††thanks: contributed equally††thanks: contributed equally

# Mechanical excitation and marginal triggering during avalanches in sheared
amorphous solids

D. Richard  Institute for Theoretical Physics, University of Amsterdam,
Science Park 904, Amsterdam, Netherlands  Department of Physics and
BioInspired Institute, Syracuse University, Syracuse, NY 13244  Univ. Grenoble
Alpes, CNRS, LIPhy, 38000 Grenoble, France  A. Elgailani  Northeastern
University, Boston, Massachusetts 02115, USA  D. Vandembroucq  PMMH, CNRS UMR
7636, ESPCI Paris, PSL University, Sorbonne Université, Université de Paris,
F-75005 Paris, France  M.L. Manning  Department of Physics and BioInspired
Institute, Syracuse University, Syracuse, NY 13244  C.E. Maloney  Northeastern
University, Boston, Massachusetts 02115, USA

###### Abstract

We study plastic strain during individual avalanches in overdamped particle-
scale molecular dynamics (MD) and meso-scale elasto-plastic models (EPM) for
amorphous solids sheared in the athermal quasi-static limit. We show that the
spatial correlations in plastic activity exhibit a short lengthscale that
grows as $t^{3/4}$ in MD and ballistically in EPM, and is generated by
mechanical excitation of nearby sites not necessarily close to their stability
thresholds, and a longer lengthscale that grows diffusively for both models
and is associated with remote marginally stable sites. These similarities in
spatial correlations explain why simple EPMs accurately capture the size
distribution of avalanches observed in MD, though the temporal profiles and
dynamical critical exponents are quite different.

Many driven disordered solids, ranging from glasses to granular matter to
magnetic systems, respond via complex avalanches that are difficult to predict
[1, 2, 3]. A better understanding of these dynamics, even in simple model
systems, would aid in avalanche detection and material design.

We focus here on amorphous solids subject to slowly imposed shear, which fail
via a broad spectrum of avalanches of plastic deformation caused by
redistribution of stress after local yielding events [4, 5, 6, 7]. The highly
anisotropic and long-ranged nature of the stress redistribution leads to a
characteristic structure of the avalanches, where correlations are strongest
along the directions of maximum imposed shear. Most previous work has focused
on the distribution of avalanche sizes [8, 9, 10, 11, 7, 12, 13] and the
spatial correlations of plastic strain which develop over the course of
successive avalanches [14, 15, 16, 17, 18]. More recently, the dissipation
rate as a function of time during individual avalanches was studied in
experiments on bulk metallic glass (BMG) pillars [19], and in a computer model
[11]. The temporal response was observed to be similar to that in previously
explored dynamically critical systems [4], and explained using a mean-field
theory [19]. However, surprisingly, no work, in either experiment or
simulation, has yet characterized how the individual avalanches proceed in
time _and space_.

To address this question we turn to computational models, including particle-
based simulations, such as molecular dynamics (MD) and related energy
minimization techniques, which have been the workhorse for modeling sheared
amorphous solids for decades, as well as elasto-plastic models (EPMs). EPMs
assume an amorphous solid is composed of meso-scale regions that will yield
when the local stress reaches a specified threshold. There are many different
versions of EPMs [20] that differ in how they introduce disorder, evolve
propogating stress fields, etc. In perhaps the simplest class of EPM [21, 8,
22, 7, 23] the system is evolved quasi-statically, and, after any instability,
the stresses are fully equilibrated over all space – effectively
instantaneously – before allowing for any subsequent yielding. This is in
contrast to MD simulations where, after a local rearrangement, the stress
change propagates continuously in time and space – diffusively for overdamped
systems [24] and ballistically for underdamped systems – away from the plastic
instability.

Despite the fact that the quasi-static EPMs are completely devoid of any
realistic description of the dynamics of stress redistribution, they capture
the critical scaling exponents observed in overdamped MD simulations [9, 25,
26]. They are therefore, in some sense, unreasonably good, and the reason for
their fidelity demands an explanation.

In this letter, we compare two systems: an overdamped, gradient-descent MD
simulation and a quasi-static EPM. We show that while the temporal profile of
the avalanches is quite different in the two models – the EPM agrees with
previously known mean field results while the MD does not – the spatial
structure of the correlations that develop is strikingly similar. The spatial
correlation function can be characterized by two distinct lengthscales: a
short length, $\xi_{\text{excite}}$, and a longer one,
$\xi_{\text{marginal}}$. $\xi_{\text{excite}}$ grows ballistically in time for
the EPM, and we argue that the reason for this ballistic propagation is the
“mechanical excitation” mechanism suggested by Idema and Liu [27]: one event
generates a stress redistribution that causes nearby sites to exceed their
threshold for stability, triggering new events. If the timescale over which an
unstable site transforms is short compared to the propagation of the stress,
then the resulting dynamics are reminiscent of toppling dominos, with a
ballistic wavespeed equal to the spacing of the dominos divided by the
“toppling time” – the time it takes one, once destabilized, to fall onto its
neighbor. $\xi_{\text{excite}}$ grows less quickly in time for the MD, but it
is still strongly super-diffusive, suggesting that a similar mechanism is at
play and hinting that the “toppling time” in MD is more complicated than in
EPM. In contrast, $\xi_{\text{marginal}}$ grows diffusively in time for both
models, and it has a pronounced system-size dependence. At long distance the
low positive and negative contributions of the quadrupolar stress interactions
add up as a mechanical noise and are expected to drive local zones close to
marginality [28, 29, 30, 31, 32]. The observed size dependence suggests that
$\xi_{\text{marginal}}$ is the length scale at which the weakest sites in the
system are close enough to the triggering event to be destabilized.

Our MD glass former consists of a generic two dimensional binary mixture [33].
Samples are deformed using standard athermal quasistatic shear protocol [34].
At the onset of each instability, we trigger the avalanche by affinely
deforming the system and subsequently let the system relax via gradient
descent dynamics. More details are given in the SM. In order to break up large
avalanche into a sequence of individual plastic events, we employ a recently
proposed persistent homology method [35], detailed in the SM [36].

For the EPM, we use the same shear transformation based model in [37] with the
same initialization and evolution rules but with different loading – forward
shearing instead of cyclic, described in more detail in the SM [36]. The
dynamical update rules under an applied global shear strain are (i) for a
given stress field, synchronously allow all sites over threshold to yield and
recompute the stress field everywhere; (ii) repeat (i) until all sites are
below threshold; and (iii) advance the globally applied total strain until
precisely one site is at its stability threshold. The synchronous update of
unstable sites defined in step (i) sets the time unit of the model.

Figure 1: Spatio-temporal plastic evolution. Typical spatio-temporal map of an
avalanche in Molecular Dynamics (MD) with gradient descent dynamics (left) and
in the Elasto-plastic Model (EPM) with synchronous dynamics (right). Each
plastic event is colored according to its normalized birth time $t/T$, with
$T$ the avalanche duration. Insets show the particle resolved plastic field
(left) and EPM local stress redistribution (right)

We first qualitatively describe a typical avalanche in both models. Fig. 1
highlights individual sites that have yielded, colored according to the time
at which the site yielded. The pattern which emerges is a set of clusters,
where all sites within a cluster are nearly the same color, extend along a
direction of maximum shear, and almost continuously fill space.

