our simulation including, $T_{com}$ and $\gamma_{com}$ because
the energy change at the collision depends on the relative velocities of the
particles colliding, as understood from Eq. (23), which are affected by the
global bath and the packing fraction. This will not change the physical
picture given in this article nonetheless, we should keep it in mind. This
peculiarity is further discussed in Appendix F.

When $T_{com}=0$, the dynamics given by Eq. (23) and (21) conserves the
position of the COM if the initial total momentum of the system is equal to 0
(otherwise, it exponentially relaxes to 0). Hyperuniformity is thus expected
[26] and we recover the system studied in Ref. 38 with an absorbing state at
low density (a feature absent in the lattice model because $T_{loc}$ is
independent on $T_{com}$ and $\gamma_{com}$). Indeed, at low density or
packing fraction $\phi=N\pi\sigma^{2}/4L^{2}$, the active collisions cannot
compensate for the energy dissipated by the drag during the free flight
leading the system to an arrested state. Denser systems attain an active
steady state where the energy injection at collision is compensated by the
dissipation of the damping.

###  III.2 Hyperuniformity and long-range order

Figure 2: Long-range behavior of three different crystalline systems with
$T_{com}=0$. ”Equilib.” corresponds to pure hard disks ($\alpha=1$,
$\Delta/\sigma\hat{\tau}^{-1}=0$ and $\gamma_{com}=0$), ”$\mathit{\Delta}$”
corresponds to hard disks with active collisions ($\alpha=0.95$,
$\Delta/\sigma\hat{\tau}^{-1}=0.015$ and $\gamma_{com}/\hat{\tau}=0$) and
”$\mathit{\Delta+\gamma_{com}}$” corresponds to hard disks with active
collisions and global damping ($\alpha=0.95$,
$\Delta/\sigma\hat{\tau}^{-1}=0.015$ and $\gamma_{com}\hat{\tau}=0.02$). a)
Structure factor of the three systems for three different system sizes in log-
log scale. The dashed line represents a power law scaling of $k^{-2}$. The
inset is the same plot for the static velocity factor. b) Value of the plateau
of the MSD of the three systems as a function of the number of particles,
averaged over five different realizations in semi-log. The dashed line is a
fit of the logarithmic increase of the equilibrium curve. c) Translational
correlation function of the three systems as a function of the distance in
log-log with $N=440538$. The dashed line corresponds to the expected power law
scaling of $g_{\bm{G}}$ extracted from the logarithmic increase of the
equilibrium curve in b).

To investigate the crystalline order in this system, we simulate crystalline
configurations at $\phi=0.75$ which sets the lattice spacing $a$. This packing
fraction corresponds to systems with long-range bond orientational order at
equilibrium [109] but can give rise to a hexatic or a liquid phase in highly
dissipative granular systems [110]. To avoid these scenarios, we choose a
relatively high $\alpha\geq 0.95$ and always check that the crystalline phase
is stable. For simulations at $T_{com}=0$, we stay away from the absorbing
region by choosing $\gamma_{com}$ and $\Delta$ such that $\phi\gg\phi_{c}$,
ensuring a sufficiently high energy injection at collision.

Figure 3: Quasi-long-range order dependence on COM temperature. a) Simulation
of a system of active hard disks at fixed $\Delta/\sigma\hat{\tau}^{-1}=0.04$,
$\gamma_{com}\hat{\tau}=0.3$ and $\alpha=0.95$ for different values of the
global Langevin bath temperature $T_{com}$ at fixed $\gamma_{com}$. We assume
$T_{loc}$ is fixed and given by Eq. (48). The main figure shows the dependence
on $T_{com}$ (color) of the plateau of the MSD for various system sizes. $T$
is the kinetic energy of the system (see Eq. (20)). The coefficient in front
of the logarithmic increase of the MSD $\kappa$ is shown in the inset as a
function of $T_{com}/T$. A linear scaling is given by a grey dashed line as a
guide. b) Scaling of the logarithmic increase of the MSD for multiple systems
with various $\Delta$ and $\gamma_{com}$ at fixed
$T_{com}/m(\sigma\hat{\tau}^{-1})^{2}=0.0025$. The underlying finite size
analysis on the MSD is performed for 30 systems ranging from $N=29946$ to
$N=1000680$ for each point. A linear scaling is given by a grey dashed line as
a guide.

