# Collective dynamics in a glass-former with Mari-Kurchan interactions

Yoshihiko Nishikawa  Laboratoire Charles Coulomb (L2C), Université de
Montpellier, CNRS, 34095 Montpellier, France  Graduate School of Information
Sciences, Tohoku University, Sendai 980-8579, Japan  Atsushi Ikeda  Graduate
School of Arts and Sciences, The University of Tokyo, Tokyo 153-8902, Japan
Ludovic Berthier  Laboratoire Charles Coulomb (L2C), Université de
Montpellier, CNRS, 34095 Montpellier, France  Yusuf Hamied Department of
Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK

###### Abstract

We numerically study the equilibrium relaxation dynamics of a two-dimensional
Mari-Kurchan glass model. The tree-like structure of particle interactions
forbids both non-trivial structural motifs and the emergence of a complex
free-energy landscape leading to a thermodynamic glass transition, while the
finite-dimensional nature of the model prevents the existence of a mode-
coupling singularity. Nevertheless, the equilibrium relaxation dynamics is
shown to be in excellent agreement with simulations performed in conventional
glass-formers. Averaged time-correlation functions display a phenomenology
typical of supercooled liquids, including the emergence of an excess signal in
relaxation spectra at intermediate frequencies. We show that this evolution is
accompanied by strong signatures of collective and heterogeneous dynamics
which cannot be interpreted in terms of single particle hopping and emerge
from dynamic facilitation. Our study demonstrates that an off-lattice
interacting particle model with extremely simple structural correlations
displays quantitatively realistic glassy dynamics.

##  I Introduction

Glasses are disordered solids formed by cooling a liquid across the glass
transition [1]. The resulting amorphous nature of glasses yields physical
properties fundamentally different from the ones of crystals [2]. The dynamics
of liquids approaching the glass transition is also different from the one of
simple liquids, although their structures appear quite similar. The drastic
increase of the structural relaxation time with a slight decrease of the
temperature is accompanied by remarkable dynamic signatures [3, 4, 5].
Research on the glass transition aims at determining the microscopic origin of
these unique dynamics in order to find the best theoretical framework to
interpret them [6].

Various theoretical approaches have attempted to interpret the glassy dynamics
using thermodynamics, starting with the Adam-Gibbs theory [7]. The random
first-order transition (RFOT) theory, which is constructed upon the mean-field
theory for the glass transition [8, 9, 10, 11, 12] suggests similarly that the
evolution of a complex free-energy landscape controls glassy dynamics. In the
mean-field limit, a dynamic transition takes place when long-lived metastable
states first appear, which bears similarity with the transition predicted by
mode-coupling theory (MCT) [8, 13, 14]. In finite dimensions, infinitely long-
lived metastable states cannot exist, and the MCT transition is avoided [9,
15]. Below this crossover, RFOT theory proposes that cooperative thermally
activated events take place over a lengthscale of thermodynamic origin which
grows at low temperatures and possibly diverges at the Kauzmann transition [9,
15]. In other structural approaches, real space structures [16] and geometric
frustration [17, 18] play preponderant roles in the description of glassy
dynamics.

A different physical picture is provided by dynamical facilitation theory [19,
20, 21], which explains glassy dynamics from a purely kinetic perspective
based on particle motion. In this view, the relaxation of supercooled liquids
is controlled by a population of dynamical defects, defined as particles that
are much more mobile than the bulk. In addition, mobile particles facilitate
the displacement of their neighbors to create, at much later times, a mobility
field that is correlated over a lengthscale that grows upon lowering the
temperature [22, 19, 23]. Here, no static lengthscale is invoked to explain
the correlated relaxation dynamics. This view is largely inspired by
kinetically constrained glass models [24, 25, 26].

Despite their different underlying microscopic pictures, these viewpoints
provide reasonable descriptions of a range of experimental and numerical data.
It has repeatedly proven difficult to detect experimental signatures which can
sharply favor one view against another. As a result, a complete and unified
theoretical understanding of glassy dynamics is still lacking [27, 28].

Computer simulations are by definition well suited to test these theoretical
concepts as the motion of all particles can be followed at all times. However,
it is difficult to simulate systems at low enough temperatures to cleanly
disentangle the role of local structures, mode-coupling crossover,
thermodynamic fluctuations and dynamic facilitation in the simulated
trajectories for finite-dimensional glass-formers.

This problematic situation changed recently with the development of the swap
Monte Carlo algorithm [29, 30, 31, 32] which permits easy access to
equilibrated configurations at essentially any of the temperatures that
experiments can also explore. Whereas several novel features of ultrastable
glasses have been revealed [33, 34, 35, 36, 37, 38], the remarkable speedup
offered by swap Monte Carlo also provoked a debate regarding the relative role
played by kinetic constraints and thermodynamics in glassy dynamics [39, 40,
41]. Swap Monte Carlo also opens a new window into the dynamics of deeply
supercooled liquids, but again support for both thermodynamic [42] and kinetic
[43] pictures were recently reported.

Another approach is to intentionally engineer artificial situations where only
some of the ingredients assumed by theory survive or dominate. One can then
ask whether and how dynamics is affected. If the dynamics is truly different
from the one of conventional glass models, this suggests that the excluded
factor plays an important role in realistic situations. There are many
previous examples of this strategy where, for instance, the curvature of space
[44, 45], the number of space dimensions [46, 47, 48], the number and geometry
of freely-moving particles [49, 50, 51, 52], or the range and strength of pair
interactions [53, 50, 54, 55] have been varied in some arbitrary way to inform
the physics.