In contrast, these larger clusters are typically separated from one another by
gaps of material. Nearby clusters are not necessarily triggered sequentially
in time, suggesting that gaps between clusters correspond to regions of
material that are stable enough to survive the increased local stresses at the
edges of the growing cluster. The same intermittent activity is found in
depinning avalanches with long-ranged interactions [38, 39].

Figure 2: Plastic activity profile. (a) Typical dissipation rate $-dU/dt$ in
MD plotted as a function of the reduced time $t/T$ for duration $1500<T<3000$.
(b) Scatter plot of avalanche size versus duration. The Blue and black empty
symbols are running means $\langle S\rangle\sim T$ and $\langle T\rangle\sim
S$, respectively. Solid and dashed lines indicate the scaling $\langle
S\rangle\sim T^{\delta}$, with exponent $\delta=0.65$ and $\delta=4/3$,
respectively. (c) Average normalized activity for $400<T<600$ (orange region
in (b)). (d) Average duration $\langle T\rangle$ at a fixed avalanche size $S$
for different system size. Insets show the same data collapsed using $\langle
T\rangle L^{-d\nu}$, with $\nu\simeq 0.4$ and $d$ the spatial dimension.
Panels (e), (f), (g) and (h) show the same results as in (a), (b), (c), and
(d) but for the EPM where $-dU/dt$ is measured as the energy dissipation per
sweep.

Given this qualitative similarity in the spatio-temporal structure, and the
utility of mean-field models in predicting temporal dynamics in depinning and
other disordered critical phenomena, we next quantitatively analyze the
temporal activity profiles in MD and EPM. In Fig.2(a) and (e), we plot the
dissipation rate $-dU/dt$ as a function of the normalized duration $t/T$ for a
few typical avalanches. In MD, we find that the activity exhibits intermittent
bursts of activity followed by long inactive periods, so that the activity
magnitude spans six decades. Inactive regions are associated with marginal
sites that are only weakly unstable. For a typical saddle node bifurcation, we
expect that the waiting time $\Delta t_{w}$ before activation will scale as
$\Delta t_{w}\sim(\sigma_{i}-\sigma_{i}^{\rm th})^{-1/2}$ [40], with
$\sigma_{i}$ and $\sigma_{i}^{\rm th}$ the local stress and local yield stress
of the unstable site $i$. In contrast, unstable sites in our EPM transform
instantaneously regardless how far they are over threshold, termed a uniform
activation rate. As a consequence, activity fluctuations in EPM (Fig.2(e))
vary by less than an order of magnitude.

Another difference between the MD and EPM can be seen by plotting the
avalanche size $S$ versus duration $T$, Fig.2(b) for MD and (f) for EPM. We
find that MD data scatter much more than our EPM. This is similar to previous
results comparing an EPM with a uniform activation rate to one with a so-
called progressive rate model, where the activation rate is a function of the
overshoot $\sigma_{i}-\sigma_{i}^{\rm th}$ [41, 32]. In both MD and EPM, the
average size for a fixed duration show a power law $\langle S\rangle\propto
T^{\delta}$, where the dynamical exponent $\delta=d_{f}/z$ with $d_{f}$ and
$z$ the static and dynamical fractal exponent, respectively. Independent
estimation of $d_{f}$ and $z$ are reported in the Supplemental Material (SM)
[36] and consistent with previous works [13, 11]. The EPM gives a larger
$\delta$ than MD as the dynamical exponent $z\simeq 0.6$ is lower than MD
($z\simeq 1.55$), although it is consistent with previous EPM-type models
[11], which used continuous-in-time dynamics rather than the automaton rules
we apply here.

More surprisingly, in MD we find a discrepancy between average avalanche size
for a fixed duration $\langle S\rangle_{T}$ and the average duration for a
fixed size $\langle T\rangle_{S}$. We find that the average duration $\langle
T\rangle_{S}$ is system-size dependent and grows with $L$ as $L^{d\nu}$, see
Fig.2(d). We can understand this size effect in terms of the large waiting
time $\Delta t_{w}$ associated with unstable sites that are just barely over
threshold, $\sigma_{i}\gtrsim\sigma_{i}^{\rm th}$. We expect that the
overshoot $\langle\sigma_{i}-\sigma_{i}^{\rm th}\rangle$ will decrease with
system size as $\sim L^{-\frac{d}{1+\eta}}$. This scaling of the
characteristic overshoot for unstable sites is different from the previously
reported scaling for thresholds associated with marginal stable sites [42, 43,
26, 44]. The average waiting time caused by weak triggering will follow
$\langle\Delta t_{w}\rangle\sim L^{\frac{d}{2(1+\eta)}}\sim L^{d\nu}$. We find
a good collapse of MD data for $\nu\simeq 0.4$, giving $\eta\simeq 1/4$. A
similar finite-size effect, though with a different exponent, was previously
noticed in progressive rate EPMs [41, 32].

In Fig.2(c) and (g), we show the normalized average activity profile $\langle
A\rangle/\langle A\rangle_{\rm max}$ (with $A=-dU/dt$) plotted against the
reduced time $t/T$ in the MD and EPM respectively. For the EPM and avalanches
within the scaling regime, we find that profiles are symmetrical and well
modelled by $f(u)=(u(1-u))^{\delta-1}$, with $u=t/T$, consistent with previous
works and mean field theories [45, 11]. For large avalanches in EPM, with long
duration beyond the scaling regime where system size effects become relevant,
one finds a profile which departs from that scaling and is skewed with more
activity at early times. A similar transition to skewed profiles for large
evens was recently observed in granular flows [46]. In MD, the profile
deviates strongly from the $f(u)=(u(1-u))^{\delta-1}$ form. This is due to
inactive periods with almost zero dissipation. The average activity at the
mid-point of the avalanche is systematically lower than at its beginning, the
latter being by definition always active. We speculate that one would recover
a symmetrical activity profile by introducing inertia in the dynamics, which
would facilitate barrier crossing in weakly unstable regions.

Figure 3: Spatio-temporal plastic propagation. (a) particle-based (MD)
2-points 2-times normalized correlation function $C(x)/C_{\rm max}$ in
$\log_{10}$ for $\Delta t=4$ and $40$. White regions correspond to a negative
correlation. $\xi_{\rm excite}$ and $\xi_{\rm marginal}$ correspond to the
short length scale decay and negative crossing, respectively. $w$ indicates
the width of the plastic localization. (b) Correlation function along the main
stress redistribution for different delay time $\Delta t=4$, $10$, $20$, $40$,
$60$, and $100$. Inset shows the correlation in semi-log scale. Solid lines
are fits of the form $C(x)/C_{\rm max}\sim e^{-x/\xi_{\rm excite}}$. (c)
$\xi_{\rm excite}$ (red) and $\xi_{\rm marginal}$ (blue) plotted against
$\Delta t$. The blue line corresponds to the diffusive response associated
with the stress field generated by a single event at $\Delta t=0$. The red
line indicates a superdiffusive regime with $\sim t^{3/4}$. Green data are the
plastic width $w$ extracted from fitting the normal correlation $C(y)$ (away
from the center) by $C(y)\sim e^{-(y/w)^{2}/2}$. The green line indicates a
subdiffusive regime with $\sim t^{1/10}$. (d) $\xi_{\rm excite}$ and $\xi_{\rm
marginal}$ for $\Delta t=t_{\rm STZ}=6$ plotted as a function of the system
size $L$. $\xi_{\rm marginal}(L)$ scales as $\sim L^{1/(1+\nu)}$, with $\nu=2$
an effective pseudo-gap exponent. (e), (f), (g), and (h) are the same results
as in (a), (b), (c) and (d) but for our mesoscale model (EPM) with $\Delta
t=1$, $2$, $4$, $6$, $8$, and $12$. The solid red line in (g) indicates
$v_{0}t$, with $v_{0}=3$. Results in (h) are for $\Delta t=1$ where $\xi_{\rm
marginal}(L)$ scales as $\sim L^{1/(1+\nu)}$, with $\nu=2/3$. $\xi_{\rm
excite}(L)$ is fairly constant for both MD and EPM.

