Cooperative atomic motion during shear deformation in metallic glass

Domain: physics
Published: 2026-01-12
Editorial outcome: minor_revision

Abstract
Elucidating mechanical deformation in glassy materials at the atomic level is challenging due to their disordered atomic structure. Using our frozen-atom analysis of the simulation data, we reveal that an elastic deformation in CuZr metallic glasses is fundamentally driven by cooperative atomic motions of tens of atoms elastically linked to one another, forming trigger groups. They initiate localized rearrangements, which can cascade into plastic flow. These cores show no clear structural or elastic precursors in the initial configuration, challenging the idea that deformation occurs in defective regions. Instead, deformation events are highly stochastic and transient, driven by collective atomic motion. This finding not only reshapes our understanding of glassy material deformation mechanisms but also highlights cooperative motion as a key factor in avalanche-like phenomena governing the behavior of disordered systems across multiple scales.

Introduction
Unraveling the microscopic mechanisms behind plastic and anelastic deformation in glassy materials, such as metallic glasses, remains an enduring mystery over half a century1–3. In crystalline materials, lattice defects, such as dislocations, are carriers of plasticity, playing a crucial role in determining mechanical properties. However, the absence of long-range atomic ordering in glassy materials makes it difficult to define defects. Moreover, in crystals, dislocation density and pinning effects are highly sensitive to microstructural variations, leading to significant differences in mechanical strength4,5. In contrast, metallic glasses display remarkably universal mechanical responses across different compositions and processing conditions6, suggesting that the conventional paradigm linking structural defects to mechanical behavior may not directly apply to metallic glasses. This raises fundamental questions: if defects are not the primary deformation carriers, what governs mechanical behavior in glassy materials? Identifying these underlying mechanisms could lead to a more comprehensive understanding of deformation behavior in strongly disordered solids.

We show, through computational simulations of shear deformation in metallic glasses, that the fundamental deformation elements in these materials are atomic groups exhibiting cooperative motion. Numerous studies have investigated atomic cooperativity in glasses and supercooled liquids, both under applied stress7–11 and in stress-free conditions12–17. Yet, even in simulations, cooperative movements have thus far been inferred only from the displacements of atomic groups during stress relaxation events. A widely used framework for describing the plastic deformation of metallic glasses is based on shear transformation zones (STZs), which are localized, collective atomic rearrangements6,7,9,18. In the framework of self-organized criticality, the interaction and propagation of STZs in an avalanche-like manner have been widely discussed19–21. These discussions suggest that plastic deformation is governed by strongly interacting elements exhibiting collective behavior across multiple length scales. However, a detailed microscopic understanding of these processes—particularly the triggers for deformation events and the nature of their interactions—remains incomplete. We identified this cooperative behavior as the essential core of STZs, which we refer to as the STZ core. We analyze their size, structural, and mechanical characteristics, while also revealing the presence of cascading STZ cores that initiate the early stages of avalanches. This analysis enhances the understanding of deformation mechanisms in metallic glasses and other amorphous materials.

Cooperativity in glassy materials is concealed within their complex atomic dynamics, making its extraction challenging. To address this, we introduce frozen-atom analysis, a computational technique that explores what-if scenarios. In real life, it is impossible to go back in time, but in a simulation, we can trace time back, like in a time-machine, and change history. We identify a local plastic deformation event, and go back in time to the instance just before the start of the event. Then we artificially immobilize an individual atom. If this atom is involved in cooperative motion, the deformation event will not happen when the simulation is continued with the atom immobilized. We applied this conditional freezing to each atom in the system and identified the atoms involved in cooperative motion, thereby revealing the STZ core. Through the comparison of parallel worlds enabled by frozen-atom analysis, we uncover the fundamental nature of STZs, distinct from the conventional perspectives that rely upon the post-mortem analysis of the event. We found that the STZ core consists of tens of atoms, which engage in cooperative motion. This value may differ among various glass systems; nevertheless, the frozen-atom analysis provides an unambiguous determination of it. These atoms lack distinct structural features, suggesting that the STZ core can emerge anywhere. Furthermore, it induces plastic deformation among the surrounding atoms and acts as a trigger, initiating cascades that drive avalanches. These findings represent a drastic departure from the conventional view that STZs correspond to defects or soft spots, paving the way toward deeper insights into the physical principles governing the deformation mechanisms of glassy materials.

Results
Frozen-atom analysis
The frozen-atom analysis enables the unambiguous identification of atomic groups involved in cooperative motion, specifically STZ cores. The athermal quasi-static (AQS) method22,23 is a widely used computational approach for modeling critical slip phenomena under quasi-static shear in glassy materials. In this method, a model glass undergoes incremental affine shear deformation, followed by athermal relaxation, transitioning through mechanically stable configurations. A characteristic feature of this process is the sudden shear stress drop during relaxation, signaling a plastic deformation event involving atomic rearrangements (Fig. 1a). Frozen-atom analysis is applied at this critical stress drop to pinpoint the STZ core. The method starts from the atomic configuration just before the deformation event, applies an affine deformation, and artificially freezes one atom while allowing the rest to relax. If the plastic deformation event does not occur, the frozen atom is identified as a part of the STZ core, confirming its essential role in the deformation event. Figure 1b schematically illustrates that freezing a single atom disrupts the cooperative motion of the entire group, emphasizing their interdependence, although this representation is oversimplified because a three-dimensional glass has much higher degrees of freedom.

We implement this method as follows: during the relaxation process of a stress-drop event in the AQS procedure, one atom from a system of N atoms is selected, and its motion is artificially frozen. The remaining atoms are allowed to relax, and the total displacement of the entire system is recorded. This is referred to as the Df parameter (in Å) here.

Df = \sqrt{\sum_{i=1}^{N} d_i^2}, (1)

where di represents the displacement vector of atom i during the structural relaxation and N denotes the total number of atoms in the simulation box. This process is repeated for every atom, generating a Df parameter value for each atom. This Df parameter clearly distinguishes between atoms that participate in cooperative motion and those that do not: a Df value of zero indicates that freezing the atom’s motion disrupts the cooperative displacement, showing that the atom with Df = 0 is a part of the deformation element. Conversely, a non-zero Df value indicates that the atom’s displacement is not critical to the cooperative motion. This method appears similar to the artificial manipulations, such as the atomic pinning approach to probe atomic cooperativity in flow24,25 or other related interventions26,27, but our approach is quite different in that atomic freezing is used merely to probe the role of each atom in deformation, and it does not affect the flow itself (see Supplementary Note 1 for details).

In conventional simulations and experiments, pinpointing the atomic-level interactions or mechanisms that drive observed plastic deformation is quite challenging. Here, by deliberately controlling the motion of specific atoms, frozen-atom analysis enables us to virtually modify their movements and directly probe causal relationships between an atom and the flow. Comparing multiple “parallel worlds,” where different atoms are immobilized, makes it possible to extract the hidden cooperative motion embedded within the disordered structure of glassy materials.

Identification of atomic group in cooperative motion
The histogram of the Df parameter obtained from a stress-drop event is shown in Fig. 1c as an example, from which three groups of atoms can be identified, as illustrated on the right side of the figure. The first group consists of atoms with Df values close to zero (visualized as Df ≤ 0.5 in the figure on the right). As mentioned earlier, these are the atoms frozen during the relaxation process that halt the motion of other atoms. In other words, this group represents atoms critically needed for cooperative motion. As shown in the Methods section, such cooperative motion was observed in all 60 stress relaxation events investigated in this study. Therefore, it can be concluded that all STZ cores observed here are the atomic groups critical for cooperative motion.

The second group of atoms is located near the peak of the histogram. Even when their motion is frozen, the cooperative motion occurs during the relaxation process, meaning they are not directly involved in the activation of the deformation event. The peak position of the histogram corresponds to the Df value when no atoms are frozen. As shown in the figure on the right (0.5 < Df ≤ 1.3 case), most atoms in the system belong to this group. The third group, found in the tail of the histogram and located near the STZ group (Df > 1.3 case), experiences large displacements due to cooperative motion, but this does not affect the deformation event. When their motion is frozen, all other atoms move to reproduce the original displacement, leading to large Df values for this group, indicating that they do not stop the cooperative motion.

A remarkable feature of the Df parameter lies in the gap observed in the histogram. As shown in Fig. 1c, this gap clearly distinguishes the group of atoms belonging to the STZ core from other atoms. For comparison, Fig. 1d and e show histograms of conventional indicators: the non-affine displacement measure D2min and the change in atomic von Mises stress during relaxation. The atoms in the tail of the distribution in Fig. 1d, namely, those with large non-affine deformation, are often assumed to represent the central region of the STZ. Similarly, in Fig. 1e the atoms exhibiting the most pronounced changes in local von Mises stress are also regarded as part of this region. In both cases, although these descriptors capture atoms involved in the event, the size of the atomic group they identify can vary significantly depending on the chosen threshold, which makes their interpretation ambiguous. In contrast, the frozen-atom analysis provides an unambiguous distinction between the indispensable STZ core (the group close to Df) and the surrounding atoms. The sizes of the STZ cores obtained in this study were up to 116 atoms, with an average of about 40 atoms, which is consistent with the sizes of STZs reported in previous studies1. However, in previous studies, the number of atoms constituting the STZ depends on arbitrary parameters, such as the threshold value. In contrast, the frozen-atom method with the Df parameter identifies the STZ core without ambiguity.