Here we follow this general strategy and analyse the dynamics of a two-
dimensional Mari-Kurchan (MK) model [56, 57] over a broad range of
temperatures. The MK model is characterised by short-range pair interactions
in a continuous space in finite dimension similar to conventional models, but
interparticle distances are defined using an infinite-range random shift which
suppresses many-body correlations [58, 57]. This produces an artificial off-
lattice particle model with short-range interactions in finite dimensions
where: 1) the dynamic mode-coupling transition is strongly avoided [59]; 2)
the thermodynamics is very simple and the model cannot display a Kauzmann
transition [57]; 3) there is no crystalline and locally-favored structures and
geometric frustration should play a limited role; 4) equilibrium
configurations of the model can easily be produced numerically at any
temperature using a planting technique [59]. Among the aforementioned physical
ingredients, only dynamic facilitation can a priori survive in the MK model.
Of course, new relaxation channels with no finite dimensional analog could
also emerge. It has in particular been argued that single particle hopping
could become the relevant pathway for structural relaxation in the MK model
[59, 60, 61].

We find that the glassy dynamics of the MK model is extremely similar to
results obtained in conventional models. It displays for instance a super-
Arrhenius increase of the relaxation time, heterogeneous dynamics, and time
correlation functions nearly indistinguishable from conventional glass-
formers. We find in particular that particle mobility becomes increasingly
correlated as the dynamics slows down, in a way that is incompatible with
single particle hopping. We conclude that dynamic facilitation plays a central
role in this model, and that dynamic facilitation alone can then lead to
consistent physical behaviour.

The rest of the paper is organized as follows. In Sec. II, we introduce the
model and describe our simulation methods. We discuss time correlation
functions, relaxation times, and relaxation spectra in Sec. III. We then
discuss the heterogeneous dynamics at the single-particle level in Sec. IV. In
Sec. V, we report the dynamical susceptibility and spatial correlation
functions characterizing correlated particle dynamics. We summarize and
discuss our results in Sec. VI.

##  II Model and methods

We study a two-dimensional Mari-Kurchan model with Hamiltonian [57]

$H(\\{\boldsymbol{r}_{i}\\})=\sum_{i<j}V\left(|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}-\boldsymbol{A}_{ij}|\right),$
(1)

where $\boldsymbol{r}_{i}$ is the position of particle $i$, $\boldsymbol{A}$
is a real antisymmetric matrix with components drawn from a uniform
distribution $\text{Unif}(0,L_{\text{box}})$, which we call the random shifts
throughout the paper. We denote by $L_{\text{box}}$ the linear dimension of
the system and fix the volume fraction
$\phi=N\pi(\sigma/2)^{2}/L_{\text{box}}^{2}=2$, with $N$ the number of
particles and $\sigma$ the particle diameter. We use a soft-sphere repulsive
power law potential

$V(r)/\varepsilon=\left(\frac{\sigma}{r}\right)^{n}+c_{0}+c_{2}\left(\frac{r}{\sigma}\right)^{2}+c_{4}\left(\frac{r}{\sigma}\right)^{4},$
(2)

if $r\leq r_{\text{cutoff}}$ and $V(r)=0$ otherwise. Here, we set $n=4$ and
the constant values $c_{i}$ ($i=0,2,4$) are chosen so that
$V(r)=V^{\prime}(r)=V^{\prime\prime}(r)=0$ at the cutoff distance
$r_{\text{cutoff}}=2.5\sigma$. With these parameters, mean-field replica
theory [40, 62] yields the dynamical transition temperature $T_{d}\simeq
0.5940$. We show numerical data of the system with $N=1024$ particles in all
figures, unless the system size is explicitly mentioned.

We simulate the underdamped Langevin dynamics at temperature $T$

$m\frac{d\boldsymbol{v}_{i}}{dt}=-\frac{\partial
H}{\partial\boldsymbol{r}_{i}}-m\gamma\boldsymbol{v}_{i}+\sqrt{2m\gamma
T}\boldsymbol{\eta}_{i}(t),$ (3)

with $\boldsymbol{v}_{i}$ the velocity of particle $i$, $m$ the particle mass,
$\gamma$ the friction coefficient, and $\boldsymbol{\eta}_{i}$ the random
force acting on particle $i$ having
$\langle\boldsymbol{\eta}_{i}(t)\rangle=\boldsymbol{0}$ and
$\langle\eta_{i\mu}(0)\eta_{i\nu}(t)\rangle=\delta_{ij}\delta_{\mu\nu}\delta(t)$,
where $\eta_{i\mu}(t)$ is the $\mu$-component of $\boldsymbol{\eta}_{i}(t)$
($\mu=x,y$). We measure time in the time unit of the dynamics
$\tau_{0}=\sqrt{m\sigma^{2}/\varepsilon}$, temperatures in the unit of
$\varepsilon$ and distances in the unit of $\sigma$, thus they are
dimensionless in the following. We set $\gamma\tau_{0}=0.1$, which allows a
stable numerical solution. In Eq. (3), the average particle velocity scales as
$\sim T^{1/2}$, and the microscopic time scale obviously slows down as $\sim
T^{-1/2}$. We thus rescale the time $t$ by $T^{1/2}$ and plot time-dependent
quantities as a function of $T^{1/2}t$ when comparing different temperatures.

We can easily generate equilibrium configurations of the MK model at arbitrary
temperature using the planting technique [63, 64, 65]. In practice this
amounts to sampling the matrix $\boldsymbol{A}_{ij}$ for a given particle
configuration $\\{\boldsymbol{r}_{i}\\}$ so that it becomes an equilibrium
configuration of the Hamiltonian Eq. (1). The mismatch between the isotropic
soft-sphere potential and the square box in which the particles reside
complicates direct sampling of $\boldsymbol{A}_{ij}$ from the Boltzmann
distribution. To deal with this complication, we use a Markov-chain Monte
Carlo method with a simple Metropolis algorithm to sample
$\boldsymbol{A}_{ij}$ [62]. We perform $200$ Monte Carlo sweeps per pair of
particles to ensure convergence to the Boltzmann distribution, and take the
last configuration of $\boldsymbol{A}$ as the random shift for the chosen
particle configuration.