We first perform a similar analysis as the one done in Ref. 32, fixing the COM
by setting $T_{com}=0$. The results for three different systems are depicted
in Fig. 2. Panels a), b) and c) show respectively the structure factor (and
the static velocity factor in inset) as a function of $k$, the long-time
behavior of the mean square displacement (MSD) for different system sizes and
finally the translational correlation function as a function of the distance.
The three quantities contain the same information expressed differently. It is
illuminating to analyze all three together to better understand the theory
given in Sec. II and verify its consistency. We note that the MSD of a
particle in a finite crystal reaches a plateau at large times. The value of
this plateau is of course dependent on the size of the cage made by
neighboring particles but also on the collective excitations that induce
vibrations on the crystal. From the Fourier transform of the displacement
$\tilde{\bm{u}}_{k}(w)$, it can be shown that the correlation
$\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle$ for an infinite system corresponds
to MSD$(t\rightarrow\infty)$ for a system of size $(a|\bm{m}-\bm{n}|)^{2}$ in
the large system size limit. We chose to use the MSD instead of the
displacement correlation to check relations derived in the theory part
involving the displacement correlation, the former being simpler to measure.

We first focus on the curves ”Equilib” which correspond to an equilibrium
system of pure hard disks ($\alpha=1$, $\Delta=0$ and $\gamma_{com}=0$). As
expected, its structure factor is approximately flat (it has in reality an
Ornstein-Zernike shape due to the non linear interactions [70]) and reaches a
well defined value at small $q$. Its large time MSD grows as a unique
logarithm of the particle number as predicted by Eq. (15) and thus its
correlation function decreases as a power law with an exponent linked to the
logarithmic decrease of the MSD as given by (16) and illustrated with the
dashed lines. The system displays quasi-long-range order.

We now turn to the $\Delta$ model, a system with only active collision and no
global damping ($\alpha<1$, $\Delta\neq 0$ but $\gamma_{com}=0$). Its
structure factor has a transient hyperuniform scaling before reaching a
plateau at small $q$. The same behavior is found for the static velocity
factor. This is incompatible with the modelization of the active collision as
an FDT respecting bath at temperature $T_{loc}$ and damping
$\gamma_{loc}\omega_{k}^{2}$. Indeed, if the active collisions were acting on
a coarse grained level like such an equilibrium bath, the structure factor
should have resembled the Equilib one. It can be shown that the fast
temperature field is highly out-of-equilibrium and acts as a genuinely non-
equilibrium bath for the slow velocity field. This effect induces the observed
decay. This is however not the main concern of this paper and does not change
the phenomenology observed when an additional global damping is added. Hence,
the study of this peculiar bath is left for future investigations. The large
time MSD of the $\Delta$ model displays two distinct logarithmic increases
corresponding to the two plateaus reached at large and low $q$ by the
structure factor. While the mechanism behind this double logarithmic scaling
differs from the one arising due to the difference of temperature between a
global and a local COM conserving bath in Eq. (15) (here we recall that there
is no global bath since $\gamma_{com}=0$), the mathematics remain the same, as
the low $k$ structure factor is related to the SDF and consequently the MSD,
through Eq. (17). At small wavectors, we conclude that any 2D isotropic
harmonic crystal displaying a hyperuniform scaling in $k\in[k_{1},k_{2}]$ and
otherwise a constant structure factor, will exhibit this double logarithmic
scaling on the displacement correlation; one in the region $r\gg 2\pi/k_{1}$
and one in $r\ll 2\pi/k_{2}$ with a transition region in between.
$S(k\rightarrow 0)$ and $S(k\rightarrow\infty)$ will respectively be
proportional to the prefactor of the long and short range logarithmic increase
of the MSD. If the hyperuniformity extends to $k_{1}=0$ the plateau of the MSD
eventually reaches a constant. The translational correlation function of the
$\Delta$ model exhibits as well a double power law decrease, in line with the
double logarithmic increase of the plateau of the MSD. The system displays
quasi-long-range order as well.

Finally, by introducing a global damping $\gamma_{com}\neq 0$ to the $\Delta$
model but still with $T_{com}=0$, we obtain the
”$\mathit{\Delta+\gamma_{com}}$” system which exhibits hyperuniformity on
large length scales with a power law consistent with $S(k)\sim k^{2}$, as
predicted in Eq. (13). From the static velocity factor, we confirm that the
energy is not evenly distributed across scales but concentrated at fine ones
and depleted at larger ones with a $k^{2}$ power law decrease. From the MSD we
observe an initial logarithmic increase followed by a plateau, in line with
Eq. (16) when $T_{com}=0$. This implies the reaching of a constant value for
the correlation function at large distances, which is the hallmark of
translational long-range order and the breakdown of the HMW theorem, as
expected when the COM is conserved in the presence of global damping.