We next analyze the spatial correlations that build up during the avalanche.
To do this, we define a 2-points 2-times correlation function
$C(r_{0},r_{0}+\vec{r},t_{0},t_{0}+\Delta t)=\langle\overline{\Delta
P(r_{0},t_{0})\Delta P(r_{0}+\vec{r},t_{0}+\Delta t)}\rangle$, with $\Delta
P=P-\overline{P}$ and where $\overline{...}$ and $\langle...\rangle$ represent
a spatial and ensemble average, respectively. Here, $P(t)$ corresponds to the
incremental plastic field measured at a given time $t$. In order to improve
the statistics of the particle-based correlations, we assume time-
translational invariance and average over $t_{0}$. Data supporting this
assumption for both MD and EPM is shown in SM Sec. 6 [36].

Fig. 3(a) shows the MD normalized correlation $C(x,y)/C_{\rm max}$ for delay
times $\Delta t=4$ (left) and $40$ (right). There are positive correlations
along the directions of imposed shear (the $x$ and $y$ axes). The spatial
extent of the region of positive correlation grows with $\Delta t$ [47]. The
same data are shown for the EPM in Fig. 3(e) for $\Delta t=2$ and $6$ where we
observe the same qualitative behavior.

In Fig. 3 (b) and (f), we plot $C(x)$ along the $x$-axis at various $\Delta
t$. $C(x)$ decays exponentially at small $x$, and we use the exponential decay
rate to define a short-range lengthscale $\xi_{\text{excite}}$. Despite the
initial exponential decay, $C(x)$ crosses through zero at a finite $x$,
indicating anti-correlation, and we use this zero crossing to define a second,
larger lengthscale, $\xi_{\text{marginal}}$. We also look at correlation in
the transverse direction, $C(y)$ at a small, fixed $x$ ($x_{\rm MD}=50$ and
$x_{\rm EPM}=40$). We find that $C(y)$ is well modeled by a Gaussian,
$C(y)\sim e^{-(y/w)^{2}/2}$ (results not shown), which allows us to quantify
the width $w$.

In Fig. 3 (c) and (g), we show that these three lengths exhibit a power-law
evolution with $\Delta t$. The trends are similar in both MD and EPM.
$\xi_{\text{excite}}$ has the strongest $\Delta t$ dependence: it is scales
like $\Delta t^{3/4}$ in the MD and is nearly ballistic in the EPM.
$\xi_{\text{marginal}}$ scales diffusively as $\Delta t^{1/2}$ in both the MD
and EPM. $w$ has a very weak dependence on $\Delta t$ which indicates that for
any given avalanche, the correlations get more an more constrained to lie
along the axes of shear as time goes during the course of the avalanche.
Finally, in Fig. 3 (d) and (h) we show how $\xi_{\text{marginal}}$ and
$\xi_{\text{excite}}$ depend on system size for a fixed, small $\Delta t$. In
both MD and EPM, $\xi_{\text{excite}}$ has relatively little dependence on
system size, while $\xi_{\text{marginal}}$ has a more pronounced system size
dependence, scaling approximately like $L^{1/3}$ and $L^{2/3}$ in the MD and
EPM respectively.

In this letter, we demonstrate that the spatio-temporal evolution of
avalanches in particle-based simulations in the overdamped limit share
important similarities and differences with an elasto-plastic model governed
by synchronous dynamics. We show that in both models, avalanches are driven by
localized clusters of activity, and that the spatial correlation function of
activity exhibits two lengthscales that grow differently in time and with
system size.

One lengthscale, $\xi_{excite}$, is mechanically excited where a diffusively
propagating stress field triggers nearby sites that generate their own stress
fields. In the EPM models, this leads to standard ballistic propagation as
seen in other excitable media, while in MD simulations we find the front
propagates as $t^{3/4}$. In both cases, our data suggests this scale is
independent of system size, as expected. An interesting question for future
work is what generates this non-standard exponent in MD simulations.

A second, longer lengthscale, $\xi_{marginal}$, propagates diffusively and
scales with system size. This suggests that it is governed by the weakest
spots in the disordered solid, as far-field stress fluctuations anywhere in
the system are sufficient to trigger events, and the stress magnitude required
to trigger the weakest spot scales as a power law with system size.

Another difference between the two types of simulations is the statistics of
their temporal dynamics. In MD simulations, the stress overshoot – the
difference between the triggering stress field and the stress threshold of an
excitable site – governs how fast the system departs a saddle and the waiting
times between localized bursts of deformation in an avalanche. These effects
generate dynamical exponents that depart significantly with mean-field
predictions, and cause duration-size curves to vary with system size. In
contrast, our simple EPM has no such mechanism; the dynamical exponent is much
closer to mean-field and the duration-size curves do not depend on system
size. Recent progressive rate EP models [41, 32] include a proxy for this
waiting time, and qualitatively reproduce this physics, but do not
quantitatively match the exponents we find in the MD. A nice feature of EPMs
is that it is possible to disambiguate the consequences of various choices for
the stress propagation, waiting times, and disorder. Future work could focus
on adjusting properties of EPMs to improve quantitative agreement with MD.
Another important avenue is understanding how these models behave in the
presence of inertia [9, 25, 48].

We have focused here on MD and EPM simulations for a relatively ductile
material, where the stress required for each site to be triggered is
relatively small. It will be very interesting to revisit similarities and
differences in brittle systems that undergo shear banding instabilities, which
provide an even stricter test of models of the spatio-temporal evolution of
disordered solids. More broadly, the tools developed here could be used to
characterize spatio-temporal dynamics of avalanches in materials with
complicated interactions (irregular shapes, friction, realistic molecular
potentials) and boundary conditions.

Acknowledgments We acknowledge support of the Simons Foundation for the
“Cracking the Glass Problem Collaboration” Award No. 348126 (D.R) and No.
454947 (A.E. and M.L.M). M.L.M. acknowledges support from NSF-DMR-1951921.