The numerical method used to prepare equilibrium configurations is efficient
and easy, so that preparing initial conditions remains extremely fast compared
to the time needed to numerically study the low-temperature dynamics of the
system. Our numerical strategy is thus to prepare a large number of
independent initial conditions at each temperature, typically a hundred, from
which we run the equilibrium dynamics. This allows us to analyse the
equilibrium dynamics of the model down to arbitrarily low temperatures,
including conditions where the equilibrium relaxation time is much longer than
the simulated time. Conceptually, the planting technique advantageously
replaces for the MK model the swap Monte Carlo technique [30, 31] used in
finite dimensional models to analyse the low-temperature equilibrium dynamics
[43].

A drawback of the MK model is the long-range nature of the matrix
$\boldsymbol{A}_{ij}$. While this is key to endow the MK model with mean-field
structure, this also penalises the numerical integration of the equations of
motion Eq. (3) in two key places. First it slows down the creation of neighbor
lists which requires the entries of the large matrix $\boldsymbol{A}_{ij}$ to
be accessed. Second, storing the matrix itself may create memory problems for
very large systems.

Finally, the random shifts in the Hamiltonian (1) imply that a snapshot of the
system using the actual particle positions ${\boldsymbol{r}}_{i}$ would
faithfully represent the positions of each particle, but would incorrectly
represent interparticles distances which are not given by
${\boldsymbol{r}}_{i}-{\boldsymbol{r}}_{j}$ but instead by the shifted
quantities ${\boldsymbol{r}}_{i}-{\boldsymbol{r}}_{j}-{\boldsymbol{A}}_{ij}$.
It is therefore impossible to visualise all particle interactions
simultaneously in real space. In particular, we will not be able to propose
meaningful visual illustrations of the spatially heterogeneous dynamics of the
MK model.

##  III Correlation functions in time and Fourier domains

We first discuss the averaged dynamic behaviour of the MK model by following
the evolution of correlation functions in both time and Fourier domains.

###  III.1 Time correlation functions

To study the glassy dynamics of the model, we measure the overlap between an
initial configuration at $t=0$ and a configuration at time $t>0$ later

$\left\langle
C(t)\right\rangle=\left\langle\frac{1}{N}\sum_{i}C_{i}(t)\right\rangle,$ (4)

where $C_{i}(t)=\theta(|\boldsymbol{r}_{i}(t)-\boldsymbol{r}_{i}(0)|-\ell)$
with $\theta(x)$ the Heaviside step function. We also measure the self-
intermediate scattering function

$\left\langle
F_{\text{s}}(t)\right\rangle=\left\langle\frac{1}{dN}\sum_{i}\sum_{\alpha}\cos\left[k(r_{i\alpha}(t)-r_{i\alpha}(0))\right]\right\rangle$
(5)

where $d=2$ is the spatial dimension, $k$ is a wavevector set to
$k=\pi/(2\ell)$, $r_{i\alpha}$ denotes the $\alpha$-component ($\alpha=x,y$)
of the position of particle $i$. We denote the average over initial
equilibrium configurations by the brackets $\left\langle\cdots\right\rangle$.
We shall take particles having displacements larger than $1$ as ‘mobile’
throughout this paper, which will be justified below in terms of the cage size
observed in the mean-squared displacements and the van Hove distribution
functions. Accordingly, we set $\ell=1$ for the overlap function $C_{i}(t)$ in
Eq. (4) and for the choice of wavevector in the self-intermediate scattering
function in Eq. (5).

Figure 1: Time dependence of the overlap $\left\langle C(t)\right\rangle$ (a),
the self-intermediate scattering function $\left\langle
F_{\text{s}}(t)\right\rangle$ (b), and the MSD
$\left\langle\Delta^{2}(t)\right\rangle$ (c) for various temperatures at
thermal equilibrium. All functions develop a clear two-step decay below the
onset temperature $T_{\rm onset}\approx 1.2$. A slow plateau dynamics can be
observed at intermediate timescales in all cases.

We show in Fig. 1 the overlap $\left\langle C(t)\right\rangle$ and the self-
intermediate scattering function $\left\langle F_{\text{s}}(t)\right\rangle$
of the MK model over a broad range of temperatures. At low temperature, both
$\left\langle C(t)\right\rangle$ and $\left\langle
F_{\text{s}}(t)\right\rangle$ show a clear two-step relaxation while they
decay quickly at high temperatures. At $T<T_{d}$, the time correlation
functions have a clear plateau, just as observed in finite-dimensional glass-
formers. Closer inspection reveals that this is not strictly a constant
plateau, as $\left\langle C(t)\right\rangle$ and $\left\langle
F_{\text{s}}(t)\right\rangle$ decay very slowly at intermediate times between
microscopic relaxation and structural relaxation. We observe the same slow
decay in a larger system size $N=4096$, and conclude that this is not a finite
size effect. This slow decay much before the structural relaxation indicates
that a small fraction of particles are not well confined in cages and escape
their initial position much before the majority of particles. This observation
already suggests that the dynamics of the MK should be highly heterogeneous. A
similar slow decay in the plateau region has recently been observed in finite-
dimensional models [43], and associated with the emergence of excess wings in
relaxation spectra.

We also measure the mean-squared displacement (MSD)

$\left\langle\Delta^{2}(t)\right\rangle=\left\langle\frac{1}{N}\sum_{i}|\boldsymbol{r}_{i}(t)-\boldsymbol{r}_{i}(0)|^{2}\right\rangle,$
(6)

see Fig. 1(c). The overall evolution is again consistent with finite-
dimensional systems with a diffusive regime at long times that becomes very
slow at low temperatures and the development of a plateau at intermediate
times. In agreement with the overlap and self-intermediate scattering
function, the MSD also shows a slow increase with time in the plateau region.
Again, this slow time dependence is observed in the larger system and should
exist in the thermodynamic limit. The plateau height of the MSD at $T\simeq
T_{d}$ is $\left\langle\Delta^{2}\right\rangle\simeq 0.4$ corresponding to a
cage size $\simeq 0.63$, justifying our choice for the value $\ell=1$ above
for the definition of the overlap and the self-intermediate scattering
function.

