On the geometry of topological defects in glasses

Domain: physics
Published: 2025-12-05
Editorial outcome: major_revision

Abstract
Recent studies have revealed a series of connections between the topological features of structural glasses and their material properties. These findings show a striking resemblance to results observed in quantum physics that underscore the significance of the nature of the wavefunction. However, so far the structural arrangement of the topological defects in glasses has remained elusive. Here, we investigate numerically the geometry and statistical properties of the topological defects related to the vibrational eigenmodes of a prototypical three-dimensional glass. We find that at low frequencies these defects form scale-invariant, quasi-linear structures and dictate the morphology of plastic events when the system is subjected to a quasi-static shear, i.e., the eigenmode geometry shapes the plastic behavior in 3D glasses. Our results indicate the presence of a profound connection between the topology of eigenmodes and plastic energy dissipation in disordered materials, thus generalizing the known link identified in crystalline materials.

Introduction
The vibrational and electronic energy spectra of materials have been the focus of interest of many condensed matter physics studies since they are directly accessible in experiments, while the properties of the wave-functions, especially their phase, have received for long times much less attention1–3. However, recently things have changed dramatically, since it has been realized that these functions contain important information about the interference and superposition of quantum states, which allows to gain deep insight into the properties of condensed quantum materials4–9. Interestingly, this shift of paradigm starts to occur also in the domain of glass physics. While in the past many studies focused on characterizing the vibrational density of states to rationalize the anomalous thermodynamic properties of glasses10, only a few investigations have looked into the nature of the eigenmodes. Most of these studies concentrated on the question whether or not these modes are localized or extended, a problem which is relevant for understanding the nature of the so-called boson-peak11–17. However, very recently it has been suggested that also the topological properties of a glass can give important insight into the thermodynamic and kinetic features of the material18,19. One example is the demonstration that in two dimensions the geometry of the topological defects of the eigenvectors is closely correlated to the plastic response in amorphous matter20, which has recently been verified experimentally in a 2D colloidal glass21,22. Probing the topological properties of disordered materials is thus a promising approach to advance our understanding of these complex systems that have so far defied to reveal how their macroscopic behavior is connected to their microscopic structure23.

It is important to realize that for a glass there is no unique way to define topological defects (TD) and so far it is not clear which definition is the most useful for gaining insight into a specific macroscopic property of the sample20,24–27. Probing the influence of TDs on material properties is also hampered by the fact that some TD definitions are applicable only to two-dimensional systems and cannot be generalized in a straightforward manner to D=3. TDs arise, e.g., in the fields defined by the vibrational eigenvectors at the points in which the amplitude of the eigenfunction vanishes and hence its phase becomes undefined. Another example are TDs occurring in the displacement field generated when the sample is sheared, since then the trajectories of the particles can form loops around topological vertices that can be transient or continuously reshaped by particle motion. For the cases of active matter and living systems, it has, e.g., been well documented that topological defects in the displacement fields can influence the dynamics or pattern formation of the systems28–32. Note that these two types of TDs, i.e., in eigenvectors or displacement fields, differ not only in the nature of the field used to define them, but also by the fact that with the first definition one considers a mechanically stable configuration, while the second one deals with an out-of-equilibrium response. Understanding the relevance of these differences is therefore crucial for exploring topological quantities in the study of amorphous matter.

In this study, we numerically investigate for a prototypical three dimensional glass the geometry and statistical properties of eigenvectors associated with the vibrational eigenmodes, with a special focus on the TDs generated by these modes. Although the definition of these TDs is related to the one considered in an earlier study of 2D systems, we point out that the generalization to 3D is not a trivial step. We find that at low frequencies the spatial arrangement of these TDs form one-dimensional structures that have a scale-invariant geometry with a power-law exponent around −5/3, a value that can be rationalized from the quasi-linear structure of the filaments formed by the TDs, and hence is expected to be universal. When subjected to quasi-statics shear, the resulting plastic events are strongly correlated with defects carrying a negative topological charge, revealing a close connection between TDs and plastic yielding. Remarkably, the spatial distribution of plastic events under shear mirrors the scaling behavior observed for the topological defects, reinforcing the idea that the eigenmode geometry plays a fundamental role for the plastic behavior of amorphous solids.

Results
System
We study a binary mixture of Lennard-Jones particles, using a model that has been extensively investigated in the past33, applying a small modification to avoid crystallization at the lowest temperatures34. In the following, we will use reduced units, reporting length and energy in terms of the size of the large particles and the well depth of the Lennard-Jones potential, respectively. The system, containing N=800,000 particles in a box of size (87.36)3, was equilibrated at a low temperature (T=0.43) before it was cooled down to zero temperature. More details on the model and simulations are given in the Methods section.

Topological defects
The first 10,000 vibrational normal modes of the system are obtained by diagonalizing the Hessian matrix using ARPACK, see Methods for details. The lowest frequency is ω=0.280, while the highest one is 2.24, slightly below the main peak in the vibrational density of states, see Fig. 1 and Supplementary Fig. 1. To identify the topological defects of a given eigenvector with components (ex,ey,ez), i=1, …N, we first define a continuous vector field u(R) by interpolating between the positions of the particles:
u(R)= ∑ w(R-ri)ei = ∑ w(R-ri)ei, (1)
where ri is the location of particle i, ei is the eigenvector component for particle i, and w is a Gaussian weighting function, w(R-r)=exp(-|R-r|2/Δ2), with a width of Δ=1. (The exact value of Δ is not crucial as long as it is on the order of the size of a particle.) The topology of this vector field is then analyzed by projecting it on a cubic lattice of size 88×88×88 superimposed to the sample (the length of the unit cell is thus ≈1). This lattice is subsequently used to identify the topological defects as follows, see Inset of Fig. 1: For each of the six square sides of a cubic unit cell, we search for a signature of a topological defect by examining the structure of the phase of the field u(R), after having it projected on the square, on the border of the plaquette via a line integral. If this integral gives a net change of ±2π, one has a vortex or an anti-vortex, while a charge zero corresponds to a field with no singularity. In practice, we assign to each plaquette (having its corners at A, B, C, and D and being, e.g., in the xy-plane) a winding number νx,y defined as
νx,y = (1/2π) ∮ ∇θ ⋅ dl, (2)
Here, ΔθA,B is the change of the phase θ from point A to B, and the angle θ at point A is given by angle formed by the vector (ux,uy) with the x-axis. Each plaquette contains either a vortex, an anti-vortex, or it is neutral, if νx,y is equal to 1, −1, or 0, respectively. The search for topological defects is done for each of the six faces of the cubes of the lattice. For each box that is part of a vortex line, the vortex must enter from one face of the box and exit from another face. Therefore, if the resolution of the discretization of the field u(R) is sufficiently high, there will be two faces, not necessarily opposite on the cube, that display a net phase change of 2π (or −2π), i.e., knowledge of whether two vortex points in neighboring planes form part of the same line is only limited by the spatial resolution. Ambiguities in the counting of the TD related to, e.g., non zero winding on more than two faces of a voxel, can in principle be resolved by using a finer mesh. However, it turns out that with the mesh size we use, the number of such cases is in practice very small and hence they do not alter the overall analysis in a significant manner. In the Supplementary Information (SI), this procedure is applied to simple synthetic vector fields, showing that it does indeed allow to identify vortex or anti-vortex lines. It is important to note that our detection procedure for the TDs is agnostic with regard to the geometrical arrangement of the defects, and in particular does not impose a line-like morphology for this arrangement. The procedure consists of two decoupled steps: We first locate zeros of the eigenvector field amplitude and assign a local topological index by evaluating the phase winding on each plaquette of the voxels. This step uses only local information and introduces no geometric bias. We then reconstruct connectivity by linking neighboring cells whose local indices are consistent. This reveals the intrinsic dimensionality of the zero set: Isolated zeros remain point-like; if zeros continue coherently across adjacent cells, a one-dimensional locus (TD line) emerges.

In Fig. 1 we show the number of topological defects per particle as a function of frequency ω. We observe that, at low ω, i.e., in the Debye-regime up to ω≈1.0, one has the same ω2-dependence found in a 2D system20. This result indicates that as long as the system can be considered as an elastic continuum, the ω2-scaling between number of TDs and frequency is independent of the system dimension. To rationalize this ω2-dependence we show in Fig. 2 the TDs of the eigenmodes at different ω. For the lowest ω the eigenvectors are a superposition of transverse plane waves and this gives rise to TDs that align in (windy) one-dimensional structures, as can be recognized in the figure for ω=0.28. (Note that at finite ω the disorder of the system makes that these plane waves couple with each other, giving rise to the structures seen at low ω, see Supplementary Fig. 5 for a snapshot of the eigenvectors.) With increasing frequency, the TD-lines become more and more complex and windy, but positive and negative lines are still clearly separated, indicating that they repel each other. At around ω=0.567 one sees the appearance of some clusters, i.e., positive and negative TD start to interfere with each other. Despite this interaction, the separation between positive and negative TDs can be seen even at frequencies as high as ω=1.21, reflecting the local geometry of the acoustic modes. This regularity leads to a distance between lines of the order of the wavelength ~ω-1, and to a scaling of the line density of ω2, rationalizing the ω2-dependence presented in Fig. 1, see also ref. 20. In general, a superposition of plane waves can produce a variety of TD morphologies, depending on how the wave vectors are chosen35. In the low-frequency regime, the vibrational modes are predominantly transverse, and when the field is effectively a two-component (complex transverse) vector field, in three dimensions, the generic zero set of such a field has codimension 2 and thus forms line-like singularities. Consequently, the filamentary nature of TDs and the observed ω2-scaling emerge naturally from codimension and spectral arguments: the typical inter-line spacing scales as ~ω-1, yielding a line-length density ~ω2. By contrast, isolated point singularities are rare events and do not significantly influence the scaling behavior. For the highest frequencies, the TD no longer form long lines but complex patterns since their mutual interactions becomes very strong.

Spatial structure of the topological defects
To quantify the spatial arrangement of the TD lines, we define a spatial correlation function of the topological defects of type α,β∈{±1} at a given ω as,
gαβ(r;ω)= Nω-1 ∑κ gκ,αβ(r)δ(ω-ωκ): (3)
Here, gκ,αβ(r) is the pair correlation function characterizing the structure of TDs in the eigenmode κ, and Nω is the number of modes in the system whose frequency lies in the range ω±Δω, using Δω=0.086. (This averaging is done in order to improve the statistics, and the exact value of Δω is not crucial.) Fig. 3 demonstrates that, at fixed frequency, g++(r;ω) and g--(r;ω) are quite similar in that they display pronounced peaks at r=1 and √2, positions that are independent of ω since they are given by the lattice, panels (a) and (b). These peaks become less pronounced with increasing frequency (Supplementary Fig. 8 shows these curves in a lin-lin plot) and the ω-dependence of the height of the main peak, gmax, is presented in panel (d), demonstrating that for low ω the data for ++ and -- indeed track each other very well. Also notable is the fact that at small frequencies, the height of the first peak is very large and seems to diverge for ω → 0, a consequence of the pronounced (quasi-linear) arrangement of the TDs. One intriguing feature of g++(r;ω) and g--(r;ω) is that the correlators exhibit within a significant range of r a scale-invariant behavior, i.e., a power-law, with an exponent around −5/3, independent of ω. Such a decay corresponds to a fractal dimension of df=4/3 since the scaling of g(r) for a fractal object of fractal dimension df is g(r) ∝ rdf-d. Since the difference between 4/3 and 1.0 is not large, the data of g(r) indicates that in our 3D system, the TDs form quasi-one-dimensional lines if ω is small, consistent with the snapshots of Fig. 2. (See Supplementary Fig. 9 for additional tests on the decay behavior of gαβ(r; ω).) The exponent df ≈4/3 reported here quantifies the line geometry of the TD set in 3D. In contrast, this arrangement has no direct analog in 2D systems, in which topological defects are point-like. As argued in ref. 20, in 2D the nature of the eigenvectors makes that the topological singularities are isolated points if the frequency is low. Thus, these isolated zeros of the eigenvector field lack the extended connectivity required to form filamentary or self-affine structures. With increasing ω the range of scale-invariant behavior decreases, as indicated by the leveling-off of the correlation functions. The distance at which this leveling-off occurs defines a cross-over length ξ and in Fig. 3e we present ξ as a function of ω. One finds that for ω < 0.604, the length ξ, which can be interpreted as characterizing the spacing of the TDs lines, is proportional to ω, in agreement with the presence of the acoustic modes with the ω2-scaling of the defects, as discussed above. We note that the slope of this linear part is given by the velocity of sound, supporting the view that at low ω this length scales is directly related to the acoustic modes. For intermediate frequencies, one finds a much weaker dependence of ξ on ω, since in this ω-range one has a mixture between TDs lines that are well separated and zones in which the TDs strongly interact (see Fig. 2). For ω>1.5 the slope of the curve increases quickly, since the separation between the TD becomes very small.

