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# Hedgehog topological defects in 3D amorphous solids

Arabinda Bera  Alessio Zaccone  Department of Physics “A. Pontremoli”,
University of Milan, via Celoria 16, 20133 Milan, Italy  Matteo Baggioli
Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China and Shanghai Research Center for Quantum
Sciences, Shanghai 201315.

###### Abstract

The underlying structural disorder renders the concept of topological defects
in amorphous solids difficult to apply and hinders a first-principle
identification of the microscopic carriers of plasticity and of the regions
more prone to structural rearrangements (“soft spots”). Recently, it has been
proposed that well-defined topological defects can still be identified in
glasses, and correlated to local and global plasticity, by looking at the
eigenvector field or the particle displacement field. Nevertheless, all the
existing proposals and analyses are only valid in two spatial dimensions. In
this work, we propose the idea of using hedgehog topological defects to
characterize the plasticity of 3D glasses and to geometrically predict the
location of their soft spots. We corroborate our proposal by simulating a
Kremer-Grest 3D polymer glass, and by using both the normal mode eigenvector
field and the displacement field around large plastic events. Our results
confirm that a topological characterization of plasticity in glasses is
feasible, and provide a concrete realization of this program in 3D amorphous
systems.

Introduction – Topological defects, such as dislocations and disclinations,
play a pivotal role for the description of mechanical failure in crystalline
solids since they provide the microscopic origin of their plasticity Taylor
(1934); Polanyi (1934); Orowan (1934). Topological defects also provide the
building blocks for the theory of crystal melting in two dimensions Kosterlitz
and Thouless (1973); Halperin and Nelson (1978); Young (1979). Despite several
attempts Spaepen (1978); Cotterill (1979); Blackett (1979); Morris (1979);
Popescu (1984); Egami and Vitek (1984); Shi (1993); Chaudhari  _et al._
(1979); Egami  _et al._ (1980); Acharya and Widom (2017); Nelson (1983);
Steinhardt and Chaudhari (1981), topological defects associated to
translational symmetry (dislocations and disclinations) cannot be formally
defined within the structure of amorphous solids such as glasses, because of
the absence of translational long-range order. This difficulty remains the
main cause for the lack of a microscopic theory of plasticity and mechanical
deformations in glasses Hufnagel  _et al._ (2016); Rodney  _et al._ (2011);
Zaccone (2023).

More in general, the definition and identification of “defects” constitute the
main obstacle for establishing a robust causal relation between structure and
dynamics in amorphous solids (see e.g. Wang (2019); Cheng and Ma (2011)). So
far, most of the approaches to solve this problem are still based on ad-hoc
coarse-grained descriptions Nicolas  _et al._ (2018), vibrational soft modes
Rottler  _et al._ (2014); Manning and Liu (2011); Ding  _et al._ (2014); Chen
_et al._ (2011); Kapteijns  _et al._ (2020); Tanguy  _et al._ (2010);
Smessaert and Rottler (2014a), shear transformation zones revisited under the
lens of anharmonic quasi-localized modes Zylberg  _et al._ (2017); Richard
_et al._ (2021); Gartner and Lerner (2016); Ruan  _et al._ (2022); Xu  _et
al._ (2018, 2021), phenomenological structural indicators Richard  _et al._
(2020); Widmer-Cooper  _et al._ (2008) or machine-learning algorithms Fan and
Ma (2021); Ciarella  _et al._ (2023).

Two recent works Baggioli  _et al._ (2021); Wu  _et al._ (2023) (see also
Baggioli (2023)) have shown that not only topological concepts can be applied
(albeit with proper generalizations) to amorphous solids, as done for vortices
in superfluids or dislocations in crystals, but that they can be also
extremely useful to predict their mechanical properties both on a local and a
global scale.

A first approach Baggioli  _et al._ (2022) is based on the idea of continuous
Burger vector Kleman and Friedel (2008) applied to the displacement field, and
has been shown to be able to accurately locate plastic events under
deformation and the yielding point Baggioli  _et al._ (2021). It is likely
that this concept is related to the geometric charges previously discussed as
the carriers of plastic screening in glasses Lemaître  _et al._ (2021). The
second approach Wu  _et al._ (2023) relies on the identification of vortex-
like defects in the eigenvector field that strongly correlate with the plastic
spots and display tight connections with the widely used concept of shear
transformation zones Desmarchelier  _et al._ (2024). This second type of
vortex-like topological defects have been recently experimentally identified
in a 2D colloidal glass Vaibhav  _et al._ (2024), confirming the simulation
results.