Characteristics of STZ core
The characteristics of STZs, including their size, shape, distribution, and hardness, have been subjects of long-standing debate1,28. We studied the atomic-level characteristics of the STZ core atoms in the zero-strain initial configuration before any shear was applied, in order to test whether precursors of STZ cores exist. The nature of the STZs identified in this study differs significantly from those previously discussed. Figure 2a illustrates the spatial distribution of STZ cores detected in the 60 events considered. These STZ cores are widely spread across the system, indicating that STZ cores do not form in a spatially confined region but rather emerge throughout the material. The color scale shows how often each atom was classified as part of an STZ core. The low occurrence of overlaps suggests that these events rarely occur repeatedly in the same location.

Figure 2b plots the features of Voronoi polyhedra, specifically the number of faces and the volume in the initial configuration, for the 100,000 atoms randomly sampled from 960,000 atoms considered in this calculation. The circle size represents the Df parameter, where larger circles basically correspond to atoms identified as participating in cooperative motion. Despite the widespread use of Voronoi polyhedra as descriptors for characterizing the atomic structure of defects, including STZs, the nearly uniform distribution of atoms with low Df values within the STZs suggests that extracting STZs based solely on Voronoi features may not be a reliable approach. Figure 2c compares histograms of atomic shear modulus at zero strain for STZ core atoms detected in each shear direction with those of non-core atoms. The obtained histograms and modulus values are consistent with those reported in previous studies29,30. No significant difference in hardness was observed between STZ core atoms and other atoms. To further examine the elastic properties, we conducted coarse-grained evaluations of atomic elasticity based on the STZ cores. Even in the undeformed configuration, variability was evident: some STZ cores were stiffer than the cell average, while others were softer. The average shear moduli of the STZ cores were 40.5 ± 3.8 GPa (xy), 40.7 ± 3.1 GPa (yz), and 42.0 ± 3.9 GPa (zx), compared with the overall averages of 41.8, 41.9, and 41.7 GPa, respectively. This indicates that STZ cores in the initial state are neither soft spots as traditionally perceived nor hard spots; in other words, they do not exhibit any distinct characteristics in terms of elastic constant.

In the “Methods” section, we present the evolution of the average atomic energy and the von Mises stress within each STZ core during the AQS process. Even within the apparent elastic regime, energy and stress responses exhibit strong variation with applied stress. This indicates that the mechanical response of this system is highly nonlinear, and atoms are possibly influenced by interactions with other atoms within the STZ core and in other activated STZ cores. This underscores the role of intrinsic structural inhomogeneity in driving localized structural and mechanical changes under external stimuli, challenging predictions based on the initial state.

Figure 3 explores the relationship between the STZ core size and the magnitude of stress drops observed in the 60 events considered. The histogram of stress drop magnitude follows a power-law distribution (Fig. 3a), whereas the histogram of the STZ core differs from it, exhibiting a shape similar to a Poisson distribution (Fig. 3b). Whereas it is widely believed that the size of the STZ is proportional to the magnitude of the stress drop, the histograms of stress drop magnitude and STZ core size in this study show otherwise. Even though some weak correlations may be found in Fig. 3c between the magnitude of stress drop and the macroscopic strain at which the event occurred, the size of the STZ core has no correlation with either.

These results suggest that the atoms in STZ cores are not in special environments that could be identified as defects, in agreement with the earlier study31. It is likely that any atom can be involved in STZ cores. This may explain why the critical strength is so universal among various metallic glasses6.

Local configurational excitation in STZ core
To further examine the nature of the STZ core thus defined at the atomic level, we investigated the local configurational excitation (LCE) that occurs during the stress-drop process. LCE refers to the action of losing or gaining nearest-neighbor atoms around a particular atom. This change in the atomic connectivity network due to bond switching is known to govern the macroscopic viscosity of high-temperature liquids (details are provided in ref. 32 and the “Methods” section). Figure 4a compares the number of bond formations and breakages that occur in a single atom during the stress relaxation process between STZ core atoms and non-STZ core atoms. It was found that most LCEs involving three or more bond breakages occur in atoms belonging to the STZ core.

We observe that many atoms in the close vicinity of the STZ core are also displaced due to the activation of the core, as exemplified by the third atom group with Df > 1.3 in Fig. 1c. LCEs that occur in the STZ core induce non-affine displacements in its surroundings, and these displacements can sometimes drive other STZ cores. Figure 4a shows that some atoms, despite being classified as non-STZ core, exhibit a significant number of LCEs. Figure 4b illustrates the corresponding event. While the Df parameter identifies an STZ core, other parameters suggest that other atoms in the vicinity are also involved in the deformation event. Of the atoms involved in LCE, only 60% are included in the STZ core, while the remainder are either surrounding the STZ core or part of events triggered by the core. The LCE analysis revealed that eight out of the 60 events considered exhibited cascade-like behavior (see the “Methods” section for details). These results imply that the activation of one STZ core triggers a local avalanche of deformation, which is more extended and involves a larger number of atoms. The deformation event, or the STZ, is complex and involves the trigger and subsequent large local deformation in its vicinity. This distinction between triggering and triggered STZ regions becomes clear through the frozen-atom analysis, which enables their separation.

Methods
Simulationsetup
We conducted quasi-static shear deformation on a periodic three-dimensional metallic glass Cu50Zr50, comprising 16,000 atoms. The molecular dynamics simulations were carried out using the LAMMPS software package41 with the embedded atom method (EAM) potential for Cu-Zr42. The metallic glass was prepared by melting and quenching a B2 CuZr alloy. A velocity distribution corresponding to 2000 K, which is well above the glass transition temperature of this alloy (~750 K, see Supplementary Fig. 2), was applied to the B2 structure, and the system was directly cooled to 0 K at a rate of 109 K/s in an NVT ensemble using the Nosé–Hoover thermostat43,44 without an intermediate annealing period. After cooling, the atomic structure and simulation cell were relaxed using the conjugate gradient method until all stress components converged to below 0.002 MPa. The final simulation box had a characteristic length of ~65.4 Å and was nearly cubic but with slight tilts in the cell vectors, as shown in Supplementary Table 1.

The athermal quasi-static (AQS) method22,23 was then employed to apply quasi-static shear deformation. This involved applying a small shear strain to the cell vectors of the glass and affinely deforming the atomic structure in accordance with the strain, followed by energy minimization at 0 K to ensure zero force on all atoms. This procedure was repeated 15,000 times to induce deformation until the engineering shear strain reached 15%, with a strain increment of 10−5 per step. A total of 60 plastic events were sampled, including ten events for each of the six shear directions (±xy, ±yz, ±zx), as shown in Fig. 5.

Frozen-atom analysis to identify STZ core
Various models, such as the shear transformation zone (STZ) model6,7,9,18, have been proposed to describe deformation behavior, and a number of attempts have been made to identify the deformation mechanism through molecular dynamics simulations23,35,37,45–48. In the typical approach, the AQS method, incremental uniform (affine) shear deformation is applied to a model glass structure, and the atomic structure is relaxed athermally at each deformation step. When the shear stress drops during the relaxation, it is assumed that a plastic deformation event occurs through some rearrangement in atomic configuration. For convenience, we call the area with strong plastic deformation an STZ. The changes in certain geometric or physical quantities before and after the relaxation are used to extract features of local atomic behavior. To identify the STZ, various metrics have been introduced. These include the non-affine squared displacement (D2min)18,47,49, Voronoi-based metrics50,51, local stress drops52, local shear elasticity53, von Mises strain, atomic displacements54, and soft modes obtained through Hessian diagonalization55. Among these, D2min has been particularly widely used. However, this method suffers from significant problems. In defining the non-affine displacement, it is assumed that the affine strain is uniform, whereas the actual stress field is rather inhomogeneous in space. For instance, when an STZ is activated, local shear softening occurs, producing an elastic field around the STZ2,56. Furthermore, to identify the STZ, a threshold value for D2min is set57. However, the choice of this value significantly affects the spatial distribution of the deformation elements, as shown later, introducing large ambiguity. In order to elucidate the deformation mechanism, in this work, we take a different approach and re-examine the definition of the deformation zones, such as STZs.

In the frozen-atom analysis, we start with the atomic configuration just a step before a cooperative deformation event (see Fig. 1a). Then, after applying an affine deformation that will result in the deformation event, we artificially freeze an atom, and relax the rest during the relaxation process. If the cooperative motion ceases as a result of this freezing, we identify this atom as belonging to the atomic group we define as the STZ core, because the movement of this atom is essential for the deformation event to take place. Figures 6 and 7 present 20 events under −xy and +xy shear as examples, illustrating results similar to those in Fig. 1. Note that the center of the gravity of the atomic groups under cooperative motion is centered in these cells. Consistent with the results in the main text, the Df parameter is obtained following the procedure described in Fig. 1. Additionally, changes in D2min and von Mises stress are calculated by determining atomic displacements or differences in atomic von Mises stress before and after relaxation at the strain step where the event occurs. In all cases, the Df parameter histogram successfully isolates deformation units that exhibit cooperative motion. Similar results were obtained for the other 40 cases not shown here. For most cases, the histogram of the Df parameter takes the form shown in Fig. 1c: the STZ core appears at Df = 0, a gap opens clearly, and a peak of the Df parameter emerges. Some cases exhibit different histogram shapes (for example, the first and second events under −xy shear), but these correspond to cascades involving the excitation of other STZ cores, as illustrated in Fig. 4. Details are discussed in the next section. Uniquely defining STZ cores enables the calculation of their local physical properties. Figure 8 exemplifies the evolution of average atomic energy and von Mises stress within each STZ core during the AQS process. Atomic energy is computed using the EAM potential, while von Mises stress is derived from the average stress components within each STZ core. The response of individual STZ cores to shear strain at the system level varies significantly, exhibiting distinct behaviors across different cores. Furthermore, the activation of one STZ core strongly influences the properties of others. Even in regions that appear to undergo elastic deformation between stress-drop events, local STZ core responses deviate considerably from the averaged behavior inferred from system-wide values. These findings demonstrate that physical quantities within STZ cores exhibit inherently nonlinear and transient characteristics.