###  III.3 Arbitrary enhancement of quasi-long-range order via non-equilibrium
effects

We now turn to the analysis of our full system with $T_{com}\neq 0$ and will
show that the relevant temperature is, as expected from the theory, the
temperature of the COM and not the temperature of the system $T$ or the
temperature of the active collisions $T_{loc}$. Results for a range of
temperatures $T_{com}$ of the global bath are presented in Fig. 3a). In the
main figure, we show various curves of the plateau of the MSD versus the
number of particles in the system $N$ for different $T_{com}$ in a semi-log
scale. We ensure $N$ is sufficiently large to only observe the MSD scaling
associated with the COM temperature. As expected from Eq. (15), the increase
of the MSD is logarithmic and is greater the larger is $T_{com}$. Of course,
we recover a true long-range order in the limit $T_{com}\rightarrow 0$ with a
flat plateau of the MSD. In the inset, we extract the coefficient $\kappa$ in
front of the logarithmic increase of the MSD as a function of $T_{com}$. That
is:

$\textrm{MSD}_{T_{com}}(t\rightarrow\infty)=\kappa(T_{com})\log(N).$ (25)

This quantity could have equivalently been obtained from the exponent of the
correlation function or from $S(0)$.

For a harmonic crystal, $\kappa$ corresponds, up to a constant factor, to the
exponent of the power law decay of the translational correlation function (see
Eq. (15) and (16)) and is given by

$\kappa\sim T_{com}/K.$ (26)

However, this simple linear increase of $\kappa$ with $T_{com}$ holds only for
harmonic crystals, not for a crystal of active hard disks. The elastic modulus
of an equilibrium hard-disk crystal scales trivially with the temperature of
the system $T$ [111, 112, 113]:

$K(T)=\bar{K}T.$ (27)

This implies that the scaling of the MSD of an equilibrium hard-disk system is
not expected to grow with the temperature since $T_{com}=T$. While our system
is a non-equilibrium hard-disk solid, we make the assumption that Eq. (27) is
still a good approximation for the effective elastic constant, thus predicting
the following scaling:

$\kappa\sim\dfrac{T_{com}}{T}.$ (28)

This scaling is well verified in our case, as observed in the inset of Fig. 3
where the dashed line represents a linear increase of $\kappa$ with
$T_{com}/T$.

In panel b), we performed the same simulations as for panel a), except that we
varied $\Delta$ and $\gamma_{com}$ at fixed $T_{com}$. We observe again
approximately a linear scaling of $\kappa$ as a function of $T_{com}/T$
implying that the effective constant of the lattice was roughly constant.

These two analyses indicate that as predicted from the theory, for active hard
sphere crystals the relevant parameter to manipulate in order to slow down the
decay of the translational quasi-long-range is $T_{com}/T$.
Counterintuitively, this implies that increasing the kinetic energy of the
system at a fixed center of mass temperature increases the quasi-long-range
order. Indeed, we can obtain the same enhancement of quasi-long-range order
given by the reduction of $T_{com}$ by keeping it fixed while varying the
driving and dissipative parameters $\Delta$ and $\gamma_{com}$.

##  IV Conclusions and perspectives towards realistic systems

In this paper, we studied the Hohenberg-Mermin-Wagner theorem’s applicability
to non-equilibrium systems. By examining a theoretical model of a harmonic
crystal coupled to a local and a global thermal-like bath, we find that
translational long-range order is controlled by the temperature of the center
of mass, rather than the overall kinetic energy, as in equilibrium systems.
This result also offers a way to better rationalize the violation of the HMW
theorem observed in the random organization model [32] via a $k$-dependent
temperature. Moreover, through numerical simulations of a hard-disk solid
driven out-of-equilibrium by active collisions we showed the possibility to
suppress density fluctuations and enhance quasi-long-range translational order
without neglecting or fine-tuning thermal fluctuations. These insights provide
a valuable theoretical understanding of crystalline phases in non-equilibrium
systems.