Importantly, all the proposals and analyses so far apply only to two-
dimensional (2D) amorphous solids and lack direct application to more
realistic 3D scenarios. In Cao  _et al._ (2018), disclination-like defects,
defined using polytetrahedral order Nelson and Spaepen (1989), were considered
in a 3D experimental granular matter system and were shown to correlate with
plasticity. In Bera  _et al._ (2024), a 2D slicing method has been used to
study vortex-like topological defects in a 3D simulation polymer glass
revealing a strong clustering tendency of negative charged topological defects
right before large plastic events.

As of today, a full-fledged definition of topological defects in 3D glasses
and a direct proof of their connection to plasticity in three dimensions are
still lacking. In this work, we present a solution to these problems.

Hedgehog topological defects – Given the large variety of topological defects
that can be defined within a three-dimensional vector field (point defects,
line defects and surface defects) Kleinert (1989); Nelson (2002), we suggest
that hedgehog topological defects (HTD) are the simplest choice to start with.
We adapt the definition of HTDs from nematic liquid crystals Kleman and
Lavrentovich (2003, 2006); Hölbl  _et al._ (2023) and Heisenberg magnets
Fumeron and Berche (2023); Berg and Lüscher (1989); Holm and Janke (1994);
Berg and Lüscher (1981) to the eigenvector and displacement fields of a 3D
simulation glass.

For a generic 3D continuous vector field $\vec{s}$, we define the hedgehog
topological charge as Kleman and Lavrentovich (2003, 2006); Hölbl  _et al._
(2023)

$Q=\frac{1}{4\pi}\oiint_{\Sigma}\vec{s}\cdot\left[\frac{\partial\vec{s}}{\partial
x_{1}}\times\frac{\partial\vec{s}}{\partial x_{2}}\right]dx_{1}dx_{2},$ (1)

where $\Sigma$ is an arbitrary closed 2D surface parametrized by coordinates
$(x_{1},x_{2})$. $Q$ represents a topological invariant in the sense that it
does not change under continuous and smooth deformations of the manifold.

Figure 1: (a) Discrete vector at a simple cubic lattice corners illustrating a
radially outward field for a monopole with $Q=+1$ (red). A vertex $x^{*}$ is
shown where $\vec{s}_{1}$, $\vec{s}_{2}$, $\vec{s}_{3}$ and $\vec{s}_{4}$
field at four corners of the vertex. (b) Continuous vector field (left) and
discrete vector field (right) for a monopole. (c) Radially inward field
representing an anti-monopole with $Q=-1$ (blue). (d) Continuous (left) and
discrete (right) vector fields for $Q=-1$.

From our simulations, we first obtain a discrete vector field
${\vec{s}}_{l,m,n}$ at each lattice grid point $(l,m,n)$ in a simple cubic
lattice. More details are provided in the Supplementary Material (SM). Let
$\vec{s}_{1}$, $\vec{s}_{2}$, $\vec{s}_{3}$, and $\vec{s}_{4}$ be the unit
vectors of the field associated with one face of the cube, where
$\vec{s}_{1}$, $\vec{s}_{2}$, and $\vec{s}_{3}$ are positioned at the corners
of triangle $i$ on that face, chosen such that the sequence $1-2-3-1$
corresponds to a counterclockwise rotation along the outward normal to the
surface of the triangle. See Fig. 1(a) for a representation of these vectors
on the example of a radially outgoing field (monopole). The topological charge
$Q$ enclosed by the unit cube is then expressed in a discrete fashion as Holm
and Janke (1994); Berg and Lüscher (1981),

$Q=\sum_{x^{*}}q(x^{*}),$ (2)

where the sum is carried over all $6$ vertices of the unit cube and $x^{*}$ is
the vertex associated with the four vectors $\vec{s}_{1}$, $\vec{s}_{2}$,
$\vec{s}_{3}$ and $\vec{s}_{4}$ (as shown in Fig. 1(a)). Here, $q(x^{*})$ is
expressed as