Local configurational excitation
Local configurational excitation (LCE) represents the reconfiguration of atomic bonding networks, as illustrated in Fig. 9, and has been discussed in the literature as a mechanism that determines the macroscopic viscosity of high-temperature liquids on an atomic level32. When LCE occurs—signified by the breaking and forming of atomic bonds—the balance of internal forces changes, leading to alterations in atomic-level stress. Additionally, atoms displace to establish a new equilibrium. Thus, LCE is the phenomenon itself, driving the observed changes. Associated stress relaxation events and atomic displacements are not separate processes but are direct consequences of LCE.

In this study, to examine the relationship between cooperative motion identified by the Df parameter and LCE, we analyzed the extent to which atoms undergoing LCE were included in groups exhibiting cooperative motion. The results are shown in Fig. 4a of the main text. Determining atomic bonding network reconfiguration is challenging in molecular dynamics; however, we assessed it by tracking atoms entering or leaving the first neighbor shell, following the method outlined in the literature3 (see Fig. 9). Specifically, for configurations before structural relaxation in stress drop events, we first determined bond presence—whether atoms were in the first neighbor shell—using the pair density function. In this study, Zr–Zr, Cu–Cu, and Zr–Cu bonds were considered part of the first neighbor shell if their interatomic distances were within 4.35, 3.05, and 3.75 Å, respectively. Bond breaking was defined as an atom moving outside the first neighbor shell with an interatomic distance increase beyond a set threshold after relaxation, whereas bond formation was defined as an atom moving into the first neighbor shell with a decrease in interatomic distance beyond the same threshold. The threshold was set at 0.075 Å for this study. We evaluated the total bond breakings and formations per atom across 60 events (totaling 960,000 samples) and generated the histogram shown in Fig. 4.

LCE analysis is useful for capturing cascade events. Among the 60 stress-drop events examined in this study, five representative cases in which cascades were likely to have occurred are shown in Fig. 10, together with the histograms of the Df parameter and the atomic structures of STZ cores, atoms involved in LCE, and atoms exhibiting large non-affine displacements. Cascade-like events were identified in 8 of the 60 cases, and the results displayed in the figure encompass all representative patterns observed. Since LCE analysis and non-affine deformation highlight atomic groups distinct from the STZ cores, these observations suggest that cascades are driven by the STZ cores. Across these cases, four characteristic histogram patterns were identified. The most common was the canonical pattern, in which the STZ core was located at Df = 0, a clear gap opened, and a peak emerged corresponding to the majority of non-core atoms (Fig. 1c). A second pattern, often associated with cascade events, exhibited a small group of atoms appearing before this peak, indicating the involvement of subsequently activated STZ cores (e.g., the first and fifth events under −xy shear and the second event under +xy shear, as exemplified in Fig. 10). In another case, the second event under −xy shear (Fig. 10), the initial STZ core and the subsequently triggered one could not be clearly separated, although such an occurrence was found only once among the 60 events. By contrast, in the 10th event under −xy shear (Fig. 10), the STZ core shifted away from Df = 0, corresponding to a situation in which a single STZ core induced two subsequent STZ cores. These diverse histogram shapes demonstrate multiple modes of interaction among STZ cores, reflecting the complexity of cascade formation mechanisms and highlighting an intriguing direction for future research.