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Supplementary material: “Mechanical excitation and marginal triggering during
avalanches in sheared amorphous solids”

## 1\. Description of molecular dynamics and synchronous elastoplastic model

Our MD glass former consists of a standard two dimensional 50:50 binary
mixture of ‘large’ and ‘small’ particles of equal mass interacting via a
short-range non-additive inverse-power-law potential (see details in Ref.
[33]). We have considered different system sizes $L=80,160,320$, and $640$,
respectively. Glasses are athermal and quasistatically sheared (AQS) up to 50%
of strain (with strain step $\Delta\gamma=10^{-4}$), and data reported here
correspond to strains ranging from 20 to 50 %. Variations in strain step are
discussed below. At the onset of each instability, we trigger the avalanche by
affinely deforming the system and subsequently let the system relax via
gradient descent dynamics according to $\dot{\mathitbf r}=-D\nabla U$, with
coordinate positions $\mathitbf r$ and potential energy $U$. During the
minimization process, we monitor the plastic deformation (i.e. the incremental
non-affine displacement) in the standard way by computing the $D^{2}_{\rm
min}$ field [49] between snapshots separated by $2\tau_{MD}$, where the
microscopic time is defined as $\tau_{MD}=\sigma^{2}/D$ with $\sigma$ a
typical particle diameter and $D$ the bare translational diffusion
coefficient. We have collected 20k and 1.2k avalanche dynamics for the
smallest ($L=80$) and largest ($L=640$) system, respectively.

Here we provide a synopsis of the elasto-plastic model used in this paper,
which was developed in previous work [37]. A 2D plane is discretized into
sites. Each site has an elastic strain $\epsilon_{e}$ and a plastic strain
$\epsilon_{p}$, with the total strain defined as the sum of the two:
$\epsilon_{t}=\epsilon_{e}+\epsilon_{p}$. The stress $\sigma$, is proportional
to $\epsilon_{e}:\sigma=2\mu\epsilon_{e}$, where $\sigma$ is defined in terms
of the Cartesian components of the Cauchy stress:
$\sigma=(\sigma_{xx}-\sigma_{yy})/2$ and, similarly,
$\epsilon_{e}=(\epsilon_{e}^{xx}-\epsilon_{e}^{yy})/2$. We set $\mu=1$
throughout this study, so $\sigma$ and $\epsilon_{e}$ are numerically equal.
When the magnitude of the stress at a site exceeds the plastic threshold, set
to 1 here, we increment the plastic strain at the site by in the same
direction as $\sigma$ ( +2 for $\sigma\geq 1$, -2 for $\sigma\leq-1$), and
update the stress at the remaining sites according to the rules of linear
elasticity analogous to Eshelby’s classical solution for a plastic inclusion
in an elastic matrix. The increment value 2 is chosen to ensure a single-
valued strain energy function. The total number of avalanches collected is
36.8K; obtained from shearing 50 different systems (L=256).

## 2\. Persistent homology methodology

In this section, we describe a persistent homology method, similar to that
recently proposed by Stanifer and Manning [35], employed to break up an
avalanche into a time sequence of individual events (as shown in Figure 1 of
the main text). In the main text, we refer to the location of each individual
event as a ”site”. During each avalanche, we record the incremental
$D^{2}_{min}$ plastic field which generates a representation of the activity
on a grid of space-time regions, see Fig. S1(a). The spatial grid size is set
to 2 particles diameters and the temporal grid size is set to $2$ MD time
units. The $D^{2}_{min}$ of a cell corresponds to the sum over all
$D^{2}_{min}$ particle values within the cell. Our goal is to agnostically
break-up high local maxima of this space-time field. To this end, we employ a
persistent homology framework. Here, we first sort all $D^{2}_{min}$ values
from the largest to the lowest up to $10^{-3}$ (from which below no actual
plastic rearrangements occur). We then prune this sorted list and create a new
cluster for each local maxima that do not belong to an existing cluster, where
the clustering distance criterion is set to the first neighbors on the space-
time grid. When a new cluster is born we record its $D^{2}_{min}$ value as its
birth. When two existing clusters merge we record their $D^{2}_{min}$ value as
their death. We can then place each cluster in a birth-death diagram (Fig.
S1(b)) and compute their persistent $p$ as the distance of the cluster to the
line $y(\rm death)=x(\rm birth)$. Events that corresponds to short spatime-
time fluctuation will be close to this line. When looking at the probability
distribution of persistence $P(p)$ average over many avalanches (Fig. S1(c)),
we observe a bimodal distribution that indicates two different populations of
individual events. Selecting events that have a large persistence $p>0.1$ and
superimposing their birth time on the avalanche activity time profile (energy
dissipation $-dU/dt$ or maximum of the incremental $D^{2}_{min}$ field), we
find that persistent cluster match perfectly the time location of large
dissipation. For each persistent cluster, we have access to its spatial
position in addition to its birth time. This allows us to construct the same
spatio-temporal map of plastic activity as in our EPM, with individual events
occurring at ”sites” corresponding to these persistent peaks. In Fig. S1(d),
we show for one avalanche the superposition of the total cumulative plastic
field (grayscale colormap) and each individual event colored according to
their normalized birth time $t/T$, where $T$ is the total duration of the
avalanche.

Figure S1: Persistent homology method. (a) Space-time map of the plastic
activity monitored by the $D^{2}_{\rm min}$ field for a typical avalanche in a
glass with $N=160\times 160$. (b) Birth-death phase diagram of individual
local maxima of our $D^{2}_{\rm min}$ grid. The persistent $p$ of a given
event is given as the distance to the line $y(\rm death)=x(\rm birth)$. (c)
Probability distribution of the persistent averaged over our ensemble of
avalanches. The red area indicates persistent events with $p>0.1$. (d)
Superposition of the actual dissipation signal and birth time of individual
events with $p>0.1$. (e) Reconstruction of the spatio-temporal evolution of an
avalanche as a sequence of individual events compared to the actual cumulative
$D^{2}_{min}$ plastic field (gray colormap). Colorbar indicates the normalized
birth time $t/T$ of each event with $T$ being the avalanche duration.

## 3\. Effect of AQS strain step on the avalanche statistics and dynamics

The results presented in the main text are exclusively for AQS dynamics with a
strain step $\Delta\gamma=10^{-4}$, which is lower than the average strain
distance to the next instability $\langle\Delta\gamma_{\rm min}\rangle=8\times
10^{-4}$. In Fig.S2, we show how a large strain step
($\Delta\gamma>\langle\Delta\gamma_{\rm min}\rangle$) can affect the energy
drop statistics and avalanche dynamics. In Fig. S2(a), we show the energy drop
statistics $P(S)$ for various $\Delta\gamma$. For small $\Delta\gamma$, we
found a $S$ range which is consistent with $P(S)\sim S^{-\tau}$, where
$\tau=1.3$ such as found in previous studies [12, 11]. In contrast, as
$\Delta\gamma$ increases we observe a depletion of small avalanches in favor
of large ones. Here, the effect of a large strain steps is similar as applying
a finite strain rate where distinct avalanches, in the true AQS limit, are
being glued together. Looking at the avalanche-duration relation $\langle
S\rangle\sim T$ shown in Fig. S2(b), we observe that avalanches of the same
average size have a shorter duration. This trend is linked to marginal sites
that are being pushed further above their threshold, and therefore being
activated on a shorter time scale. In Fig. S2(c), we present the normalized
average activation profile $\langle A\rangle=\langle-dU/dt\rangle$ for
avalanches of size $40<S<100$. We find a significant depletion of activity for
$t/T>10\%$ for $\Delta\gamma=5\times 10^{-3}$. In Fig. S2(e), (f), and (g), we
show the spatio-temporal evolution of a typical avalanche (with $S\simeq 100$)
for different strain steps. For $\Delta\gamma=10^{-4}$, we always find that
the avalanche dynamics is triggered by a single event. In contrast, for
$\Delta\gamma=5\times 10^{-3}$ (Fig. S2(g)), we find many spatially
uncorrelated plastic events at the beginning of the avalanche. Nonetheless,
investigating the incremental plastic spatio-temporal propagation and plotting
the plastic length $l_{c}=\xi_{\text{excite}}$ as a function of the delay time
$\Delta t$, we do not observe a significant influence of the imposed strain
step. Finally, comparing the plastic propagation in the smaller system
($L=160$) with the larger one ($L=320$) we do not observe significant finite
size effects, see Fig. S2(d). Similar results have been found in our EPM model
by driving the system with a finite strain step instead of the true AQS limit
for which the strain increment is always equal to the next lowest threshold in
the system.