$q(x^{*})=\frac{1}{4\pi}[(\sigma_{s}A)(\vec{s}_{1},\vec{s}_{2},\vec{s}_{3})+(\sigma_{s}A)(\vec{s}_{1},\vec{s}_{3},\vec{s}_{4})],$
(3)

where $(\sigma_{s}A)(\vec{s}_{1},\vec{s}_{2},\vec{s}_{3})$ is the signed area
of the spherical (non-Euclidean) triangles with corners $\vec{s}_{1}$,
$\vec{s}_{2}$ and $\vec{s}_{3}$. The sign of the area can be calculated as
${\sigma_{s}=\rm{sign}}\\{\vec{s}_{1}\cdot[\vec{s}_{2}\times\vec{s}_{3}]\\}$.
Finally, the value of $A$ is calculated as Holm and Janke (1994); Berg and
Lüscher (1981),

$\cos\left(\frac{A}{2}\right)=\frac{1+\vec{s}_{1}\cdot\vec{s}_{2}+\vec{s}_{2}\cdot\vec{s}_{3}+\vec{s}_{3}\cdot\vec{s}_{1}}{\sqrt{2(1+\vec{s}_{1}\cdot\vec{s}_{2})(1+\vec{s}_{2}\cdot\vec{s}_{3})(1+\vec{s}_{3}\cdot\vec{s}_{1})}}.$
(4)

A visual representation of this algorithm for the case of hedgehog defects
with charges $Q=\pm 1$ (monopole and anti-monopole) is presented in Fig. 1.

3D polymer glass model – We simulate a 3D polymer glass system using the
Kremer-Grest model Kremer and Grest (1990) with $N=10{,}000$ monomers of
alternating masses of $m_{1}=1$ and $m_{2}=3$ Kriuchevskyi  _et al._ (2020)
interacting via the Lennard-Jones (LJ) potential ($u_{\text{LJ}}$) and the
finite extensible nonlinear elastic (FENE) potential ($u_{\text{FENE}}$). We
set the dihedral bending rigidity of the bonds to zero, as appropriate for
flexible chains Palyulin  _et al._ (2018). More details can be found in the
SM.

We define the $3N\times 3N$ Hessian matrix, with elements written in terms of
the total potential energy $u(r)=u_{\text{LJ}}(r)+u_{\text{FENE}}(r)$,

$H_{ij}^{\alpha\beta}=\frac{1}{\sqrt{m_{i}m_{j}}}\frac{\partial^{2}u(\vec{r})}{\partial{r}_{i}^{\alpha}\partial{r}_{j}^{\beta}}.$
(5)

Here, $i$ and $j$ are particle indices, while $\alpha$ and $\beta$ denote the
spatial directions ($x$, $y$, $z$). Here $m_{i}$ represents the mass of the
$i$-th particle, and $r_{i}^{\alpha}$ is the $\alpha$-component of the
position vector of the $i$-th particle.

The eigenvalues $\lambda_{k}$ (or equivalently eigenfrequencies
$\omega_{k}=\sqrt{\lambda_{k}}$) and their associated eigenvectors are
obtained by direct diagonalization of the Hessian matrix. In the rest of the
manuscript, we will use the symbols $\omega_{k}$ and $\omega$ interchangeably.

Soft spots from the eigenvector field – We start by applying our algorithm to
the 3D eigenvector field of the simulated polymer glass system. To determine
the HTDs, we derived the smoothed eigenvector field on a simple cubic lattice
(see SM for details). For direct visualization, we present a snapshot of the
smooth eigenvector field that exhibits an hedgehog anti-monopole and a
monopole in the SM, Fig. S2.

By scanning the eigenvector field at different frequencies, we have verified
that the total number of HTDs scales at low frequency (up to $\omega\approx
4$) as $N_{\text{HTD}}\propto\omega^{2}$ (shown in SM, Fig. S3(b)) and
correlates with the corresponding density of states, which exhibits Debye
scaling, $D(\omega)\propto\omega^{2}$ (see SM, Fig. S3(a)) at low frequency.
This result is consistent with previous findings in 2D simulation and
experimental works Wu  _et al._ (2023); Vaibhav  _et al._ (2024).