By design, frozen-atom analysis can capture only the initial STZ core, and therefore, the secondary atomic groups observed in some histograms cannot be directly detected by this method. Nevertheless, as shown above, their presence suggests the existence of additional STZ cores whose activation is suppressed when the motion of their constituent atoms is frozen. Taken together, these observations support the following scenario for cascade formation: an STZ core induces non-affine deformation via the LCE it contains, which subsequently triggers another STZ core. This chain reaction, with each step serving as an elementary process, ultimately results in an avalanche that drives large-scale plastic deformation.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the submitted manuscript, using numerical simulations of a model metallic glass (CuZr alloy) the authors propose a\nmethod for identifying the core of a plastic event occurring by employing a pinning construction. The analysis is done in the\nathermal quasistatic limit, wherein a plastic event is initially identified and then one reverts back to the state prior to energy\nminimization and pins individual atoms one by one, and thereafter monitor whether this intervention inhibits the relaxation\nprocess by measuring a quantity Df. P(Df) exhibits a distinct gap, which is particularly interesting as it suggests the existence\nof a subpopulation of particles whose pinning significantly suppresses relaxation. This behavior is used to define the core of\nthe shear transformation zone (STZ).\n1) The approach is interesting. One should note that there are prior studies on the effect of pinning on mechanical response\n(Bhowmik et al 123, 185501 2019) where it was shown how plastic activity can be influenced by pinning. In the AQS limit, it\nis possible to know, using the Hessian approach, which eigenmode will lead to failure. It is intriguing that pinning a particle\n(within the eigenmode) that is predicted to largely participate in the failure can completely suppress the process. This needs\nto be properly analyzed. While this study gives us a glimpse of what is possible, I still feel more analysis across diverse\nmodels is needed to justify the efficacy of this construction.\n2) The authors study N=2000 particles. It is not clear what is the lateral size of the simulation box. For the EAM potential, we\nknow that there are long range effects. Hence, one needs to study bigger system sizes, I feel. Also different cooling histories\nneed to be studied, to understand the efficiency of the construction.\n3) Also, I would prefer to study a 2d system, even if it is a LJ model, to better visualize the effect of pinning on the eigenmode\nand compare with the full relaxation (i.e. without pinning).\n4) In the usual scenario, once the occurrence of plastic event is identified in an AQS scenario, the distribution of\ndisplacements during the event would have the core in the tail, from where one can already identify the participants. That\nseems to me a quicker analysis than this pinning construction where one has to go through each and every particle, do the\nrelaxation step and then identify the core. This is O(N) process, which for a large enough system size would be time\nconsuming. Is this cost beneficial enough? I can see the usefulness in the case of large cascades to understand where the\ncore was, but there are now other methods to follow the cascade process and identify individual events.\nConsidering all these, I do not feel that the work presented in this manuscript can be considered for publication in Nature\nCommunications.\nSome other feedback -\n(i) The manuscript is poorly written in several sections, especially in the discussion of figures. For instance, the content of\nFig. 1d,e is not clearly explained in the main text; readers are forced to refer to the Methods section for clarification.\n(ii) The description of the amorphous state preparation is sketchy. Without an estimate of the glass transition temperature, it\nis not clear what a temperature of 2000 K implies. Was the system equilibrated at 2000 K prior to quenching? After cooling,\ndoes the kinetic energy effectively reach zero? While this can be inserted in LAMMPS input scripts, we need the the\ntransient data to clarify what was actually achieved after the cooling. Furthermore, the statement \"the atomic structure and\ncell were relaxed structurally\" is vague and needs more precise explanation. Important parameters such as system density\nand lateral box size are missing—these are critical for reproducibility.\n(iii) In the AQS protocol, what is the magnitude of the shear strain used? Were smaller strain increments tested to examine\nthe robustness of the identified plastic events and the proposed pinning-based method?\n(iv) The cartoon in Fig.1b is misleading. In reality, we usually study 2d or 3d systems, where pinning of a single point still\nallows for relaxation, since other pathways are available.\n(v) There is already a \"frozen atom\" construction in the literature (ideated by P. Sollich; see Barbot et al Physical Review E\n97, 033001 (2018)), which does the complementary step of the freezing everyone except a sub-population within a specific\nzone. Such a construction also allows one to identify the first plastic event using an appropriate size of the non-frozen zone.\nIn fact, Barbot et al identify the first few events via their construction which shows that events are not repeated at the same\nplace -- so, we already have such examples.\n(vi) In the plot for P(Df), it would be useful for having x-axis in log-scale to know the overall displacements during the pinning\nconstruction.", "reviewer_2": "This paper is concerned with the physical mechanisms underlying shear stress relaxation in metallic glasses subjected to\nshear strain. This is a topic of practical interest for the design of stronger materials, as well as of fundamental scientific\ninterest as it connects to ideas in complex systems undergoing avalanche-like behavior.\nThe authors start there work with a well-established computer simulation protocol using the LAMMPS code and a well-\nestablished interaction potential for copper-zirconium metallic glass.\nThe novelty of the paper lies in the specific diagnostic employed to detect atoms that are part of the so-called Shear\nTransformation Zone (STZ) for specific stress relaxation events. Previous methods to identify STZ atoms have proven\nunreliable, with results often ambiguous and sensitive to the choice of various threshold values in the analysis. In contrast,\nthe conceptually simple \"frozen atom analysis\" developed in this work provides a very clear distinction between STZ and\nnon-STZ-atoms. Importantly, the identification of STZ atoms is unambiguous, with the appropriate choice of threshold\nemerging from the analysis.\nI predict that this \"frozen atom analysis\" will be quickly adopted by other groups and will lead to better understanding of\nmany stress relaxation simulations in glassy systems, and perhaps extending to non-glassy systems.\nThe paper is exceedingly well written and the presentation of results is clear and compelling. I see no real weaknesses.\nI have one suggestion for future work, probably outside the scope of this paper. Both the simulation protocol and the analysis\nare thermal and quasi-static. In contrast, real glasses always have some small amount of thermal motion. I wonder to what\nextent the STZ is affected by thermal noise. This could be tested by introducing random displacements of a given average\namplitude to all the atoms or just the frozen atom atom prior to the minimization step.", "reviewer_3": "Review report on Manuscript#: NCOMMS-25-25276-T\nThis manuscript deals with a long-standing issue about the nature of plastic deformation in amorphous solids with a new\ncomputational methodology. In order to do this, the authors propose a novel analysis method – the frozen atom analysis --\nbased on simulations of the athermal quasistatic shear in a CuZr metallic glass sample. This method is based on analysis\non the frozen atoms after affine shear back to a state just before the plastic event. The logic is natural, if the frozen atom\nblocks formation of the shear transformation zone (STZ) upon proceeding shear deformation, it is categorized into the STZ\ncores; otherwise, it is glassy matrix. With this method it is allowed to recognize two critical steps of deformation, the shear\ntransformation event and the following cascade deformation with STZ as a trigger of activation. The main finding is that the\nsize of STZ core is of about 40 atoms. The STZs are highly stochastic and there is no clear structural feature in them. The\nSTZs deformation acts as triggers where atoms move collectively due to elastic constraints, which induces following large\nscale deformation cascade that drives strain avalanches as many studied has revealed. In principle, this is a solid study with\nbroad interest in the community of materials science and physics. The proposal method is new, and the conclusion drawn is\nimpactful in physics. However, there are still a couples of concerns about novelty in science and robustness of conclusions\nmade with the current data at hand. It is suggested to conduct extra studies to enhance reliability of the study and generalize\nthe conclusion to all amorphous solids in terms of plastic deformation mechanism. The suggestions about revisions are\nlisted below.\nIt is true that the STZs are highly stochastic and can happen everywhere, however their occurring probability is highly\ndependent on the local environment of atoms. After sufficient spatial and temporal analysis, there is possibility to reveal a\nsort of structure-property relationship for amorphous solids, see the work by Tong and Tanaka\n(https://journals.aps.org/prx/abstract/10.1103/PhysRevX.8.011041). The utility of ‘structure’ (e.g., Voronoi cell) in property of\nglasses has also been quantitatively estimated in both metallic glass and Kob-Anderson model glassy. The indication there is\nthe same to the present conclusion that local structure is irrelevant (Wei et al, J. Chem. Phys. 150, 114502 (2019)).\nHowever, there exist a physically meaningful length scale for the correlation between ‘defects’ and STZs, which is of roughly\nsub-nano-meter scale (Ref [41] Wei et al., Physical Review B 99, 014115 (2019)). That means statistics is always\nnecessitated when discussing the structural trigger of STZs in amorphous materials from a physical perspective.\nIs there a coarse-graining size in computation of the atomic shear modulus? Or instead, the modulus is just estimated by the\nVoronoi volume a specific atom? If the modulus is defined at atomic scale, it is not important at all, as the authors have\npointed out that there is no correlation between STZ and structural feature. However, if the atomic-scale modulus is spatially\naveraged to some extent, it is quite possible that the peak positions of STZc and non-STZc will be shifted in Fig. 2c. In other\nwords, the referee doubts on the strong conclusion made without careful statistical analysis to some proper space range. As\na result, the following statements are overclaims.\n“This indicates that STZ cores are neither the “soft spots” as traditionally perceived nor “hard spots”; in other words, they do\nnot exhibit any distinct characteristics in terms of hardness.”\n“These results suggest that the atoms in STZ cores are not in special environments that could be identified as defects”.\n“…the formation of STZ cores is a highly stochastic and transient process: unlike conventional deformation units such as\ndislocations, they do not possess distinct structural features and can form anywhere.”\nThe conclusions are too strong with the limited discussion at hand. It is possible that other method like the topological\nanalysis can provide some hint after taking count of some higher dimensional structural information.\nIt is claimed that the size of STZ core includes about 40 atoms. However, the referee argues that this size should be strongly\naffected by the thermodynamic state of the glass model. Recently, Ding et al. (https://doi.org/10.1073/pnas.2213941119) has\nprepared ultrastable glasses with swap Monte Carlo, in which it is found that the size of STZs is about 10 atoms. This is\nmuch smaller than claim of several decades here. The referee does not mean that 40 is incorrect, but this quantity is strongly\ndependent on the structural and energy status of a glass sample. In philosophy, it is again that the nature of STZ is relevant\nto structure in a broader sense.