Figure S2: Effect of AQS strain step on the avalanche statistics and dynamics
in MD. (a) Energy drop statistics $P(S)$ for various AQS strain steps
$\Delta\gamma$ in the small system $N=160\times 160$. The dashed line
indicates power law $P(S)\sim S^{-\tau}$, with $\tau=1.3$. (c) Relation
between avlanche size and avalanche duration. The dashed line indicates the
relation $\langle S\rangle\sim T^{-\delta}$, with $\delta=0.65$. (d) Plastic
length $l_{c}=\xi_{\text{excite}}$ as a function of the delay time $\Delta t$
extracted from the incremental correlation function (see main text for
definition). Different colors correspond to the same strain step values as in
(a) and gray points correspond to the data set of a large system with $L=320$.
(e), (f), and (g) show typical spatio-temporal evolution of avalanche with
$S\simeq 100$ for different strain step $\Delta\gamma$.

## 4\. Static and dynamical exponents

In this section, we extract estimates for the static and dynamical exponents
associated with the distribution of avalanche size and duration. We find that
our MD avalanche size and duration distributions for different system sizes
can be well collapsed using exponents from Ref. [13] , see Fig. S3(a) and Fig.
S3(b). Namely the avalanche statistic follows $P(S)\sim S^{-\tau}$ with
$\tau\simeq 1.3$ and cutoff $S_{c}\sim L^{d_{f}}$, with static fractal
dimension $d_{f}\simeq 0.95$ and the duration statistic follows $P(T)\sim
T^{-\alpha}$ with $\alpha\simeq 1.18$ and cutoff $T_{c}\sim L^{z}$, with
dynamical fractal dimension $z\simeq 1.55$. Our EPM shows similar static
exponents $\tau\simeq 1.3$ and $d_{f}\simeq 1$, but different dynamical
exponent $\alpha\simeq 1.44$ and $z\simeq 0.6$, see Fig. S3(c) and Fig. S3(d).
The latter is consistent with previous estimations using uniform rate and
instantaneous stress propagation, see Ref. [11].

Figure S3: Static and dynamical exponents for MD and EPM. (a) System size
collapse of the MD avalanche size distributions. (b) Same as (a) but for the
MD avalanche duration distributions. (c) and (d) show the same results as in
(a) and (b) but for our EPM.

## 5\. Typical STZ duration

In contrast to our mesoscale model, there is no analogue to a sweep time in
particle-based simulations. Here, we propose to extract a typical duration
$t_{\rm STZ}$ associated with an individual STZ. To do so, we select
avalanches that are composed of one single peak of activity, as shown in Fig.
S4(a). We compute the duration of an event as the time for which the activity
signal $A(t)$ drops by an order of magnitude from its max $A_{\rm max}$. In
Fig. S4(b), we show the distribution of duration $P(t_{\rm STZ})$. We estimate
a typical STZ rearranges over $6$ units of Brownian time.

Figure S4: Activity profile of STZ singlets. (a) Normalized activity profile
of several avalanches composed of a single STZ. (b) Distribution of the
duration of singlets.

## 6\. Time translational invariance and size effects

For both MD and our EPM, we have checked whether or not time translational
invariance can be applied to our incremental correlation function, i.e
$C(t_{0}=0)=C(t_{0}>0)$ where $t_{0}$ is the reference time of the incremental
plastic field that we correlate with subsequent deformation at $t_{0}+\Delta
t$. For both MD (Fig. S5(a)) and our EPM (Fig. S5(b)), we find that time
translational invariance hold relatively well in the early exponential decay.
We find a slightly larger likelihood to find more plasticity further in space
for $t_{0}>0$. This trend is explained by the progressive accumulation of
stress redistribution across the system, which makes it more likely for
marginal sites to yield away from the original source.

In addition, we show for MD (Fig. S5(c)) and our EPM (Fig. S5(d)) the
influence of the system size for a fixed delay time $\Delta t$. We clearly
find for both MD and EPM that the exponential decay does not strongly depend
on the system size. In contrast, the negative crossing does increase with the
box size $L$, as discussed in more details in the main text.

Figure S5: Time translational invariance and size effects in MD and EPM. (a)
Comparison of the MD incremental correlation $C=\langle\overline{\Delta
f(r_{0},t_{0})\Delta f(r_{0}+\vec{r},t_{0}+\Delta t)}\rangle$ for $t_{0}=0$
(green) and $t_{0}\geq 0$ (black). (b) Same data as in (a) for our EPM but
with $t_{0}=0$ (green) and $t_{0}/T=1/5$ (black). (c) MD correlation function
at $\Delta t=6$ for different system sizes with box length $L$. (d) Same data
as in (c) but for our EPM at $\Delta t=3$

## 7\. Finite size study of $\xi_{\rm excite}$ and $\xi_{\rm marginal}$

In the main text, we only provide the change of $\xi_{\rm excite}$ and
$\xi_{\rm marginal}$ as a function of the delay time $\Delta t$ for a fixed
system size ($L=320$ for MD and $L=256$ for EPM). Here, we provide additional
data for different system sizes. We find that $\xi_{\rm excite}$ does not vary
much with $L$. In contrast, $\xi_{\rm marginal}$ grows as the system size
increases.

Figure S6: Finite size study of $\xi_{\rm excite}$ and $\xi_{\rm marginal}$.
(a) MD data for $\xi_{\rm excite}$ plotted as a function of the delay time
$\Delta t$ for different system size. The dashed line indicates the
superdiffusive scaling $t^{3/4}$. (b)$\xi_{\rm marginal}$ as a function of
$\Delta t$. The dashed line indicates the diffusive scaling $t^{1/2}$. (c) and
(d) are the same as (a) and (b) but for our EPM model. The dashed lines in (c)
and (d) indicate $t$ and $t^{1/2}$, respectively.


$\overline{D}\ni y\mapsto F(y)\in X$ in $\Gamma$ with

$\max_{y\in\overline{D}}E(F(y))<\delta.$

For $\delta<\delta_{1}(M)$ sufficiently small, it follows from the lemma that
$\operatorname{dist}_{b}(h(A),\Lambda)<\pi$ for every couple $(u,A)=F(y)$ in
the family. In particular, since the assignment $(u,A)\mapsto h(A)$ gives a
continuous map $X\to\mathcal{H}^{1}(M)$, and since $h(A)=A=0$ for
$y\in\partial\overline{D}$, it follows that $0$ is the nearest point in the
lattice $\Lambda$ to $h(A)$ for every $y\in\overline{D}$, and the estimate
therefore becomes

$\|h(A)\|\leq C\delta^{1/2}.$

In particular, combining this with (7.7), we see now that

(7.16) $\|A\|_{W^{1,2}}\leq C\delta^{1/2}$

for every couple $(u,A)=F(y)$ in the family.

