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# Continuum limit of the vibrational properties of amorphous solids

Edited by Andrea J. Liu, University of Pennsylvania, Philadelphia, PA, and approved September 25, 2017 (received for review June 1, 2017)

## Significance

The thermal properties of crystalline solids follow universal laws that are explained by theories based on phonons. Amorphous solids are also characterized by universal laws that are, however, anomalous with respect to their crystalline counterparts. These anomalies begin to emerge at very low temperatures, suggesting that the vibrational properties of amorphous solids differ from phonons, even in the continuum limit. In this work, we reveal that phonons coexist with soft localized modes in the continuum limit of amorphous solids. Importantly, we discover that the phonons follow the Debye law, whereas the soft localized modes follow another universal non-Debye law. Our findings provide a firm theoretical basis for explaining the thermal anomalies of amorphous solids.

## Abstract

The low-frequency vibrational and low-temperature thermal properties of amorphous solids are markedly different from those of crystalline solids. This situation is counterintuitive because all solid materials are expected to behave as a homogeneous elastic body in the continuum limit, in which vibrational modes are phonons that follow the Debye law. A number of phenomenological explanations for this situation have been proposed, which assume elastic heterogeneities, soft localized vibrations, and so on. Microscopic mean-field theories have recently been developed to predict the universal non-Debye scaling law. Considering these theoretical arguments, it is absolutely necessary to directly observe the nature of the low-frequency vibrations of amorphous solids and determine the laws that such vibrations obey. Herein, we perform an extremely large-scale vibrational mode analysis of a model amorphous solid. We find that the scaling law predicted by the mean-field theory is violated at low frequency, and in the continuum limit, the vibrational modes converge to a mixture of phonon modes that follow the Debye law and soft localized modes that follow another universal non-Debye scaling law.

The low-frequency vibrational and low-temperature thermal properties of amorphous solids have been a long-standing mystery in condensed matter physics. Crystalline solids follow universal laws, which are explained in terms of phonons (1, 2). Debye theory and phonon-gas theory predict that the vibrational density of states (vDOS) follows

A number of theoretical explanations for these anomalies have been proposed, and these explanations substantially differ. One approach (10⇓⇓⇓⇓–15) assumes that an inhomogeneity in the mechanical response at the nanoscale (16⇓–18) plays the central role. In this approach, the heterogeneous elasticity equation is solved by using the effective medium technique to predict the BP and anomalous acoustic excitations (10, 11).

Another approach is the so-called soft potential model (19⇓⇓⇓⇓–24), which is an extension of the famous tunneling two-level systems model (25). This theory assumes soft localized vibrations to explain the anomalous thermal conductivity and the emergence of the BP. Soft localized vibrations have been numerically observed in a wide variety of model amorphous solids (26⇓⇓⇓⇓–31) and in the Heisenberg spin glass (32). Interestingly, these localized modes are also argued to affect the dynamics of supercooled liquids (33) and the yielding of glasses (34⇓–36).

Recently, a quite different scenario has been emerging. This scenario is based on studies of the simplest model of amorphous solids (37), which is randomly jammed particles at zero temperature interacting through the pairwise potential,

Importantly, the mean-field theory analysis of this model is now considerably advancing, thereby providing a new way to understand the anomalies of amorphous solids (38⇓⇓⇓⇓⇓⇓⇓⇓⇓–48). Previous theoretical (38, 39) and numerical (49, 50) works have clearly established that the vDOS of this model exhibits a characteristic plateau at

Considering these different theoretical arguments, it is absolutely necessary to numerically observe the nature of the low-frequency vibrations of amorphous solids and determine the laws that such vibrations obey. This task is not trivial because the lower is the frequency that we require, the larger is the system that we need to simulate. Here, we perform a vibrational mode analysis of the model amorphous solid defined by Eq. **1**, composed of up to millions of particles (

## Results and Discussion

We first study the vibrational modes of the 3D model system at a pressure (density) above the unjamming transition (*Materials and Methods*). Fig. 1*A* (top image, red circles) presents the reduced vDOS **5**). The reduced vDOS clearly exhibits a maximum (i.e., the BP). Because our data cover a wide range of frequencies, we can precisely identify the position of the BP

To characterize the modes, we calculate three different parameters (*Materials and Methods*). First, the second image from the top in Fig. 1*A* presents the phonon order parameter **6** and **7**), for each mode k. *A* plots the participation ratio **9**). *Inset* shows that the nonphonon modes (small *A* presents the normal and tangential vibrational energies, **10**) (52). Again, the modes are split into two groups. The phonon modes follow the scaling behavior

We now perform more stringent tests of the nature of these two types of modes. In Fig. 2*A*, in the leftmost image, the phonon order parameter *Materials and Methods*), and we display them as vertical lines in the figure. Indeed, the phonon modes (modes with large *A* show the eigen-vector field

We note that the soft localized modes present an extended vibrational character. This character is best observed in the spatial correlation function **8**), as shown in second image from the left of Fig. 2*A*. Here, we calculate the spatial correlation of the eigen-vector field *Materials and Methods*). This function exhibits a nice sinusoidal shape not only for the phonon mode, but also for the localized mode. This result indicates that the localized mode has disordered vibrational motions in the localized region; however, these motions are accompanied by extended phonon vibrations in the background. This feature is very similar to the quasilocalized modes in defect crystals, which are produced by hybridization of the extended phonon and the localized defect modes (2, 28). Consistent with this extended character, we observe that the participation ratios of the localized modes are independent of the system size N at a fixed ω (26, 28). However, as the recent work (31) demonstrated, the localized modes lying below the lowest phonon mode exhibit N-dependent participation ratios. We speculate that these localized modes are affected by phonon vibrations so weakly that only a tiny extended character is attached. Indeed, this is true for the localized modes (below the lowest phonon) in defect crystals (28).

