Universal Non-Debye Scaling in the Density of States of Amorphous Solids

At the jamming transition, amorphous packings are known to display anomalous vibrational modes with a density of states (DOS) that remains constant at low frequency. The scaling of the DOS at higher packing fractions remains, however, unclear. One might expect to find a simple Debye scaling, but recent results from effective medium theory and the exact solution of mean-field models both predict an anomalous, non-Debye scaling. Being mean-field in nature, however, these solutions are only strictly valid in the limit of infinite spatial dimension, and it is unclear what value they have for finite-dimensional systems. Here, we study packings of soft spheres in dimensions 3 through 7 and find, away from jamming, a universal non-Debye scaling of the DOS that is consistent with the mean-field predictions. We also consider how the soft mode participation ratio evolves as dimension increases.

Figure 1

DOS for a single configuration in $d=4$ for $N=8192$ with rescaled $\omega$. The collapsed data are fitted to Eq. (3) (solid line). Inset: DOS for $\mathrm{\Delta }\varphi /{\varphi }_{\mathrm{J}}=9.1\times {10}^{-8}$, $9.2\times {10}^{-7}$, $9.2\times {10}^{-6}$, $9.2\times {10}^{-5}$, $9.0\times {10}^{-4}$, $8.7\times {10}^{-3}$, $8.0\times {10}^{-2}$, and $9.4\times {10}^{-1}$, from left to right. Errors are smaller than the symbol size, except at very low $\omega$, where they are comparable to the data scatter.

Figure 2

DOS for $N=8192$ in $d=3$ to 7 averaged over 30 to 50 configurations for (a) moderately compressed $\mathrm{\Delta }\varphi /{\varphi }_{\mathrm{J}}\approx 0.2$ and (b) highly compressed $\mathrm{\Delta }\varphi /{\varphi }_{\mathrm{J}}\approx 1.4$ above jamming. The solid line gives Eq. (3) with the fit parameters obtained in Fig. 1. Inset: the fraction $f$ of rattlers in the system (and hence the fraction of trivial zero modes in the DOS) vanishes around $\mathrm{\Delta }\varphi \approx 0.1$, and is exponentially suppressed with dimension [27]. Errors are smaller than the symbol size, except at very low $\omega$, where they are comparable to the data scatter.

Figure 3

Evolution of IPR with frequency in $d=4$ for $N=8192$ averaged over 50 configurations for $\mathrm{\Delta }\varphi /{\varphi }_{\mathrm{J}}=1.5\times {10}^{-2}$, $4.7\times {10}^{-2}$, $1.5\times {10}^{-1}$, $5.1\times {10}^{-1}$ and 1.6. Statistical errors are comparable to the data scatter.

Figure 4

IPR for $N=8192$ particles in $d=4$–7, averaged over 30 to 50 configurations, (a) moderately compressed $\mathrm{\Delta }\varphi /{\varphi }_{\mathrm{J}}\approx 0.2$ and (b) highly compressed $\mathrm{\Delta }\varphi /{\varphi }_{\mathrm{J}}\approx 1.4$ above jamming. Statistical errors are comparable to the data scatter.