Now, for $(u,A)=F(y)$, our structural assumption (G) on $W(u)$ gives

$\|u\|_{L^{p}}^{p}\leq C+E(u,A)\leq C+\delta,$

which together with the smallness

$\|A\|_{L^{2^{*}}}\leq C\|A\|_{W^{1,2}}\leq C\delta^{1/2}$

of $A$ in $L^{2^{*}}$ (recalling that $p>n$) gives

$\int_{M}|uA|^{2}\leq C\delta.$

Combining this with the fact that $\int|du-iuA|^{2}\leq E(u,A)<\delta$ by
assumption, we then deduce that

$\int_{M}|du|^{2}\leq C\delta$

as well.

Finally, by (7.2) and the Poincaré inequality, we have

$\displaystyle 1-\Big{|}\frac{1}{\operatorname{vol}(M)}\int_{M}u\Big{|}$
$\displaystyle\leq
C\int_{M}|1-|u||+C\int_{M}\Big{|}u-\frac{1}{\operatorname{vol}(M)}\int_{M}u\Big{|}$
$\displaystyle\leq
C\Big{(}\int_{M}W(u)\Big{)}^{1/2}+C\Big{(}\int_{M}|du|^{2}\Big{)}^{1/2}$
$\displaystyle\leq C\delta^{1/2}.$

As a consequence, we find that $\int_{M}u_{y}$ is nonzero for all
$(u_{y},A_{y})=F(y)$ in the family. But then the averaging map

(7.17) $\displaystyle\overline{D}\to\mathbb{C},\qquad
y\mapsto\frac{\int_{M}u_{y}}{|\int_{M}u_{y}|}.$

gives a retraction $\overline{D}\to\partial\overline{D}$, whose nonexistence
is well known. This gives the desired contradiction. ∎

Having shown positivity $\omega_{\epsilon}(M)>0$ of the min-max energies, we
can now deduce the lower bound in (7.1) from the following simple fact.

######  Proposition 7.10.

There exists $c(M)>0$ and $\epsilon_{0}(M)>0$ such that the following holds,
for $\epsilon<\epsilon_{0}$. If $(u,\nabla)$ is critical for the functional
$E_{\epsilon}$, then either $E_{\epsilon}(u,\nabla)\geq c$ or
$E_{\epsilon}(u,\nabla)=0$.

######  Remark 7.11.

For future reference, we make the obvious observation that the trivial case
$E_{\epsilon}(u,\nabla)=0$ can only occur when the bundle $L$ is trivial.

###### Proof.

As discussed in the appendix, it is straightforward to see that critical
points are smooth up to change of gauge. We claim that, whenever
$E_{\epsilon}(u,\nabla)>0$, $u$ has to vanish at some point $x_{0}\in M$. Once
we have this, Corollary 4.4 implies that
$r^{2-n}E_{\epsilon}(u,\nabla,B_{r}(x_{0}))$ has a lower bound independent of
$\epsilon$ for any $\epsilon<r<\operatorname{inj}(M)$, and the statement
follows.

Indeed, if the claim fails, then $u$ is nowhere vanishing, so $L$ must be
trivial and we can use the section $\frac{u}{|u|}$ to identify $L$
isometrically with the trivial line bundle $M\times\mathbb{C}$, equipped with
the canonical Hermitian metric. Under this identification, $u:M\to\mathbb{C}$
takes values into positive real numbers. Writing $\nabla=d-iA$ and observing
that $\langle\nabla u,iu\rangle=-|u|^{2}A$, (2.5) becomes

$\displaystyle\epsilon^{2}d^{*}dA+|u|^{2}A=0.$

Integrating against $A$ we get
$\int_{M}(\epsilon^{2}|dA|^{2}+u^{2}|A|^{2})=0$, so $A=0$ and $\nabla$ is the
trivial connection. At a minimum point $y_{0}$ for $u$, (3.4) gives

$\displaystyle
0\leq\frac{1}{2}\Delta|u|^{2}=|du|^{2}-\frac{1}{2\epsilon^{2}}(1-|u|^{2})|u|^{2}=-\frac{1}{2\epsilon^{2}}(1-u^{2})u^{2},$

which forces $u(y_{0})\geq 1$ and thus $u=1$ everywhere, giving the
contradiction $E_{\epsilon}(u,\nabla)=0$. ∎

Finally, we turn to the uniform upper bound. In the next statement, $L\to M$
is a Hermitian line bundle with a fixed Hermitian reference connection
$\nabla_{0}$. We identify any other Hermitian connection $\nabla$ with the
real one-form $A$ such that $\nabla s=\nabla_{0}s-iA\otimes s$ for all
sections $s$.

######  Proposition 7.12.

Given a smooth section $u:M\to L$, we can find a smooth couple
$(u^{\prime},A^{\prime})$ such that

(7.18) $\displaystyle E_{\epsilon}(u^{\prime},A^{\prime})\leq
C\left(\epsilon^{-2}\operatorname{vol}\Big{(}\Big{\\{}|u|\leq\frac{1}{2}\Big{\\}}\Big{)}+(1+\epsilon^{2}\|\nabla_{0}u\|_{L^{\infty}}^{2})\int_{\\{|u|\leq\frac{1}{2}\\}}|\nabla_{0}u|^{2}+\epsilon^{2}\int_{M}|\omega_{0}|^{2}\right)$

for a universal constant $C$.

###### Proof.

On $\\{u\neq 0\\}$ we let

$\displaystyle w:=\frac{u}{|u|},\qquad A\otimes iw:=\nabla_{0}w.$

The compatibility of $\nabla_{0}$ with the Hermitian metric on $L$ forces
$\langle\nabla_{0}w,w\rangle=0$, so that $A$ is a real one-form. Equivalently,
viewing $w$ as a map from $M$ to the circle bundle $U(L)$ of $L$, which is a
principal $S^{1}$-bundle with induced connection form
$\varpi\in\Omega^{1}(U(L),\mathbb{R})$, we have $A=\varpi\circ dw$.

We fix a smooth function $\rho:[0,\infty]\to[0,1]$ with

$\displaystyle\rho(t)=0\text{ for }t\leq\frac{1}{4},\qquad\rho(t)=1\text{ for
}t\geq\frac{1}{2}$

and we set

$\displaystyle(u^{\prime},A^{\prime}):=\rho(|u|)(w,A)$

(the right-hand side is meant to be zero on $\\{u=0\\}$). Writing
$F_{\nabla_{0}}=-i\omega_{0}$, observe that

$\displaystyle|F_{A}|=|dA+\omega_{0}|=0\quad\text{on }\\{u\neq 0\\},$

so that $e_{\epsilon}(u^{\prime},A^{\prime})=0$ on $\\{|u|>\frac{1}{2}\\}$.
From the estimates $|A|=|\nabla_{0}w|\leq 2|u|^{-1}|\nabla_{0}u|$ and
$|d|u||\leq|\nabla_{0}u|$, it follows that also