After confirming the validity of the 3D algorithm presented above, we
investigate the physical properties of the HTDs, and in particular whether
they bear any connection with plastic soft spots. In order to do so, we follow
Smessaert and Rottler (2014b) and we identify the soft spots in the 3D model
glass system using the softness parameter

$\phi_{i}=\frac{1}{N_{m}}\sum_{j=1}^{N_{m}}\frac{|{\bf
e}^{(i)}_{j}|^{2}}{m_{i}\omega_{j}^{2}},$ (6)

where $N_{m}$ is the number of low frequency modes used in the analysis and
${\bf e}^{(i)}_{j}$ is the eigenvector of the $j$-th mode, corresponding to
the $i$-th particle. To define $N_{m}$, we have considered modes up to
$\omega=5$. Finally, the plastic soft spots are identified using the locations
of the $5\%$ of particles having higher values of $\phi_{i}$.

Figure 2: (a) The color map visualizes the 3D average topological charge
density. The black symbols are the locations of soft spots, identified using
the softness parameter, Eq. (6). (b)-(f) Pair correlation functions between
HTDs: monopoles (p) and anti-monopoles (n), and plastic soft spots (s) for
different frequencies. Note that the curves for $\omega>1.1$ are shifted
upward by constant factors for better visualization in this and all other
similar data sets. See SM, Fig. S4, for extended data.

In Fig. 2(a), we show a 3D color map indicating the intensity of the average
topological charge density (see SM for details). Red and blue colored regions
represent therefore areas with higher concentrations of positive and negative
topological charge $Q$, respectively. In the same box, we provide the
locations of the plastic soft spots using black symbols. In order to test the
correlation between HTDs and plasticity, in Fig. 2(b)-(g), we present various
pair correlation functions between HTDs (p), anti-monopole HTDs (n) and
plastic soft spots (s) for different frequencies.

In Fig. 2(b)-(c), the correlation between HTDs and soft spots is shown. Our
results reveal that both positive and negative defects exhibit strong
correlations with soft spots at short and intermediate distances at low
frequency, that nevertheless vanish quickly by increasing $\omega$ (see Fig.
S4 in the SM). This outcome presents marked differences with the 2D case Wu
_et al._ (2023), where soft spots predominantly correlate with negative
defects at short distances. This might be attributed to the different nature
of point defects in 2D and 3D systems Kleman and Lavrentovich (2003).

Moreover, HTDs with same charge also display a peak in their spatial
correlation at approximately the same distance, see Fig. 2(d) and Fig. 2(f),
indicating a widespread tendency to clustering of HTDs. This may be explained
with the fact that HTDs tend to nucleate more in regions of significant
nonaffine motions Baggioli  _et al._ (2022), which are spatially localized
DiDonna and Lubensky (2005). Hence, it is statistically more likely to have
charges, even with same sign, next to each other. Finally, in Fig. 2(e), we
observe a very strong short-distance correlation between monopole and anti-
monopole defects that suggests that defects with opposite charges attract each
other and tend to pair together to minimize the local topological charge. This
correlation is very strong at low frequency and becomes weaker at higher
$\omega$.

Predicting plasticity from the displacement vector – Inspired by the proposal
of looking at topological defects in the dynamical displacement field
$\vec{u}$ Baggioli  _et al._ (2021), rather than in the eigenvector field, we
now apply the same 3D algorithm to find HTDs in $\vec{u}$. We notice that
there is no direct relation between the displacement field and a single
eigenvector $\mathbf{e}_{k}$, despite $\vec{u}$ can be decomposed onto the
complete basis set of eigenvectors. Therefore, a correlation between
topological defects in the eigenvector field and in the displacement field is
not obvious. Nevertheless, around plastic events, it is known that the
dynamical response is dominated by a few localized and low frequency modes
(see e.g. Yang  _et al._ (2020)), providing a possible connection between the
topology of the low-frequency eigenvectors and that of the displacement field.