\nThe technique of frozen analysis is fine, but the novelty of this method is somehow question. In the literature, Ning\nXu (https://doi.org/10.1073/pnas.2304974120) has proposed a concept of “stabilizing bonds” that is quite related to instability of\namorphous solids, which is quite similar to the frozen atom notion here. It is informative if the author could discuss them\ntogether.\nWhat is the meaning of ‘N’ is equation (1). It is not clear whether it is the number of atoms in the simulation box, or the STZ\ncore region.\nThe abbreviation “local configurational excitations (LCEs)” has been defined a couple of times.\nIt is not clear how large is the strain step used in the athermal quasistatic shear, which might slightly affect the definition of\nSTZ core."}, "author_response": "Ref: NCOMMS-25-25276-T\nSeptember 20, 2025\nResponse to reviewers\nWe sincerely thank the reviewers for their careful reading of our manuscript and for the constructive comments\nthat have greatly improved our work. In response, we have undertaken a substantial revision of the paper.\nSpecifically, we have extended the molecular dynamics calculations from the original 2,000-atom system to a\n16,000-atom system, and all simulation results have been recomputed accordingly. In addition, the number of\ndeformation events analyzed has been nearly doubled. Importantly, the core novelty of our approach is\npreserved in this extended analysis, while the results have become physically more robust and convincing.\nThus, the reviewers’ suggestions have significantly strengthened the study.\nWe provide a point-by-point response below. The submission consists of three components: the marked\nmanuscript (differences from the previous version), the clean manuscript, and the Supplementary Materials.\nBlue highlighting indicates revisions/additions in the marked manuscript; line numbers cited correspond to\nthat marked version.\n-----------------------------------------\nReviewer #1 (Remarks to the Author):\nIn the submitted manuscript, using numerical simulations of a model metallic glass (CuZr alloy) the\nauthors propose a method for identifying the core of a plastic event occurring by employing a pinning\nconstruction. The analysis is done in the athermal quasistatic limit, wherein a plastic event is initially\nidentified and then one reverts back to the state prior to energy minimization and pins individual\natoms one by one, and thereafter monitor whether this intervention inhibits the relaxation process by\nmeasuring a quantity Df. P(Df) exhibits a distinct gap, which is particularly interesting as it suggests\nthe existence of a subpopulation of particles whose pinning significantly suppresses relaxation. This\nbehavior is used to define the core of the shear transformation zone (STZ).\nQ1-1:\n1) The approach is interesting. One should note that there are prior studies on the effect of pinning\non mechanical response (Bhowmik et al 123, 185501 2019) where it was shown how plastic activity\ncan be influenced by pinning. In the AQS limit, it is possible to know, using the Hessian approach,\nwhich eigenmode will lead to failure. It is intriguing that pinning a particle (within the eigenmode) that\nis predicted to largely participate in the failure can completely suppress the process. This needs\nto be properly analyzed. While this study gives us a glimpse of what is possible, I still feel more analysis\nacross diverse models is needed to justify the efficacy of this construction.\nWe thank the reviewer for the positive assessment of our approach. We agree that pinning is known to be a\npowerful approach to probe the nature of mechanical deformation and the glass transition. While related in\nspirit, our frozen-atom analysis is fundamentally different from the well-known random pinning: we freeze\nonly the motion of a single atom just before each stress drop and apply this procedure independently to all\natoms, which makes our approach completely new and promising as a method to identify the STZ cores, the\natomic groups in cooperative motion. As the reviewer suggested, applying our frozen-atom analysis to\nparticipating particles identified from the Hessian matrix might indeed yield similar results and could stimulate\nfurther studies along this line. We believe this possibility highlights the novelty of our approach. At this stage,\nour emphasis is on detecting STZ cores and clarifying their statistical properties with our new method, while\nwe regard its relation to eigenmodes as a promising direction for future studies.\nInstead, we have substantially strengthened the present study by enlarging the system size from 2,000 to 16,000\natoms and by recomputing all simulation results and repeating the analysis to ensure much greater robustness.\nThe details are provided in A1-2 and the subsequent responses. We hope that these extensive revisions\nadequately address the reviewer’s main concerns.\nQ1-2:\nThe authors study N=2000 particles. It is not clear what is the lateral size of the simulation box. For\nthe EAM potential, we know that there are long range effects. Hence, one needs to study bigger\nsystem sizes, I feel. Also different cooling histories need to be studied, to understand the efficiency\nof the construction.\nA1-2:\nWe thank the reviewer for this valuable comment. The lateral size of the simulation box with 2,000 atoms is\nabout 33 Å, and the interaction range of EAM potential used is 6.5 Å. Therefore, we did not expect the 2,000-\natom system to cause serious computational artifacts, and indeed similar system sizes are commonly used in\nstudies employing the activation–relaxation technique (Fan 2014, Fan 2015). Nevertheless, we recognize that\nthe displacement field generated by the excitation of an STZ can extend over long distances, and the 2,000-\natom simulation cell might not have been sufficiently large. In the present revision, we have enlarged the\nsystem volume by a factor of eight, corresponding to 16,000 atoms, which is comparable to those employed in\nearlier STZ studies (e.g., Ding 2014; Zhang 2022), and therefore provides a more robust basis for the present\nanalysis. The changes are summarized as follows:\n• System size expansion: The MD model was enlarged from 2,000 atoms to 16,000 atoms, and all AQS\nsimulations were recomputed (Methods, “Simulation Setup” in Supplementary materials).\n• Increased number of events: The analyzed stress-drop events were increased from 36 to 60, and all\nrelated figures and tables were updated (Main Text Figs. 1–8, 10; Supplementary Figs. S1-S2, Table S1).\n• Strengthened statistics (Voronoi / LCE analyses): The statistics of Voronoi features, atomic elasticity,\n2\nand bond formation/breaking were expanded from 72,000 samples to 960,000 samples (60 events ×\n16,000 atoms), and histograms were reconstructed (Fig. 2, Fig. 4a).\n• Cascade analysis extended: Out of 60 events, 8 exhibited cascade-like behavior, and three characteristic\nhistogram patterns were summarized with representative examples (Marked manuscript: Lines 449-484,\nFig. 10).\nDespite the significant expansion of system size and statistical sampling, the core findings regarding the STZ\ncores remain unchanged, confirming the robustness of our conclusions.\nHaving considered the reviewer’s comment, in the 2,000-atom simulations, we realized that the displacement\nfield of a single STZ core was large enough to influence the entire cell (as shown in Fig. 8 of the previous\nversion). This indicates that in small cells, STZ cores may interact with their own periodic images. In contrast,\nwith 16,000 atoms such finite-size artifacts are no longer observed. This is demonstrated in the revised Fig. 8,\nwhere only a limited number of other STZ cores are affected by the excitation of a given STZ core. As a result,\nthe results have become physically more robust and convincing.\nThe reviewer also requested simulations at different cooling rates. We acknowledge that this is indeed an\ninteresting and important question. We consider this to be a separate research topic on how the stability of glass\naffects the properties of STZ cores, as previous work has demonstrated that cooling rate can significantly\nmodify the size and population of STZs (Zhang 2022). Such an investigation will therefore be better addressed\nin future work.\nReference:\nFan, Y., Iwashita, T., & Egami, T. (2014). How thermally activated deformation starts in metallic glass. Nature\nCommunications, 5, 5083.\nFan, Y., Iwashita, T., & Egami, T. (2015). Crossover from localized to cascade relaxations in metallic glasses.\nPhysical Review Letters, 115(4), 045501.\nDing, J., Patinet, S., Falk, M. L., Cheng, Y., & Ma, E. (2014). Soft spots and their structural signature in a\nmetallic glass. Proceedings of the National Academy of Sciences, 111(39), 14052-14056.\nZhang, Z., Ding, J., & Ma, E. (2022). Shear transformations in metallic glasses without excessive and\npredefinable defects. Proceedings of the National Academy of Sciences, 119(48), e2213941119.\nQ1-3:\nAlso, I would prefer to study a 2d system, even if it is a LJ model, to better visualize the effect of\npinning on the eigenmode and compare with the full relaxation (i.e. without pinning).\nA1-3:\nWe appreciate the reviewer’s suggestion of studying a two-dimensional (2D) system and acknowledge its\nadvantage for easier visualization. At the same time, we believe this would open up a very different line of\ninvestigation. In 2D, the elastic energy of dipolar stress fields diverges, which leads to screening effects and\n3\npotentially inducing spurious atomic behavior (Egami 2011). As a result, the fundamental physics is qualitatively\ndifferent from that of 3D metallic glasses. More recent simulation work has also demonstrated that the\nstructural ordering mechanisms in 2D metallic glasses differ essentially from those in 3D, for example, crystal-\nlike hexagonal order dominates in 2D, whereas icosahedral order governs the 3D case (Hu 2017). These\nfindings highlight that 2D and 3D systems raise distinct sets of questions. For this reason, we consider that\npursuing the 2D case, while interesting, would be beyond the scope of the present study.\nEgami, T. (2011). Atomic level stresses. Progress in Materials Science, 56(6), 637-653.\nHu, Y. C., Tanaka, H., & Wang, W. H. (2017). Impact of spatial dimension on structural ordering in metallic\nglass. Physical Review E, 96(2), 022613.\nQ1-4:\nIn the usual scenario, once the occurrence of plastic event is identified in an AQS scenario, the\ndistribution of displacements during the event would have the core in the tail, from where one can\nalready identify the participants. That seems to me a quicker analysis than this pinning construction\nwhere one has to go through each and every particle, do the relaxation step and then identify the\ncore. This is O(N) process, which for a large enough system size would be time consuming. Is this\ncost beneficial enough? I can see the usefulness in the case of large cascades to understand where\nthe core was, but there are now other methods to follow the cascade process and identify individual\nevents.\nA1-4:\nOur central claim is that an STZ should be regarded as consisting of two parts: the indispensable atomic group\nthat acts cooperatively to trigger the excitation (the STZ core) and the surrounding atoms whose displacements\nare induced by it. We appreciate the reviewer’s suggestion that examining the displacement distribution during\na plastic event may provide a quicker way to identify the active atoms. We would like to point out that this\napproach is essentially equivalent to analyzing the D2 field, since D2 effectively captures the displacement\nmin min\ndistribution. As shown in Fig. 1(d), D2 indeed highlights the spatial zone of activity, yet it provides only an\nmin\napproximate indication of the core and does not allow us to identify the indispensable atoms that constitute the\ncore itself. By contrast, the frozen-atom analysis enables us to extract the STZ core directly and without\nambiguity. As noted in Q1-5, this distinction was not sufficiently emphasized in the previous version, and we\nhave now added explicit statements in the revised manuscript as explained in A1-5.\nOn the other hand, we would like to thank the reviewer for drawing our attention to the important issue of\ncomputational cost. Inspired by the reviewer's comment, we recognized that using atomic displacement as an\ninitial screening step made it possible to greatly reduce the computational cost of the frozen-atom analysis.