###  III.2 Relaxation times and characteristic temperatures

We now extract the $\alpha$-relaxation time $\tau_{\alpha}$ which we define as
the time where $\left\langle C(t)\right\rangle$ reaches $1/e$. For the lowest
temperatures at which $\left\langle C(t)\right\rangle$ clearly decays from its
plateau but does not reach $1/e$ in our simulation time window, we assume that
time-temperature superposition (TTS) holds to estimate $\tau_{\alpha}$ [66].
We report the results of this analysis in Fig. 2(a).

Figure 2: (a) Temperature evolution of the normalised relaxation time,
together with various fits. High temperature Arrhenius regime,
$\sim\exp(\Delta E/T)$ with $\Delta E=5.3$; power-law divergence
$\sim(T-T_{d})^{-\gamma}$ with $T_{d}=0.5940$ and $\gamma=2.97$. Low
temperatures are equally well fitted with the parabolic law
$\sim\exp((T_{\text{o}}/T-1)^{2})$ with $T_{\text{o}}=1.49$, and the VFT law
$\sim\exp(A/(T-T_{0}))$ with $T_{0}=0.22$. Using $\left\langle
C(t)\right\rangle$ at temperature $T<0.6$, we estimate the relaxation time
using time-temperature superposition, see black circles. (b) Temperature
evolution of the stretching exponents $\beta$ for $\left\langle
C(t)\right\rangle$ and $\left\langle F_{\text{s}}(t)\right\rangle$, converging
to $\beta\approx 0.6$ at low $T$ (dashed line).

At low temperatures where the two-step relaxation becomes pronounced, we
observe that $\tau_{\alpha}$ increases rapidly and shows a super-Arrhenius
dependence on the temperature. Fitting the high temperature regime to an
Arrhenius law provides an estimate of the onset of glassy dynamics for this
system near $T_{\rm onset}\approx 1.2$, below which the Arrhenius description
does not hold.

We recall that the mean-field replica theory yields the dynamical transition
temperature $T_{d}\simeq 0.5940$. The mean-field theory of the glass
transition predicts that the relaxation time shows a critical divergence at
$T_{d}$. However, around this temperature, the relaxation time is almost
independent of the system size and no critical divergence is observed, see
Fig. 2(a). This indicates the MCT transition is strongly avoided for this
system [59].

On the other hand, the relaxation time is well fitted by the parabolic law
[20, 21] with an onset temperature $T_{\text{o}}\simeq 1.49$, which appears
consistent with a dynamic facilitation description. Interestingly, even though
the MK model has no Kauzmann transition at any finite temperature, the Vogel-
Fulcher-Tammann (VFT) law [6, 67] with a critical temperature $T_{0}\simeq
0.22$ fits the relaxation time just as well as the parabolic law, see Fig.
2(a). This implies that, similarly to finite-dimensional systems, it is
impossible to distinguish between these two functional forms reliably using
solely the temperature dependence of the relaxation time [68].

We also characterize the shape of the $\alpha$-relaxation observed in the
functions $\left\langle C(t)\right\rangle$ and $\left\langle
F_{s}(t)\right\rangle$ by fitting their long-time decay to a stretched
exponential form $\sim\exp(-(t/\tau)^{\beta})$. As shown in Fig. 2(b), the
measured exponent $\beta(T)$ for these two correlations converge to a similar
low-temperature value $\beta\simeq 0.6$ when the temperature decreases below
$T_{\text{onset}}$. A common interpretation for the stretching exponent
$\beta<1$ is that the underlying dynamics is highly heterogeneous [3]. Note
that in the high-temperature limit where the dynamics is expected to be
homogeneous and diffusive, the self-intermediate function becomes a simple
exponential with $\beta\approx 1$, but the overlap $\left\langle
C(t)\right\rangle$ is never a simple exponential. This is a direct consequence
of its mathematical definition involving the Heaviside function $\theta(x)$
which leads to a power law decay for pure Fickian dynamics, explaining the low
$\beta$ values extracted at high temperatures for the overlap. Both functions
become however equivalent when they are controlled by a broad underlying
distribution of relaxation times. Also, the convergence of the stretching
exponent $\beta$ at low temperatures to a nearly constant value justifies our
use of time-temperature superposition to extrapolate the relaxation time at
lower temperatures, as shown in Fig. 2(a). Similar findings were recently
reported for several glass-forming models [43, 42, 66].

###  III.3 Spectra in the frequency domain

To analyze more precisely the plateau dynamics detected in the time
correlation functions, it is convenient to perform this analysis in the
frequency domain because the large constant value of the plateau automatically
disappears and only the small, interesting time dependence contributes.

