of 8
Short-range correlation of stress
chains near solid-to-liquid transition in
active monolayers
Siavash Monfared
1
, Guruswami Ravichandran
2
, José E. Andrade
2
and Amin
Doostmohammadi
1
1
Niels Bohr Institute, University of Copenhagen, Kobenhavn, 2100, Denmark
2
Division of Engineering and Applied Science, California Institute of Technology, CA, 91125, USA
SM,
0000-0002-7629-7977
Using a three-dimensional model of cell monolayers, we study the spatial
organization of active
stress chains
as the monolayer transitions from a solid
to a liquid state. The critical exponents that characterize this transition
map the isotropic stress percolation onto the two-dimensional random
percolation universality class, suggesting short-range stress correlations
near this transition. This mapping is achieved via two distinct, independent
pathways: (i) cell–cell adhesion and (ii) active traction forces. We unify our
findings
by linking the nature of this transition to high-stress
fluctuations,
distinctly linked to each pathway. The results elevate the importance of the
transmission of mechanical information in dense active
matter
and provide
a new context for understanding the non-equilibrium statistical physics of
phase transition in active systems.
1. Introduction
The transition between solid and liquid states in living cells is of fundamen-
tal relevance to a range of biological processes, including cancer metastasis
[1–3], wound healing [
4–6] and tissue morphogenesis [
7–9]. To study this
transition, the concept of jamming [
10,11
] has been applied to inherently
out-of-equilibrium living cells [
12–14
]. This is despite the geometrical roots
of the jamming transition and its correspondence to the zero temperature
and zero activity limit of a glass transition [
15,16
]. Notwithstanding these
seminal contributions, the universality of the transition between active solid
and liquid phases in cellular systems and its broader applicability are yet to
be established [
7,17–26
].
Long-range correlations of mechanical information, i.e. force or stress,
prevail in both passive glasses [
27,28
] and most athermal jammed granular
systems [
29]. In such passive systems, the long-range force or stress correla-
tions are
attributed
to mechanical equilibrium [
30–33
]. However, in some
non-equilibrium systems, long-range mechanical correlations can also emerge
in two seemingly disparate systems: (i) active
matter
[ 34], where local energy
injection results in the breaking of time-reversal symmetry and detailed
balance [
35], and (ii) boundary-driven shear of granular systems [
36]. Whether
a broad framework to understand these two systems exists remains to be
established [
37].
Here, using a three-dimensional model of cell monolayers, we provide
evidence for the emergence of active stress chains (
figure
1
a,b
) in the
monolayers and show stress percolation criticality near the solid-to-liquid-like
transition (
figure
1
c,d
). Importantly, we establish this criticality by driving
the transition via two distinct paths by independently increasing (i) cell–cell
adhesion strength and (ii) active traction forces, well-established axes on the
© 2024 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License
http://creativecommons.org/licenses/by/4.0/
, which permits unrestricted use, provided the original
author and source are credited.
Research
Cite this article:
Monfared S, Ravichandran G,
Andrade JE, Doostmohammadi A. 2024 Short-
range correlation of stress chains near solid-to-
liquid transition in active monolayers.
J. R. Soc.
Interface
21
: 20240022.
https://doi.org/10.1098/rsif.2024.0022
Received: 10 January 2024
Accepted: 8 March 2024
Subject Category:
Life Sciences–Physics interface
Subject Areas:
biophysics
Keywords:
active matter, random percolation, stress
correlation, soft matter, critical phenomena
Authors for correspondence:
Siavash Monfared
e-mail:
siavash.monfared@nbi.ku.dk
Amin Doostmohammadi
e-mail:
doostmohammadi@nbi.ku.dk
Electronic supplementary material is available
online at
https://doi.org/10.6084/
m9.figshare.c.7209354
.
phase boundary for this transition [
38,39
]. Using
finite-size
scaling analyses, we further demonstrate that critical exponents
from each path correspond to the two-dimensional random percolation universality class near the solid-to-liquid transition.
Remarkably, this points to the short-range nature of stress correlations. To explain this, we provide a mechanistic basis by
linking the universality class mapping to high
fluctuations
in stress
fields
with distinct signatures associated with each path. We
further discuss the implications for the short-range nature of stress correlations in active monolayers and its relationship with
non-equilibrium glasses, granular and biological systems.