\nBy implementing this strategy, we were able to reduce the computational complexity of the frozen-atom analysis\nfrom O(N²) to O(N). Furthermore, this remains substantially lower than the O(N³) scaling required for a full\neigenmode analysis, providing a significant computational advantage especially for large-scale simulations.\nChanges in the manuscript:\n(Supplementary Materials, Section \"Scalability of frozen-atom analysis\")\nIn terms of big-O notation with the number of atoms N, conventional molecular dynamics simulations scale as\nO(N), whereas performing frozen-atom analysis by freezing every atom scales as O(N²). To reduce the\ncomputational burden, we applied the frozen-atom analysis only to atoms that exhibited a displacement norm\nabove a certain threshold (e.g., 0.01 Å) during the stress-drop event. Using atomic displacement as an initial\nscreening step made it possible to greatly reduce the computational cost of the frozen-atom analysis. While an\nSTZ core contains at most about 100 atoms, the threshold was chosen so that an atomic group roughly ten\ntimes larger was included for analysis. An illustrative example is shown in the figure below, where the STZ\ncore is fully contained within the selected group. Test calculations for 20 events under +xy and –xy shear\nconfirmed that the STZ cores identified by this efficient frozen-atom analysis are identical to those obtained\nby full analysis. Because the size of the STZ core is independent of the total number of atoms in the system, it\nis evident that freezing all atoms is unnecessary. With this refinement, the frozen-atom analysis can be regarded\nas an O(N) algorithm, which makes the method scalable to large systems.\n0.0 D 2.5 z\nf\nx y\n(a) (b) (c)\nD <1.0 |d|>0.01\nf\nFig. Example of the computational cost reduction in frozen-atom analysis, illustrated for the second event\nunder +xy shear. (a) The STZ core (96 atoms) identified as the group of atoms with D. (b) The set of atoms\nf\nthat exhibited displacement norm | d | greater than 0.01 Å (1204 atoms). (c) Superposition of (a) and (b). As\nseen in (c), the STZ core can be determined by applying the frozen-atom analysis only to the subset of atoms\nwith relatively large displacements.\nQ1-5:\nThe manuscript is poorly written in several sections, especially in the discussion of figures. For\ninstance, the content of Fig. 1d,e is not clearly explained in the main text; readers are forced to refer\nto the Methods section for clarification.\nA1-5:\nWe thank the reviewer for this pointing this out and apologize for the lack of clarity in the previous version.\nParticularly, Fig. 1d and 1e were not sufficiently clear in the main text. In the revised manuscript we have\n5\nrevised the explanations in the main text as follows, so that readers can now understand the contents without\nhaving to refer to the Methods section. In addition, brief explanatory sentences were also added for Figs. 2a\nand 2b to clarify how to interpret the color scale and circle size, respectively.\nChanges in the manuscript:\n(Marked manuscript: Lines 155-168)\nFor comparison, Figures 1d and 1e show histograms of conventional indicators: the non-affine displacement\nmeasure D2 and the change in atomic von Mises stress during relaxation.\nmin\nThe atoms in the tail of the distribution in Figure 1d, namely, those with large non-affine deformation, are often\nassumed to represent the central region of the STZ. Similarly, in Figure 1e the atoms exhibiting the most\npronounced changes in local von Mises stress are also regarded as part of this region. In both cases, the\nhighlighted atoms overlap considerably with those identified by the D2 parameter in Fig. 1c. This indicates\nmin\nthat, although conventional descriptors capture atoms involved in the event, the size of the atomic group they\nidentify can vary greatly depending on the chosen threshold, which makes their interpretation ambiguous. In\ncontrast, the frozen-atom analysis provides an unambiguous distinction between the indispensable STZ core\n(the group close to D2 ) and the surrounding atoms.\nmin\n(Marked manuscript: Lines 180-184)\nFigure 2a illustrates the spatial distribution ... throughout the material. The color scale shows how often each\natom was classified as part of an STZ core.\n(Marked manuscript: Lines 186-190)\nFigure 2b plots the features of ... this calculation. The circle size represents the D parameter, where larger\nf\ncircles basically correspond to atoms identified as participating in cooperative motion.\nQ1-6:\nThe description of the amorphous state preparation is sketchy. Without an estimate of the glass\ntransition temperature, it is not clear what a temperature of 2000 K implies. Was the system\nequilibrated at 2000 K prior to quenching? After cooling, does the kinetic energy effectively reach\nzero? While this can be inserted in LAMMPS input scripts, we need the the transient data to clarify\nwhat was actually achieved after the cooling. Furthermore, the statement \"the atomic structure and\ncell were relaxed structurally\" is vague and needs more precise explanation. Important parameters\nsuch as system density and lateral box size are missing—these are critical for reproducibility.\nA1-6:\nWe thank the reviewer for these constructive comments. In the revised Methods section, we have clarified the\npreparation of the amorphous state in detail as follows. We have also added the missing parameters required\nfor reproducibility, including the system density and the box size. In addition, the transient data of the\nquenching process have been included in the Supplementary Materials. From the figure below, we find that\nthe glass transition temperature is about 750 K, indicating that 2000 K is sufficiently high to ensure\nequilibration with a cooling rate of 109 K/s.\n(Marked manuscript: Lines 351-362)\nThe metallic glass was prepared by melting and quenching a B2 CuZr alloy. A velocity distribution\ncorresponding to 2000 K, which is well above the glass transition temperature of this alloy (approximately 750\nK, see Supplementary Fig. S1), was applied to the B2 structure, and the system was directly cooled to 0 K at\na rate of 109 K/s in an NVT ensemble using the Nosé-Hoover thermostat[44,45] without an intermediate\nannealing period. After cooling, the atomic structure and simulation cell were relaxed using the conjugate\ngradient method until all stress components converged to below 0.002 MPa. The final simulation box had a\ncharacteristic length of approximately 65.4 Å and was nearly cubic but with slight tilts in the cell vectors, as\nreported in the Supplementary Materials.\n(Supplementary Materials, Section \"Preparation of the model glass structure\")\nAs described in the Methods section, in this study a velocity distribution corresponding to 2000 K was applied\nto the B2 structure of Cu Zr , and the system was directly cooled to 0 K at a rate of 109 K/s without an\n50 50\nintermediate annealing stage. The transient data of the average total energy per atom and the average potential\nenergy per atom obtained during this process are shown in Supplementary Fig. S2. The potential energy\nexhibited a clear change in slope at around 750 K, from which the glass transition temperature T of this model\ng\nglass was estimated to be approximately 750 K. Since the initial temperature of 2000 K is well above this\nvalue, the system can be regarded as having been quenched from a fully equilibrated liquid state. At 0 K, the\ntotal and potential energies coincide, indicating that the kinetic energy has vanished (less than 2 meV/atom).\nAfter cooling, the atomic structure and simulation cell were relaxed using the conjugate gradient method until\nall stress components converged to below 0.002 MPa. The final cell configuration of the Cu Zr glass after\n50 50\nquenching and relaxation is provided in Table S1 to ensure reproducibility. The simulation box has an edge\nlength of approximately 65.4 Å and is nearly cubic, but with small tilt factors, indicating a slightly triclinic\ngeometry. The system density was 7.38 g/cm3 after quenching and relaxation. The system density and T\ng\nobtained here are consistent with those reported in Ref. [4] where the same Embedded-Atom Method (EAM)\npotential was employed, showing no significant discrepancy.\nPotentialenergy\nQuenching\nTemperature[K]\nFigure S2. Preparation of the model glass: evolution of per-atom total and potential energies during rapid\nquenching. The glass transition temperature, T is estimated to be ~750 K from the change in slope of the\ng\npotential-energy curve.\nTable S1. Final cell parameters of the Cu Zr glass after quenching and relaxation\n50 50\nQ1-7:\nIn the AQS protocol, what is the magnitude of the shear strain used? Were smaller strain increments\ntested to examine the robustness of the identified plastic events and the proposed pinning-based\nmethod?\nA1-7:\nWe thank the reviewer for raising this important question. We agree that the choice of strain increment in the\nAQS protocol is critical to avoid multiple plastic events at the same time. In the previous version of the\nmanuscript, we used a strain increment of 5×10⁻⁵ per step. In the revised work, following the practice in the\nrelevant literature (Zhang2022), we have reduced the increment to 1×10⁻⁵ per step. This clarification has been\nexplicitly added to the Methods section.\nChanges in the manuscript:\n(Marked manuscript: Line 367-369)\nThis procedure was repeated 15,000 times to induce deformation until the engineering shear strain reached\n8\n]mota/Ve[\nygrenemotareP\n-4.4\n-4.6\n-4.8\nT\ng\n-5.0\n22000000 1500 1000 500 0\n15%, with a strain increment of 10-5 per step.\nReference:\nZhang, Z., Ding, J., & Ma, E. (2022). Shear transformations in metallic glasses without excessive and\npredefinable defects. Proceedings of the National Academy of Sciences, 119(48), e2213941119.\nQ1-8: The cartoon in Fig.1b is misleading. In reality, we usually study 2d or 3d systems, where pinning of a single point still\nallows for relaxation, since other pathways are available.\nA1-8:\nWe thank the reviewer for pointing out the possible misunderstanding caused by the schematic cartoon in Fig.\n1b. We agree that the illustration is an oversimplification, since a three-dimensional glass indeed has many\nmore degrees of freedom. The intent of the cartoon is only to convey the conceptual basis of our method:\nfreezing the motion of a single atom can disrupt the cooperative motion of the entire group, thereby\nhighlighting the interdependence of atoms within the STZ core. To avoid confusion, we have revised the text\nto explicitly state this limitation:\nChanges in the manuscript:\n(Marked manuscript: Line 104-108)\n“Figure 1b schematically illustrates that freezing a single atom disrupts the cooperative motion of the entire\ngroup, emphasizing their interdependence, although this representation is oversimplified because a three-\ndimensional glass has much higher degrees of freedom.”\nWe hope this clarification makes clear that the figure is not meant as a literal description of the system but as\na conceptual aid. For this reason, we would prefer to retain the cartoon. Nevertheless, we remain open to\nremoving it should the reviewer still find it inappropriate.\nQ1-9:\nThere is already a \"frozen atom\" construction in the literature (ideated by P. Sollich; see Barbot et al\nPhysical Review E 97, 033001 (2018)), which does the complementary step of the freezing everyone\nexcept a sub-population within a specific zone. Such a construction also allows one to identify the\nfirst plastic event using an appropriate size of the non-frozen zone. In fact, Barbot et al identify the\nfirst few events via their construction which shows that events are not repeated at the same place -\n- so, we already have such examples.\nA1-9:\nBecause of the multidimensional nature of glassy response, many approaches have been proposed to reveal the\nessential physics by constraining part of the motion. Our frozen-atom analysis is distinct in that it directly\nidentifies the number of atoms that cooperatively trigger a stress drop, rather than serving as a method to detect\n9\nthe plastic events themselves. Nevertheless, it is important to clarify how our method should be positioned\nwithin this broader context, and we are grateful to the reviewers for drawing our attention to this point. In\nresponse to their comments, we have added a discussion in the Supplementary Materials introducing the related\nstudies noted by Reviewer #1 (Bhowmik 2019, Barbot 2018) and Reviewer #3 (Xu 2023), and explicitly\ndescribing how our method differs from them, thereby underscoring the novelty of the present work.