We calculate the analog of the imaginary part of the dielectric loss, i.e. the
relaxation spectrum [69, 70],

$\chi^{\prime\prime}(\omega)=-\int d\log\tau
G(\log\tau)\frac{\omega\tau}{1+(\omega\tau)^{2}},$ (7)

where $G(\log\tau)$ is a timescale distribution satisfying

$\left\langle C(t)\right\rangle=\int d\log\tau
G(\log\tau)\exp\left(-\frac{t}{\tau}\right).$ (8)

Unfortunately, estimating $G(\log\tau)$ directly from $\left\langle
C(t)\right\rangle$ requires an inverse Laplace transform, which is notoriously
difficult and numerically unstable. In the same vein, direct Fourier transform
of $\langle C(t)\rangle$ is plagued by important statistical noise [43].
Following earlier work [71, 43], we make the following assumption

$G(\log\tau)\simeq-\frac{d\left\langle C(\tau)\right\rangle}{d\log\tau}$ (9)

and calculate

$\chi^{\prime\prime}(\omega)=\int d\log\tau\frac{d\left\langle
C(\tau)\right\rangle}{d\log\tau}\frac{\omega\tau}{1+(\omega\tau)^{2}},$ (10)

which is an excellent approximation to the desired relaxation spectrum. We
numerically checked the approximation works quite well by comparing Eq. (10)
and Eq. (7) at small $\omega$. In addition, since we only collect data in a
finite window of maximum duration $t_{\text{sim}}$, it is obvious that the
precision of Eq. (10) is limited to $\omega\gg t_{\text{sim}}^{-1}$. At low
temperatures $T=0.575$, $0.6$, and $0.65$ where $\left\langle
C(t)\right\rangle$ reached $1/e$ within the simulation time and the exponent
$\beta$ is well approximated as $\beta=0.6$, we concatenate $\left\langle
C(t)\right\rangle$ and the fitted stretched-exponential decay to obtain
$\chi^{\prime\prime}$ down to $\omega=10^{-5}$. For the other temperatures, we
show the numerical data over a range limited to $\omega\geq
10^{2}/t_{\text{sim}}$.

Figure 3: Relaxation spectra $\chi^{\prime\prime}(\omega)$ obtained from the
overlap $\left\langle C(t)\right\rangle$ and Eq. (10). The peaks at
$\omega\simeq 1$ and lower $\omega\simeq\tau_{\alpha}^{-1}(T)$ correspond to
microscopic and structural relaxations, respectively. Dashed lines are
obtained using stretched exponential fits to the overlap $\left\langle
C(t)\right\rangle$ in Eq. (10). A clear excess signal appears at intermediate
frequencies at low temperatures.

In Fig. 3 we show $\chi^{\prime\prime}(\omega)$ at several temperatures. As a
reference, we also plot $\chi^{\prime\prime}(\omega)$ at much lower $\omega$
by applying Eq. (10) to the stretched exponential fits of $\left\langle
C(t)\right\rangle$ used to estimate the relaxation times at very low
temperatures. The two-step decay of the time correlation functions gives rise
to relaxation spectra in the frequency domain characterised by two well-
separated peaks. The first peak at large frequency $\omega\simeq 1$
corresponds to the fast relaxation at microscopic times. The second peak
shifts to lower frequencies at lower temperatures and corresponds to the
structural $\alpha$-relaxation occurring near
$\omega\simeq\tau_{\alpha}^{-1}(T)$ [72].

In the frequency domain intermediate between these two peaks, the spectra at
low temperatures are significantly larger than the linear superposition of
these two processes. In particular, when comparing the measured spectra with
the high-frequency extrapolation of the structural relaxation peak, we observe
that all measured spectra show a signal which is significantly in ‘excess’ of
the main peak, see Fig. 3. This intermediate frequency regime corresponds, in
the time domain, to the plateau regime where $\left\langle C(t)\right\rangle$
decays very slowly on timescales that are much shorter than the structural
relaxation fitted with a stretched exponential decay. These measurements
confirm that in the plateau regime at times $t\ll\tau_{\alpha}$, there is
considerably more relaxation dynamics than expected from the short-time
expansion of the streched exponential fit to the $\alpha$-relaxation.

Although the signal that we measure at intermediate frequencies does not have
a simple power-law dependence on $\omega$ at low temperatures, this excess
signal is qualitatively similar to the excess wing observed in various
experiments in molecular liquids [73, 74, 75, 76, 77, 78] and very recently
detected numerically in glass-formers in two and three dimensions [43]. From
Fig. 3 we observe that the excess signal could be fit with a very small power
law that would seem to decrease with decreasing temperature, as observed in
experiments [73, 74]. This is maybe consistent with the power law distribution
of escape times measured numerically in Ref. [59]. For the lowest temperature,
the time evolution on the plateau of $\langle C(t)\rangle$ becomes nearly
logarithmic, which would correspond to a nearly flat spectrum, as indeed
observed in Fig. 3. One would need to simulate lower temperatures for longer
times to fully assess the frequency dependence and temperature evolution of
the excess signal in the relaxation spectra of the MK model.

##  IV Dynamic heterogeneity: Single particle dynamics

We now move to the analysis of the heterogeneous nature of the dynamics,
starting at the single particle level. To this end, we measure the probability
distribution function of single particle displacements, which is the self-part
of the van Hove distribution function [79],

$P_{t}(\Delta
x)=\left\langle\frac{1}{dN}\sum_{i}\sum_{\alpha}\delta\left[\Delta
x-\left(r_{i\alpha}(t)-r_{i\alpha}(0)\right)\right]\right\rangle.$ (11)

At very high temperatures, where the system behaves as a simple fluid with
diffusive behaviour, we expect $P_{t}(\Delta x)$ to be Gaussian with a
variance increasing linearly with time. In glass-formers, deviations from this
Gaussian behaviour typically indicate that the system is dynamically
heterogeneous [80, 81, 82].

Figure 4: The van Hove distribution function $P_{t}(\Delta x)$ at (a)
$T=0.675$ and (b) $T=0.5$. Insets show the distributions scaled using the
measured variance with the dashed line indicating the normal distribution. The
structural relaxation time is $\tau_{\alpha}\simeq 2.5\times 10^{5}$ at
$T=0.675$ and $\simeq 5\times 10^{7}$ at $T=0.5$.