In our study, we apply quasi-static shear to our system using the athermal
quasi static (AQS) protocol Lemaître and Maloney (2006) with strain step
$\delta\gamma=10^{-3}$. The stress-strain curve is presented in the SM (Fig.
S1(c)) and displays a yielding instability at $\gamma^{*}\approx 0.099$. For
presentation purposes, here we focus on four plastic events, including the
yielding point, and consider the non-affine displacement field at these
specific strain values. In order to test the universality of our findings, we
define the soft spots using two different methods: (I) the softness parameter
defined in Eq. (6) (and used in the previous analysis) and (II) the value of
the particle level non-affine displacement field $\vec{u}_{\text{NA}}$. Within
this second method, we identify the plastic soft spots by looking at the
locations of the $5\%$ of particles having higher values of
$|\vec{u}_{\text{NA}}|$.

Figure 3: (a)-(d) Spatial correlation functions between soft spots (identified
using the softness parameter, Eq. (6)) and HTDs within the non-affine
displacement field across four plastic events at different $\gamma$.
Correlations between soft spots and HTDs with $Q=+1$($-1$) are shown in red
(blue).

In order to test the predictability of the plastic regions from pure static
configurations, we obtain the soft spots using method (I) applied just before
the plastic events. In Fig. 3(a)-(d) we show the spatial correlation between
HTDs within non-affine displacement field (see SM for details) and the soft
spots identified with the softness parameter $\phi_{i}$ (method I) for four
plastic events at $\gamma=0.099,~{}0.120,~{}0.126$ and $0.154$ (see Fig. S1(c)
for associated stress drop). From there, it is evident that both positive and
negative HTDs with displacement field display a clear correlation with the
soft spots that is rather localized around $r\approx 1$. Once again, no marked
distinction between negative and positive defects is observed.

We then move on to perform the same analysis using method (II), based on the
non-affine displacement magnitude. In Fig. 4, we present our results at
$\gamma=0.099$, $0.12$, and $0.154$, for which there are significant stress
drops (see stress-strain curve in SM Fig. S1(c)). In the 3D boxes, we indicate
the location of the $+1$ and $-1$ defects, respectively, with red and blue
color. Additionally, we highlight the soft spots with yellow color. From
there, the strong spatial correlation between these objects is already
evident.

In order to confirm this result, in the bottom panels of Fig.4, we show the
spatial correlation function between monopoles in the displacement field and
soft spots (red data) and between anti-monopoles in the displacement field and
soft spots (blue data). For all the three values of strain considered, we
observe a very strong spatial correlation between the HTDs in the displacement
field and the soft spots, with a pronounced peak appearing at $r\approx 1$.
Interestingly, the distance at which this correlation emerges is compatible
with the low-frequency results presented in Fig. 2, where the HTDs have been
identified in the eigenvector field, and the results in Fig. 3 as well. This
confirms our initial speculation that, around plastic events, the topology of
the low-frequency eigenvectors, and even its spatial characteristics, are
compatible, and in fact very similar, to those observed in the displacement
vector.

Finally, we notice that there is no apparent difference between negative
($Q=-1$) and positive ($Q=+1$) HTDs, or at least in their correlation with
plasticity. This is independent of the methods employed to define the HTDs and
also the soft spots, and therefore rather universal. Interestingly, this
represents a dissimilarity with respect to the analysis using the eigenvector
field in 2D (or in the 2D-slicing method for 3D systems), where negative
defects seem to have a privileged role Wu  _et al._ (2023); Desmarchelier  _et
al._ (2024); Bera  _et al._ (2024). In addition, we notice that the results
shown in Fig. 4, using the non-affine displacement field, display a rather
long-range structure in which spatial correlations survive up to large
distances, $r\approx 5$. This very long correlation length might depend on
several parameters and deserves further investigation. Moreover, the spatial
structure of the HTDs and the spatial decay of their correlation with soft
spots might provide useful insights to achieve a microscopic definition of the
elastic screening length proposed in other macroscopic approaches to amorphous
solids Lemaître  _et al._ (2021).

Figure 4: (a)-(c) The location of HTDs with charges $Q=+1$ and $Q=-1$ in the
displacement vector field at different $\gamma$ are shown in red and
colorblueblue, respectively. The yellow regions are the plastic soft spots,
identified using the softness parameter, Eq. (6). (d)-(f) The spatial
correlation between between monopoles (in the displacement field) and soft
spots (red data) and anti-monopoles (in the displacement field) and soft spots
(blue data) for the same values of the strain $\gamma$. For this analysis,
soft spots were identified from the top $10\%$ of highest
$|\vec{u}_{\text{NA}}|$ values.