The behavior of the function g+-(r;ω) is different, panel (c). The correlation at r=1 is about three times lower than the one of the TDs self-correlation, see panel (d), and also the amplitude of the oscillations at short distance is smaller, i.e., the short-range order is less pronounced. Interestingly, the height of the correlator at small frequencies shows a weaker ω-dependence than the one found for the two other correlators. Furthermore, Fig. 3d reveals a crossover behavior of gmax(ω) at around ω=1.121 in that for low frequencies the quantity tracks gmax++(ω) while at high frequency it follows the gmax+-(ω) curve. This cross-over frequency corresponds to the point in which the positive and negative TD start to interfere strongly, thus affecting the total number of TD at a given frequency.

Spatial structure of plastic events induced by shear
To probe the mechanical properties of the glass we have applied, at constant volume, an athermal quasi-static simple shear (using a strain increment of Δγ=0.0005) and determined the stress-strain curve, Fig. 4a. We note that this curve shows no drops before reaching the very sharp yielding point at around strain γ=0.098, indicating that the sample is large enough to avoid such finite size effects36,37. Also included in the graph is the result if the strain increment is doubled, and one sees no qualitative difference. Supplementary Fig. 13 demonstrates that the sharp drop at global yielding gives rise to the formation of a very narrow shear band, which indicates that the system is rather brittle.

For a sheared configuration at strain γ, we calculate the non-affine displacement D2min38 of the particles between two consecutive configurations that have a difference in strain of Δγ = 0.0005. For each particle i, we determine the largest non-affine displacement δmax(γ)=max{D2min(i,Δγ), D2min(i,2Δγ), D2min(i,3Δγ), ..., D2min(i,γ)} that a particle i has made up to the strain γ. We then select the particles having the top 0.5% of δmax(γ) and define them as plastic event (PE) particles at this γ. Figure 4b shows these PE particles for three different γ, sheared parallel to the xy-plane. At small γ, the PE particles are highly clustered, while with increasing strain, the distribution in space becomes more random. (Note that once the system has yielded, the PE particles are concentrated in a shear band.)

In Fig. 4c we show the selected PE particles at strain γ=0.01 after shearing in three different directions, and one recognizes that the three sets of particles are spatially highly correlated. This demonstrates that for the local yielding behavior, the strain direction is not a relevant parameter, i.e., predicting the location of the PE can be done via a scalar quantity, an insight that is useful for developing theoretical approaches of yielding. This correlation can be quantified via the pair correlation function gAB(r) for the particles in the three sets, where A,B∈{x,y,z} represents the set of PE particles generated under different shear directions, Fig. 4d. One recognizes that there is a strong spatial overlap between the PE particles in the three sets in that the peak height at small r exceeds 2⋅103, confirming the visual impression of panel (c). Interestingly, one finds that this correlation function decays at intermediate distances with a power-law with an exponent that is compatible with −5/3, i.e., the value we found for the algebraic decay of the TD, see Fig. 3. This indicates that the PE particles and the TD are correlated to each other and form a network with the same spatial arrangement. Furthermore, the information is obtained by determining the center of mass of the clusters formed by these particles and the resulting radial pair correlation function of these centers. For the cluster analysis, we use a cutoff distance of 1.35, i.e., the location of the first minimum in the pair correlation function of the particles. Once more one finds, see Fig. 4e, that this correlation function shows a power-law decay and that the exponent is the same as the one obtained for the correlation between the TD. This remarkable agreement provides compelling evidence for a fundamental, previously unrecognized connection between microscopic eigenmode geometry and macroscopic plastic deformation. Thus, while the fact that the fractal dimension is close to 1.0 itself is not surprising, its coincidence with the fractal dimension characterizing the spatial patterns of plasticity substantially enhances our understanding of how eigenmode structures govern the mechanical behavior of amorphous solids. This correspondence, rather than the absolute value of df, is another important physical insight which indicates that the morphology of plastic rearrangements is organized by the pre-existing topology of low-frequency eigenmodes. This is thus strong evidence that the TDs and PE particles are closely related to each other and in the next section, we will probe this connection in more detail.

Correlation between topological defects and plastic events
Previous studies have shown that TDs carrying negative topological charge are characterized by eigenvector flow fields corresponding locally to saddle-point configurations20. Physically, these configurations imply local mechanical instability: Under external perturbations such as shear, the delicate force balance at these points is readily disrupted, potentially triggering localized plastic rearrangements, and hence one expects a noticeable correlation between such TDs and PE. The location of the PE particles can be correlated with the position of the topological defects of a given eigenmode κ by means of a corresponding radial distribution function gκ,αPE(r), where α∈{±1} denotes the nature of the TD (gκ,αPE(r) is defined in Methods). In order to see how this spatial correlation function evolves with ω, we define an average correlation gαPE(ω;r) as
gαPE(r;ω)= Nω-1 ∑κ gκ,αPE(r)δ(ω-ωκ): (4)
Here Nω is the number of modes in the system whose frequencies are in the range ω±Δω using Δω=0.086. (Also, here the exact choice of Δω is not crucial.) Fig. 5a demonstrates that at low frequencies (ω≤0.43) the PE particles and the positive TDs are uncorrelated, while there is a noticeable short-range correlation with the negative TDs, results that are compatible with earlier findings in a two-dimensional systems20 and reflect the 3D arrangement of the TD lines at low frequency discussed in Fig. 2. This correlation increases with ω and shows a broad maximum before peaking at around ω=1.1, i.e., there is a strong correlation between PEs and −1 TDs in the frequency interval of 0.604<ω<1.294, see Fig. 5b. Interestingly, this range coincides with the frequencies in which the TD lines start to fragment, i.e., wherein in Fig. 3e the ω-dependence of the length scale characterizing the size of the TD lines becomes less steep. We also note that at intermediate frequencies one observes a correlation between the PE particles and the positive TDs, the origin of which is the spatial correlation between the former and the −1 TDs, see Fig. 3c. For ω≈0.6 the g+PE(r;ω) has a peak at around r≈0.7, signaling that +1 and −1 TD pair together and create a dipole structure, a phenomenon also observed 2D20. In 3D, these dipoles correspond to a local structure of TDs where two TDs with opposite topological charge occupy two different faces on the same unit cell of the coarse-graining grid.

The ω-dependence of gmax suggests that the correlation between the TDs and the PE particles can be revealed best by averaging gαPE(r) in the range 0.6<ω<1.3. For this, we weight the modes with 1/ω2, since this gives an equivalent weight to different frequencies. The resulting correlation functions gav+PE(r) and gav-PE(r) (see “Methods” for a definition) are shown in Fig. 5c and from this graph one clearly recognizes that at strain γ=0.01 the negative TDs are significantly correlated with the PE in that the peak at small r rises above 2.1, which mean that there is a two-fold increase of the probability that a PE occurs close to a TD as compared to a uniform distribution. Also included in Fig. 5c is the γ-dependence of the correlation functions, and one sees that at small γ the correlation is strong because the weakest spots in the sample are plastically deformed, and that upon approaching the yielding strain, γ=0.098, the correlation is lost since the TD determined at γ=0 are no longer a good predictor for the PEs.

In Fig. 5d we show the smoothed 3D topological charge density field Ω(R) (see Methods) at which the charge density is smaller than −1.25⋅10−4. The snapshot shows the intricate pattern the negative TDs form in space, indicating that the zones at which a plastic event occurs is highly complex. (Supplementary Fig. 11 shows the same type of graph for the TD field with high positive charge.) Figure 5e presents two representative slices from the corresponding topological charge density field, allowing to get a visual impression of the correlation between the location of the TDs and the PEs (marked as white spots superimposed to the maps). The graph demonstrates the strong correlation between zones with a high density of −1 TDs and the PEs. As already argued in ref. 20, this correlation between −1 TDs and PEs is related to the fact that for such charges the local geometry of the normal mode field has a hyperbolic structure, see Supplementary Figs. 2 and 3, i.e., a small disturbance of the particle positions due to the shear will give rise in a significant change of the motion of the particle and hence most likely result in a PE.

Methods
MDSimulation
The simulations were carried out for a system with N=800,000 and we used a potential that is a small modification of the standard Kob-Andersen potential. This modification (an addition of a linear term to the original potential) was proposed by Schrøder and Dyre and has the effect that crystallization is (so far) completely avoided34. We equilibrated the system at T=0.430 (T_MCT=0.436) for 30 million steps (density was 1.200 and step size was 0.005) and then cooled the sample to T=0 within 30 million time steps. This slow cooling schedule ensures that the resulting samples are well-annealed and representative of stable glasses, as supported by previous studies36 and confirmed by our stress-strain response (Supplementary Fig. 11), which exhibits sharp yielding transitions characteristic of well-annealed, mechanically stable glasses63. These features indicate the robustness and reproducibility of the mechanical and topological analyses reported in this work.

Normal modes
A conjugate gradient energy minimization process was used to get the inherent structure of the configuration, and subsequently, the vibrational normal modes were obtained by diagonalizing the dynamical matrix D which is defined as
D = 1/√(mi mj) ∂2U(rN)/∂ri∂rj. (5)
Here, U(rN) is the total potential energy of the system, and mi is the mass of particle i, and all masses were set to 1.0. We calculated the first (lowest frequency) 104 eigenmodes for the system with N=800,000 with ARPACK. The vibrational density of states D(ω) were calculated as
D(ω)=(1/3N) ∑κ δ(ω-ωκ): (6)

Correlation function between the TDs and the PEs
For each eigenmode κ=1,2,...,3N, we define the radial pair correlation function gκ,αPE(r) between the TDs and PEs, with α∈{−1,+1}, as
gκ,αPE(r)= L3 / (4πr2 NTD NPE) ∑i=1NTD ∑j=1NPE δ(r-|rij|): (7)
Here, NTD and NPE are the number of TDs of the mode κ and the number of particles associated to PEs, respectively, and r is the distance between the TD i and the PE j. Since the number of TDs increases quadratically with ω, the average correlation function gav,αPE(r) is then defined by
gav,αPE(r)= ∑κ gκ,αPE(r) / ∑κ 1 = ∑κ gκ,αPE(r) ⋅ ω2κ / ∑κ ω2κ, (8)
where the sum over κ runs over the ω-range defined in the main text.

