Spatial structure of quasilocalized vibrations in nearly jammed amorphous solids

The low-temperature properties of amorphous solids are widely believed to be controlled by the low-frequency quasilocalized modes. However, what governs their spatial structure and density is unclear. We study these questions numerically in very large systems as the jamming transition is approached and the pressure $p$ vanishes. We find that these modes consist of an unstable core and a stable far-field component. The length scale of the core diverges as $p^{-1/4}$ and its characteristic volume diverges as $p^{-1/2}$. These spatial features are precisely those of the anomalous modes that are known to cause the boson peak in the vibrational spectra of weakly coordinated materials. From this correspondence, we deduce that the density of quasilocalized modes must follow $g_{\mathrm{loc}}\left(\omega \right)\sim {\omega }^4/p^2$, which is in agreement with previous observations. Thus, our analysis demonstrates the nature of quasilocalized modes in a class of amorphous materials.

Figure 1

Vibrational displacement field in the quasilocalized vibrations at (a)–(c) high ($p=0.05$) and (d)–(f) low ($p=0.001$) pressures (three samples for each pressure). We show the particles' vibrational displacements (denoted by arrows) that are larger than 1% of the largest vibrational displacement, and the particle with the largest displacement is positioned at the center of the box. These modes are chosen from the lowest-frequency modes in each configuration.

Figure 2

The average energy versus the average norm of each particle. Both axes are normalized by the values of the particle with the largest displacement. Error bars are shown only for pressure $p=0.05$; the other errors are comparable. The symbols are connected by lines to guide the eye.

Figure 3

(a) Radial energy distribution functions $\delta E\left(r\right)$, which are normalized by their values at the origin, for various values of pressure $p$. We define the length ${\xi }_1$ where the functions initially become positive and plot it for $p=0.05$. (b) Integrated radial energy distribution functions $\mathrm{\Lambda }\left(r\right)$, which are normalized by their minimum values. We also defined the length ${\xi }_2$ at which the functions attain their minimum value and ${\xi }_3$ at which the functions first become positive. For visibility in (a) and (b), we show error bars only for $p=0.001$; the other error bars are comparable.

Figure 4

(a) The pressure dependences of three lengths, namely, ${\xi }_1,{\xi }_2,$ and ${\xi }_3$, which are defined in the main text and Fig. 3. Since ${\xi }_3$ is much larger than the other two lengths, we present ${\xi }_3$ divided by 8. The dashed line indicates the power-law dependence of $\propto p^{-1/4}$. (b) The pressure dependence of the volume. The dotted line indicates the dependence of $\propto p^{-1/2}$.