Conclusions – In this work, we proposed to use point-like hedgehog topological
defects (HDTs) to characterize the plasticity of 3D amorphous solids and to
locate their soft spots using well-defined mathematical concepts. First,
borrowing from the physics of liquid crystals Kleman and Lavrentovich (2003,
2006); Hölbl  _et al._ (2023) and Heisenberg magnets Fumeron and Berche
(2023); Berg and Lüscher (1989); Holm and Janke (1994); Berg and Lüscher
(1981), we have outlined a concrete and practical procedure to identify HDTs
in 3D glasses – a method that is suitable for experimental discrete data sets
as well.

We have then applied this proposed algorithm to both the eigenvector field and
the displacement vector of a simulated 3D polymer glass model and verified the
feasibility of this method. For the HTDs identified in the eigenvector field,
we have proved that the number of HTDs as a function of frequency follows the
same trend as in previous 2D studies, and directly correlates with the
vibrational density of states at low frequency. We have also shown a strong
tendency in forming oppositely charged monopole/anti-monopole pairs of
topological defects and even clusters with larger number of defects. Moreover,
we have revealed a very strong spatial correlation between the hedgehog
topological defects and the plastic soft spots at short distances.

Finally, we considered HTDs in the 3D displacement vector field for a quasi-
static shear deformation and for different large plastic events, including the
yielding point. In this analysis, we have found that HTDs correlate very
strongly with the extended plastic soft spots from which the plastic
instability originates, that are defined using two different methods based
respectively on the softness parameter Smessaert and Rottler (2014b) and the
amplitude of the non-affine displacement field. This result proves that the
location of HTDS, and their yet undisclosed dynamics, might play a pivotal
role to achieve a microscopic understanding of plasticity and yielding in 3D
glasses.

Interestingly, the analysis of topological defects in 3D amorphous solids
revealed that the correlation with the plastic soft spots is independent of
the charge of the defect, unlike previous findings in 2D systems Wu  _et al._
(2023); Desmarchelier  _et al._ (2024); Bera  _et al._ (2024) and in the
2D-slicing of 3D systems Bera  _et al._ (2024). At present, we do not have an
explanation for this intriguing phenomenon that requires further study.

Before concluding, we also notice that the correlation between plasticity and
topological defects appears stronger when the latter are defined using the
displacement field, rather than the low-frequency eigenvector fields. This is
promising since, apart from few exceptions (e.g., Vaibhav  _et al._ (2024)),
the eigenvector field is not a quantity directly accessible via experiments,
whereas the displacement vector is.

On a general ground, our work proves that the idea of using geometric topology
to describe, predict and rationalize plastic properties of amorphous solids
can be successfully extended to three dimensional systems, such as real
materials. Aside from the theoretical part of this development, this result
provides a formidable occasion to test these ideas on realistic experimental
amorphous solids for which the method we proposed can be directly applied. For
example, three dimensional sheared granular systems Cao  _et al._ (2018)
constitute a possible candidate platform to test these ideas.

Acknowledgments – M.B. thanks W. Kob, Z. Wu, J. Zhang, Y. Wang, C. Jiang, Z.
Zheng, H. Tong for useful discussions. We would like to thank A. Liu and T.
Petersen for ongoing collaborations and endless discussions about this topic
as well. M.B. acknowledges the support of the Shanghai Municipal Science and
Technology Major Project (Grant No.2019SHZDZX01) and the sponsorship from the
Yangyang Development Fund. A.Z. gratefully acknowledges funding from the
European Union through Horizon Europe ERC Grant number: 101043968 “Multimech”.
A.Z. gratefully acknowledges the Niedersächsische Akademie der Wissenschaften
zu Göttingen in the frame of the Gauss Professorship program. A.Z. and A.B.
gratefully acknowledge funding from US Army Research Office through Contract
No. W911NF-22-2-0256.

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## Supplementary Material

In this Supplementary Material (SM), we provide further details about the
numerical and analytical computations presented in the main text. Moreover, we
show further analysis to corroborate our findings.