Generating of 3D topological charge density field
To get more insight into the geometry of the eigenmodes, it is useful to introduce a density field that is a weighted average over the topological charges in the system. Since the TD are defined via the plaquettes given in Eq. (2), i.e., they have orientational information on the eigenvector field, we keep track of this information by taking the average only in the plane of the plaquette. The weighted density field of the TDs at position R is thus given by
Ω(R)=Ωxy(R)+Ωxz(R)+Ωyz(R), where, e.g., the field Ωxy is defined by
Ωxy(R)= 1/N ∑κ 1/ω2κ ∑i(κ) ν i,xy / |ri - R| ⋅ Π(zi - Z): (9)
(The fields Ωxz and Ωyz are defined analogously.) Here, R=(X,Y,Z), N is the number of particles contained in the system, ωκ is the frequency of mode κ and ν i,xy is the topological charge (+1 or −1) of the i-th TD, located at the point (xi,yi,zi), in the xy-plane of mode κ. The function Π(zi - Z) is a rectangle function (with a width of the discretization of the field) which assures that the TDs are only visible in the xy-plane, but not outside of this plane. Thus, essentially each TD is multiplied with a planar 1/r-type convolution kernel function, which smooths it spatially over some length scales in the plane it belongs to. This smoothening process quantifies the influence of a topological charge ν placed at a distance r, which in our case has the physical meaning of measuring how much the local orientation of the field varies on a circular circuit of radius r around the defect core, hence taking into account the interference effect between different TDs. In practice, we set the range of action of the 1/r-kernel as 1<|r|<L/2, where L is the size of the simulation box.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The report is attached", "reviewer_2": "I strongly recommend the publication of this manuscript. The current work makes very substantial contributions to the understanding and quantitative analysis of glassformers in terms of the system eigenvectors and defects and their admixture. Earlier approaches suggesting global many-body eigenstate and topological defect descriptions of supercooled liquids have been made a decade ago by now. However, these approaches did not define nor address topological defects in a manner as lucid as the current manuscript does. The introduced definition and characterization of defects that the authors of the current manuscript advance is very novel and insightful. The relation to plasticity that is subsequently found in this manuscript was also very illuminating to read and learn of. I have only several minor comments below. These do not constitute a critique of any kind but are rather suggestions. Whether or not any particular particular item below should be changed is left up to the authors.\n(1) The authors discovered a numerical correlation between negative topological defects and plastic events. It would be good to augment this by further possible physical underpinnings rationalizing this. Is it possible, say, to predict the onset of plastic events from the existence of these defects and further illustrate how this prediction contrasts/correlates with participation ratio metrics and non-affine displacements. This could be bolstered by statistical metrics. This might make the observed correlation more compelling.\n(2) The authors state that the defects are scale-invariant and that the defects form quasi-linear structures at low frequencies. It might be good to add to discuss more the statistical distribution of defect lengths, angles, and densities. Does the defect density change with aging? Similar to the other numbered items in this list, since exploring this may be time consuming, if the results are not known to the authors, whether or not such an additional analysis should be undertaken is up to the authors.\n(3) If possible, it might be good to explicitly state how robust the results are when different cooling rates are used and the initial conditions are varied.\n(4) There have been works attempting to detect and quantify structure and defects in glassformers using machine learning approaches--\nboth of the supervised and earlier unsupervised (clustering or community detection) types. Unlike the current manuscript, the detection of structure and defects with machine learning approaches has not been as simple and transparent. However it might be good to note these other approaches so as to further situate the manuscript.", "reviewer_3": "In “On the geometry of topological defects in glasses” by Wu, Barrat and Kob, the authors present a generalisation/extension of their analysis of vortex defects in the vibrational eigenmodes and shear displacement fields of glasses from 2D to 3D. The original 2D analysis was published recently (Wu, Z. W., Chen, Y., Wang, W.-H., Kob, W. & Xu, L. Topology of vibrational modes predicts plastic events in glasses. Nat. Commun. 14, 2955 (2023)). The original report investigated the correlations between +1 and -1 vortex defects in 2D in both the vibrational eigenmodes and displacement field under shear. The manuscript under consideration generalises the identification of defects to 3D and examines correlations between defects and plastic zones. Here, they observe a curious -5/3 scaling behaviour in the spatial pair-correlations for defects found in the low frequency eigenmodes. Overall, the work is interesting and a useful extension of the previous work to 3D. However, a deeper discussion of the significance of these vortex defects is highly desirable.\nIn particular, the addition of plane waves forms vortex and anti-vortex arrays, a well-known observation in optics (O’Holleran, Padgett, and Dennis, \"Topology of optical vortex lines formed by the interference of three, four, and five plane waves,\" Opt. Express 14, 3039-3044 (2006)) and recently also noted for phonons in glass simulations (“Testing the heterogeneous-elasticity theory for low-energy excitations in structural glasses”, Lerner and Bouchbinder, Phys. Rev. E 111, L013402, (2025)). The 3D network of vortex/anti-vortex defect lines that is formed depends on the wavelength and direction of the plane waves. Examining Figure 2 it appears that the vortex/anti-vortex arrays may be related to the superposition of fundamental vibrational excitations of the cubic simulation cell. Considering this, the authors should discuss the deeper significance of these defects for glass properties, especially plasticity and yielding and the origin of the intriguing scaling relationship.\nTopologically non-trivial vortex defects are well-known in condensed matter systems that have spontaneous symmetry breaking (for example, liquid crystals, quantum condensates, magnetic materials). Here the ordered phase consists of a magnitude which can be the same everywhere (outside the vortex core where it is zero) and a direction that varies as a function of location. In these systems, vortex defects are detected from the direction alone, as the magnitude need not vary and does not contribute to the scalar winding number. This is the same as the analysis in this work, in which eigenvector and displacement magnitudes are not used to find vortex defects, only the direction. However, magnitude clearly plays a role, for example in identifying plastic zones in the case of the sheared glass. The authors should justify the reason for ignoring the magnitude of the field when identifying these defects.\nIn summary, to strengthen the article the authors should add an in-depth discussion of the significance of these defects and their correlations in relation to vortex networks and topological vortex/anti-vortex defects observed in other systems.\nSmall suggested improvements:\n- Fig 3 – label frequencies in graphs a,b,c\n- Suggest confirming scaling behaviours for larger simulations and smaller steps in strain.", "reviewer_1_attached": "Response to the Reviewer’s Comments for\nNCOMMS-25-30828\nZhen Wei Wu, Jean-Louis Barrat, Walter Kob\nResponse to Reviewer #1\nThe authors investigate the statistical and geometrical properties of topological line\ndefects in the normal mode eigenvectors of a simulated three-dimensional glassy system.\nThey further show that the features of these topological structures are strongly correlated\nwith plastic events occurring under quasi-static shear, thereby proposing a fundamental\nconnection between the topology of eigenvector fields and plastic deformation in disordered\nsolids.\nOverall, the manuscript presents a thorough and well-executed study with technically\nsound results. However, I have reservations regarding the significance and originality of\nthe work. In particular, the current study closely follows the methodology and analysis\nintroduced in [Nat. Commun. 14, 2955 (2023)], essentially reproducing those results in a\nthree-dimensional setting. The methods, observables, and analysis pipeline appear nearly\nidentical, and the main contribution—demonstrating that the approach extends to 3D–\nwhile valuable, is not in itself sufficiently novel or conceptually transformative to warrant\npublication in Nature Communications.\nIn conclusion, although the computational effort required to carry out this study in\nthree dimensions is appreciated, I do not believe the current manuscript meets the thresh-\nold of conceptual advance or broader impact expected for this journal. I recommend that\nthe authors consider submitting this work to a more specialized venue.\nWe thank the reviewer for acknowledging the thoroughness and technical quality of our\nwork. We appreciate the concerns of the referee regarding novelty and agree that merely\nextending previous 2D results (Nat. Commun. 14, 2955, 2023) to three dimensions would\nbe insufficient to justify publication in Nature Communications. We emphasize, however,\nthat the present study is clearly distinct from such incremental work. The previous study\nof a two-dimensional system was the first to establish the existence of topological defects\n(TDs) in the vibrational eigenmodes of glasses and proposed a well-defined method to\nidentify such defects, focusing mainly on the spatial correlations between TDs and plastic\neventsinducedwhenthesampleissheared. Incontrasttothis, ourcurrent3Dinvestigation\nfocuses on the spatial arrangement of the TDs in 3D, revealing the existence of unexpected,\nintrinsically three-dimensional, properties of these defects. (We also note that coming up\nwith a sensible definition of TD in 3D was not straightforward and hence this constitutes\n1\nan advancement that will be of interest for the glass-science community.) Specifically, we\ndiscover that TDs obtained from the eigenmodes form scale-invariant, i.e., fractal like,\nspatial structures with a non-trivial power-law exponent around −5/3. Furthermore, we\nfind that also the spatial distribution of plastic events exhibits a scaling behavior, with an\nexponent that is identical to the one of the topological defects. This novel correlation un-\nderscoresthefundamentalroleofeigenmodegeometryingoverningplasticityofamorphous\nsolids (with likely consequences for the fracture of the sample). We believe this geometric\nand scaling perspective clearly distinguishes our current study from incremental progress\nand represents a genuinely novel contribution to the field, in particular since the geomet-\nricalarrangementoftheplasticeventsisapredictionthatcanbetestedinrealexperiments.\nSpecific comments\n1. Abstract speculation: The final sentence of the abstract appears highly specu-\nlative, and no supporting evidence is presented in the manuscript. The authors should\nconsider whether this statement is necessary or appropriate within the scope of the current\nwork.\nWe agree with the reviewer that this sentence was speculative and hence we have re-\nmoved it from the abstract and only mention it now as an outlook at the end of the main\ntext. However, it is noteworthy that we have already obtained some very promising re-\nsults, which give evidence that the dynamical heterogeneity of supercooled liquids can be\npredicted from the structure of topological defects in the eigenmodes.\n2. Introduction and broader context: The opening paragraph of the introduction\ndiscusses wavefunction properties in a broad context, particularly emphasizing quantum\ncondensed matter systems. However, in my opinion, several claims do not reflect the c ur-\nrent understanding in the field and may be misleading. For example, stating that the\nwavefunction and its phase have received limited attention is surprising given the central\nroletheyplayintopologicalmaterials(e.g., Berrycurvature, Chernnumber). Furthermore,\nthe wavefunction is not inherently a many-body observable. In fact, Berry curvature and\nrelated topological quantities are routinely derived from single-particle Bloch wavefunc-\ntions. The study of topological properties in wavefunctions is, therefore, not exclusively a\nmany-body pursuit, and it is certainly not new nor unexplored.\nWeacknowledgethereviewer’sconcernandappreciatehis/herinsightfulcomments. We\nwould like to clarify that our original intention was precisely to highlight that the essence\nof topological physics involves shifting the primary focus from eigenvalue structures to\neigenvector properties. Traditionally, quantum physics has emphasized eigenvalues, as\nthey directly correspond to measurable physical quantities such as energies, momenta,\nand spin. In contrast, eigenfunctions, representing probabilities of configurations, were\nhistoricallyconsideredlessdirectlyrelevant,mainlyservingasintermediariesforcalculating\nexpectation values and transition rates. However, recent developments, particularly in the\ncontext of topological physics, have significantly altered this perspective by underscoring\n2\nthe structural and topological properties inherent to wavefunctions. So, in the original\ntext we wrote that the wave-functions have received less attention (with respect to the\neigenvalues/spectra), but we did not want to imply that the attention was limited, as the\nreviewer writes, since in recent times there have in fact been many paper on the subject\nof topological physics in crystalline systems.\nCrucially, although wavefunction topology has become central to the study of many\ncondensed matter systems, we emphasize that its investigation in the context of glasses\nand amorphous solids, especially through vibrational eigenmodes, has so far remained very\nlittle explored and represents a key innovative aspect of our study, and this is also what\nsets our work apart, offering a genuinely novel insight into the field. We have now modified\nthe sentences in order to express more clearly and precisely our view on this point.\nConcerning the Reviewer #1’s remark about the distinction between single-particle and\nmany-body wavefunctions: We agree that the exact answer will depend on the context.\nSince this point is in fact not relevant for the present work we have reformulated this part\nof the text.\nLine 21:\nThe vibrational and electronic energy spectra of materials have been in the focus of interest\nof many condensed matter physics studies since they are directly accessible in experiments,\nwhile the properties of the wave-functions, especially their phase, have received for long\ntimes much less attention [1-3]. However, recently things have changed dramatically, since\nit has been realized that these functions contain important information about the inter-\nference and superposition of quantum states, which allows to gain deep insight into the\nproperties of condensed quantum materials [4-9].\n3. Related works in 3D: There are already two existing studies addressing topo-\nlogical defects in three-dimensional glasses: [PNAS Nexus, Volume 3, Issue 9, September\n2024] and [arXiv:2407.20631]. It would strengthen the manuscript to discuss these works\nand clearly delineate methodological and physical differences with the present study.