\nChanges in the manuscript:\n(Marked manuscript: Lines 120-124)\nThis method appears similar to the artificial manipulations such as atomic pinning approach to probe atomic\ncooperativity in flow[24, 25] or other related interventions[26, 27], but our approach is quite different in that\natomic freezing is used merely to probe the role of each atom in deformation, and it does not affect the flow\nitself (see Supplementary Materials for details).\n(Supplementary Materials, in Section \" Frozen-atom analysis and artificial perturbations \")\nBefore the development of frozen-atom analysis, several studies introduced artificial manipulations to probe\nthe physics of disordered structures. Pinning is one such approach: Berthier, et al. constrained the motion of\nrandomly selected atoms under NVT conditions and thereby revealed hidden static order (point-to-set\ncorrelations) in glass forming liquids[2]. Bhowmik, et al. applied pinning during AQS deformation of a model\namorphous solid, showing that frozen atoms suppress avalanche-like plastic events[3]. Barbot, et al. instead\nconstrained the surroundings of a local region inside model amorphous solids and imposed affine deformation\nto forcibly trigger STZs, providing a map of local yielding propensity[1]. Xu, et al. proposed to remove the\ncontribution of specific pair bonds from the force constant matrix to identify those whose elimination directly\ntriggers instabilities[5].\nFrozen-atom analysis likewise employs artificial manipulation, but differs from pinning method in that it does\nnot suppress deformation indiscriminately throughout the AQS process. Compared with Barbot's method, the\ndistinction lies in whether the constraint is applied inside or outside the STZ, while Xu's approach identifies\ncritical bonds related to Local Configurational Excitation (LCE) rather than cooperative cores. The essential\nnovelty of frozen-atom analysis is its ability to extract cooperative motion directly, a feature not accessible by\nprevious methods. This indicates that frozen-atom analysis and these methods are complementary, and their\ncombination may yield further insights into glass physics.\nReferences:\nBarbot, A., Lerbinger, M., Hernandez-Garcia, A., García-García, R., Falk, M. L., Vandembroucq, D., & Patinet,\nS. (2018). Local yield stress statistics in model amorphous solids. Physical Review E, 97(3), 033001.\nBhowmik, B. P., Chaudhuri, P., & Karmakar, S. (2019). Effect of pinning on the yielding transition of\namorphous solids. Physical review letters, 123(18), 185501.\nXu, D., Zhang, S., Liu, A. J., Nagel, S. R., & Xu, N. (2023). Discontinuous instabilities in disordered\nsolids. Proceedings of the National Academy of Sciences, 120(34), e2304974120.\nQ1-10:\nduring the pinning construction.\nA1-10:\nWe thank the reviewer for this suggestion. We have tested plotting the distribution P(Df) with a logarithmic x-\naxis. In presenting our main claim, namely that the frozen-atom analysis enables a simple separation between\nthe STZ core and the surrounding atoms, we found that the current linear-scale representation is more\nstraightforward and easier for readers to follow. At the same time, we agree that a logarithmic representation\nmay provide additional insights when probing the detailed structure of the STZ core. We therefore regard this\nas a useful complementary approach and have noted it as a direction for future work.\n-----------------------------------------\nReviewer #2 (Remarks to the Author):\nThis paper is concerned with the physical mechanisms underlying shear stress relaxation in metallic\nglasses subjected to shear strain. This is a topic of practical interest for the design of stronger\nmaterials, as well as of fundamental scientific interest as it connects to ideas in complex systems\nundergoing avalanche-like behavior.\nThe authors start their work with a well-established computer simulation protocol using the LAMMPS\ncode and a well-established interaction potential for copper-zirconium metallic glass.\nThe novelty of the paper lies in the specific diagnostic employed to detect atoms that are part of the\nso-called Shear Transformation Zone (STZ) for specific stress relaxation events. Previous methods\nto identify STZ atoms have proven unreliable, with results often ambiguous and sensitive to the\nchoice of various threshold values in the analysis. In contrast, the conceptually simple \"frozen-atom\nanalysis\" developed in this work provides a very clear distinction between STZ and non-STZ-atoms.\nImportantly, the identification of STZ atoms is unambiguous, with the appropriate choice of threshold\nemerging from the analysis.\nI predict that this \"frozen-atom analysis\" will be quickly adopted by other groups and will lead to better\nunderstanding of many stress relaxation simulations in glassy systems, and perhaps extending to\nnon-glassy systems.\nThe paper is exceedingly well written, and the presentation of results is clear and compelling. I see\nno real weaknesses.\nQ2-1:\nI have one suggestion for future work, probably outside the scope of this paper. Both the simulation\nprotocol and the analysis are thermal and quasi-static. In contrast, real glasses always have some\n11\nsmall amount of thermal motion. I wonder to what extent the STZ is affected by thermal noise. This\ncould be tested by introducing random displacements of a given average amplitude to all the atoms\nor just the frozen atom prior to the minimization step.\nA2-1:\nWe sincerely thank the reviewer for the positive evaluation of our work and for highlighting its novelty. As\npointed out, we also hope that this method will prove useful to many researchers in materials science. We are\nfurthermore grateful for the constructive suggestion regarding the influence of thermal fluctuations on the\nstability of STZ cores. We recognize that clarifying the relationship between the characteristics of the STZ\ncore and its stability, as well as exploring how the frozen-atom analysis can be applied to dynamic phenomena,\nrepresents an important and intriguing direction for future research. We intend to address these issues in our\nforthcoming studies.\n-----------------------------------------\nReviewer #3 (Remarks to the Author):\nThis manuscript deals with a long-standing issue about the nature of plastic deformation in\namorphous solids with a new computational methodology. In order to do this, the authors propose a\nnovel analysis method – the frozen-atom analysis -- based on simulations of the athermal quasistatic\nshear in a CuZr metallic glass sample. This method is based on analysis on the frozen atoms after\naffine shear back to a state just before the plastic event. The logic is natural, if the frozen atom blocks\nformation of the shear transformation zone (STZ) upon proceeding shear deformation, it is\ncategorized into the STZ cores; otherwise, it is glassy matrix. With this method it is allowed to\nrecognize two critical steps of deformation, the shear transformation event and the following cascade\ndeformation with STZ as a trigger of activation. The main finding is that the size of STZ core is of\nabout 40 atoms. The STZs are highly stochastic and there is no clear structural feature in them. The\nSTZs deformation acts as triggers where atoms move collectively due to elastic constraints, which\ninduces following large scale deformation cascade that drives strain avalanches as many studied\nhas revealed. In principle, this is a solid study with broad interest in the community of materials\nscience and physics. The proposal method is new, and the conclusion drawn is impactful in physics.\nHowever, there are still a couples of concerns about novelty in science and robustness of\nconclusions made with the current data at hand. It is suggested to conduct extra studies to enhance\nreliability of the study and generalize the conclusion to all amorphous solids in terms of plastic\ndeformation mechanism. The suggestions about revisions are listed below.\nQ3-1:\nIt is true that the STZs are highly stochastic and can happen everywhere, however their occurring\nprobability is highly dependent on the local environment of atoms. After sufficient spatial and\ntemporal analysis, there is possibility to reveal a sort of structure-property relationship for amorphous\nsolids, see the work by Tong and Tanaka (https://journals.aps.org/prx/abstract/10.1103/\nbeen quantitatively estimated in both metallic glass and Kob-Anderson model glassy. The indication\nthere is the same to the present conclusion that local structure is irrelevant (Wei et al, J. Chem. Phys.\n150, 114502 (2019)). However, there exist a physically meaningful length scale for the correlation\nbetween ‘defects’ and STZs, which is of roughly sub-nano-meter scale (Ref [41] Wei et al., Physical\nReview B 99, 014115 (2019)). That means statistics is always necessitated when discussing the\nstructural trigger of STZs in amorphous materials from a physical perspective.\nIs there a coarse-graining size in computation of the atomic shear modulus? Or instead, the modulus\nis just estimated by the Voronoi volume a specific atom? If the modulus is defined at atomic scale, it\nis not important at all, as the authors have pointed out that there is no correlation between STZ and\nstructural feature. However, if the atomic-scale modulus is spatially averaged to some extent, it is\nquite possible that the peak positions of STZc and non-STZc will be shifted in Fig. 2c. In other words,\nthe reviewer doubts on the strong conclusion made without careful statistical analysis to some proper\nspace range. As a result, the following statements are overclaims.\n“This indicates that STZ cores are neither the “soft spots” as traditionally perceived nor “hard spots”;\nin other words, they do not exhibit any distinct characteristics in terms of hardness.”\n“These results suggest that the atoms in STZ cores are not in special environments that could be\nidentified as defects”.\n“…the formation of STZ cores is a highly stochastic and transient process: unlike conventional\ndeformation units such as dislocations, they do not possess distinct structural features and can form\nanywhere.”\nThe conclusions are too strong with the limited discussion at hand. It is possible that other method\nlike the topological analysis can provide some hint after taking count of some higher dimensional\nstructural information.\nA3-1:\nWe thank the reviewer for this valuable comment, which helped us clarify the scope of our study. First, we\nagree with the reviewer’s opinion that the probability of STZ activation is highly dependent on the local\nenvironment of atoms. We understand that the “local environment” in this context refers to the current\nconfiguration immediately before the event, rather than the initial configuration at zero strain. This distinction\nis crucial, because if no such features are present in the initial configuration, it indicates that STZs are not\nembedded defects but transient and stochastic excitations, and our definition of the STZ core makes it possible\nto discuss this point unambiguously for the first time. In this paper we focused on whether features in the initial\nconfiguration could provide precursors of STZ cores—in other words, whether STZ cores exist as intrinsic\ndefects in the structure—and addressed this question in Fig. 2.\nRegarding the reviewer’s concern about the definition of the atomic shear modulus, we performed additional\ncalculations in direct response to this comment. Specifically, the coarse-graining was carried out for the atomic\ngroups belonging to the identified STZ cores, each consisting of several tens of atoms. Within these groups,\n13\nwe averaged the shear atomic elasticity, which effectively smooths out atomic-scale noise and yields a\nmeaningful local shear modulus. Even with this coarse-graining, some STZ cores appeared stiffer and others\nsofter than the cell average, with no systematic trend. Combined with the results of Voronoi indices and Voronoi\nvolumes, these new calculations confirmed that STZ cores cannot be identified as defects in the initial\nconfiguration. The discussion on these results is now described in the manuscript as shown below.