To conclude, we outline a possible application of our results to suppress
density fluctuations in a realistic granular system. From the analysis
provided in the previous sections, it is clear that an equilibrium-like global
thermal noise prevents long-range translational order and hyperuniformity in
both the harmonic and the hard-disk driven solid. In these systems, only the
exponent of the power-law decay of the correlation function can be tuned.
However, true hyperuniformity or equivalently long-range order are
unachievable. One might be tempted to think that athermal systems, where the
effects of thermal fluctuations are negligible, could provide a suitable
playground for the actual realization of a two-dimensional long-range ordered
crystal. However, things are not so simple. Indeed, once global thermal
fluctuations are neglected, it remains practically very difficult to realize a
COM-preserving athermal driving mechanism. Moreover, the effect of any
confinement will inevitably influence the COM dynamics. Once this intrinsic
difficulty in realizing true long-range order is accepted, one could still
hope to find a way to arbitrarily enhance the quasi-long-range order of a
system by tuning some experimental parameters following a methodology similar
to that used for the simulation of the coarse-grained model in Sec. III.

Quasi-2D vibrated granular fluids [114, 115, 116, 117, 118, 119, 38] represent
realization of the coarse grained model discussed in section III . In these
systems, macroscopic grains are confined in a box of size $L\times L\times h$
with $h\ll L$ the height of the box. The box is sinusoidally vibrated which
imparts vertical energy to the grains. Collisions between particles transfer
this vertical energy to the $xy$ components of the velocity. Moreover,
tangential friction resulting from collisions with the top or bottom plates
slows down the beads in the $xy$ plane. When viewed purely in the $xy$ plane,
this system behaves similarly to the coarse-grained model discussed in section
III. Indeed, collisions between particles inject energy into the $xy$ plane
and play the same role as the $\Delta$ in Eq. (23) while the damping
$\gamma_{com}$ during the free flight in the coarse-grained model is
equivalent to the energy loss due to collisions with the roof and the bottom
plate in the quasi-2D system. Moreover, the roughness of the two horizontal
confining plates implies that each grain-plate collision introduces some
randomness to the velocities (without even considering the angular momentum of
the grains). Remarkably, this effect has been found to act as a small
homogeneous thermal noise [42] which can be mapped into an effective global
temperature $T_{com}$. The fact that this realistic setup is correctly
described by the coarse grain model suggests the possibility of enhancing the
quasi-long-range order by tuning the experimental parameters (driving
amplitude, frequency, material properties, …) to reduce as much as possible
the ratio between the effective $T_{com}$ caused by the plate roughness and
the horizontal kinetic energy of the system $T$. Of course, the specific way
in which the experimental parameters influence $T_{com}$ and $T$ can be very
complex, but after some preliminary analysis, one might in principle be able
to find the right combination of them to vary in order to arbitrarily reduce
$T_{com}/T$. We look forward to applying these ideas to realistic granular
quasi-2D systems in future studies.

###### Acknowledgements.

We are grateful to Giuseppe Foffi and Frank Smallenburg for carefully reading
and commenting on the manuscript. We also thank Andrea Puglisi and Lorenzo
Caprini for fruitful scientific discussions. This work has been done with the
support of Investissements d’Avenir of LabEx PALM (Grant No.
ANR-10-LABX-0039-PALM).

##  Appendix A Discrete real space conserving noises for vectorial quantities

In Sec. II, we introduced a momentum conserving noise. On a general lattice,
such noise can be written as:

$\begin{split}m\ddot{\bm{u}}_{n}=-\gamma_{loc}\Lambda(\dot{\bm{u}}_{n})+\sqrt{2\gamma_{loc}T_{loc}}(\bm{\nabla}\cdot\bm{\Xi}_{{n}}).\end{split}$
(29)

The first term is a discrete Laplacian and the second is a discrete divergence
of a rank 2 random tensor. Once the discrete differential operators are well
defined, this equation can be understood as a discrete analogue of the model B
field theory. Note however that the global conservation of momentum depends on
boundary conditions.