We show the van Hove function $P_{t}(\Delta x)$ in Fig. 4 at temperatures
$T=0.675$ and $0.5$. Measuring the variance $\lambda^{2}(t)$ of these
distributions, we also replot $\lambda P_{t}$ as a function of $\lambda\Delta
x$ in the insets of Fig. 4. These rescaled distributions become the normal
distribution whenever $P_{t}(\Delta x)$ is Gaussian. This is the case at all
timescales at high temperatures, $T\gg 1$, as expected.

At low temperatures instead, the distributions exhibit broad tails except at
very small and very large times. Qualitatively, the van Hove distributions
develop nearly exponential tails already in the plateau region, see Fig. 4.
Such exponential tails decay much more slowly than the Gaussian prediction for
Fickian dynamics, which reveals the existence of a small concentration of
particles that have moved much farther than the majority after a given time.
This non-Gaussian shape and exponentially decaying tails in the van Hove
function have been universally observed in finite-dimensional glass-forming
liquids [83].

For the lowest temperature, $T=0.5$, the largest time $t=10^{6}$ is much
shorter than the estimated relaxation time, $\tau_{\alpha}\sim 5\times 10^{7}$
and most of these distributions thus correspond to the very extended plateau
regime seen at this temperature. In this plateau regime, the distribution is
almost invariant in its nearly Gaussian core, but the tails become broader and
more populated with increasing time. The population of faster than average
particles thus increases with time and moves, on average, over larger
distances. This population of very fast particles is therefore responsible for
the slow time decay of $\left\langle C(t)\right\rangle$ in the plateau time
region and of the excess relaxation signal detected in the relaxation spectra.
This observation is again consistent with the microscopic interpretation of
excess wings in finite dimensional glass-formers [43, 84].

##  V Dynamic heterogeneity: Collective dynamics

###  V.1 Four-point dynamic susceptibility

The analysis of the van Hove distributions in Sec. IV shows the coexistence of
fast and slow moving particles at any given time. As is well known, however,
the distribution of single particle displacements contains no information
about possible correlations between particle displacements. Here, we analyse
observables that can reveal this information and allow us to discuss the
collective nature of the relaxation dynamics in the MK model.

Using multiple independent trajectories, we can calculate the four-point
dynamical susceptibility defined by the variance of the spontaneous
fluctuations of the dynamic overlap [85, 86, 87]:

$\chi_{4}(t)=N\frac{\left\langle C^{2}(t)\right\rangle-\left\langle
C(t)\right\rangle^{2}}{\left\langle C(t)\right\rangle(1-\left\langle
C(t)\right\rangle)}.$ (12)

In Eq. (12), $\chi_{4}(t)$ is normalized such that $\chi_{4}(t)=1$ when the
dynamics of all pairs of particles is completely uncorrelated.

Figure 5: Dynamical susceptibility $\chi_{4}(t)$ at several temperatures. The
dashed line is $\chi_{4}(t)=1$, the trivial value observed for uncorrelated
dynamics. Inset: maximal value $\chi^{*}_{4}$ as a function of rescaled
structural relaxation time $T^{1/2}\tau_{\alpha}$, with dashed line indicating
the power law $\chi_{4}^{*}\sim(T^{1/2}\tau_{\alpha})^{1/\Delta}$ fitted to
the numerical data with $\Delta=4.9(2)$.

In Fig. 5 we show the dynamical susceptibility $\chi_{4}(t)$ at several
temperatures. At very short time $T^{1/2}t\simeq 1$, $\chi_{4}(t)\simeq 1$,
consistent with uncorrelated microscopic motion at microscopic times. On the
time scale where time correlation functions have a plateau, $\chi_{4}(t)$
grows monotonically with time, possibly as a power law with a small exponent
near 0.2. Therefore the increasing number of mobile particles at intermediate
times revealed by the van Hove distribution cannot be interpreted as the
independent hopping of particles out of their respective cages. Instead, the
dynamics of mobile particles become increasingly correlated with time, and
particle hopping is actually a correlated, many-body process.

At times $t\simeq\tau_{\alpha}(T)$, $\chi_{4}(t)$ reaches a maximum value
$\chi_{4}^{*}(T)$ which is significantly larger than $1$, before decreasing
towards unity at much longer time scales $t\gg\tau_{\alpha}$. This time
dependence is in agreement with the typical behaviour observed in finite-
dimensional glass-forming liquids [87]. This similarity implies that the
dynamics of the MK model is just as collective as the one of conventional
glass models. Remarkably, even below the avoided dynamical transition
temperature $T_{d}$, $\chi_{4}(t)$ grows with time and becomes much larger
than $1$, contrary to the proposed picture of a relaxation process governed by
the independent hopping of individual particles [59].

In the inset of Fig. 5, we show the parametric evolution of $\chi_{4}^{*}(T)$
with the rescaled relaxation time $T^{1/2}\tau_{\alpha}(T)$ together with an
algebraic dependence
$\chi_{4}^{*}\sim\left(T^{1/2}\tau_{\alpha}\right)^{1/\Delta}$ with
$\Delta=4.9(2)$. The exponent $\Delta$ is somewhat larger than that reported
for finite-dimensional models [23, 87, 88, 6, 50], $\Delta\approx 2.5$, but
close to that of the hard-sphere MK model [57], $\Delta\approx 4.5$. This may
be due to the fact that the mode-coupling crossover, which would yield the
prediction $\chi_{4}^{*}\sim\tau_{\alpha}^{1/\gamma}$, is very strongly
suppressed in the present MK model. As a result, the growth of $\chi_{4}^{*}$
may be solely due to collective activated events for which much slower growth,
possibly logarithmic, is expected on general grounds [87] and indeed found in
several cases [71, 89]. In any case, the present data unambiguously reveals
that the low-temperature dynamics of the MK model is actually governed by
strong many-body correlations which do not stem from (avoided) mode-coupling
criticality.