2. Methods
We consider a cellular monolayer consisting of
N
= 400
cells on a rigid substrate with its surface normal
e
n
=
e
z
=
e
x
×
e
y
and periodic boundaries in both
e
x
and
e
y
, where
e
x
,
e
y
,
e
z
constitute the global orthonormal basis. Each cell
i
is represen-
ted by a three-dimensional phase
field,
φ
i
=
φ
i
x
,
t
and initialized with radius
R
0
. The phase
field
φ
i
is evolved through
Model A
[40] type dynamics with an extra advective term,
(2.1)
t
φ
i
+
v
i
⋅∇
φ
i
=
Γ
δ
δφ
i
,
i
= 1, ...,
N
,
where
v
i
is the velocity of cell
i
,
Γ
represents mobility,
is the free energy functional that stabilizes cell interface and includes
mechanical properties such as cell
stiffness
(
E
) as well as compressibility (
μ
), and puts a soft constraint on the cell volume [
41–
44] around
V
0
= (4/3)π
R
0
3
. Additionally, the free energy comprises gradient contributions (
φ
) that account for, and distinguish
between, cell–cell (
ω
cc
) and cell–substrate (
ω
cw
) adhesions, as introduced recently [
45],
(2.2)
=
i
N
E
λ
2
dx
{4
φ
i
2
(1
φ
i
)
2
+
λ
2
(
φ
i
)
2
}
+
i
N
μ
1
1
V
0
dx
φ
i
2
2
+
i
N
j
i
κ
cc
λ
dx
φ
i
2
φ
j
2
+
i
N
j
i
ω
cc
λ
2
dx
φ
i
⋅∇
φ
j
+
i
N
κ
cw
λ
dx
φ
i
2
φ
w
2
+
i
N
ω
cw
λ
2
dx
φ
i
⋅∇
φ
w
,
where
k
captures repulsion between cell–cell (subscript ) and cell–substrate (subscript ) and denotes a static phase
field
representing the substrate. Based on this free energy function, the interior and exterior of cell corresponds to and , respectively,
connected by a
diffuse
interface of length . To resolve the forces generated at the cellular interfaces, we consider an over-dam-
ped dynamics, , where denotes traction [
18,46,47
] and contains both active and passive contributions, is substrate friction and
represents an active polar force that captures the self-propulsion associated with cell polarity, constantly pushing the system out
of equilibrium, with characterizing the strength of the self-propulsion force. The dynamics of cell polarity are introduced based
on contact inhibition of locomotion [
48,49
] by aligning the polarity of the cell to the direction of the total interaction force acting
on the cell [
50]. The polarization dynamics is given by , where is the counterclockwise angle of cell polarity measured from ,
and is the Gaussian white noise with zero mean, unit variance, is rotational
diffusivity,
is the angle between and , with positive
1
–1
n
sima
0
(
a
)
(
b
)
(
c
)
(
d
)
Figure
1.
An example showing stress chains in an active monolayer. Snapshots of isotropic stress
fluctuations,
δσ
iso
=
σ
iso
σ
iso
, normalized by maximum
compression for (
a
) solid and (
b
) liquid states outlining both compressive (blue) and tensile (red) chains. The trajectories of individual cells during total simulation time
steps (
n
sim
) corresponding to (
c
) a solid-like state and (
d
) a liquid-like state. The colour bar indicates simulation time. (
c
) and (
d
) correspond to simulations shown in
(
a
) and (
b
), respectively.
2
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J. R. Soc. Interface
21:
20240022
constant
setting
the alignment time scale between them (see electronic supplementary material for simulation parameters).
Previously, this model was tuned directly against experiments [
51] and
specifically
chosen given its success in reproducing
stress
fields
around nematic defects [
45,47
].
In passive, athermal granular systems, the jamming transition is intimately related to the packing fraction (density) of that
system. However, for living cells at
confluence,
the collective behaviour can change from a liquid to a solid-like behaviour
at a constant density. A recent study suggests density has a second-order
effect
on this transition [
12]. Consequently, here
we keep the cell density constant in our simulations, i.e. no cell proliferation nor extrusion, while ensuring
confluency.
For
each considered pathway, we perform large-scale simulations by incrementally increasing the dimensionless cell–cell to cell–
substrate adhesion ratio
ω
~
=
ω
cc
/
ω
cw
[0.1, 0.5]
for the
first,
and the dimensionless traction force
α
~
=
ατ
pol
/
ξR
0
,
α
~
[0, 0.8]
, for
the second pathway. We consider three distinct realizations for each case. As we show next, the considered ranges for
ω
~
and
α
~
capture the transition from a solid-like to a liquid state.