On a general basis, any lattice can be understood as a graph $\mathcal{G}$
with a set of vertices $\mathcal{V}$ and edges $\mathcal{E}$ connecting them.
The usual notion of continuous field living on $\mathbb{R}^{d}$ (or more
generally on a manifold) is replaced by the notion of field living on vertices
and mapping vertex to vectors. For example in our case
$\bm{u}:\mathcal{V}\to\mathbb{R}^{2}$. The complexity of dealing with discrete
calculus on lattices lies in the fact that the discrete gradient maps fields
living on vertices to fields living on the edges while the discrete divergence
maps fields living on the edges to fields living on the vertices [120, 121].
Hence, the Laplacian being the composition of the divergence with the gradient
maps fields leaving on the vertices to fields living on the vertices. This
immediately implies that the divergence in Eq. (29) is ill defined since we
are applying it on a (random) field living on the vertices and not on the
edges. Nonetheless, following Ref. 66, we will see that this term can be
represented through a finite difference of random vertices field and be made
well defined. Moreover, in order for the dynamics to be equilibrium-like, the
discretization scheme of the differential operators as well as the variance of
the noise must be well chosen in order to respect the FDT.

Indeed if we start from a definition of the derivative as a finite difference
of terms. We can write the divergence of a tensor $\bm{\nabla}\cdot\bm{a}_{n}$
as finite differences between neighboring points. Then, the corresponding
discrete Laplacian $\Lambda$ takes the form
$\Lambda(\bm{a}_{n})=\bm{\nabla}\cdot(\bm{\nabla}\bm{a}_{n})$ where
$\bm{\nabla}$ is a gradient defined from the chosen finite difference used for
the derivative. With these choices, taking $\bm{\Xi}$ with unit variance leads
to a dynamics that correctly respects the FDT.

As an example, if we define the derivative on a square lattice as the forward
finite difference, then:

$\bm{\nabla}\cdot\bm{\Xi}_{i,j}=\begin{pmatrix}\Xi_{i+1,j}^{x,x}-\Xi_{i,j}^{x,x}+\Xi_{i,j+1}^{x,y}-\Xi_{i,j}^{x,y}\\\
\Xi_{i+1,j}^{y,x}-\Xi_{i,j}^{y,x}+\Xi_{i,j+1}^{y,y}-\Xi_{i,j}^{y,y}\end{pmatrix},$
(30)

where we replaced the usual lower vectorial index $n$ by 2 indexes $i$ and $j$
denoting the $x$ and $y$ position on the lattice. Then, we can check that if
$\bm{\Xi}$ has unit variance:

$\langle\bm{\Xi}_{i,j}^{\alpha,\mu}(t)\bm{\Xi}_{k,l}^{\beta,\nu}(t^{\prime})\rangle=\delta(t^{\prime}-t)\delta^{\alpha,\beta}\delta^{\mu,\nu}\delta_{i,k}\delta_{j,l},$
(31)

then:

$\begin{split}\langle(\bm{\nabla}\cdot\bm{\Xi}_{i,j}(t))^{\alpha}(\bm{\nabla}\cdot\bm{\Xi}_{k,l}(t^{\prime}))^{\beta}\rangle=&\delta^{\alpha,\beta}\delta(t-t^{\prime})(4\delta_{i,k}\delta_{j,l}-\\\
&\delta_{i+1,k}\delta_{j,l}-\delta_{i,k}\delta_{j+1,l}-\\\
&\delta_{i-1,k}\delta_{j,l}-\delta_{i,k}\delta_{j-1,l}).\end{split}$ (32)

Where we recognize the discrete Laplacian found for simple forward finite
differences, a sum over first neighbors. A different discretization of the
derivative (Eq. (30)) would have given us a different correlation
corresponding to the adequate Laplacian to use in order for the system to
respect the FDT. For example, a symmetric representation of the derivative
would have given rise to a Laplacian with second neighbors [48]. Conservative
discrete equations in real space have been used in a plethora of domains [122,
123, 48, 66, 51] and naturally arise when a stochastic partial differential
equation is discretized to be solved numerically.

##  Appendix B Fourier transform convention

We choose the following convention for the semi-discrete Fourier transform:

$\begin{split}\tilde{f}_{k}(w)&=\int_{-\infty}^{\infty}\mathrm{d}t\frac{1}{N}\sum_{\bm{n}}e^{iwt+ia\bm{k}\cdot\bm{n}}f_{n}(t)\\\
f_{n}(t)&=\dfrac{1}{(2\pi)^{3}}\int_{-\infty}^{\infty}dw\dfrac{1}{N}\sum_{\bm{k}}e^{-iwt-
ia\bm{k}\cdot\bm{n}}\tilde{f}_{k}(w)\end{split}$ (33)

with $\bm{k}=2\pi\bm{n}/L$ and $n^{\alpha}\in-\sqrt{N}/2+1,\dots,\sqrt{N}/2$.
When the spatial continuous limit is taken, we obtain the following Fourier
series representation of the real lattice:

$f_{n}(t)=\dfrac{1}{(2\pi)^{3}}\int_{-\infty}^{\infty}dw\dfrac{a^{2}}{(2\pi)^{2}}\int_{-\pi/a}^{\pi/a}\mathrm{d}\bm{k}e^{-iwt-
ia\bm{k}\cdot\bm{n}}\tilde{f}(\bm{k},w).$ (34)

This follows directly from taking the limit $a\rightarrow 0$ while keeping $L$
fixed (or $L\rightarrow\infty$ at fixed $L/\sqrt{N}=a$) and corresponds to an
integration in the first Brillouin zone.

##  Appendix C Computation of the displacement correlation function

The displacement correlation function is computed in the infinite size limit
where the discrete Fourier transform over wavevectors is approximated as a
continuous Fourier transform (see Appendix B):

$\begin{split}\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle&=\dfrac{a^{4}}{(2\pi)^{4}}\int\mathrm{d}\bm{k}\mathrm{d}\bm{q}(e^{ia\bm{k}\cdot\bm{n}}-e^{ia\bm{k}\cdot\bm{m}})\\\
&\times(e^{ia\bm{q}\cdot\bm{n}}-e^{ia\bm{q}\cdot\bm{m}})\langle\tilde{\bm{u}}(k,t=0)\tilde{\bm{u}}(q,t=0)\rangle\\\
&=\dfrac{a^{2}}{2\pi^{2}}\int\mathrm{d}\bm{k}(1-\cos(a\bm{k}\cdot(\bm{n}-\bm{m}))C_{uu}(\bm{k})\end{split}$
(35)

Since we only are interested in the (quasi-)long-range behavior of our system,
we compute the long distance limit of this object. We decompose the integral
into two intervals. From $0$ to $1/a|\bm{n}-\bm{m}|$, if $T_{com}\neq 0$ the
integral is bounded by a term in $1/\omega_{k}^{2}$ and can be neglected. For
the second interval from $1/a|\bm{n}-\bm{m}|$ to $2\pi/a$, the oscillating
part averages out. Similar arguments work for $T_{com}=0$. Up to a constant,
at long distances, Eq. (35) is thus approximated as:

$\begin{split}\langle(\bm{u}_{n}-\bm{u}_{m})^{2}\rangle&\simeq\dfrac{a^{2}}{\pi}\int_{\frac{1}{a|\bm{n}-\bm{m}|}}^{\frac{\pi}{a}}\mathrm{d}kC_{uu}(k)k\\\
&\simeq\left[\vphantom{2T_{com}\log(\pi|n-m|)}(T-T_{com})\log\dfrac{\gamma_{com}/\gamma+\pi^{2}}{\gamma_{com}/\gamma+1/|\bm{n}-\bm{m}|^{2}}\right.\\\
&~{}~{}~{}\left.+~{}2T_{com}\log(\pi|\bm{n}-\bm{m}|)\vphantom{(T-T_{com})\log\dfrac{\pi^{2}}{\gamma_{com}/\gamma+1/|\bm{n}-\bm{m}|^{2}}}\right]/\pi
K\\\ &\simeq\left\\{\begin{array}[]{ll}\dfrac{2T_{com}}{\pi
K}\log(\pi|\bm{n}-\bm{m}|)\quad|\bm{n}-\bm{m}|\gg\delta^{-1}\\\ \\\
\dfrac{2T_{loc}}{\pi
K}\log(\pi|\bm{n}-\bm{m}|))\quad|\bm{n}-\bm{m}|\ll\delta^{-1}.\end{array}\right.\end{split}$
(36)

which gives Eq. (15).

##  Appendix D Entropons in a two temperature delta correlated bath

Entropons, as discussed in Caprini et al.’s recent work [67, 95] are general
non-equilibrium collective excitations linked to the spectral entropy
production and are a consequence of a Harada-Sasa like relation [124, 125,
95]. Indeed, for a generalized Langevin equation:

$\ddot{\tilde{{u}}}_{k}(t)=-K\omega_{k}^{2}\tilde{{u}}_{k}-\int_{-\infty}^{\infty}\mathrm{d}t^{\prime}\Gamma(k,t-t^{\prime})\dot{\tilde{u}}_{k}(t^{\prime})+F+\tilde{{\mathcal{E}}}_{k}(t),$
(37)

with $\Gamma(t)$ the friction kernel and
$\langle\tilde{\mathcal{E}}_{k}(t^{\prime})\tilde{\mathcal{E}}_{q}(t)\rangle=\tilde{\nu}(k,t-t^{\prime})\delta_{k,-q}$,
we can prove that, if the force $F$ is even under time reversal the following
relation holds [126]:

$\dfrac{C_{uu}(k,w)}{\nu(k,w)}=\dfrac{\Im(R(k,w))}{w\Re(\Gamma(k,w))}+\dfrac{\sigma^{g}(k,w)}{2\Re(\Gamma(k,w))^{2}w^{2}},$
(38)

where

$\sigma^{g}(k,w)=2\Im\langle\tilde{u}(k,w)F(-k,-w)\rangle\Re(\Gamma(k,w))\nu^{-1}(k,w)w$
(39)

is the entropy production of the generalized Langevin equation,
$R(\mathbf{k},w)$ is the response of the displacement to an instantaneous
force while $\Re$ and $\Im$ are the real and imaginary part of the function in
front of it. Note that this formula holds independently on the choice of the
noise correlation and damping. Notably in the case we are interested,
$\Gamma(k,w)=\gamma_{com}+\gamma_{loc}\omega_{k}^{2}$ and
$\nu(w)/\Gamma(k,w)=2\tilde{T}_{k}$, without additional forces, the spectral
entropy production $\sigma^{g}$ is equal to 0 and the non-equilibrium
character of the system is only included in the effective temperature
$T_{\text{eff}}(k,w)=\nu(k,w)/(2\Gamma(k,w))$. In this sense, we might
consider that entropons are absent and only phonons at an effective
temperature exist in our system since the dynamic SDF reads:

$C_{uu}(k,w)=\dfrac{2\tilde{T}_{k}}{w}\Im(R(k,w)).$ (40)

The entropy production equal to 0 might come as a surprise since it is clear
that as long as $T_{com}\neq T_{loc}$, heat flows from one reservoir to
another and thus entropy is continuously created. However, the particle itself
is oblivious to these heat transfers between baths because they collectively
act as a unique equilibrium-like reservoir with an effective temperature and
viscosity [127, 128, 129]. From the point of view of the particle, the system
is thus too coarse-grained to probe these non-equilibrium effects and heat
fluxes between reservoirs are invisible to dynamical observables [130, 131].

To circumvent this issue, we can rewrite our model in an equivalent form for
the particle, with an auxiliary variable [132, 133, 134, 135, 136, 137, 138,
139, 131]. This auxiliary variable provides enough information about the
system to probe the heat fluxes between reservoirs and compute the spectral
heat flow between them. The heat flux for Eq. (6) is given by [132]:

$m\dot{Q}(k,w)=\dfrac{4\gamma_{com}\gamma_{loc}\omega_{k}^{2}(T_{com}-T_{loc})w^{2}}{(K\omega^{2}_{k}-mw^{2})^{2}+(\gamma_{com}+\gamma_{loc}\omega_{k}^{2})^{2}w^{2}}.$
(41)

From which we can prove in our system without additional forces that:

$C_{uu}(k,w)=\dfrac{2T_{loc}}{w}\Im(R(k,w))+\dfrac{1}{2w^{2}\gamma_{loc}\omega_{k}^{2}}\dot{Q}(k,w),$
(42)

or

$C_{uu}(k,w)=\dfrac{2T_{com}}{w}\Im(R(k,w))-\dfrac{1}{2w^{2}\gamma_{com}}\dot{Q}(k,w).$
(43)

The first term represents the usual equilibrium phonons at temperature
$T_{loc}$ while the second term is clearly a non-equilibrium term, vanishing
when $T_{com}=T_{loc}$ and proportional to the heat transferred between
reservoir. This new term, arising from a rewriting of the effective
temperature is very similar to entropons except that it is better written as
proportional to the heat transfer instead of the entropy production due to
Fourier’s law. Note however that, contrary to entropons arising from active
forces, this new term might be negative according to which bath we chose as
the one with the reference temperature in front of the linear response
function.

The equivalence between Eqs. (42)-(43) and Eq. (40) can be proved more
generally with multiple exponentially correlated reservoirs and external non-
conservative forces using the markovian auxiliary variable representation of
the generalized Langevin equation. Hence, for non-interacting systems, the
effective temperature in $\bm{k},w$ space, obtained from the ratio between the
autocorrelation and the response function can be expressed as the temperature
of a bath plus terms proportional to the heat flow between all the other
baths. We look forward to developing these ideas in future work.