$\displaystyle|\nabla_{0}u^{\prime}|$ $\displaystyle\leq C|\nabla_{0}u|,$
$\displaystyle|A^{\prime}|$ $\displaystyle\leq C|\nabla_{0}u|,$
$\displaystyle|dA^{\prime}|$ $\displaystyle\leq|\rho^{\prime}(|u|)d|u|\wedge
A|+|\omega_{0}|\leq C|\nabla_{0}u||d|u||+|\omega_{0}|,$

and the statement follows immediately. ∎

###### Proof of (7.1).

The method used in [33, Section 3] gives a continuous map $H:\overline{D}\to
W^{1,2}\cap C^{0}(M,\mathbb{C})$ such that $H(y)\equiv y$ for $y\in\partial D$
and

(7.19) $\displaystyle\|dH(y)\|_{L^{\infty}}\leq
C\epsilon^{-1},\quad\int_{\\{|H(y)|\leq\frac{3}{4}\\}}|dH(y)|^{2}\leq
C,\quad\operatorname{vol}\Big{(}\Big{\\{}|u|\leq\frac{1}{2}\Big{\\}}\Big{)}\leq
C\epsilon^{2}$

for all $y\in\overline{D}$ (the full Dirichlet energy having a worse bound
$\int_{M}|dH(y)|^{2}\leq C\log\epsilon^{-1}$, which is the natural one in the
setting of Ginzburg–Landau). By approximation, we can assume that $H$ takes
values in $C^{\infty}(M,\mathbb{C})$, continuously in $y$, and still satisfies
the same uniform bounds (7.19) (possibly increasing $C$ and replacing
$\frac{3}{4}$ with $\frac{1}{2}$).

To each section $H(y)$ of the trivial line bundle, Proposition 7.12 assigns in
a continuous way an element $F(y)\in X$. From the way $F(y)$ is constructed,
it is clear that $F\in\Gamma$. Finally, applying Proposition 7.12 with (7.19)
gives

$\displaystyle\omega_{\epsilon}(M)\leq\max_{y\in\overline{D}}E_{\epsilon}(F(y))\leq
C.\qed$

###  7.2. Minimizers for nontrivial line bundles

Suppose now that $L$ is a nontrivial line bundle, equipped with a Hermitian
metric. Fix a smooth Hermitian connection $\nabla_{0}$ and identify any other
Hermitian connection $\nabla$ with the real one-form $A$ such that

$\displaystyle\nabla=\nabla_{0}-iA.$

We can define $\widehat{X}$ and $X$ as in the previous subsection. With this
notation, observe that the curvature of $\nabla$ is given by

$\displaystyle F_{\nabla}=F_{\nabla_{0}}-idA.$

Hence, writing $F_{\nabla_{0}}=-i\omega_{0}$, we have

$\displaystyle E_{\epsilon}(u,\nabla)=\int_{M}|\nabla_{0}u-A\otimes
iu|^{2}+\epsilon^{-2}\int_{M}W(u)+\epsilon^{2}\int_{M}|\omega_{0}+dA|^{2}.$

######  Definition 7.13.

For a fixed $n<p<\infty$, we define $\widehat{X}$ to be the Banach space of
couples $(u,A)$, where $u:M\to L$ is an $L^{p}$ section and
$A\in\Omega^{1}(M,\mathbb{R})$, both of class $W^{1,2}$, with the norm

$\displaystyle\|(u,A)\|:=\|u\|_{L^{p}}+\|\nabla_{0}u\|_{L^{2}}+\|A\|_{L^{2}}+\|DA\|_{L^{2}}.$

We let $X:=\\{(u,A)\in\widehat{X}:d^{*}A=0\\}$.

The analogous statements to Remark 7.4 and Propositions 7.5 and 7.6 hold, with
identical proofs (replacing $du$ and $uA$ with $\nabla_{0}u$ and $A\otimes u$,
respectively).

Arguing as in the proof of Proposition 7.6, it is easy to see that a
minimizing sequence of couples for $E_{\epsilon}$ converges—in the appropriate
Coulomb gauge—to a global minimizer $(u_{\epsilon},A_{\epsilon})$. We now show
that the energy of these minimizers enjoys uniform upper and lower bounds as
$\epsilon\to 0$.

###### Proof of (7.1).

The lower bound in (7.1) follows directly from Proposition 7.10. In order to
obtain the upper bound, pick a smooth section $s:M\to L$ transverse to the
zero section (see, e.g., [24, Theorem IV.2.1]) and let $N:=\\{s=0\\}$, which
is a smooth embedded $(n-2)$-submanifold of $M$. Proposition 7.12 applied to
$\epsilon^{-1}s$ gives a couple
$(u_{\epsilon}^{\prime},A_{\epsilon}^{\prime})$ with

$\displaystyle E_{\epsilon}(u_{\epsilon}^{\prime},A_{\epsilon}^{\prime})\leq
C\epsilon^{-2}\operatorname{vol}\Big{(}\Big{\\{}|\epsilon^{-1}s|\leq\frac{1}{2}\Big{\\}}\Big{)}+C\epsilon^{2}\int_{M}|\omega_{0}|^{2}.$

By transversality of $s$, the set $\\{|s|\leq\frac{\epsilon}{2}\\}$ is
contained in a $C(s)\epsilon$-neighborhood of $N$, whose volume is bounded by
$C(s)\epsilon^{2}$. We infer that

$\displaystyle E_{\epsilon}(u_{\epsilon},A_{\epsilon})\leq
E_{\epsilon}(u_{\epsilon}^{\prime},A_{\epsilon}^{\prime})\leq\epsilon^{-2}\operatorname{vol}\Big{(}\Big{\\{}|s|\leq\frac{\epsilon}{2}\Big{\\}}\Big{)}\leq
C.\qed$

######  Remark 7.14.

$N$ can be oriented in such a way that $[N]\in H_{n-2}(M,\mathbb{R})$ is
Poincaré dual to the Euler class $e(L)\in H^{2}(M,\mathbb{R})$ of the line
bundle, which equals the first Chern class $c_{1}(L)$. The fact that the
energy of our competitors concentrates along $N$ suggests that, given a
sequence of global minimizers $(u_{\epsilon},A_{\epsilon})$, up to
subsequences the corresponding energy concentration varifold is induced by an
integral mass-minimizing current whose homology class is Poincaré dual to
$c_{1}(L)$. Theorem 6.10 provides the natural candidate $\Gamma$, which also
satisfies $|\Gamma|\leq\mu$.

## Appendix. Interior regularity in the Coulomb gauge

In this short appendix, we describe the essential ingredients needed to
establish local regularity in the Coulomb gauge for finite-energy critical
points $(u,A)$ of the ($\epsilon=1$) abelian Higgs energy $E(u,A)$, collecting
some estimates which will be of use elsewhere in the paper.

Consider the manifold with boundary $(\overline{\Omega}^{n},g)$ given by a
smooth, contractible domain $\Omega^{n}\subset\mathbb{R}^{n}$ equipped with a
$C^{2}$ metric $g$, and let $L\cong\mathbb{C}\times\Omega$ be the trivial line
bundle over $\Omega$, with the standard Hermitian structure. With respect to
the metric $g$, we then define the Yang–Mills–Higgs energies

$E(u,A):=\int_{\Omega}e(u,A)=\int_{\Omega}|du-iA|^{2}+|dA|^{2}+W(u)$

as in the preceding section. By Proposition 7.5 in the preceding section, it
is easy to see that a pair $(u,A)$ in $W^{1,2}$ with

(A.1) $|u|\leq 1$

is a critical point for $E$ (with respect to smooth perturbations supported in
$\Omega$) if and only if the equations

(A.2) $\displaystyle d^{*}dA$ $\displaystyle=\langle du-iAu,iu\rangle,$ (A.3)
$\displaystyle\Delta u$ $\displaystyle=2\langle
idu,A\rangle+|A|^{2}u-\frac{1}{2}(1-|u|^{2})u-i(d^{*}A)u$

are satisfied distributionally in $\Omega$, where all geometric quantities and
operators are defined with respect to the metric $g$.