\nWethankthereviewerforhighlightingthetwostudiesabove. Intherevisedmanuscript,\nwe have now have added a discussion of these works and clarified the methodological and\nphysicaldifferenceswithrespecttoourpresentstudy. Belowwesummarizethedistinctions:\n1) Focus on Eigenmodes vs. Shear Displacement: The previous studies considered\nmainly topological defects defined via the deformation field induced by shear. In contrast,\nour work centers on the vibrational eigenmodes of the unstressed glass as the field used\nto identify defects, i.e., our approach leverages the “built-in” topological defect structure\nof the glass prior to any deformation. This is a key difference: Rather than identifying\ndefects created or activated by shear, we reveal that the glass already contains a network\nof topological defects in its vibrational modes. We then connect this inherent defect net-\nwork to subsequent plastic events. So our approach is able to predict the properties of the\nplastic events (location, spatial arrangement,...) while the method using the displacement\n3\nwill just measure these properties once they have been observed.\n2)Nature of Defects-Point Charges vs. Vortex Lines: Bothcitedstudiesdealwithpoint-\nlike defects carrying an integer charge (±1). Bera et al. identify and count point defects\n(monopoles or 2D singularities) in 3D. In our work, we find that these point singularities\nare not isolated; instead, at low frequencies the defects organize into connected, quasi-\nlinear structures spanning the sample, and we explain why this can be expected to be the\ngeneric behavior. In other words, we observe extended “vortex lines” or line-defect clus-\nters instead of isolated singular point defects. This result has the important consequence\nthat we predict that the soft regions in the glass form an extended thread or network of\ntopological irregularity, reminiscent of a dislocation line in a crystal (though here highly\ndisordered and fractal in shape). This geometric arrangement was not detected in the\npoint-defect analyses used in the two other papers. By mapping out entire defect lines, our\nstudy generalizes thus the concept of a topological defect in glasses from points to line-like\n(one-dimensional) objects, providing a more complete picture of the defect geometry in\n3D. Notably, these line defects in our simulations can, at low frequencies, be surprisingly\nextended, implying the collective nature of imminent plastic events, whereas point-defect\napproaches did not capture this highly non-trivial connectivity of defects.\n3)Scale-Invariant Statistics: Oneimportantinsightfromourstudyisthatin3Dglasses\nthe geometrical arrangement of the topological defect is scale-invariant. We find that the\nnetwork of defect lines in low-frequency modes follows a power-law distribution, indicating\nfractalcharacteristicsoverasubstantialrangeofscales. NeitherBeraetal. ([PNASNexus)\nnor Bera et al. (arXiv) examined the scaling law of defect arrangements in this manner.\nThis observation suggests that the glass’s inherent topological disorder has a critical-like\norganization, and we hope that future studies will identify the reason for this unexpected\nbehavior.\n4) Relation to Plasticity: All three studies agree that topological defects are intimately\nlinked to plastic deformation in glasses. The differences lie in how this link is established.\nThe PNAS Nexus paper showed a temporal correlation: A drop in the number of defects\n(especially annihilation of negative defects) coincides with a plastic yield event, implying\nthat these defects “screen” or relieve stress during yielding. The paper considering hedge-\nhog defects demonstrated a spatial correlation: The locations of (hyperbolic) point defects\ncoincide with the place at which shear transformations occur, identifying these defects as\nprecursors to plastic rearrangements. In our work, we unify and extend our/these ideas.\nWe not only correlate the presence of defects with plastic events, but also show that the\nspatial arrangement of the plastic events is dictated by the geometry of the defect struc-\nture. In practice, this means that when the sample is sheared, the cluster of particles that\nundergo a rearrangement (a shear transformation zone) aligns along the underlying topo-\nlogical vortex line related to the low-frequency mode. Our results thus provide a direct\ncausal link between these two quantities. This level of detailed correspondence was not\ndrawn in earlier works, which focused on counting the number of defect and their loca-\n4\ntion, but did not establish the one-to-one mapping of defect structure to the geometrical\narrangement of plastic events. By clearly delineating that the topology of the eigenmodes\nshapes plastic energy dissipation, our study thus generalizes the established concept of\ncrystal plasticity concept (where dislocation lines dictate slip geometry) to the case of dis-\nordered solids.\n5) Methodological Advances: We also point out the practical differences in methodol-\nogy. The approach of Bera et al. (PNAS Nexus) relies on projecting the displacement\nfield onto 2D planes to find winding numbers, which is an indirect approach that could\nmiss genuine 3D structures. The hedgehog algorithm (arXiv:2407.20631) overcame this\nproblem by computing 3D topological charges on a lattice, but it still characterizes defects\nas dimensionless points. Our method uses the 3D eigenmode field and identifies defect lines\nby tracing the alignment of multiple singular points. This allows us to capture the spatial\ncontinuity of the defects through the material. In essence, earlier works identified “where\nare the defects?” whereas we also answer “how are the defects connected and distributed\nin space?”. Moreover, we believe that the mentioned distinction of the specific algorithm\nis not just a question of technique, but does instead lead to a different understanding re-\ngarding the implication of the TD for the properties of amorphous materials.\nIn summary, we have now added in the Discussion section of the manuscript a compar-\nison between the contributions of Bera et al. (2024, PNAS Nexus) and Bera et al. (2024,\narXiv) that clearly delineates how our work differs in both approach and findings. While\nthese pioneering studies demonstrated that topological defects can be defined in 3D glasses\nand are relevant to plasticity, our work provides a significantly more detailed framework\nwhich allows to make predictions of the mechanical properties of the sample without per-\nturbing it.\nAdded text in the Discussion: Line 339:\nTwo recent studies have highlighted the use of TD in amorphous systems. Bera et al. [26]\nidentified isolated point defects using the non-affine displacement field in 2D slices of a 3D\nsample, and observed a temporal correlation with plastic flow. Subsequently this group\nintroduced a 3D “hedgehog defect” [27], and found that at low frequencies the number of\nsuch defects increases like ω2 and that the location of these defects correlates well with\nnon-affine displacements and plastic events of the sheared system, in qualitative agreement\nwith our findings. However, instead of focusing on isolated point defects, we reconstruct\ntheir connectivity into extended, quasi-linear TD lines that capture the spatial continuity\nand network geometry of TDs, features that are inaccessible to slice-based or point-wise\nanalyses. Our insight into the spatial arrangement of the TD allows us to reveal that\nthese TD networks are scale-invariant at low frequencies, with an inter-line spacing scaling\nas ∼ ω−1 and line density ∼ ω2, consistent with acoustic wave interference in disordered\nmedia.\n4. Generalization to 3D:Themanuscriptstatesthatgeneralizingtothreedimensions\n5\nis “nontrivial.” Could the authors clarify whether this challenge is primarily conceptual or\ncomputational?\nThis is indeed an important point. We have now explicitly clarified in our manuscript\nthat extending the study of topological defects from two- (2D) to three-dimensional (3D)\nsystems is challenging, both computational and conceptual. Conceptually: Once one has\nmade the choice regarding the field one wants to consider (displacement fields, vibrational\nmodes, gradients of various scalar quantities...), one needs a procedure to define and de-\ntect the topological defects (hedgehogs, plaquettes,... ). While in 2D systems one can have\nonlypointdefects, inthree-dimensionalsystems, defectscanoccurintheformofpoint-like,\nline-like, or even surface-like entities, depending intricately on the specific properties and\nstructure of the underlying vector fields. For our present work we thus had to carefully\naddress these additional complexities and establish an appropriate framework for system-\natically characterizing topological defects in three-dimensional amorphous materials.\nThe computational challenge: In order to avoid strong finite size effects, one needs a\nlinear extension of the simulation box, L, that is relatively large. So going from 2D to 3D\nwill require a numerical effort that is by a factor of L larger. Usually this is not a very big\nproblem, since L ≈ 100 is already a decent size. The real challenge might instead come\nfrom the computational cost to get the TD. If these defects are defined via a displacement\nfield, then it is not difficult to determine the field and subsequently the TD. However, if\none defines the TD via the eigenmodes as we do, one needs to diagonalize the dynamical\nmatrix of the system. This is an effort that scales like N3 ∝ L9, i.e., for large systems this\nbecomes numerically very expensive. This is the reason why we have focused on the 104\nsmallest eigenvalues, which, fortunately, are enough to reveal the relevant physics.\n5. Dimensionality of identified structures: The manuscript claims that topolog-\nical defects form one-dimensional structures. However, it seems the method is explicitly\ndesigned to detect line defects, as stated around line 89 where vortex lines are mentioned.\nGiven this, is the dimensionality truly a result or merely a consequence of the detection\nscheme? For instance, could this approach identify point-like or surface-like defects? If\nnot, this point should be carefully qualified.\nThis is an important point and it seems that the original text was not sufficiently clear\nabout this. We emphasize that our method to identify TD does not impose any predefined\ngeometry onto the detected defects. In particular it does not preferentially “force” de-\nfects into line-like structures. Instead, our approach searches without bias for zeros in the\namplitude of the eigenvector field. Only in a second step we reconstruct the connectivity\nbetween the TD and this is based solely on local field configurations. Consequently, if the\nTDs form extended linear structures in the system, the algorithm identifies these as lines;\nconversely, isolated zeros will appear explicitly as point-like defects. Thus, the line-like\nTD structures we observe reflect indeed the inherent physical properties of the eigenvector\nfields, and they are not artifacts of our analysis method. Importantly, this unbiased ap-\nproach can naturally and consistently detect topological defects of different dimensionality,\n6\ni.e., point-like, line-like, or even surface-like, if they exist in the system under investigation.\nEvidence that the method does indeed work as expected, is given in Figs. S2–S4. These\nfigures demonstrate that on synthetic fields with tilted (with respect to the lattice),\nevolving, and intersecting defects the algorithm faithfully recovers both isolated singularities\nand extended structures and is robust to noise. For the real data for our glass sample,\nlow-frequency modes robustly generate extended, quasi-linear objects, whereas at higher\nfrequencies isolated point-defects appear; we find no evidence for extended surface-like de-\nfects (See SI, Fig. S6). Thus, the reported one-dimensional organization reflects a property\nof the low-frequency eigenmode fields themselves and are not artifacts of the detection\nscheme.\nTo clarify this point we have added the following text:\nLine 117:\nIt is important to note that our detection procedure for the TDs is agnostic with regard to\nthe geometrical arrangement of the defects, and in particular does not impose a line-like\nmorphology for this arrangement. The procedure consists of two decoupled steps: We\nfirst locate zeros of the eigenvector field amplitude and assign a local topological index by\nevaluating the phase winding on each plaquette of the voxels. This step uses only local\ninformation and introduces no geometric bias. We then reconstruct connectivity by linking\nneighboring cells whose local indices are consistent. This reveals the intrinsic dimension-\nality of the zero set: Isolated zeros remain point-like; if zeros continue coherently across\nadjacent cells, a one-dimensional locus (TD line) emerges.\n6. Methodological clarity and comparisons: The identification of vortex lines\nrelies on observing consistent winding numbers across opposing faces of a cube. Is this\nalways the case? Could there be instances with nonzero winding on more than two faces?\nHow are such cases treated? A comparison with the method in [arXiv:2407.20631] would\nbe helpful, as the two approaches seem quite similar.\nWe thank the reviewer for raising this question, as it allows us to clarify our vortex\nidentification method in more detail. In our analysis, vortex lines are identified by eval-\nuating phase windings on the six faces of each cubic element in our discretized 3D space.\nContrary to the impression that the Reviewer got from the text, these faces need not neces-\nsarily be opposing pairs, i.e., a vortex line may also enter and exit through adjacent faces.\nSpecifically, a cubic element containing part of a vortex line typically exhibits exactly two\nfaces with a net phase winding of ±2π, representing the entry and exit points of the vor-\ntex line within that cube. However, the reviewer rightly points out the possibility of more\ncomplex scenarios where, due to limited spatial resolution or a large number of overlapping\nwave modes, a single cubic element might display net phase windings across more than\ntwo faces, indicating that multiple vortex lines cross or merge within that element. While\nsuch cases are very rare when the analysis is done with sufficient spatial resolution, they\ncan occur when the resolution is low or if the geometry of the eigen-vector has a very high\n7\ncomplexity. But even if a small fraction of cubes contain multiple intersecting vortices, the\nstatistical averaging ensures that their contributions do not significantly bias the overall\nanalysis. Most cubic elements do only contain a single vortex segment, and thus occasional\ncomplexities have negligible impact on our results.\nIn order to clarify this important point we have added the following text:\nLine 111:\nAmbiguities in the counting of the TD related to, e.g., nonzero winding on more than two\nfaces of a voxel, can in principle be resolved by using a finer mesh. However, it turns out\nthat with the mesh size we use the number of such cases is in practice very small and hence\nthey do not alter the overall analysis in a significant manner.\n7. Scaling of defect number: The authors report a scaling of the number of defects\nwith frequency as ω2, attributing this to the acoustic nature of the modes. However, a\nna¨ıve extension from [Nat. Commun. 14, 2955 (2023)] to 3D would suggest a scaling of\nω3. Is the difference due to the codimension of the defects (i.e., from point-like in 2D to\nline-like in 3D)? A more detailed discussion is warranted.