3. Results
3.1. Solid-to-liquid transition
To characterize the solid-to-liquid-like transition, we quantify the one-dimensional pair correlation function
g
r
= (
)
1
i
j
δ
r
|
r
i
r
j
|
, where
ρ
denotes number density, shown in
figure
2 for time-averaged
configurations.
For
low values of
ω
~
and
α
~
,
g
r
shows a peak near
r
/
R
0
2
, i.e. the diameter of a single cell, indicating a solid-like state. As
ω
~
and
α
~
increase, this peak gets smaller while another one, at a shorter distance, appears, indicative of a transition. For the
time-averaged
configurations,
the peak
r
/
R
0
0
indicates
significant
overlaps between cells, relative to their initial positions,
during the simulation (see
figure
1
d
). For
ω
~
= 0.5
and
α
~
= 0.8
, the pair-correlation functions exhibit peaks at a distance much
smaller than the cell diameter, suggesting a liquid-like state. Thus, the solid-to-liquid transition takes place near
ω
~
= 0.45
and
α
~
= 0.70
, mindful of the resolution for
ω
~
and
α
~
spaces. We also quantify the two-dimensional static structure factor that further
confirms
the
identified
transition points (see electronic supplementary material for details).
In the three-dimensional phase
field
model of active cell monolayers,
stress chains
reminiscent of force chains in passive,
athermal granular systems [
52,53
] are observed (
figure
1). In contrast with passive, athermal, convex granular systems, the stress
chains in this active model exhibit both compressive and tensile stress components. This can also be the case in non-convex
granular systems with the ability to entangle [
54,55
]. To understand the
identified
transition between solid–liquid states and its
link to the spatial organization of mechanical information, we compute a stress
field
[ 56,57
]
σ
=
σ
(
x
,
t
)
that encodes both active
and passive contributions on a discretized domain, for node
i
on the complementary stress
lattice,
(3.1)
σ
i
=
1
a
0
3
j
N
i
r
ij
T
j
,
where
a
0
3
is the volume of the unit cell,
N
i
is the number of neighbours of node
i
on the original
lattice
(see electronic
supplementary material for details),
r
ij
=
x
i
x
j
. Given the
definition
of active traction for cell
i
,
T
i
, the coarse-grained stress
field
contains contributions from both active and passive forces.
Next, we compute the time-averaged isotropic stress
field
tensor,
σ
̄
x
= (1/
n
sim
)
t
n
sim
σ
x
,
t
and focus on the normalized
isotropic stress
field,
σ
̄
~
iso
x
=
σ
̄
iso
x
/
σ
̄
max
iso
with
n
sim
number of simulation time steps,
σ
̄
max
iso
the maximum value of the
time-averaged isotropic stress
field
and
σ
̄
iso
x
=
1/3
trσ
̄
x
, providing a measure for expansion (
σ
iso
> 0
) and compression
(
σ
iso
< 0
) in the cell layer.
Visual inspection of the
fluctuations
in isotropic stress
fields
and the associated
patterns
show that for both adhesion
parameter
ω
~
and traction force
α
~
, as the control parameter is increased and the system approaches a liquid-like state, a more
disordered isotropic stress
field
emerges. We quantify the spatial organization of emerging stress
patterns
by lowering a
threshold and monitoring the connectivity of the time-averaged isotropic stress
field
sites larger than the threshold, on a square
lattice.
Then, we extract the critical exponents for the stress percolation by performing
finite-size
scaling analyses [
58,59
].
We begin with quantifying the density of the spanning cluster,
P
p
, ℓ
, which is the probability of a site belonging to
a spanning cluster as a function of the occupation probability,
p
, and system size,
. The occupation probability,
p
[0, 1]
,
corresponds to sites where
σ
̄
~
iso
x
is greater than
1
p
× 100
th percentile of the isotropic stress distribution, including both
compressive and tensile stresses. In the large system size, the ‘thermodynamic limit’, i.e.
, we expect the following
power-law scaling near the percolation probability,
p
c
, for the density of a spanning cluster
P
p
characterized by the critical
exponent
β
,
P
p
p
p
c
β
. A cluster is a set of connected sites, and two sites are considered connected if they are nearest
neighbours (eight-point connectivity). Furthermore, we quantify the average cluster size,
S
p
=
s
, where
s
is the size of a
cluster. Near the percolation probability, the power-law scaling of
S
p
quantified
by critical exponent
γ
is
S
p
|
p
p
c
|
γ
and
3
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21:
20240022
critical exponent
ν
, which describes the power-law scaling of correlation length
ξ
, i.e. average distance between two sites in the
same cluster, is expected to follow
ξ
|
p
p
c
|
ν
.