Now, given a pair $(u,A)$ in $W^{1,2}$ satisfying (A.2)–(A.3) and

(A.4) $E(u,A)\leq\Lambda<\infty,$

we can select a _local Coulomb gauge_ adapted to $A$ as follows. Denote by
$\theta\in W^{2,2}(\Omega,\mathbb{R})$ the unique solution of the Neumann
problem

(A.5) $\Delta\theta=d^{*}A\text{ in
}\Omega;\quad\frac{\partial\theta}{\partial\nu}=-A(\nu)\text{ on
}\partial\Omega$

with zero mean $\int_{\Omega}\theta=0$. Then the gauge-transformed pair

$(\widetilde{u},\widetilde{A}):=(e^{i\theta}u,A+d\theta)$

continues to satisfy (A.2)–(A.3), with

$E(\widetilde{u},\widetilde{A})=E(u,A)\leq\Lambda,$

but now with the additional constraints

(A.6) $d^{*}\widetilde{A}=0\text{ on }\Omega;\quad\widetilde{A}(\nu)=0\text{
on }\partial\Omega.$

For the remainder of the section, we will assume that the pair $(u,A)$ is
already in the Coulomb gauge on $\Omega$, so that $A$ satisfies (A.6). Note
that the last term in the equation (A.3) then vanishes, so that we have

(A.7) $\Delta u=2\langle idu,A\rangle+|A|^{2}u-\frac{1}{2}(1-|u|^{2})u.$

We now establish the local regularity for critical points $(u,A)$ in the
Coulomb gauge, giving in particular estimates for $(u,A)$ in $W^{2,q}$ norms.

######  Proposition A.1.

Let $(u,A)$ solve (A.2)–(A.3) in the Coulomb gauge (A.6) on $(\Omega,g)$, with
$|u|\leq 1$. If

(A.8) $E(u,A;\Omega)\leq\Lambda$

and

(A.9) $\|g\|_{C^{2}}+\|g^{-1}\|_{C^{2}}\leq\Lambda,$

then for every compactly supported subdomain
$\Omega^{\prime}\subset\subset\Omega$ and $q\in(1,\infty)$ there exists
$C_{q}(\Lambda,\Omega,\Omega^{\prime})<\infty$ such that

(A.10) $\|u\|_{W^{2,q}(\Omega^{\prime})}+\|A\|_{W^{2,q}(\Omega^{\prime})}\leq
C_{q}.$

###### Proof.

To begin, note that (A.2), (A.6), and standard Bochner–Weitzenböck identities
give the (weak) subequation

$\displaystyle\Delta\frac{1}{2}|A|^{2}$
$\displaystyle\geq-\langle\Delta_{H}A,A\rangle+|DA|^{2}+\operatorname{Ric}(A,A)$
$\displaystyle\geq-C(\Lambda)|A|^{2}-C|du-iAu|$

for $|A|^{2}$. On the other hand, as in Section 3, we also obtain from (A.3)
the relation

$\Delta\frac{1}{2}|u|^{2}=|du-iAu|^{2}-\frac{1}{2}(1-|u|^{2})|u|^{2},$

and combining the two (recalling also that $|u|\leq 1$), we find an estimate
of the form

(A.11) $\frac{1}{2}\Delta(|A|^{2}+|u|^{2})\geq-C(|A|^{2}+|u|^{2})-C.$

Applying Moser iteration to (A.11), we see in particular that

(A.12) $\|A\|_{L^{\infty}(\Omega_{1})}^{2}\leq
C(\Lambda,\Omega_{1},\Omega)(1+\|A\|_{W^{1,2}(\Omega)}^{2})$

for any $\Omega_{1}\subset\subset\Omega$. Moreover, by standard estimates for
one-forms $A$ satisfying (A.6) (see, e.g., [20]), we have the global $L^{2}$
bound

$\|A\|_{W^{1,2}(\Omega)}\leq\|dA\|_{L^{2}(\Omega)}\leq\Lambda^{1/2},$

which together with the preceding $L^{\infty}$ estimate gives

(A.13) $\|A\|_{L^{\infty}(\Omega_{1})}\leq C(\Lambda,\Omega_{1},\Omega)$

for any subdomain $\Omega_{1}\subset\subset\Omega$.

Now, fixing some intermediate domain
$\Omega^{\prime}\subset\subset\Omega_{1}\subset\subset\Omega$ between
$\Omega^{\prime}$ and $\Omega$, the equation (A.7) together with the
$L^{\infty}(\Omega_{1})$ estimate for $A$ give pointwise bounds of the form

(A.14) $|\Delta u|\leq C(\Omega_{1},\Omega,\Lambda)(|du|+1)\text{ in
}\Omega_{1}.$

And since

$|du|\leq|du-iAu|+|A|\leq e(u,A)+C$

in $\Omega_{1}$, we obtain from the energy bound $E(u,A)\leq\Lambda$ and
(A.14) the simple estimate

$\|\Delta u\|_{L^{2}(\Omega_{1})}\leq C(\Omega_{1},\Omega,\Lambda),$

and consequently

$\|u\|_{W^{2,2}(\Omega_{2})}\leq C$

for any $\Omega^{\prime}\subset\subset\Omega_{2}\subset\subset\Omega_{1}$.
Returning to the pointwise bound (A.14), we can now employ a simple iteration
argument—combining $L^{q}$ regularity theory with the Sobolev embedding
$W^{2,r}\hookrightarrow W^{1,\frac{rn}{n-r}}$—over successive domains between
$\Omega^{\prime}$ and $\Omega$, to arrive at the desired $W^{2,q}$ estimates
for $u$.

Returning finally to the equation (A.2) for $A$, in the Coulomb gauge, we see
that

$\Delta_{H}A=\langle du-iAu,iu\rangle,$

and it therefore follows from the preceding estimates that

$\|A\|_{L^{\infty}(\Omega^{\prime\prime})}+\|\Delta
A\|_{L^{\infty}(\Omega^{\prime\prime})}\leq
C(\Omega^{\prime\prime},\Omega,\Lambda)$

for some intermediate domain
$\Omega^{\prime}\subset\subset\Omega^{\prime\prime}\subset\subset\Omega$. In
particular, this gives us upper bounds for $\|\Delta
A\|_{L^{q}(\Omega^{\prime\prime})}$ for every $q\in(1,\infty)$, and $L^{q}$
regularity theory therefore gives us the desired estimates for $A$ in
$W^{2,q}(\Omega^{\prime})$. ∎

Finally, we remark that higher regularity of $u$ and $A$ in the Coulomb gauge
follows in a standard way—e.g., via Schauder theory—from the $W^{2,q}$
estimates obtained in the preceding proposition.

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