\nWe thank the reviewer for highlighting this important point about the scaling of de-\nfect numbers with frequency. Indeed, a straightforward extension of the argument used to\nrationalize the two-dimensional (2D) results reported in [Nat. Commun. 14, 2955 (2023)]\nsuggests in 3D an ω3 scaling, if one assumes a volume-scaling of available modes in recipro-\ncal space. However, the observed scaling with ω2 arises due to the nature of the topological\ndefects in 3D. As illustrated in Fig. 2 of our manuscript, the analysis reveals that in 3D\nthe defects form extended line-like structures, and not discrete points. These TD lines\nemerge due to the transverse acoustic eigenmodes, which are the modes that have the\nlowest frequency, and when linearly superposed give rise to one-dimensional, filament-like\ntopological structures. These line defects effectively sample a two-dimensional manifold\nwithin the three-dimensional reciprocal space, leading to an inter-line spacing proportional\nto the wavelength (∼ ω−1). Consequently, the defect line density scales as the inverse\nsquare of this characteristic spacing, yielding the observed ω2 dependence. Our data (see\nFig. 3e) further confirms this interpretation. Specifically, we find that the inverse crossover\nlength scale, 1/ξ, scales linearly with frequency at low ω, consistent with an acoustic mode.\nThe slope of this linear regime corresponds to the sound velocity (see Supplementary In-\nformation), underscoring the acoustic nature and supporting the observed scaling.\nTo clarify this point we have added the following text\nLine 143:\nIn general a superposition of plane waves can produce a variety of TD morphologies,\ndepending on how the wave vectors are chosen [35]. In the low-frequency regime, the\nvibrational modes are predominantly transverse, and when the field is effectively a two-\n8\ncomponent (complex transverse) vector field, in three dimensions, the generic zero set of\nsuch a field has codimension 2 and thus forms line-like singularities. Consequently, the\nfilamentary nature of TDs and the observed ω2-scaling emerge naturally from codimension\nand spectral arguments: the typical inter-line spacing scales as ∼ ω−1, yielding a line-\nlength density ∼ ω2. By contrast, isolated point singularities are rare events and do not\ninfluence significantly the scaling behavior.\n8. Fractal exponent interpretation: The observed power-law with exponent −5/3\ncorresponds to a fractal dimension d ≈ 4/3, which is close to one. This seems to confirm\nf\nthe line-like nature of the defects rather than provide deeper insight. In essence, a set of\nquasi-one-dimensional objects embedded in 3D space would naturally yield such scaling.\nIf this is indeed the case, the authors may want to tone down claims of significance here.\nWe agree with the Reviewer that the value −5/3 we find for the observed scaling\nexponent, corresponding to a fractal dimension d ≈ 4/3, is closely related to the quasi-\nf\none-dimensional geometry of the topological defects (TDs) in 3D. However, we emphasize\nthat the significance of our result extends beyond merely confirming the line-like nature\nof these defects. Instead, our key finding is the unexpected close match between the spa-\ntial scaling exponent −5/3 characterizing the TD geometry in eigenmodes and the spatial\ndistribution of plastic events under quasistatic shear. This remarkable agreement provides\ncompelling evidence for a fundamental, previously unrecognized connection between mi-\ncroscopic eigenmode geometry and macroscopic plastic deformation. Thus, while the fact\nthat the fractal dimension is close to 1.0 itself might not be inherently surprising, its coin-\ncidence with the spatial patterns of plasticity substantially enhances our understanding of\nhow eigenmode structures govern the mechanical behavior of amorphous solids. We have\nnow carefully clarify this distinction in our revised manuscript.\nAdded text (Line 242):\nThis remarkable agreement provides compelling evidence for a fundamental, previously un-\nrecognized connection between microscopic eigenmode geometry and macroscopic plastic\ndeformation. Thus, while the fact that the fractal dimension is close to 1.0 itself is not\nsurprising, its coincidence with the fractal dimension characterizing the spatial patterns of\nplasticity substantially enhances our understanding of how eigenmode structures govern\nthe mechanical behavior of amorphous solids. This correspondence, rather than the abso-\nlute value of d , is another important physical insight which indicates that the morphology\nf\nof plastic rearrangements is organized by the pre-existing topology of low-frequency eigen-\nmodes.\n9. System size and acoustic modes: The discussion on small system sizes appears\ncounterintuitive. One might expect that in small systems, low-frequency acoustic modes\nare suppressed due to finite-size effects, allowing clearer observation of plasticity-related\ntopological features. This would, in principle, strengthen rather than weaken correlations\nwith plasticity. Clarification is needed.\nthere is a strong correlation between the location of the TD related to low frequency modes\nand the plastic events (PE). (By low frequency modes we refer here to modes with a wave-\nlength that is substantially larger than the inter-particle distance, i.e., these modes have\nacoustic nature.) If one does a similar analysis for a small system, one will of course\nfind PE and there may be some correlation with the TDs. But these PE cannot show a\ncorrelation with low frequency modes since in a small system the latter do not exist. By\nconsidering sufficiently large systems we thus allow the system decide with what TDs the\nPE are correlated since in the thermodynamic limit this will be the relevant mechanism\nfor plastic yielding. Only if the system size is large enough to accommodate low frequency\nmodes the system can pick up the relevant process. If the system is too small, yielding will\noccur, but the details will depend on the system size.\nAdded text (Line 309):\nSince well defined acoustic modes are present only in sufficiently large systems, this obser-\nvation implies that the study of the geometrical arrangement of these topological defects,\nand hence the identification of the correlation between TDs and plastic events, should not\nbe done in small systems since these do not include the acoustic-like modes that seem to\nbe relevant for the mechanical properties of a macroscopic sample.\n10. Non-locality of plastic events: The discussion claims that plastic events cannot\nbe understood locally and that non-local effects are essential. This argument should be\nelaborated more clearly. As currently phrased, it is not well-supported by the data or\nanalysis.\nWe are sorry that we didn’t explain clearly what we meant by this statement. A clas-\nsical view of plastic events is that these entities occur preferentially at special points in\nspace (often referred to as soft spots) and interact (quite weakly) via an elastic response\nof the system. This is, e.g., the basic assumption of the popular elasto-plastic models.\nWhat we discovered is that the soft spots where the PE occur show a subtle spatial cor-\nrelation (the power-law we document in Fig. 4d), i.e., they are not just arranged in a\nrandom manner. We rationalize this finding via the geometrical arrangement of the TDs,\ni.e., the fractal structure that we have identified, Fig. 3. This structure is in turn the\nresult of the interference of the acoustic modes, i.e., modes that are clearly delocalized.\nHence we conclude that the soft spots are indeed the result of non-local interference effects.\nAdded text (Line 317 in the Discussion):\nOur results show that the soft spots at which the plastic events occur possess non-trivial\nspatial correlations, so that the PE cannot be regarded as local rearrangements that nu-\ncleate independently. The two-point correlator of PE positions exhibits a scale-free regime\n(Figs.4d,e)withapower-lawdecaythatmirrorsthefractalorganizationoftheTDnetwork\n(Fig. 3), indicating that the soft spots are spatially organized by the pre-existing geometry\n10\nof the eigenmodes rather than being just a random set. The fractal TD-line structure\nitself emerges from the interference of low-frequency acoustic modes, whose wavelength\nsets the inter-line spacing (∼ ω−1) and whose phase coherence gives rise to the long-range\nstructure of the eigenvector field. It can be expected that, although the existence of a low-\ndimensional network of the PEs is due to the corresponding structure of the TDs, the PE\nstructure becomes enhanced by the presence of elastic interactions that are long-ranged,\ninducing additional correlations in plastic activity along this geometry.\n11. Fractal scaling in 2D vs. 3D: The discussion mentions that the fractal expo-\nnent 4/3 cannot be observed in 2D. Is this simply due to the absence of line defects in\ntwo-dimensional systems? If so, a clearer explanation would help the reader.\nWe thank the reviewer for pointing out the need for a clearer explanation regarding the\nabsence of the 4/3 fractal exponent in two-dimensional systems. Indeed, the reason lies in\nthe fundamental difference in the geometry of topological defects between 2D and 3D sys-\ntems. In three dimensions, the topological defects identified in the vibrational eigenmodes\nare line-like objects, and their spatial configurations can form scale-invariant, filamentary\nstructures. In contrast, in two-dimensional systems, topological defects are point-like. As\nargued in Nat. Commun. 14, 2955, 2023, in 2D the nature of the eigenvectors makes that\nthe topological singularities are isolated points if the frequency is low. Thus these isolated\nzeros of the eigenvector field lack the extended connectivity required to form filamentary\nor self-affine structures. As a result, the notion of a fractal dimension for connected defect\nlines does not apply in 2D, and the corresponding 4/3 scaling behavior is simply not rel-\nevant in this lower-dimensional setting. In order to clarify this distinction more explicitly\nin the revised manuscript to help readers better understand the dimensional dependence\nof topological defect geometry and scaling behavior, we have added the following text:\nAdded text (Line 177):\nThe exponent d ≈ 4/3 reported here quantifies the line geometry of the TD set in 3D. In\nf\ncontrast, this arrangement has no direct analogue in 2D systems, in which topological de-\nfects are point-like. As argued in Ref. [20], in 2D the nature of the eigenvectors makes that\nthe topological singularities are isolated points if the frequency is low. Thus these isolated\nzeros of the eigenvector field lack the extended connectivity required to form filamentary\nor self-affine structures.\n12. Relation to defective crystals: Inlines263–264,themanuscriptbrieflymentions\ncrystals. It would be relevant to cite [arXiv:2410.04720], which demonstrates that topolog-\nical defects in eigenvector fields correspond to conventional structural defects responsible\nfor plasticity in defective crystals. This connection further supports the universality of the\nproposed connection between eigenvector topology and plasticity.\nWe agree with the referee on this and we point out that this preprint cited in our origi-\nnal manuscript as Ref. [49], and we have explicitly discussed the important insights of this\n11\nwork on lines 300–303 of the original text.\n13. Figures in supplementary: In my opinion, the black background in the sup-\nplementary figures is not so nice. The authors might consider using a standard white\nbackground and change color scheme.\nWe thank the Reviewer for this suggestion. After some trials we feel that a black back-\nground is preferable.\n14. Figure S2: How is the case in the right of Fig.S2 interpreted? Is it interpreted as\nno vortex line is passing there?\nWe interpret it as two lines with different topology that terminate in the voxel. If the\nresolution would be increased, one would likely see a gap, since the topological signature\ncannot change in a continuous manner.\nAdded text (Caption of Fig. S2):\nIn the last case the two vortex lines are expected to terminate in the interior of the voxel.\nThis could be seen by increasing the resolution with which the field is probed, i.e., if the\nsize of the voxel is decreased.\n15. Nature of defects: If my understanding is correct, ideal plane waves would corre-\nspond to a perfectly ordered array of vortex lines, as illustrated in Fig. S5. Consequently, if\nthe same analysis were applied to an ideal three-dimensional crystal, one would expect (I)\na low-frequency scaling of defect number proportional to ω2, and (II) a spatial distribution\nof defects consistent with an exact fractal dimension d = 1. These expectations should\nf\nbe straightforward to verify through simulations of a perfect crystalline packing.\nGiven this, I wonder whether deviations from properties (I) and (II) in the glassy\nsystems studied here are indicative of truly glass-specific defects and plasticity. In particular,\nthe observation of a fractal dimension d slightly larger than 1 might signal the presence of\nf\nadditional topological structures that are absent in crystals and potentially responsible for\nplastic behavior in amorphous materials. The authors are encouraged to clarify this point\nfurther and, if possible, support their argument by performing analogous simulations in\n3D crystalline systems.\nWe appreciate the reviewer raising this insightful point. The slight deviation from an\nexact fractal dimension of 1 observed in our glassy systems (d ≈ 4/3) does indeed suggest\nf\nthe presence of additional complexity in defect structures specific to amorphous materials.\nSuch subtle but important differences likely reflect inherently glass-specific structural het-\nerogeneities and disorder-driven interactions absent in perfect crystals. We fully agree with\nthe reviewer that systematically exploring these differences via analogous simulations on\nperfect crystalline systems, and of crystalline system with a controlled number of defects,\nwould provide valuable insights. However, as performing these comparative analyses is\n12\nbeyond the current scope of our paper, we are unfortunately unable to include them at\nthis time. Nevertheless, we acknowledge the significance of this line of inquiry and intend\nto explicitly investigate it in future studies.\nWe thank the Reviewer for his/her multiple questions, suggestions, and constructive\ncomments which have allowed us to improve the clarity of the manuscript. We hope that\nour answers to the various points were satisfying and that therefore the new version of the\nmanuscript is now suitable for publications in Nature Communications.\nResponse to Reviewer #2\nI strongly recommend the publication of this manuscript. The current work makes very\nsubstantial contributions to the understanding and quantitative analysis of glassformers\nin terms of the system eigenvectors and defects and their admixture. Earlier approaches\nsuggesting global many-body eigenstate and topological defect descriptions of supercooled\nliquids have been made a decade ago by now. However, these approaches did not define nor\naddress topological defects in a manner as lucid as the current manuscript does. The intro-\nduced definition and characterization of defects that the authors of the current manuscript\nadvance is very novel and insightful. The relation to plasticity that is subsequently found\nin this manuscript was also very illuminating to read and learn of. I have only several\nminor comments below. These do not constitute a critique of any kind but are rather\nsuggestions. Whether or not any particular particular item below should be changed is left\nup to the authors.