Both the density of the spanning cluster
P
p
, ℓ
and the average cluster size
S
p
, ℓ
are shown in
figure
3 at the onset of the
transition, i.e.
ω
~
(
figure
3
a,c
) and
α
~
= 0.7
(
figure
3
e,g
) for various system sizes,
. Additionally,
figure
3
b,d,f
and
h
shows the
collapse of
P
p
, ℓ
and
S
p
, ℓ
when scaled with critical exponents obtained from the
finite-size
scaling analyses, accounting for a
diverging correlation length
ξ
near the
p
c
. Remarkably, the critical exponents corresponding to the scaling of the time-averaged
isotropic stress
field
at the onset of the solid-to-liquid transition in the active cell layer (
table 1
) lead to a reasonable collapse
and are in close agreement with those from the two-dimensional random percolation universality class [
60]. Importantly,
this agreement is obtained via two independent pathways, i.e. cell–cell adhesion and active traction force, and points to the
short-range nature of stress correlations near this transition.
3.2. Mechanism
Our results thus far do not explain why short-range correlations that are characteristic of random percolation emerge in
the solid-to-liquid transition of the studied active monolayers. To explore this, we analysed the stress
field
statistics exten-
sively and
identified
two mechanical signatures associated with each path to transition. For the
first
path, increasing cell–
cell adhesion leads to a solid-like to liquid transition. Although this counterintuitive behaviour, called the
adhesion paradox
,
was observed experimentally and captured by a two-dimensional vertex model previously [
61], the mechanism responsible
for it is yet to be explained. Our analyses suggest increasing cell–cell adhesion results in higher global mean and
field
fluctuations,
quantified
by susceptibility,
χ
( . ) =
n
.
2
.
2
with
n
denoting
field
dimensions, in the out-of-plane component
of the time-averaged stress tensor,
σ
̄
zz
x
=
e
z
σ
̄
x
e
z
. This is shown in
figure
4
a
for
σ
̄
~
zz
x
=
σ
̄
zz
/
σ
̄
zz
max
. This high mean
out-of-plane stress, neither accessible in two-dimensional models nor experiments, can lead to local buckling owing to cell
shape changes caused by adhesion increase [
62] providing a slightly more free volume needed for a transition to a liquid
state [
63]. Moreover, increased
fluctuations
are consistent with the emergence of short-range stress correlations and random
percolation at the onset of the transition. For the second path to transition, increasing active polarity and hence in-plane
active traction forces lead to an increase in both the global mean and
field
fluctuations
in the maximum in-plane shear
field,
σ
τ
x
,
t
= (1/2)(
σ
max
x
,
t
σ
min
x
,
t
)
, where
σ
max(min)
x
,
t
is a
field
that corresponds to the largest (smallest) eigenvalue of
the two-dimensional stress tensor (in-plane),
σ
2
D
x
,
t
=
σ
xx
x
,
t
(
e
x
e
x
) +
σ
yy
x
,
t
(
e
y
e
y
) +
σ
xy
x
,
t
(
e
x
e
y
+
e
y
e
x
)
.
This is shown in
figure
4
b
for the time-averaged
field,
σ
̄
τ
x
, normalized by its maximum,
σ
̄
τ
max
,
σ
̄
~
τ
x
=
σ
̄
τ
/
σ
̄
τ
max
. Thus, the
transition driven by increasing active traction leads to an increase in both mean and
fluctuations
of local, maximum shear, from
which emerges stress percolation, near the transition. We note that an important observation here is the encoding of
different
types of mechanical information in the isotropic stress
field.
In this vein, our approach has the potential to unify disparate
observations in biological systems, such as percolations based on (i) cell connectivity [
25] and (ii) edge tension network [
64,65
]
through understanding the transmission of mechanical information.
50
(
a
)
(
b
)
(
c
)
(
d
)
(
e
)
(
f
)
(
g
)
(
h
)
0
50
0
50
0
0
2
4
6
distance, r/
R
0
g (r)
0
2
4
6
50
0
Figure
2.
Characterization of solid-to-liquid transition via one-dimensional pair-correlation function, corresponding to time-averaged
configurations,
for (
a
)–(
d
)
cell–cell adhesion drive and (
e
)–(
h
) active traction forces.
Table 1.
Critical exponents obtained from
finite-size
scaling analyses compared with those from two-dimensional random percolation (RP).