## Model and numerical methods

In this study, we implement the Kremer-Grest model Kremer and Grest (1990) to
simulate a coarse-grained polymer system. This system comprises linear polymer
chains, each consisting of 50 monomers, with alternating monomer masses of
$m_{1}=1$ and $m_{2}=3$ Kriuchevskyi  _et al._ (2020). The total number of
monomers in our system was $N=10{,}000$.

Figure S1: (a) Schematic representation of the polymer chains with alternating
masses $m_{1}$ and $m_{2}$. The FENE bonds and the LJ interactions range
$r_{c}$ from a monomer (A) are shown. (b) A typical simulation configuration
is presented where monomer breads with masses $m_{1}$ and $m_{2}$ are
indicated in red and blue, respectively and FENE bonds by green colors. (c)
Stress versus strain curve. The yielding point is indicated by the dashed
black line at $\gamma=0.099$.

The interactions between monomers in our model are governed by the truncated
and shifted Lennard-Jones (LJ) potential:

$u_{\text{LJ}}(r)=\begin{cases}V(r)-V(r_{c}),&\text{for}~{}r<r_{c}=2.5\sigma,\\\
0,&\text{otherwise.}\end{cases}$ (S1)

Here, $V(r)$ represents the standard Lennard-Jones potential:

$V(r)=4{\varepsilon}\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right],$
(S2)

where $\varepsilon$ is the interaction strength, and $\sigma$ is the monomer
diameter. Here $r_{c}=2.5\sigma$ is the interaction distance.

In addition to LJ interactions, covalent bonds along the polymer chains are
modeled using the finite extensible nonlinear elastic (FENE) potential:

$u_{\text{FENE}}(r)=-\frac{Kr_{0}^{2}}{2}\ln\left[1-\left(\frac{r}{r_{0}}\right)^{2}\right].$
(S3)

We set $K=30$ and $r_{0}=1.5\sigma$ in our simulations. In Fig. S1(a) we
schematically show these interactions. The polymer chains were simulated
within a cubic box with periodic boundary conditions applied in all three
dimensions. The system was equilibrated using the LAMMPS simulation package
Plimpton (1995). We show a typical configuration in Fig. S1(b). Following
equilibration, we employed an athermal quasi-static (AQS) deformation
protocol. This involved quenching the glass sample to absolute zero
temperature and subsequently applying quasi-static shear with a strain
increment of $\delta\gamma_{xz}=0.001$. The units of mass, length, and time of
our simulation are taken as $m_{1}$, $\sigma$, and
$\tau=\sqrt{m_{1}\sigma^{2}/{\varepsilon}}$, respectively. We set $m_{1}$,
$\sigma$, and $\varepsilon$ to unity. Throughout our simulations, polymer
chains were considered fully flexible, with no angle-bending terms included in
the potential. The stress-strain curve in the shear plane is presented in Fig.
S1(c).

## Calculation of smoothed eigenvector field

Given the Hessian matrix, we compute the eigenvector field
${\mathbf{e}}_{k}=(e^{x}_{k},e^{y}_{k},e^{z}_{k})$ for each eigenmode with
eigenfrequency $\omega_{k}$ ($k=1,2,\ldots,3N$). This eigenvector field is
interpolated onto a $40\times 40\times 40$ simple cubic lattice with grid
spacing of $\sigma/2$ in all directions, superimposed on the 3D simulation
box. The eigenvector field at each lattice site $\vec{r}=(l,m,n)$ is then
obtained as Wu  _et al._ (2023)

$e^{\mu}_{k}(\vec{r})=\frac{\sum_{d_{i}<R_{\text{cut}}}w(\vec{r}-\vec{r}_{i})e^{\mu}_{i}}{\sum_{d_{i}<R_{\text{cut}}}w(\vec{r}-\vec{r}_{i})},\quad\mu=x,y,z,$
(S4)

where $\vec{r}_{i}$ is the position of particle $i$, and
$w(\vec{r}-\vec{r}_{i})$ is a Gaussian weight function given by
$w(\vec{r}-\vec{r}_{i})=\exp(-|\vec{r}-\vec{r}_{i}|^{2}/r_{c}^{2})$ with
$r_{c}=1$. The distance $d_{i}=|\vec{r}-\vec{r}_{i}|$ is between the lattice
site and the particle $i$, and $R_{\text{cut}}=4\sigma$ is the cut-off
distance for interpolation.