A clear distinction between phonon modes and soft localized modes enables us to separately consider the vDOSs of these two types of modes. We define *A* (the third image from the top); however, the results are not sensitive to the choice of *A* (the top image and *Inset* in the second image from the top). As shown,

We repeat this analysis using different packing pressures (densities). We observe that the basic features are unchanged (Fig. S3*A*). Furthermore, we find that the vDOSs at different pressures can be summarized by the scaling laws. To illustrate this result, we determine *A*). The BP is located at

Rather, at lower frequency *A*, we plot *B*. Remarkably, all of the data converge to another universal scaling law

Therefore, by collecting the results in all of the frequency regions, we can write the functional form of the vDOS that covers the continuum limit as follows:

Based on the vDOS in Eq. **2**, the heat capacity

To further discuss the origin of the nonphonon behaviors, we perform a vibrational mode analysis of the “unstressed” system. The unstressed system is defined as the system in which the particle–particle contacts of the original system are replaced with relaxed springs (*Materials and Methods*). In the present model, the unstressed system is known to be far from the marginally stable state (39, 44, 54). Thus, by observing whether a mode disappears in the unstressed system, one can evaluate whether the mode originates from the marginal stability. We observe that the scaling region for *B*). [We observe that the unstressed system begins to exhibit soft localized modes when the system is brought close to the unjamming transition (elastic instability).] This result suggests that the soft localized modes also originate from the marginal stability, although they are not captured by the current mean-field framework.

Finally, we focus on the vibrational modes of the 2D model system, where we encounter a surprisingly different situation. Fig. 1*B* shows that *B* for the system of **2** and in Fig. 4*A* (also see Eq. **S3** and Figs. S7 and S8). However, note that some soft localized modes appear even below

In conclusion, we have used a large-scale numerical simulation to observe the continuum limit of the vibrational modes in a model amorphous solid. (In this work, we have studied an amorphous system of the harmonic potential defined by Eq. **1**. However, we expect that our main results may persist in a variety of amorphous systems, e.g., the Lennard–Jones glasses, which requires further study.) In 3D, we have found that the vDOS follows the non-Debye scaling

Our results, on the one hand, provide a direct verification of the basic assumption of the soft potential model (19⇓⇓⇓⇓⇓–25). We showed the coexistence of phonon modes and soft localized modes, which is the central idea for explaining the low-T anomalies of thermal conduction and the formation of BP in this phenomenological model. However, more quantitative predictions of this theory, such as the pressure dependence of

## Materials and Methods

### System Description.

We study 3D (**1**], where *N* = 16,000, to extremely large, *N* = 2,048,000. We always remove the rattler particles that have less than d contacting particles.

Mechanically stable amorphous packings are generated for a range of packing pressures from

In the packings obtained by using the above protocol, the interparticle forces are always positive

### Vibrational Mode Analysis.

We perform the standard vibrational mode analysis (1, 2), where we solve the eigen-value problem of the dynamical matrix (

From the dataset of eigen frequencies,

In the present work, we analyze several different system sizes, ranging from *N* = 16,000 to *N* = 2,048,000. We first calculate all of the vibrational modes in the system of *N* = 16,000. We then calculate only the low-frequency modes in the larger systems of *N* > 16,000. Finally, the modes obtained from different system sizes are combined as a function of the frequency

The present system exhibits a characteristic plateau in

### Phonon and Debye vDOS.

In an isotropic elastic medium, phonons are described as

In the low-ω limit, the continuum mechanics determine the dispersion relation as

### Phonon Order Parameter.

We evaluate the extent to which the mode

If the mode k is a phonon, then

In addition, we calculate the (normalized) spatial correlation function

### Participation Ratio.

We measure the extent of vibrational localization using the participation ratio

### Vibrational Energy.

Finally, we calculate the vibrational energy

## SI Text

In the following, we report supporting data, including the onset frequency of the plateau in vDOS

Here, we summarize the power-law scalings with p of the characteristic frequencies,

## Acknowledgments

We thank H. Ikeda, Y. Jin, L. E. Silbert, P. Charbonneau, F. Zamponi, L. Berthier, E. Lerner, E. Bouchbinder, and K. Miyazaki for useful discussions and suggestions. This work was supported by Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists B 17K14369, Grant-in-Aid for Young Scientists A 17H04853, and Grant-in-Aid for Scientific Research B 16H04034. The numerical calculations were partly performed on SGI Altix ICE XA at the Institute for Solid State Physics, The University of Tokyo.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: hideyuki.mizuno{at}phys.c.u-tokyo.ac.jp.

Author contributions: H.M. and A.I. designed research; H.M. and A.I. performed research; H.M., H.S., and A.I. analyzed data; and H.M. and A.I. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1709015114/-/DCSupplemental.

Published under the PNAS license.

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