RP (2D)*
ω
~
α
~
p
c
0.5927
0.612 ± 0.004
0.590 ± 0.007
γ
2.388
2.305 ± 0.146
2.103 ± 0.423
β
0.1388
0.176 ± 0.049
0.134 ± 0.009
ν
1.333
1.422 ± 0.051
1.243 ± 0.198
*[60
]
4
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4. Discussion
The critical exponents obtained by
finite-size
scaling analyses of time-averaged isotropic stress
fields
map the solid-to-liquid
transition in active monolayers to a two-dimensional random percolation universality class. Thus, we reveal the short-range
nature of stress correlations near this transition,
attuned
to
differing
critical exponents for percolation in systems with long-
range correlated disorder [
66]. In this vein, our
findings
lend further support to the viewpoint that in passive amorphous solids,
it is the mechanical equilibrium condition that gives rise to long-range stress correlations [
30–33
]. Importantly, we establish the
criticality of the active solid-to-liquid transition via two distinct paths, which are both well established experimentally [
23,61
].
A particularly appealing feature of our approach is the implicit manifestation of physical interactions as mechanical
information in isotropic stress
fields,
given the two distinct drives considered in our study. This is important since direct
measurement of traction forces and mechanical stress
fields
is experimentally accessible [
46,67
] and could be used to
confirm
the
proposed critical behaviour. Our results are particularly illuminating since recent experiments highlight the dominant role of
traction forces in deriving the solid-to-liquid transition in epithelial tissues [
23]. Activity in our study is owing to active polarity.
The particular model we use is tuned directly by experiments [
51] and
specifically
chosen given its success in representing cell
motility by reproducing stress
patterns
around nematic defects [
45,47
]. In this vein, a recent study suggests that the active glassy
dynamics are mute to the details of such a model [
68].
The critical exponents were estimated by extracting squared subsystems of characteristic length
[20, 100]
, spanning
almost a decade of length scales, with dimensions
(ℓ + 1)
2
given the
lattice
spacing
a
0
= 1
R
0
, from the time-averaged isotropic
stress
fields
with total dimensions of
(
L
x
+ 1) × (
L
y
+ 1) =
103 041 (see electronic supplementary material for
finite-size
scaling
analyses). While our analyses can
definitely
benefit
from more statistics, we are
confident
in its robustness since we obtain
similar critical exponents from two distinct and independent paths towards solid-to-liquid transition. Additionally, we used
site percolation on a square
lattice,
given the nature of our simulations, but it is well known that critical exponents for random
percolation universality class are robust to model details [
60,66,69
], e.g. bond versus site percolation and
lattice
type. Thus, we
do not expect our results to change based on
lattice
types, e.g. triangular, excluding the Bethe
lattice.
The
finite-size
scaling analyses were performed on two-dimensional isotropic stress
fields
that embed the out-of-plane
stress component
σ
zz
. Furthermore, the three-dimensional nature of our modelling approach presents other unique advantages
including the ability to tune independently and explicitly for cell–cell and cell–substrate interactions. Most importantly, we can
1
(
a
)
(
b
)
(
c
)
(
d
)
(
e
)
(
f
)
(
g
)
(
h
)
1.5
600
800
400
0
0.4
l
= 20
l
= 40
l
= 50
l
= 60
l
= 80
l
= 100
0.2
0
0.3
0.15
300
0
0.75
0
1.5
0.75
0
0.5
P
(
p
,
l
)
P
(
p
,
l
)
S
(
p
,
l
)
S
(
p
,
l
)
l
β
/
ν
(
p
,
l
)
l
β
/
ν
(
p
,
l
)
l
γ
/
ν
S
(
p
,
l
)
l
γ
/
ν
S
(
p
,
l
)
l
1/
ν
S
(
p
p
c
)
0
1
0.5
0
0.5
p
p
–15
0
15
–25
0
25
0.5
1
–15
0
15
–25
0
25
1
0.5
1
0.5
1
Figure
3.
Finite-size scaling analyses of isotropic stress criticality. The density of spanning cluster
P
(
p
, ℓ)
(
a, e
) and the average cluster size
S
(
p
, ℓ)
(
c, g
) for
different
subsystems with characteristic length
and their collapse after performing a
finite
size scaling analysis for
P
(
p
, ℓ)
(
b, f
) and
S
(
p
, ℓ)
(
d, h
). First row
(
a–d
) corresponds to increasing dimensionless cell–cell adhesion
ω
~
=
ω
cc
/
ω
cw
and the second row (
e–h
) corresponds to the increasing strength of active traction
forces
α
~
.