##  Appendix E Hybrid event-driven/time-stepped molecular dynamics

Since the potential of hard-disks particles is discontinuous, usual time-
stepped molecular dynamics methods are not suitable for simulations of such
systems. Instead, we use event-driven methods [106] where the time before the
collision of two particles $i$ and $j$ denoted $t^{col}_{ij}$ can be
analytically computed:

$|\bm{r}_{i}(t^{col}_{ij})+\bm{r}_{j}(t^{col}_{ij})|=\dfrac{\sigma_{i}+\sigma_{j}}{2},$
(44)

with $r_{i}(t)$ the position of particle $i$ at time $t$. For particles free
flying or undergoing constant viscous drag, $t_{col}^{ij}$ can be found
exactly from the initial velocities and positions of the particles.

For viscous friction, the time before the next collision is:

$t^{col}_{ij}=-\log\left(1-\gamma_{com}\delta t_{ij}\right)/\gamma_{com},$
(45)

with:

$\delta
t_{ij}=\frac{-b-\sqrt{b^{2}-\bm{v}_{ij}^{2}(\bm{r}_{ij}^{2}-(\sigma_{i}+\sigma_{j})^{2}/4)}}{\bm{v}_{ij}^{2}}$
(46)

where $b=\bm{r}_{ij}\cdot\bm{v}_{ij}$ and $\bm{r}_{ij}$ and $\bm{v}_{ij}$ are
respectively the relative position and velocity of particles $i$ and $j$ at
the moment the subsequent collision time is computed. From every collision
time and the collision rule (23) the full dynamics of the system can be solved
exactly in the absence of $T_{com}$. When the bath has a non-zero temperature,
we must include the effect of the thermostat in the system by adding events
every $\delta t_{noise}$ where each particle is kicked by an amount
$\sqrt{2\gamma_{com}T_{com}\delta t_{noise}}/m$. The FDT is respected for
every value of $\delta t_{noise}$ because of the continuous nature of the
damping [140]. We nonetheless chose $\delta t_{noise}\ll\tau_{f}\equiv
1/\omega(\phi,T)$ the mean free time.

##  Appendix F Temperature of the active collision bath

For the model including only active collisions ($\gamma_{com}=0)$, the average
energy change during a collision can be obtained from the collision rule (24):

$\langle
E^{\prime}-E\rangle_{coll}=m\Delta^{2}+m\alpha\Delta\langle|\bm{v}_{ij}\cdot\hat{\bm{\sigma}}_{ij}|\rangle_{coll}\\\
-m\dfrac{1-\alpha^{2}}{4}\langle|\bm{v}_{ij}\cdot\hat{\bm{\sigma}}_{ij}|^{2}\rangle_{coll},$
(47)

where the average can be computed from the assumption of molecular chaos,
gaussianity and using the cross section of hard disks [141, 104, 38]. In the
steady state, on average collisions do not inject or retrieve energy from the
system. The steady-state temperature $T_{loc}$ is thus given by the zero of
Eq. (47) which is [104]:

$T_{loc}=\Delta^{2}\left(\dfrac{\epsilon+\sqrt{\epsilon^{2}+4m(1-\alpha^{2})}}{2(1-\alpha^{2})}\right)^{2}$
(48)

with $\epsilon=\alpha\sqrt{\pi m}$. This can be taken as the temperature
$T_{loc}$ of the local bath. It can be shown [38] that when damping is added
to the system, far from the absorbing state, the new temperature of the system
$T$ keeps the exact same form as Eq. (48) but with $\epsilon=\alpha\sqrt{\pi
m}-\gamma_{com}\sigma\sqrt{\pi m}/2\Delta\phi\chi(\phi)$ with $\chi(\phi)$ the
radial distribution function at contact.

This implies that the temperature of the system $T$ does not behave as the
temperature of a system subject to a Langevin bath at temperature $T_{loc}$
and damping $\gamma_{loc}\omega_{k}^{2}$ on top of which we add an additional
damping $\gamma_{com}$. This is due to the dependence of Eq. (47) on the
relative velocity of particles at collision which indicates that $T_{loc}$
should itself be dependent on the other sources of energy change in the
system. It will not cause any significant practical problem but this is a
peculiarity that should be kept in mind.

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