\nAlthough the previous version already stated that our focus is on the features of the initial configuration, this\npoint may not have been conveyed with sufficient clarity. To address this, we have revised and clarified the\nrelevant parts of the manuscript. (In the previous version, we noticed an incorrect description stating that STZ\ncores were analyzed “just prior to the deformation events,” even though other parts of the manuscript discussed\nthe initial configuration. This descriptive error has been corrected in the revised manuscript [Marked\nmanuscript, Lines 269–272], and we apologize for the confusion caused.)\nChanges in the manuscript:\n(Marked manuscript: Lines 32-33)\nThese cores show no clear structural or elastic precursors in the initial configuration, challenging the idea\nthat deformation occurs in defective regions.\n(Marked manuscript: Lines 176-179)\nWe studied the atomic-level characteristics of the STZ core atoms in the zero-strain initial configuration\nbefore any shear was applied, in order to test whether precursors of STZ cores exist.\n(Marked manuscript: Lines 186-188)\nFigure 2b plots the features of Voronoi polyhedra, specifically the number of faces and the volume in the\ninitial configuration, for the 100,000 atoms randomly sampled from 960,000 atoms considered in this\ncalculation.\n(Marked manuscript: Lines 197-203)\nTo further examine the elastic properties, we conducted coarse-grained evaluations of atomic elasticity based\non the STZ cores. Even in the undeformed configuration, variability was evident: some STZ cores were stiffer\nthan the cell average, while others were softer. The average shear moduli of the STZ cores were 40.5 ± 3.8\nGPa (xy), 40.7 ± 3.1 GPa (yz), and 42.0 ± 3.9 GPa (zx), compared with the overall averages of 41.8, 41.9, and\n41.7 GPa, respectively. This indicates that STZ cores in the initial state are neither the ``soft spots\" as\ntraditionally perceived nor ``hard spots\"; in other words, they do not exhibit any distinct characteristics in\nterms of elastic constant.\n(Marked manuscript: Lines 269-272)\nHowever, the analysis of the STZ core atoms in the undeformed initial state did not reveal any distinctive\nelastic characteristics. STZs are statistically created and disappear after deformation events, as presumed in\nthe STZ theory by Langer[34].\n(Marked manuscript: Lines 288-291)\nUnlike defects that can be identified from static configurations before deformation, STZ cores emerge\ndynamically through cooperative atomic motion in response to the nonlinear stress and strain behavior of a\nglassy structure under external stimuli.\nthrough topological analyses involving some higher-dimensional structural information. Identifying the\nfeatures of local structures in the current configuration that lead to instability, and locating similar structures\nwithin the disordered system, is important for advancing the understanding of deformation mechanisms in\nglassy materials. Methods such as graph neural networks might provide a way to detect subtle features that are\nnot readily identifiable by human inspection. Let us emphasize that such approaches can only become\nmeaningful once the boundary of the STZ core has been clearly defined, which highlights the importance of\nthe frozen-atom analysis. We have added this point to the Discussion section. We thank the reviewer for\nenabling us to emphasize more clearly the advantage of the STZ core concept, namely that it provides an\nunambiguous definition of the core boundary.\nChanges in the manuscript:\n(Marked manuscript: Lines 297-306)\nSTZ cores, or their specific characteristics, may nevertheless be extractable through topological analyses that\nincorporate higher-dimensional structural information. Here, this refers to describing atoms collectively rather\nthan individually, consistent with previous reports that collective structural diversity rather than specific local\nmotifs governs the dynamics of amorphous alloys[36, 37]. Graph neural networks, which have recently been\napplied in the context of STZs[38], may capture subtle features not readily identifiable by conventional\nanalysis. Such approaches, however, become meaningful only after the nature of the STZ is identified and the\nboundary of the STZ core is clearly defined; frozen-atom analysis provides this definition unambiguously by\ndirectly identifying the indispensable atoms whose immobilization blocks the cooperative process.\nQ3-2:\nIt is claimed that the size of STZ core includes about 40 atoms. However, the referee argues that\nthis size should be strongly affected by the thermodynamic state of the glass model. Recently, Ding\net al. (https://doi.org/10.1073/pnas.2213941119) has prepared ultrastable glasses with swap Monte\nCarlo, in which it is found that the size of STZs is about 10 atoms. This is much smaller than claim\nof several decades here. The referee does not mean that 40 is incorrect, but this quantity is strongly\ndependent on the structural and energy status of a glass sample. In philosophy, it is again that the\nnature of STZ is relevant to structure in a broader sense.\nA3-2:\nAlthough the definition of STZ size in our study differs from that used by Zhang, Ding, and Ma (Zhang2022),\nwhere atoms with a D2 value above a certain threshold are considered to belong to an STZ, we cannot rule\nmin\nout the possibility that the cooling rate affects the STZ core size. As discussed in A1-4, because the STZ core\nis embedded within a group of atoms exhibiting a certain magnitude of D2 , if the D2 values become smaller\nmin min\nunder slower cooling conditions, the STZ core could also become smaller. Therefore, we find the reviewer’s\nconcern reasonable, and in accordance with this comment we have removed the statement referring to “40\n15\natoms.” What we wish to emphasize here is that, unlike the other methods compared in Fig. 1, the frozen-atom\nanalysis makes it possible to draw a clear boundary between non-STZ core and STZ core atoms as discussed.\nWhile it is noted in the previous version, to clarify this point, we have added the following sentence to the\nrevised manuscript.\nChanges in the manuscript:\n(Marked manuscript: Lines 27-30)\nUsing our novel ``frozen-atom analysis\" of the simulation data, we reveal that anelastic deformation in CuZr\nmetallic glasses is fundamentally driven by cooperative atomic motions of tens of atoms elastically linked to\none another, forming trigger groups.\n(Marked manuscript: Lines 81-83)\nWe found that the STZ core consists of tens of atoms, which engage in cooperative motion. This value may\ndiffer among various glass systems; nevertheless, the frozen-atom analysis provides an unambiguous\ndetermination of it.\n(Marked manuscript: Lines 331-334)\nThis number may vary across different glass systems, and could depend on chemical composition and\npreparation condition. However, the frozen-atom analysis can determine this number without ambiguity.\nReference:\nZhang, Z., Ding, J., & Ma, E. (2022). Shear transformations in metallic glasses without excessive and\npredefinable defects. Proceedings of the National Academy of Sciences, 119(48), e2213941119.\nQ3-3:\nThe technique of frozen analysis is fine, but the novelty of this method is somehow question. In the\nliterature, Ning Xu (https://doi.org/10.1073/pnas.2304974120) has proposed a concept of “stabilizing\nbonds” that is quite related to instability of amorphous solids, which is quite similar to the frozen atom\nnotion here. It is informative if the author could discuss them together.\nA3-3:\nWe thank the reviewer for this valuable comment. As the reviewer points out, and as we have also emphasized\nin A1-9 of this reply, it is important to distinguish our frozen-atom analysis from other approaches that\nconstrain atomic displacements, by clarifying what is fundamentally different. To address the reviewer’s\nsuggestion, we have added a discussion in the Supplementary Materials comparing our method with the related\napproaches and highlighting the novelty of the frozen-atom analysis.\nChanges in the manuscript:\n(Marked manuscript: Lines 120-124)\nThis method appears similar to the artificial manipulations such as atomic pinning approach to probe atomic\ncooperativity in flow[24, 25] or other related interventions[26, 27], but our approach is quite different in that\natomic freezing is used merely to probe the role of each atom in deformation, and it does not affect the flow\nitself (see Supplementary Materials for details).\n(Supplementary Materials, in Section \" Frozen-atom analysis and artificial perturbations \")\nBefore the development of frozen-atom analysis, several studies introduced artificial manipulations to probe\nthe physics of disordered structures. Pinning is one such approach: Berthier, et al. constrained the motion of\nrandomly selected atoms under NVT conditions and thereby revealed hidden static order (point-to-set\ncorrelations) in glass forming liquids[2]. Bhowmik, et al. applied pinning during AQS deformation of a model\namorphous solid, showing that frozen atoms suppress avalanche-like plastic events[3]. Barbot, et al. instead\nconstrained the surroundings of a local region inside model amorphous solids and imposed affine deformation\nto forcibly trigger STZs, providing a map of local yielding propensity[1]. Xu, et al. proposed to remove the\ncontribution of specific pair bonds from the force constant matrix to identify those whose elimination directly\ntriggers instabilities[5].\nFrozen-atom analysis likewise employs artificial manipulation, but differs from pinning method in that it does\nnot suppress deformation indiscriminately throughout the AQS process. Compared with Barbot's method, the\ndistinction lies in whether the constraint is applied inside or outside the STZ, while Xu's approach identifies\ncritical bonds related to Local Configurational Excitation (LCE) rather than cooperative cores. The essential\nnovelty of frozen-atom analysis is its ability to extract cooperative motion directly, a feature not accessible by\nprevious methods. This indicates that frozen-atom analysis and these methods are complementary, and their\ncombination may yield further insights into glass physics.\nQ3-4:\nWhat is the meaning of ‘N’ is equation (1). It is not clear whether it is the number of atoms in the\nsimulation box, or the STZ core region.\nQ3-5:\nThe abbreviation “local configurational excitations (LCEs)” has been defined a couple of times.\nQ3-6:\nIt is not clear how large is the strain step used in the athermal quasistatic shear, which might\nslightly affect the definition of STZ core.\nA3-4, 3-5, 3-6:\nWe appreciate the reviewer’s attention to these points. In response, we have clarified in the revised manuscript\nthat in equation (1) N denotes the total number of atoms in the simulation box (Q3-4). We have also removed\nthe redundant definitions of “local configurational excitations (LCEs)” (Q3-5). Regarding the strain increment\nin the athermal quasistatic shear (Q3-6), as noted in A1-7 we have corrected the step size from 5×10⁻⁵ to 1×10⁻⁵,\nrecomputed the results with this more appropriate value, and revised the manuscript accordingly.\nChanges in the manuscript:\n(Marked manuscript: Line 114)\n... and N denotes the total number of atoms in the simulation box.\n(Marked manuscript: Lines 291-292)\nThis suggests that LCEs and ...\n(Marked manuscript: Lines 313-314)\nEach STZ core involves atomic bond rearrangement, specifically LCE, which generates...\n(Marked manuscript: Lines 367-369)\nThis procedure was repeated 15,000 times to induce deformation until the engineering shear strain reached\n15%, with a strain increment of 10-5 per step.\n-----------------------------------------\nOther Changes:\nIndependent of the reviewers’ comments, we have made broad textual and stylistic improvements throughout\nthe manuscript. These edits are documented in the manuscript text file with track changes, and therefore we\ndo not enumerate them in detail here.\n18"}, {"round": 2, "reviewer_comments": {"reviewer_2": "I am satisfied with the response to my comments and that of the other reviewers. I recommend publication without further\nreview.", "reviewer_3": "2nd review report\nAs I stated in my previous report, the idea of \"frozen-atom analysis\" to identify STZs is both simple and powerful. The\nproposal carries sufficient novelty to merit publication in Nature Communications. In this revision, the authors have carefully\naddressed all reviewers’ suggestions by employing a larger simulation box, a smaller strain step, and additional cases of\nSTZ analysis. The conclusions remain robust, and all necessary details have been clearly explained and well presented in\nthe manuscript. I consider the work publishable in its current form."}, "author_response": ""}]

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