2
(
a
)
(
b
)
4
2
0
4
20
10
0
2
0
zz
χ
σ
~
τ
χ
σ
~
( )
zz
σ
~
( )
τ
σ
~
0.2
0.8
1
0
0.2
0.5
ω
α
Figure
4.
Global
field
statistics for time-averaged, normalized, mechanical
fields.
(
a
) The mean and the susceptibility of time-averaged out-of-plane stress
fields
for
the cell–cell adhesion drive. (
b
) The mean and the susceptibility of the time-averaged maximum in-plane shear for the active traction drive. All values are normalized
by those corresponding to
ω
~
= 0.2
for (
a
) and
α
~
= 0.2
for (
b
).
5
royalsocietypublishing.org/journal/rsif
J. R. Soc. Interface
21:
20240022
ensure that, within the range of studied parameters, the solid-to-liquid transition is not owing to an extrusion event, altering the
confluent
state. This cannot be done with two-dimensional modelling approaches.
We also examined potential links to jamming transition in passive granular systems [
70,71
]. Interestingly, a recent study
suggests short-range force correlation in jammed granular systems [
72], in agreement with our
findings.
However, this contrasts
with other studies that point to long-range correlations in such systems [
29,53
]. This may be owing to the dependence of critical
parameters associated with jamming transition on the nature of the global load, e.g. shear versus isotropic compression, as well
as the loading rate [
73,74
]. Moreover, our results
differ
from the critical exponents associated with standard rigidity percola-
tion [
75,76
], indicating distinct universality classes. In this vein,
efforts
to link the jamming transition to rigidity percolation
remain elusive [
71,77
]. Additionally, contact connectivity percolation [
78,79
] has critical exponents that are
different
from those
for random percolation [
80,81
]. Our results also contrast with non-equilibrium, boundary-driven sheared granular systems
exhibiting long-range force correlation [
36] and delineate the
difference
between an active
matter
system, driven by local
injection of energy and boundary-driven non-equilibrium systems. In this vein, our study provides a new context to explore
potential links between dense active
matter
and glasses [
82–84
]. Exploring other pathways to transition, such as intercellular
friction [
85] and the role of mechanochemical waves [
86], are exciting avenues for near-future studies. Moreover, progress is
made towards direct measurement of traction forces and mechanical stress
fields
in three dimensions [
87] which can be used
to probe the proposed universality in cellular monolayers in experiments. Furthermore, understanding the spatial organization
of stresses and their correlations in active p-atic liquid crystal models beyond onefold (p=1) rotational symmetry, i.e. active
polarity, and coupling with active poroelasticity [
88] are the natural next steps for this study.
Ethics.
This work did not require ethical approval from a human subject or animal welfare
committee.
Data accessibility.
The current manuscript is a computational study, so no data have been generated for this manuscript. Modelling code is available
here [
89]. Data and post-processing codes can be found here [
90].
Electronic supplementary material is available online at [
91].
Declaration of AI use.
We have not used AI-assisted technologies in creating this article.
Authors’ contributions.
S.M.: conceptualization, data curation, formal analysis, investigation, methodology, software, visualization, writing—
original draft, writing—review and editing; G.R.: conceptualization, funding acquisition, methodology, resources, supervision, writing—
review and editing; J.E.A.: conceptualization, funding acquisition, methodology, resources, supervision, writing—review and editing; A.D.:
conceptualization, funding acquisition, methodology, project administration, resources, supervision, writing—original draft, writing—review
and editing.
All authors gave
final
approval for publication and agreed to be held accountable for the work performed therein.
Conflict
of interest declaration.
We declare we have no competing interests.
Funding.
S.M. is grateful for the generous support of the Rosenfeld Foundation fellowship at the Niels Bohr Institute, University of Copenhagen.
S.M., G.R. and J.A. acknowledge support for this research provided by US ARO funding through the Multidisciplinary University Research
Initiative (MURI) grant no. W911NF-19-1-0245. A.D. acknowledges funding from the Novo Nordisk Foundation (grant no. NNF18SA0035142
and NERD grant no. NNF21OC0068687), Villum Fonden grant no. 29476, and the European Union via the ERC-Starting Grant PhysCoMeT.
Acknowledgements.
S.M. is thankful for insightful feedback from Jacob Notbohm and Aboutaleb Amiri and discussions with
Matthieu
Wyart and
David R. Nelson.
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