\nWe thank the Referee for the very positive evaluation of our work and are pleased to\nread that he/she liked the manuscript and the presented results, and acknowledges our\napproach to be novel and insightful.\n(1) The authors discovered a numerical correlation between negative topological defects\nand plastic events. It would be good to augment this by further possible physical under-\npinnings rationalizing this. Is it possible, say, to predict the onset of plastic events from\nthe existence of these defects and further illustrate how this prediction contrasts/correlates\nwith participation ratio metrics and non-affine displacements. This could be bolstered by\nstatistical metrics. This might make the observed correlation more compelling.\nWe greatly appreciate the reviewer’s valuable suggestion to further explore the physical\nmechanisms underlying the numerical correlation we observed between negatively charged\ntopological defects (TDs) and plastic events. Indeed, TDs carrying negative topological\ncharge are characterized by eigenvector flow fields corresponding locally to saddle-point\nconfigurations. Physically, these configurations imply local mechanical instability: Under\nexternal perturbations such as shear, the delicate force balance at these points is readily\ndisrupted, potentially triggering localized plastic rearrangements.\nPrevious studies have shown that TDs carrying negative topological charge are charac-\nterized by eigenvector flow fields corresponding locally to saddle-point configurations [20].\nPhysically, these configurations imply local mechanical instability: Under external per-\nturbations such as shear, the delicate force balance at these points is readily disrupted,\npotentially triggering localized plastic rearrangements and hence one expects a noticeable\ncorrelation between such TDs and PE.\nThe reviewer also suggests to explore the ability of these TDs to predict the onset of\nplasticity, and comparing this prediction with established metrics such as the participa-\ntion ratio and non-affine displacements. Quantifying the accuracy of such a prediction\nand compare it with the one of other indicators proposed in the literature is indeed an\nimportant task and the paper of Richard et al. PRM, 4, 113609 (2020) has shown the\nusefulness of such a study. However, conducting a comprehensive statistical comparison\nof TD-based predictions with these alternative metrics, while extremely valuable, goes be-\nyond the primary scope of our current manuscript, which is to introduce and explore the\nconcept of topological defect structures from a geometric standpoint. So we consider that\nthe main result obtained in this work is the insight that TD have a spatial arrangement\nthat is unexpected and non-trivial. That this arrangement is correlated with the location\nof the plastic events is an interesting consequence since it implies a non-trivial spatial ar-\nrangement of the PEs. However, we do not want to claim that the location of the TD is\nthe optimal predictor for the location of plastic events. In order to make such a claim one\nwould have to repeat the work done in the mentioned paper Richard et al., a task that\nwould certainly be very valuable, but is well beyond present scope of the manuscript.\nAdded text (Line 375 in the Discussion):\nIt is also important to clarify that we do not claim that the identified correlation between\nthe location of TD and PE make the TDs the best predictor regarding the occurrence of\nPEs. The quality of such a prediction would have to be assessed by following the approach\nby Richard et al [41], work that is outside the scope of the present study. Our result do\ninstead establish a valuable connection between well-defined physics based quantities, the\nTDs, and mechanical response of the system.\n(2) The authors state that the defects are scale-invariant and that the defects form\nquasi-linear structures at low frequencies. It might be good to add to discuss more the\nstatistical distribution of defect lengths, angles, and densities. Does the defect density\nchange with aging? Similar to the other numbered items in this list, since exploring this\nmay be time consuming, if the results are not known to the authors, whether or not such\nan additional analysis should be undertaken is up to the authors.\nWe thank the reviewer for this insightful suggestion regarding a more detailed sta-\ntistical analysis of the defect geometry. The angles between defects, their densities, and\n14\ntheir potential changes with aging, are indeed interesting and important properties of the\nTD network that merit to be explored in more detail since this will significantly enhance\nour understanding of TD behavior and aging phenomena in amorphous solids. We plan\nto address these questions systematically in future studies and mention this as promising\navenues for further research in our revised manuscript.\nAdded text (Line 405 in the Discussion):\nHence it will also be of interest to characterize the details of the geometry of the structure\nformed by the TDs, such as length of TD segments, angles between such segments, local\ndensities, etc.\n(3) If possible, it might be good to explicitly state how robust the results are when\ndifferent cooling rates are used and the initial conditions are varied.\nThis is indeed an important point since the yielding behavior of glasses are known to\ndepend on the cooling rate. Although in the present study we have not investigated how\nthe cooling rate affects the results, previous studies have shown that the fracture behavior\n(and thus the plastic events) do not depend on the cooling rate once the sample are cooled\ndown from the glassy-liquid state to low temperatures withing several million time steps\n(J. Non-Cryst. Solids 532, 119895 (2020)). Since in the present work we have used 30 mil-\nlion steps to cool the sample, we are confident that the glass is well annealed and hence our\nresults are robust. In fact, our strain-stress curves presented in Fig. S11 exhibit very sharp\nyielding behavior, closely resembling that of stable, well-annealed glasses. This indicates\nthat our large and carefully annealed sample is not affected by cooling rate effects, thus\nallowing us to obtain highly reproducible results.\nAdded text (Line 432 in Methods):\nThis slow cooling schedule ensures that the resulting samples are well-annealed and rep-\nresentative of stable glasses, as supported by previous studies [36] and confirmed by our\nstress–strain response (Fig. S11) which exhibits sharp yielding transitions characteristic of\nwell-annealed, mechanically stable glasses [61]. These features indicate the robustness and\nreproducibility of the mechanical and topological analyses reported in this work.\n(4) There have been works attempting to detect and quantify structure and defects in\nglassformers using machine learning approaches–both of the supervised and earlier unsu-\npervised (clustering or community detection) types. Unlike the current manuscript, the\ndetection of structure and defects with machine learning approaches has not been as simple\nand transparent. However it might be good to note these other approaches so as to further\nsituate the manuscript.\nWe thank the Reviewer for bringing up this valuable point. We fully agree that re-\ncent studies using machine learning have offered interesting ways to detect and quantify\nstructural heterogeneities and defects in glass-forming systems. While these methods can\n15\nprovide useful insights, the Reviewer points out correctly that usually they offer less trans-\nparency in physical interpretation compared to the direct topological approach used in\nour manuscript. In line with the Reviewer’s suggestion, we have included in the revised\nmanuscript a brief discussion on representative studies employing machine learning tech-\nniques to analyze structural defects in glass-forming materials. This addition will help to\nput into context our simpler and more transparent eigenvector-based topological approach\nwithin the broader landscape of methods available to study amorphous systems.\nAdded text (Line 381 in the Discussion):\nIn recent years, several machine learning (ML) approaches, both supervised and unsuper-\nvised, have been employed to detect and quantify structural heterogeneities and defect-like\nregions in glass-forming systems [42-44]. While these methods have yielded valuable in-\nsights into dynamic heterogeneity and plasticity, they often require a large number of\nmicroscopic descriptors. In contrast, our approach leverages quantities derived from vi-\nbrational eigenmodes to identify topological defects without supervision or descriptors,\nallowing for a more transparent, physically interpretable characterization. This simplicity\nof our method facilitates to establish a clearer connection between microscopic eigenmode\ngeometry and emergent plasticity morphology. In this way, our results not only extends\nthe insights gained from ML-based frameworks but may also offer useful insights for con-\nstructingmoreinterpretable[45,46], physics-informedMLmodelsforamorphousmaterials.\nWe thank the Reviewer for his/her multiple questions, suggestions, and constructive\ncomments which have allowed us to improve the clarity of the manuscript. We hope that\nour answers to the various points were satisfying and that therefore the new version of the\nmanuscript is now suitable for publications in Nature Communications.\nResponse to Reviewer #3\nIn “On the geometry of topological defects in glasses” by Wu, Barrat and Kob, the\nauthors present a generalisation/extension of their analysis of vortex defects in the vibra-\ntional eigenmodes and shear displacement fields of glasses from 2D to 3D. The original\n2D analysis was published recently (Wu, Z. W., Chen, Y., Wang, W.-H., Kob, W. & Xu,\nL. Topology of vibrational modes predicts plastic events in glasses. Nat. Commun. 14,\n2955 (2023)). The original report investigated the correlations between +1 and -1 vor-\ntex defects in 2D in both the vibrational eigenmodes and displacement field under shear.\nThe manuscript under consideration generalises the identification of defects to 3D and ex-\namines correlations between defects and plastic zones. Here, they observe a curious -5/3\nscaling behaviour in the spatial pair-correlations for defects found in the low frequency\neigenmodes. Overall, the work is interesting and a useful extension of the previous work\nin 3D. However, a deeper discussion of the significance of these vortex defects is highly\ndesirable.\n“curious” one, and acknowledge our work to be “interesting and a useful extension of the\nprevious work in 3D”.\nIn particular, the addition of plane waves forms vortex and anti-vortex arrays, a well-\nknown observation in optics (O’Holleran, Padgett, and Dennis, “Topology of optical vortex\nlines formed by the interference of three, four, and five plane waves,” Opt. Express 14,\n3039-3044 (2006)) and recently also noted for phonons in glass simulations (“Testing the\nheterogeneous-elasticitytheoryforlow-energyexcitationsinstructuralglasses”, Lernerand\nBouchbinder, Phys. Rev. E 111, L013402, (2025)). The 3D network of vortex/anti-vortex\ndefect lines that is formed depends on the wavelength and direction of the plane waves.\nExamining Figure 2 it appears that the vortex/anti-vortex arrays may be related to the su-\nperpositionoffundamentalvibrationalexcitationsofthecubicsimulationcell. Considering\nthis, the authors should discuss the deeper significance of these defects for glass properties,\nespecially plasticity and yielding and the origin of the intriguing scaling relationship.\nWe thank the Reviewer for this suggestion and for having pointed out these references\nto us. As the Reviewer correctly notes, in disordered solids, vibrational eigenmodes may\nbe viewed as superpositions of plane-wave-like excitations that are scattered by structural\ndisorder, as has been know for some time and elaborated in more detail in the article by\nLerner and Bouchbinder that is mentioned by the Reviewer. This interference gives rise to\na rich structure of topological defects (TDs), whose density and morphology evolve with\nfrequency. At low frequencies, where the modes have acoustic character, the resulting\nvortex arrays are relatively sparse and well-organized. With increasing frequency, the\ninterference pattern becomes increasingly complex, giving rise to denser, more tangled\ndefect structures. These observations are consistent with the literature on optical vortices\nas well as the recent work on glasses by Lerner and Bouchbinder. We now cite both papers\nin the revised manuscript and have clarified the origin and implications of the vortex/anti-\nvortex structures identified in our eigenmode analysis.\nThe physical significance of these TDs extends beyond their geometric features related\nto the mentioned acoustic modes. Each vortex core corresponds to a singular point in\nthe eigenvector field which suggests the presence of locally stable/unstable configurations,\ndepending on the sign of the TD. Regions with a negative charge, i.e., a local hyperbolic\ngeometry, are structurally predisposed to respond nonlinearly under deformation, aligning\nwith the concept of “soft spots” in glasses. Our analysis shows that plastic events un-\nder shear are indeed preferentially correlated with negatively charged TDs, supporting the\nnotion that the topology of vibrational eigenmodes encodes information about incipient\nplasticity. Moreover, the observed scaling of defect density with frequency (∼ ω2) reflects\nthe inverse scaling of vortex spacing with the wavelength of the mode (∼ 1/ω), as ex-\npected from an interference-based picture, and this scaling aligns with known features of\nacoustic-like modes in disordered solids. Hence this observation reinforces the idea that\nthe geometrical structure of the eigenmodes, and not just their energies, plays a key role\nfor the mechanical response of amorphous materials. Finally, the scale-invariant organi-\n17\nzation of vortex lines, together with their strong spatial correlation with plastic events,\nsuggests a potential structural mechanism underpinning avalanche-like yielding phenom-\nena in glasses. By linking the geometry of vibrational modes to the statistical behavior of\nplasticity, our findings contribute to a broader framework that unites microscopic excita-\ntions and macroscopic mechanical properties. We have revised the manuscript to better\nexpress this connection and have included the suggested citations.\nAdded text (Line 334 in the Discussion):\nThat the superposition of acoustic modes can give rise to a complex pattern of vortices\nin the displacement vector field, has already been shown in Ref. [39], although in that\nstudy no connection with TDs was made. This observation supports thus the view that\nthe results we have obtained regarding the structural arrangements of the TDs are generic.\nAdded text (Line 300):\nAs already argued in Ref. [20], this correlation between −1TDs and PEs is related to the\nfact that for such charges the local geometry of the normal mode field has a hyperbolic\nstructure, see Fig. S2 and S3, i.e., a small disturbance of the particle positions due to the\nshear will give rise in a significant change of the motion of the particle and hence most\nlikely result in a PE.