###  V.2 Spatial correlation functions

The dynamic susceptibility $\chi_{4}(t)$ can also be represented as the
integral of the four-point correlation function $G_{4}(r,t)$ over space:

$\chi_{4}(t)=\frac{\int drG_{4}(r,t)}{G_{4}(0,t)},$ (13)

where

$G_{4}(r,t)=\frac{1}{V}\left\langle\sum_{i,j}\delta C_{i}(t)\delta
C_{j}(t)\delta(r-r^{\boldsymbol{A}}_{ij}(0))\right\rangle,$ (14)

with $\delta C_{i}(t)=C_{i}(t)-\left\langle C(t)\right\rangle$ and
$r^{\boldsymbol{A}}_{ij}(t)=|\boldsymbol{r}_{i}(t)-\boldsymbol{r}_{j}(t)-\boldsymbol{A}_{ij}|$.
Thus, a large value of $\chi_{4}(t)$ implies the existence of large
correlations between local fluctuations of the dynamics, $\langle\delta
C_{i}(t)\delta C_{j}(t)\rangle$. Recall however that the meaning of the
spatial variable $r$ in Eq. (14) does not have the ordinary interpretation as
in conventional models due to the long-range random shifts in the definition
of the Hamiltonian Eq. (1).

Figure 6: Four-point correlation function (a) $G_{4}(r,t)/2\pi r$ for
different times and (b) $G^{(m)}_{4}(r,t)/2\pi r$ at $t=10^{6}$, around which
$\chi_{4}(t)$ is maximum. The temperature is $T=0.575$ in both panels.

Even if they are difficult to visualise, correlations between the dynamics of
individual particles are anyway directly revealed in the four-point
correlation $G_{4}(r,t)$. We show $G_{4}(r,t)/2\pi r$ in Fig. 6 at $T=0.575$.
At shown in Fig. 6(a), $G_{4}(r,t)$ develops a clear peak at $r\simeq{2}$
which changes with time and temperature. At the temperature shown in Fig.
6(a), the relaxation time is $\tau_{\alpha}\simeq 10^{6}$ which is also
roughly the time when $\chi_{4}(t)$ becomes maximum. The time dependence of
$G_{4}(r,t)$ is compatible with this evolution.

However, $G_{4}(r,t)$ displays two features that are not seen in finite
dimensional models. First, $G_{4}(r,t)$ has a negative dip at short distance
$r\lesssim{1}$ at any time $t$. This may naively suggest that particles in
very close contact have anticorrelated mobility, which appears
counterintuitive at first sight. Second, $G_{4}(r,t)$ does not decay at large
distances. We discuss these two points in the following.

To understand the physical origin of the anticorrelation and the negative dip
at small $r$, we decompose $G_{4}(r,t)$ into two independent parts, that is,
mobile and immobile contributions to $G_{4}(r,t)$. Let us define

$G_{4}^{(m)}(r,t)=\frac{1}{V}\left\langle\sum_{i,j}\alpha_{i}(t)\delta
C_{i}(t)\delta C_{j}(t)\delta(r-r^{\boldsymbol{A}}_{ij}(t))\right\rangle,$
(15)

where $\alpha_{i}(t)=C_{i}(t)$ if $m=\text{mobile}$ and
$\alpha_{i}=1-C_{i}(t)$ if $m=\text{immobile}$. Then by definition, we have
$G_{4}(r,t)=G_{4}^{\text{(mobile)}}(r,t)+G_{4}^{\text{(immobile)}}(r,t)$. We
remind that particles with displacement larger than $1$ are defined as mobile
and the rest as immobile. We show the result of this decomposition in Fig.
6(b) at $T=0.575$. Clearly, at $r\simeq{1}$, the immobile contribution is
negative while the mobile one is positive, meaning that the anticorrelation
observed for the total function comes solely from immobile particles.

Figure 7: Radial distribution functions for (a) mobile and (b) immobile
particles at $T=0.575$ and different times. Mobile particles have more
neighbors at small distances $r\simeq{1}$ and less at $r\simeq{2}$, while
immobile particles have less at $r\simeq{1}$ and more at $r\simeq{2}$. This
indicates a clear correlation between structure and dynamics.

The clear difference between mobile and immobile contributions to $G_{4}(r,t)$
suggests that particle mobility originates, on average, from the local
structure formed by surrounding particles. To test this idea, we measure the
static radial distribution function separately for mobile and immobile
particles:

$g^{(m)}(r,t)=\frac{\left\langle\sum\alpha_{i}(t)\delta(r-r^{\boldsymbol{A}}_{ij}(t))\right\rangle}{V\rho\left\langle\rho^{\text{(m)}}(t)\right\rangle},$
(16)

where $\rho^{\text{(m)}}(t)$ is the number density of mobile (immobile)
particles if $m=\text{mobile}$ (immobile). This radial distribution function
characterizes the average surrounding structure of mobile or immobile
particles. We normalise $g^{(m)}(r,t)$ by the bulk pair correlation function
$g(r)$,

$g(r)=\left\langle\frac{V}{N^{2}}\sum_{i}\delta\left(|\boldsymbol{r}_{i}-\boldsymbol{r}_{j}-\boldsymbol{A}_{ij}|\right)\right\rangle,$
(17)

so that differences between the local structure and the average one are
directly revealed by a ratio that differs from unity. The results are shown in
Fig. 7. We can see that mobile particles have more neighbors on average at
short distances $r\lesssim{1}$, but less at large distances, $r\simeq{2}$. By
contrast, immobile particles have less neighbors at short distances
$r\lesssim{1}$, and slightly more at $r\simeq{2}$. We finally understand that
the negative dip in the immobile contribution to $G_{4}(r,t)$ thus comes from
the fact that immobile particles have less neighbors at short distances, and
indeed does not point to the existence of unphysical mobility
anticorrelations.

