Universal Nonphononic Density of States in 2D, 3D, and 4D Glasses

It is now well established that structural glasses possess disorder- and frustration-induced soft quasilocalized excitations, which play key roles in various glassy phenomena. Recent work has established that in model glass formers in three dimensions, these nonphononic soft excitations may assume the form of quasilocalized, harmonic vibrational modes whose frequency follows a universal density of states $D(\omega )\sim {\omega }^4$, independently of microscopic details, and for a broad range of glass preparation protocols. Here, we further establish the universality of the nonphononic density of vibrational modes by direct measurements in model structural glasses in two dimensions and four dimensions. We also investigate their degree of localization, which is generally weaker in lower spatial dimensions, giving rise to a pronounced system-size dependence of the nonphononic density of states in two dimensions, but not in higher dimensions. Finally, we identify a fundamental glassy frequency scale ${\omega }_c$ above which the universal ${\omega }^4$ law breaks down.

Figure 1

Visualizations of soft quasilocalized vibrational modes in (a) 2D, (b) 3D, and (c) a 3D projection of a 4D mode. We plot the largest 3%, 0.1%, and 0.01% components, that amount to approximately 70%, 92%, and 87% of the displayed modes’ weights in 2D, 3D, and 4D, respectively. The modes displayed have frequencies of roughly half the respective longest wavelength transverse phonon frequency or less, and were calculated in a generic model glass of purely repulsive spheres interacting via a $r^{-10}$ pairwise potential; see text for details. (d) Symbols represent the spatial decay of the squared magnitude (see [31] for details) of the modes shown in panels (a)–(c), as a function of the distance $r$ away from the modes’ respective cores.

Figure 2

Sample-to-sample average of the density of vibrational modes $D(\omega )$, in (a) 2D, (b) 3D, and (c) 4D glasses, for a variety of system sizes as indicated by the legends. We find $D(\omega )\sim {\omega }^4$ in all spatial dimensions. The intrusion of phonons into the ${\omega }^4$ scaling regime upon increasing system size is clearly observed in our 2D data. Noticeably, the prefactor of the ${\omega }^4$ scaling appears to be system-size independent in 3D and 4D, but not so in 2D, where it grows with increasing system size; see text for discussion. The frequency axes are rescaled by ${\omega }_0=1.0$, 1.6, 2.8 for 2D, 3D, and 4D, respectively, for visualization purposes. The vertical axes are rescaled such that $D(\omega )\sim 1$ for the lowest frequencies measured.

Figure 3

Running averages $\overline{e}$ of the participation ratio $e$ (see text for definition) of low-frequency vibrational modes measured in (a) 2D, (b) 3D, and (c) 4D glasses, scaled by the system size $N$, binned over and plotted against frequency $\omega$. For visualization purposes the frequency axes are rescaled by the same scales ${\omega }_0$ as reported in the caption of Fig. 2. The peaks of $\overline{e}$ correspond to the lowest frequency phonons [15].

Figure 4

Cumulative distributions $C(\omega )\equiv {\int }_0^{\omega }D({\omega }^{\prime })d{\omega }^{\prime }$, divided by ${\omega }^5$ for (a) 3D systems and (b) 4D systems. The frequency axes are rescaled by ${\omega }_0=3.3$ (2.8) in 3D (4D) for visualization purposes. The vertical lines indicate the estimated position of the first phonon band; i.e., they follow $2\pi \sqrt{G/\rho }/L$ with $G$ the athermal shear modulus. The cutoff frequency scale ${\omega }_c$ is marked by the vertical arrows, at the frequency at which $C(\omega )/{\omega }^5$ dips downwards from the low-frequency plateau, the latter reflecting the ${\omega }^4$ scaling of $D(\omega )$.