\nTopologically non-trivial vortex defects are well-known in condensed matter systems\nthat have spontaneous symmetry breaking (for example, liquid crystals, quantum conden-\nsates, magnetic materials). Here the ordered phase consists of a magnitude which can be\nthe same everywhere (outside the vortex core where it is zero) and a direction that varies\nas a function of location. In these systems, vortex defects are detected from the direction\nalone, as the magnitude need not vary and does not contribute to the scalar winding num-\nber. This is the same as the analysis in this work, in which eigenvector and displacement\nmagnitudes are not used to find vortex defects, only the direction. However, magnitude\nclearly plays a role, for example in identifying plastic zones in the case of the sheared\nglass. The authors should justify the reason for ignoring the magnitude of the field when\nidentifying these defects.\nWe thank the reviewer for raising this important question regarding the role of vector\nmagnitude in detecting topological defects (TDs). We fully agree that the magnitude of\nthe eigenvector field might indeed be an important quantity, particularly when identifying\nregions prone to plastic deformation or instability in sheared glasses. However, our pri-\nmary goal in the present study is specifically to highlight the geometric and topological\naspects associated with the eigenvectors, i.e., information that depends only on the direc-\ntion of the vector field. By focusing on this directional information, our approach draws\ndirect analogies to well-established analyses in condensed matter physics, such as the Berry\ncurvature calculations used in momentum-space studies of electronic systems or the local\norientationalfieldinliquidcrystals. Inthesecontexts, theangularstructureofeigenvectors\nalone(withoutmagnitudeinformation)issufficienttocaptureessentialtopologicalfeatures\n18\nthat fundamentally influence physical properties. Our results show that also in the case\npresented here, the directional information of the eigenvectors already allows to identify\nthe surprisingly complex structure of the spatial arrangement of the TDs and to predict\nwith good accuracy the location of plastic events. Nevertheless, we fully acknowledge that\nincluding the information of the magnitude of the field is most likely useful, especially\nwhen attempting to precisely localize plastic events or soft-mode instabilities. How this\ninformation should be included is, however, not straightforward, since one has to come\nup with a reasonable method to characterize the strength of the singularity, and hence is\nbeyond the scope of the present work. In the revised version of the manuscript we now\nmention this point and point out its interest for future work.\nAdded text (Line 371 in the Discussion):\nNote that in the present work we have only probed the correlation between the location\nof the TDs and those of the PEs. In principle it is possible to take also into account the\nstrength of the topological singularity and this might permit to improve the mentioned\ncorrelation. Such an analysis is, however, beyond the scope of the present work and is thus\nleft to future investigations.\nIn summary, to strengthen the article the authors should add an in-depth discussion of\nthe significance of these defects and their correlations in relation to vortex networks and\ntopological vortex/anti-vortex defects observed in other systems.\nWe thank the reviewer for emphasizing the importance of discussing the significance of\ntheobserveddefectsandtheircorrelationswithvortexnetworksandtopologicalvortex/anti-\nvortex defects known from other physical systems. We regret that our original version of\nthe manuscript was not sufficiently clear on this point. In fact, we briefly addressed this\nimportant aspectin the concludingparagraph of ouroriginal manuscript, where we pointed\nout that the linear arrangement of topological defects (TDs) observed in our system shares\nsimilarities with vortex-line structures commonly observed in quantum fluids and gases.\nWe emphasized that our results potentially signals a connection to critical-like, avalanche-\ndriven deformation and plasticity behaviors widely reported in the literature. Indeed,\nrecent studies have shown universal scaling laws governing plastic events and energy dis-\nsipation across crystalline, polycrystalline, and amorphous systems, independent of their\nspecific microscopic details, but instead depending primarily on the underlying topology\nand elastic properties.\nIn the revised manuscript, we have now extended our discussion, explicitly linking our\nfindings to vortex and anti-vortex structures observed in other physical systems, thereby\nclearly expressing the broader significance and potential universality of the observed topo-\nlogicalfeaturesandtheirimplicationsforthedeformationandplasticbehaviorofmaterials.\nAdded text (Line 365 in the Discussion):\nIn the context of yielding of glassy materials, in recent experiments where the samples\nwere subjected to pure shear, the TDs did form filamentary structures before condensating\n19\ninto shear bands [22]. These results hint that it is indeed possible to probe in real systems\nthe spatial arrangement of the TDs and therefore to test whether they do indeed show the\nfractal structure that we predict from our study.\nSmall suggested improvements:\n- Fig 3 – label frequencies in graphs a,b,c\n- Suggest confirming scaling behaviours for larger simulations and smaller steps in strain.\nWe thank the reviewer for these helpful suggestions.\n(1) Regarding the labeling of frequencies in Fig. 3, we kindly point out that these fre-\nquency labels were already provided on the right-hand side of panel (a) in Fig. 3. However,\nto further enhance clarity and visibility, we have now explicitly included this information\nin the figure caption of Fig. 3 in our revised manuscript.\n(2) Regarding the scaling behavior under larger simulations and smaller strain incre-\nments,wehavetesteddifferentstrainincrementsexplicitly. Asdescribedinthemanuscript,\nwe applied athermal quasi-static simple shear with a strain increment of ∆γ = 0.0005, ob-\ntaining the stress-strain curve shown in Fig. 4a. Additionally, we performed simulations\nwith a doubled strain increment (∆γ = 0.001), and observed no qualitative differences\nin the stress-strain response or the scaling behaviors (see text on line 208-209 where we\ndiscuss Fig 4a.). Moreover, the sharp yielding behavior observed around γ = 0.098 and\nthe absence of stress drops prior to yielding indicate that our simulation system size is\nsufficiently large to avoid significant finite-size artifacts, as supported by previous studies\n(Refs. [Zhang et al., 2020; Richard et al., 2021]), (see text on line 208-209 where we discuss\nFig 4a.).\nWe thank the Reviewer for his/her multiple questions, suggestions, and constructive\ncomments which have allowed us to improve the clarity of the manuscript. We hope that\nour answers to the various points were satisfying and that therefore the new version of the\nmanuscript is now suitable for publications in Nature Communications.\n20"}, "author_response": ""}, {"round": 2, "reviewer_comments": {"reviewer_2": "Dear Editor,\nMuch effort was expended on this work. This revised manuscript by Wu, Barrat, and Kob reads very well. Their reply and changes in response to the other referees have further made this work more lucid and detailed. This is, in my opinion, quite an important and original work. I very strongly recommend the publication of this manuscript as it currently stands.\nWhether or not to include the below is up to the authors. As an exceptionally minor comment regarding one of the added items in the revised manuscript, prior to Refs. [42-44], there were earlier unsupervised machine learning approaches for finding structure and defects in glass-forming systems that did not require multiple descriptors, e.g., P. Ronhovde et al., The European Physical Journal E 34, 105 (2011); Scientific Reports 2, 329 (2012). These, however, were not nearly as illuminating and transparent as what the authors have achieved here.", "reviewer_3": "Thanks to the authors for considering and addressing my feedback in a revised manuscript. Their response and revisions address my concerns.", "reviewer_1_attached": "Response to the Reviewer’s Comments for\nNCOMMS-25-30828\nZhen Wei Wu, Jean-Louis Barrat, Walter Kob\nResponse to Reviewer #1\nAfter reading the clarifications and explanations provided in the authors’ reply, I now\nhaveabetterappreciationforthevalueandnoveltyofthiswork, particularlyincomparison\nto the previously published Nature Communications paper by a subset of the present\nauthors.\nIn light of these clarifications, I have revised my opinion and now consider this paper,\nin its clearer current form, to be valuable and potentially suitable for publication in Nature\nCommunications.\nWe thank the Reviewer for the positive assessment and for acknowledging the value\nand novelty of our work.\nHowever, I still have a few remaining questions that merit further clarification before\nI can recommend publication.\n1. I now understand that the method used by the authors does not introduce any\nbias toward the geometry of the topological defects (TDs). However, I remain somewhat\nconfused by this aspect. Let us focus on panel (a) of Fig. 2, for instance. There, I observe\nmany isolated points. What do these correspond to? Are they isolated point defects?\nIf so, what is the associated topological number (in a continuous description)? I would\nappreciate clarification regarding the nature of these isolated point defects. Are those the\nHedgehog defects discussed in the other work?\nWethanktheReviewerforraisingthispoint. Ourunderstandingisthatmost“isolated”\ndots in Fig. 2 are not point singularities, but cross-sections of vortex filaments imposed\nby the discretized lattice (i.e., short filament pieces below the visualization/resolution\nthreshold). Thisissupportedbytheobservationthatthesedotsarenotrandomlydispersed\nbut seem to outline a one-dimensional structure. The finite resolution of the lattice used\nfor detecting the topological defects will result that a single filament appears as a string\nof disconnected dots. In the Supplementary Information (Fig. S4), we present an example\nillustrating how a continuous (!) tilted line becomes locally modified because the presence\nof the lattice, i.e., the continuous line becomes a sequence of discrete points. As illustrated\n1\nfurther in Fig. R1 (now included in the SI as Fig. S7), an increase of the resolution,\ni.e., decrease of the lattice spacing, has the expected effect that the points move closer\ntogether, thus outlining the one-dimensional structure better. This behavior is thus a\nstrong indication that the underlying structure is a line and not a point.\nUnit cell length=0.5 Unit cell length=1.0\nFigure R1: An illustrative example of the effect of changing resolution. The two graphs\nshow a projection of a three-dimensional TD structure onto the yz plane with high (left\npanel) and low (right panel) resolution. Increasing the spatial resolution gives rise to an\nimproved outline of the expected continuity of the TD line.\nAlso the ω-dependence of the number of defects argues against an isolated point-defect\ninterpretation. True point singularities in a 3D three-component field require three inde\npendentconstraints(codimension3),yieldinganumberdensitythatscalesask3 (∝ ω3). By\ncontrast,thedefectswefindarezerosofaneffectivelytwo-component(transverse/complex)\nfield, whose generic zero set in 3D has codimension 2 and therefore forms lines, with a line\ndensity ∝ k2 (∝ ω2). In other words, the filamentary nature of TDs and the observed\nscaling emerge naturally from the intrinsic structure of the superposition of low-frequency\nacoustic-waves, aswediscussedinthemanuscript. OurmeasureddefectdensityN(ω) ∝ ω2\nis thus consistent with a filament picture and inconsistent with a dominant population of\ncodimension-3 point defects. For these reasons, we do not think that the apparent dots\nhave the nature of “hedgehogs” (monopoles) as it was claimed in Ref. [27]. To clarify these\npoints we have added the following text:\nAdded text (Caption of Fig. 2):\nMost isolated dots in the panels are not genuine point singularities but cross-sections\nor short, sub-resolution pieces of vortex filaments introduced by the discrete grid. As\none recognizes from the figure, these dots are not randomly dispersed but outline a one-\ndimensional structure; increasing spatial resolution (decreasing voxel size) causes them to\napproach each other thus outlining the one-dimensional structure better, see Fig. S7 in the\nSI for a resolution-dependent illustration of this effect.\nAdded text in the SI:\nThe above argumentation has also been added in the SI where we discuss Fig. S7.\n2. In the same figure, I notice several vortex lines that appear to terminate within the\nbulk of the 3D sample. My understanding is that this is not topologically allowed. For\nexample, in crystalline materials, dislocation lines can only terminate at free surfaces or at\nother defects such as grain boundaries. Could the authors clarify why, in this case, vortex\nlines can end within the system?\nWe agree with the Reviewer that in an ideal, continuous vector field with periodic\nboundaries, vortex filaments either form closed loops (including straight lines that wrap\naround the box because of the periodic boundary conditions) or meet at junctions; they\ncannot terminate arbitrarily in the bulk. This follows from conservation of total topological\ncharge, i.e., that the net winding number of the whole system is zero, which our detection\nprocedure preserves. Apparent terminations of a TD-line in 3D renderings could arise from\nfinite resolution: (i) The amplitude of the field generating the filament can drop below the\nnoise level of the field within a voxel, so the visual continuity appears broken; (ii) Two\nnearby segments can reconnect inside a single voxel, creating a local junction that, if the\nresolution is not high enough, looks like a termination of the filaments. In SI Sec. II (Figs.\nS2–S4), we present several cases of defect lines that “interact” with each other to form\ncomplex TD structures, and one sees that knowledge of whether or not two vortex points\nin neighboring planes form part of the same line is limited by the spatial resolution.\nTo clarify this point we have added in the caption of Fig. 2 the following text:\nAdded text (Caption of Fig. 2):\nNote that the filaments formed by the TDs do not end within the bulk of the sample arbi-\ntrarily, since for topological reasons this is forbidden. The apparent termination of these\nfilaments is thus due to insufficient spatial resolution of the eigen-vector field or approach-\ning line junctions.\n3. As the authors mentioned in their previous reply, defects can organize into both\nlines (vortex lines) and isolated entities (as mentioned in my first question). Would it be\npossible to quantify the relative proportion of these two types of defects, and to identify\nwhich of the two plays a more dominant role in the phenomena discussed in the paper? My\nimpression is that line defects are more relevant, as they are responsible for the observed\nfractal-like arrangements similar to those found in plastic events under deformation. Does\nthis imply that the number of isolated defects is significantly smaller than that of line\ndefects? A brief analysis along these lines would be useful.\norated in our response to point #1, genuine point singularities in a three-component field\nrequire three independent constraints (codimension 3), leading to a density ∝ k3 (∝ ω3).