Figure S2: Normalized eigenvector field for a portion of the simulation box
exhibiting a monopole defect (red) and an anti-monopole one (blue). These data
correspond to $\omega=3$.

## Calculation of topological charge density

In order to compute the topological charge density, we divide the entire 3D
simulation box into cubical cells of side $\sigma/2$. Within each such cell we
count number of HTDs $N^{+}$ and $N^{-}$ with $Q=+1$ and $Q=-1$, respectively,
for $\omega_{k}<4$. The topological charge density inside each cubical cell is
defined as

$\bar{C}=\frac{\sum_{\omega_{k}}(N^{+}-N^{-})/{\omega}_{k}^{2}}{\sum_{\omega_{k}}1/{\omega}_{k}^{2}},$
(S5)

where the weight factor is motivated by the quadratic dependence of the number
of HTDs at low frequency.

## Number of HTDs and vibrational density of states

In Fig. S2, we show a snapshot of a 3D eigenvector field presented two clear
topological defects with opposite charge.

In Fig. S3(a), we present the vibrational density of states (vDOS) of the
polymer glass model at $T=0$. As highlighted in the inset, at low frequency
the vDOS exhibits a distinct Debye scaling, $D(\omega)\propto\omega^{2}$. In
panel (b) of the same figure, we present the number of hedgehog topological
defects in the eigenvector field $N_{\text{HTD}}$ as a function of frequency
$\omega$. At low frequency, the number of defects also scales as $\omega^{2}$,
implying a correlation with the vDOS.

Figure S3: (a) The vibrational density of states $D(\omega)$ of the polymer
glass model at $T=0$ extracted from the direct diagonalization of the Hessian
matrix. The inset highlights the Debye scaling of the vDOS at low frequency.
(b) The number of hedgehog topological defects in the eigenvector field
$N_{\text{HTD}}$ as a function of frequency $\omega$. The inset zooms on the
low frequency region and highlights the $\omega^{2}$ scaling.

## Calculation and interpolation of non-affine displacement field on a cubic
lattice

We derive the displacement field as Baggioli  _et al._ (2021); Bera  _et al._
(2024) $\vec{u}_{i}={\vec{r_{i}}}(\gamma)-{\vec{r}_{i}}(\gamma-\delta\gamma)$,
where $\vec{r}_{i}(\gamma)$ and $\vec{r}_{i}(\gamma-\delta\gamma)$ represents
the locations of the particle $i$ at strains $\gamma$ and
$\gamma-\delta\gamma$, respectively. The affine component for the simple shear
scenario is calculated as $\vec{u}_{A}=z\delta\gamma\hat{x}$. Finally we
obtain the non-affine displacement field by subtracting the affine component
from the total displacement, $\vec{u}_{\text{NA}}=\vec{u}_{i}-\vec{u}_{A}$
Baggioli  _et al._ (2021); Bera  _et al._ (2024). To obtain HTDs within the
non-affine displacement field, we interpolated the discrete non-affine
displacement field $\vec{u}_{\text{NA}}$ onto a simple cubic lattice grid with
a side length $\sigma/2$ using the standard Python package griddata with
linear interpolation.

##  Appendix A Extended data and correlation between soft spots identified
with different methods

In Fig. S4, we provide extended data for the spatial correlation between
between soft spots (s) identified using the softness parameter, Eq. (6), and
negative (n) and positive (p) HTDs. Low-frequency data are provided in Fig. 2
in the main text. From Fig. S4, the vanishing of the correlation at high
frequency ($\omega\gtrapprox 2$) is evident.

Finally, for completeness, in Fig. S5 we show the spatial correlation between
the plastic soft spots identified with method I (softness parameter, Eq. (6))
and method II (non-affine displacement field). From there, we observe that the
two definitions provide a strong correlation at short distance that,
nevertheless, does not survive for large length scales.

Figure S4: Extended data showing the spatial correlation between soft spots
(s) identified using the softness parameter, Eq. (6), and negative (n) and
positive (p) HTDs for several higher frequencies. See Fig. 2 in the main text
for low-frequency data. Figure S5: Spatial correlation between soft spots
identified using the softness parameter, Eq. (6), and the non-affine
displacement field. Different lines correspond to different values of strain,
located around major plastic events along the stress-strain curve, Fig. S1(c).