Let us offer some physical interpretation of these findings. In supercooled
liquids, particles are confined within a cage comprised of neighboring
particles, and a particle becomes mobile by escaping this cage. In this view,
the radial distributions for mobile particles indicating that they have more
neighbors at short distance, and are thus more confined than average appears
counterintuitive. We believe that the reason is that particles interacting
with their neighbors at short distances are then subject to large forces,
which potentially leads to a stronger force imbalance which can induce rapid
displacements. In addition, Fig. 7(a) shows that mobile particles have less
particles at $r\simeq{2}$. Thus, once they escape their first shell of
neighbors $r\lesssim{1}$, they are not efficiently pushed back to their
original position by particles further away. The opposite effects apply to the
immobile particles, see Fig. 7(b).

Regarding the absence of decay at large $r$, we recall that the distance $r$
used in the definition of $G_{4}(r,t)$ is meaningful for neighboring particles
which effectively interact at short shifted distances
$r_{ij}^{\boldsymbol{A}}$. However, the physical meaning of large shifted
distances is lost. In particular, it is not possible to extract a dynamic
correlation length from the four-point function $G_{4}(r,t)$. Still, the large
value in $G_{4}(r,t)$ at large distances is consistent with the large value of
$\chi_{4}(t)$, and it proves that dynamics is collective and involves multi-
body correlations. This implies that the two-particle correlations detected by
$G_{4}(r,t)$ at short distances must propagate in space and time to form large
correlated clusters at $t\simeq\tau_{\alpha}$, even though we cannot visualise
them in real space.

To discuss the correlations to second and third neighbors and extract a
dynamic correlation length, a more advanced analysis considering the tree-like
structure of interactions would be needed, which we leave for future work.

##  VI Summary and Discussion

To summarize, we studied the equilibrium dynamics of a two-dimensional Mari-
Kurchan model over a wide range of temperatures. Using the planting technique,
we prepared equilibrium configurations down to arbitrarily low temperatures at
essentially no cost, well below the MCT crossover temperature, and studied
several time correlations functions and their fluctuations.

We have shown that both $\left\langle C(t)\right\rangle$ and $\left\langle
F_{\text{s}}(t)\right\rangle$ develop a two-step decay separated by a plateau
regime when the glassy dynamics emerges. The final decay occurs even far below
the mode-coupling crossover $T_{d}$, as observed in finite-dimensional glass-
formers. The MCT transition is thus strongly avoided in the MK model. Since a
Kauzmann transition does not occur either, RFOT theory has little to say about
the glassy dynamics in that model, even though it was initially introduced as
a glass model with mean-field character.

Overall, the observed glassy dynamics is quantitatively consistent with the
phenomenology of supercooled liquids. The final decay of time correlation
functions can be described by a stretched exponential, and a clear excess
signal appears at intermediate frequencies which resembles the excess wings
found in molecular and simulated liquids. Finally, structural relaxation is
accompanied by increasing multi-point dynamic correlations characterising the
emergence of collective dynamics at low temperatures.

Our results show that the MK model displays dynamical properties that are
extremely similar to the ones of finite-dimensional systems despite the mean-
field nature of its local structure. Because the MK model has no MCT
transition, the emergence of collective dynamics cannot be described or
attributed to the dynamic criticality. In addition, the thermodynamic Kauzmann
transition is forbidden by the random nature of the MK interactions, and it is
also impossible to invoke collective activated dynamics over a correlation
lengthscale controlled by a decreasing configurational entropy, as envisioned
in RFOT theory.

The sole candidate to explain the emergence of correlated dynamics in the MK
model is dynamic facilitation. We expect that the most mobile particles are
able to escape their initial position much before the bulk, and this motion
then helps neighboring particles (in the shifted representation) to relax more
easily themselves. If this process repeats itself, it directly leads to the
emergence of dynamic correlations that become very large when structural
relaxation finally takes place. In fact, the existence of a small population
of particles that relax much faster than the bulk naturally opens a broad time
window where dynamic correlations can be built via dynamic facilitation.
Moreover, our results suggest that the particles playing the role of the
localised defects hypothesized in kinetically constrained models can be
statistically identified in the MK model based on their local structure which
is, on average, different from the bulk.

The MK model then emerges as an interacting particle model with short-range
interactions which, in effect, captures very well the physics of kinetically
constrained models. It appears more realistic than the original constrained
models themselves which have no Hamiltonian and where defects are introduced
by hand. Here, the defects are instead an emergent physical property. The MK
model is also different from plaquette models [90] where interactions do not
resemble the ones found in liquids, and the interactions in the MK model are
also much less artificial than in models of infinitely-thin hard needles [91,
92].

Since the MK model resembles so closely conventional glass-forming models, it
could become a useful tool to better understand how dynamic facilitation can
operate in off-lattice models and how to best quantify its consequences.
However, it could be that despite the similarities revealed by our work, some
more subtle differences emerge after more careful examination. For instance,
it is possible that the nature of the particle motion observed at times much
shorter than the bulk is simpler in the MK model than in finite-dimensional
models where non-trivial thermodynamic fluctuations exist that can influence
the slow dynamics. Future efforts should aim at determining whether such
differences exist.

In future work, it would also be interesting to test whether the correlation
that we have revealed between structure and dynamics is only obeyed at the
average level, or whether it applies particle by particle at the microscopic
level. Due to the simplified nature of the structure in the MK model, it again
appears as a good model to test current efforts aimed at identifying at the
microscopic level the correlation between structure and dynamics, which has
seen a surge of interest recently due to the advent of machine learning
techniques [93, 94, 95].

###### Acknowledgements.

We thank P. Charbonneau and F. Zamponi for discussions. L.B. acknowledges
support from the Simons foundation (No. 454933 L. B.). A.I. acknowledges
support from JSPS KAKENHI (Grants No. 18H05225, 19H01812, 20H01868, 20H00128).

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