\nBy contrast, the defects we observe are zeros of an effectively two-component transverse\nfield, whose generic zero set in 3D has codimension 2 and thus forms filaments with line-\nlength density ∝ k2 (∝ ω2). Although from these phase space arguments isolated point\ndefects might seem to be more important than linear arrangements of the TD, the numer-\nical results show that this is not the case. The robust ω2 scaling we measure is therefore\ndirect evidence that TD are mainly arranged in the form of filamentary structures; true\npoint singularities, while possible, are statistically subdominant and do not control the ob-\nserved statistics. Crucially, also the spatial organization of plastic events under quasistatic\nshear exhibits a power-law with exponent -5/3, which matches that of the TD network ex-\ntracted from low-frequency eigenmodes. This correspondence is difficult to reconcile with\na picture dominated by truly isolated point defects and instead supports the view that\nplasticity is guided by the pre-existing filamentary topology of the vibrational modes.\n4. Finally, as a matter of curiosity, do the authors expect the observed power law to\nbe universal, or do they anticipate that it depends on the specific properties of the glass\nsystem under investigation?\nWe thank the reviewer for this important question regarding the universality of the\nobserved power-law scaling. We believe that the observed ω2 scaling of defect line density\nis generic for 3D disordered solids with acoustic vibrational modes. This is because the\nscaling derives from the general geometric and spectral properties of acoustic modes, i.e.,\ntheir long-wavelength character, delocalization, and codimension-2 nodal structure in the\ntransverse components.\nRegarding the −5/3 exponent, our simulations consistently find the same power-law\nfor both the TD network and the plastic events. We present this as a prediction. To the\nbest of our knowledge, emerging experimental measurements of the spatial organization of\nplastic events report a power-law close to −5/3, consistent with this prediction (Wang et\nal, unpublished studies). This experimental observation gives additional evidence for the\nrobustness and physical relevance of the scaling laws established in our work, and rein-\nforces the proposed link between the eigenmode geometry (overall structure of the TDs)\nand plastic deformation in glasses. We agree, however, that its broader universality across\ndifferent glass families and interaction models remains to be experimentally and numer-\nically tested and obtaining more insight on this will advance our understanding of the\nmechanical behavior of glasses.\nWe thank the Reviewer for having brought up these interesting questions and hope to\nhave answered them in a satisfactory manner.\nMuch effort was expended on this work. This revised manuscript by Wu, Barrat, and\nKob reads very well. Their reply and changes in response to the other referees have further\nmade this work more lucid and detailed. This is, in my opinion, quite an important and\noriginal work. I very strongly recommend the publication of this manuscript as it currently\nstands.\nWhether or not to include the below is up to the authors. As an exceptionally minor\ncommentregardingoneof theaddeditems intherevised manuscript, priortoRefs. [42-44], there were earlier unsupervised machine learning approaches for finding structure and defects in glass-forming systems that did not require multiple descriptors, e.g., P. Ronhovde et al., The European Physical Journal E 34, 105 (2011); Scientific Reports 2, 329 (2012). These, however, were not nearly as illuminating and transparent as what the authors have achieved here.\nWe thank the Reviewer again for the high evaluation, strong recommendation of our\nwork and for highlighting its originality, clarity, and transparency. We are also grateful for\nthe historical references to earlier unsupervised ML-based approaches, which we have now\ncited in the revised text to acknowledge the development of structural detection methods\nin glass physics.\nResponse to Reviewer #3\nThankstotheauthorsforconsideringandaddressingmyfeedbackinarevisedmanuscript.\nTheir response and revisions address my concerns.\nWe thank the Reviewer again for his/her positive evaluation and for confirming that\nall prior concerns have been addressed.\n5"}, "author_response": "Response to the Reviewer’s Comments for\nNCOMMS-25-30828A\nZhen Wei Wu, Jean-Louis Barrat, Walter Kob\nResponse to Reviewer #1\nAfter reading the clarifications and explanations provided in the authors’ reply, I now\nhaveabetterappreciationforthevalueandnoveltyofthiswork, particularlyincomparison\nto the previously published Nature Communications paper by a subset of the present\nauthors.\nIn light of these clarifications, I have revised my opinion and now consider this paper,\nin its clearer current form, to be valuable and potentially suitable for publication in Nature\nCommunications.\nWe thank the Reviewer for the positive assessment and for acknowledging the value\nand novelty of our work.\nHowever, I still have a few remaining questions that merit further clarification before\nI can recommend publication.\n1. I now understand that the method used by the authors does not introduce any\nbias toward the geometry of the topological defects (TDs). However, I remain somewhat\nconfused by this aspect. Let us focus on panel (a) of Fig. 2, for instance. There, I observe\nmany isolated points. What do these correspond to? Are they isolated point defects?\nIf so, what is the associated topological number (in a continuous description)? I would\nappreciate clarification regarding the nature of these isolated point defects. Are those the\nHedgehog defects discussed in the other work?\nWethanktheReviewerforraisingthispoint. Ourunderstandingisthatmost“isolated”\ndots in Fig. 2 are not point singularities, but cross-sections of vortex filaments imposed\nby the discretized lattice (i.e., short filament pieces below the visualization/resolution\nthreshold). Thisissupportedbytheobservationthatthesedotsarenotrandomlydispersed\nbut seem to outline a one-dimensional structure. The finite resolution of the lattice used\nfor detecting the topological defects will result that a single filament appears as a string\nof disconnected dots. In the Supplementary Information (Fig. S4), we present an example\nillustrating how a continuous (!) tilted line becomes locally modified because the presence\nof the lattice, i.e., the continuous line becomes a sequence of discrete points. As illustrated\n1\nfurther in Fig. R1 (now included in the SI as Fig. S7), an increase of the resolution,\ni.e., decrease of the lattice spacing, has the expected effect that the points move closer\ntogether, thus outlining the one-dimensional structure better. This behavior is thus a\nstrong indication that the underlying structure is a line and not a point.\nUnit cell length=0.5 Unit cell length=1.0\nFigure R1: An illustrative example of the effect of changing resolution. The two graphs\nshow a projection of a three-dimensional TD structure onto the yz plane with high (left\npanel) and low (right panel) resolution. Increasing the spatial resolution gives rise to an\nimproved outline of the expected continuity of the TD line.\nAlso the ω-dependence of the number of defects argues against an isolated point-defect\ninterpretation. True point singularities in a 3D three-component field require three inde\npendentconstraints(codimension3),yieldinganumberdensitythatscalesask3 (∝ ω3). By\ncontrast,thedefectswefindarezerosofaneffectivelytwo-component(transverse/complex)\nfield, whose generic zero set in 3D has codimension 2 and therefore forms lines, with a line\ndensity ∝ k2 (∝ ω2). In other words, the filamentary nature of TDs and the observed\nscaling emerge naturally from the intrinsic structure of the superposition of low-frequency\nacoustic-waves, aswediscussedinthemanuscript. OurmeasureddefectdensityN(ω) ∝ ω2\nis thus consistent with a filament picture and inconsistent with a dominant population of\ncodimension-3 point defects. For these reasons, we do not think that the apparent dots\nhave the nature of “hedgehogs” (monopoles) as it was claimed in Ref. [27]. To clarify these\npoints we have added the following text:\nAdded text (Caption of Fig. 2):\nMost isolated dots in the panels are not genuine point singularities but cross-sections\nor short, sub-resolution pieces of vortex filaments introduced by the discrete grid. As\none recognizes from the figure, these dots are not randomly dispersed but outline a one-\ndimensional structure; increasing spatial resolution (decreasing voxel size) causes them to\napproach each other thus outlining the one-dimensional structure better, see Fig. S7 in the\nSI for a resolution-dependent illustration of this effect.\nAdded text in the SI:\nThe above argumentation has also been added in the SI where we discuss Fig. S7.\n2. In the same figure, I notice several vortex lines that appear to terminate within the\nbulk of the 3D sample. My understanding is that this is not topologically allowed. For\nexample, in crystalline materials, dislocation lines can only terminate at free surfaces or at\nother defects such as grain boundaries. Could the authors clarify why, in this case, vortex\nlines can end within the system?\nWe agree with the Reviewer that in an ideal, continuous vector field with periodic\nboundaries, vortex filaments either form closed loops (including straight lines that wrap\naround the box because of the periodic boundary conditions) or meet at junctions; they\ncannot terminate arbitrarily in the bulk. This follows from conservation of total topological\ncharge, i.e., that the net winding number of the whole system is zero, which our detection\nprocedure preserves. Apparent terminations of a TD-line in 3D renderings could arise from\nfinite resolution: (i) The amplitude of the field generating the filament can drop below the\nnoise level of the field within a voxel, so the visual continuity appears broken; (ii) Two\nnearby segments can reconnect inside a single voxel, creating a local junction that, if the\nresolution is not high enough, looks like a termination of the filaments. In SI Sec. II (Figs.\nS2–S4), we present several cases of defect lines that “interact” with each other to form\ncomplex TD structures, and one sees that knowledge of whether or not two vortex points\nin neighboring planes form part of the same line is limited by the spatial resolution.\nTo clarify this point we have added in the caption of Fig. 2 the following text:\nAdded text (Caption of Fig. 2):\nNote that the filaments formed by the TDs do not end within the bulk of the sample arbi-\ntrarily, since for topological reasons this is forbidden. The apparent termination of these\nfilaments is thus due to insufficient spatial resolution of the eigen-vector field or approach-\ning line junctions.\n3. As the authors mentioned in their previous reply, defects can organize into both\nlines (vortex lines) and isolated entities (as mentioned in my first question). Would it be\npossible to quantify the relative proportion of these two types of defects, and to identify\nwhich of the two plays a more dominant role in the phenomena discussed in the paper? My\nimpression is that line defects are more relevant, as they are responsible for the observed\nfractal-like arrangements similar to those found in plastic events under deformation. Does\nthis imply that the number of isolated defects is significantly smaller than that of line\ndefects? A brief analysis along these lines would be useful.\norated in our response to point #1, genuine point singularities in a three-component field\nrequire three independent constraints (codimension 3), leading to a density ∝ k3 (∝ ω3).\nBy contrast, the defects we observe are zeros of an effectively two-component transverse\nfield, whose generic zero set in 3D has codimension 2 and thus forms filaments with line-\nlength density ∝ k2 (∝ ω2). Although from these phase space arguments isolated point\ndefects might seem to be more important than linear arrangements of the TD, the numer-\nical results show that this is not the case. The robust ω2 scaling we measure is therefore\ndirect evidence that TD are mainly arranged in the form of filamentary structures; true\npoint singularities, while possible, are statistically subdominant and do not control the ob-\nserved statistics. Crucially, also the spatial organization of plastic events under quasistatic\nshear exhibits a power-law with exponent -5/3, which matches that of the TD network ex-\ntracted from low-frequency eigenmodes. This correspondence is difficult to reconcile with\na picture dominated by truly isolated point defects and instead supports the view that\nplasticity is guided by the pre-existing filamentary topology of the vibrational modes.\n4. Finally, as a matter of curiosity, do the authors expect the observed power law to\nbe universal, or do they anticipate that it depends on the specific properties of the glass\nsystem under investigation?\nWe thank the reviewer for this important question regarding the universality of the\nobserved power-law scaling. We believe that the observed ω2 scaling of defect line density\nis generic for 3D disordered solids with acoustic vibrational modes. This is because the\nscaling derives from the general geometric and spectral properties of acoustic modes, i.e.,\ntheir long-wavelength character, delocalization, and codimension-2 nodal structure in the\ntransverse components.\nRegarding the −5/3 exponent, our simulations consistently find the same power-law\nfor both the TD network and the plastic events. We present this as a prediction. To the\nbest of our knowledge, emerging experimental measurements of the spatial organization of\nplastic events report a power-law close to −5/3, consistent with this prediction (Wang et\nal, unpublished studies). This experimental observation gives additional evidence for the\nrobustness and physical relevance of the scaling laws established in our work, and rein-\nforces the proposed link between the eigenmode geometry (overall structure of the TDs)\nand plastic deformation in glasses. We agree, however, that its broader universality across\ndifferent glass families and interaction models remains to be experimentally and numer-\nically tested and obtaining more insight on this will advance our understanding of the\nmechanical behavior of glasses.\nWe thank the Reviewer for having brought up these interesting questions and hope to\nhave answered them in a satisfactory manner.\nMuch effort was expended on this work. This revised manuscript by Wu, Barrat, and\nKob reads very well. Their reply and changes in response to the other referees have further\nmade this work more lucid and detailed. This is, in my opinion, quite an important and\noriginal work. I very strongly recommend the publication of this manuscript as it currently\nstands.\nWhether or not to include the below is up to the authors. As an exceptionally minor\ncommentregardingoneof theaddeditems intherevised manuscript, priortoRefs. [42-44], there were earlier unsupervised machine learning approaches for finding structure and defects in glass-forming systems that did not require multiple descriptors, e.g., P. Ronhovde et al., The European Physical Journal E 34, 105 (2011); Scientific Reports 2, 329 (2012). These, however, were not nearly as illuminating and transparent as what the authors have achieved here.\nWe thank the Reviewer again for the high evaluation, strong recommendation of our\nwork and for highlighting its originality, clarity, and transparency. We are also grateful for\nthe historical references to earlier unsupervised ML-based approaches, which we have now\ncited in the revised text to acknowledge the development of structural detection methods\nin glass physics.\nResponse to Reviewer #3\nThankstotheauthorsforconsideringandaddressingmyfeedbackinarevisedmanuscript.\nTheir response and revisions address my concerns.\nWe thank the Reviewer again for his/her positive evaluation and for confirming that\nall prior concerns have been addressed.\n5"}]

Decision letter