Finite-size effects in the nonphononic density of states in computer glasses

The universal form of the density of nonphononic, quasilocalized vibrational modes of frequency $\omega$ in structural glasses, $\mathcal{D}\left(\omega \right)$, was predicted theoretically decades ago, but only recently revealed in numerical simulations. In particular, it has been recently established that, in generic computer glasses, $\mathcal{D}\left(\omega \right)$ increases from zero frequency as ${\omega }^4$, independent of spatial dimension and of microscopic details. However, it has been shown [Lerner and Bouchbinder, Phys. Rev. E 96, 020104(R) (2017)] that the preparation protocol employed to create glassy samples may affect the form of their resulting $\mathcal{D}\left(\omega \right)$: glassy samples rapidly quenched from high-temperature liquid states were shown to feature $\mathcal{D}\left(\omega \right)\sim {\omega }^{\beta }$ with $\beta <4$, presumably limiting the degree of universality of the ${\omega }^4$ law. Here we show that exponents $\beta <4$ are seen only in small glassy samples quenched from high-temperature liquid states—whose sizes are comparable to or smaller than the size of the disordered core of soft quasilocalized vibrations—while larger glassy samples made with the same protocol feature the universal ${\omega }^4$ law. Our results demonstrate that observations of $\beta <4$ in the nonphononic density of states stem from finite-size effects, and we thus conclude that the ${\omega }^4$ law should be featured by any sufficiently large glass quenched from a melt.

Figure 1

Sample-to-sample mean athermal shear modulus $G$, plotted against the equilibrium parent temperature $T_p$ from which our glassy samples were instantaneously quenched. In this work we express temperatures in terms of the onset temperature of the high-$T_p$ plateau of $G$, and elastic moduli in terms of the high-$T_p$ plateau of $G$ itself.

Figure 2

The low-frequency regime of the total density of states $D\left(\omega \right)$, measured in our ensembles of glassy samples quenched from high temperatures of (a) $N=2048$, (b) $N=8192$, (c) $N=32768$, and (d) $N=131072$ particles. The vertical dashed lines mark the first phonon frequency $2\pi c_{\infty }/L$. These data show that $D\left(\omega \right)\sim {\omega }^{\beta }$ with $\beta <4$ is a finite-size effect.

Figure 3

(a) Decay functions $c\left(r\right)$ calculated on the responses to local force dipoles, as defined in the text, plotted against the distance $r$ from the applied force dipole, for different system sizes as reported in the legend. We expect to see a crossover to the continuum $r^{-6}$ scaling at $r\gtrsim {\xi }_g$, with ${\xi }_g$ representing the QLMs' core size. (b) Scaling the decay functions by $r^6$ shows that the smallest systems employed here are too small to exhibit the crossover to the expected continuum scaling.