Statistics and Properties of Low-Frequency Vibrational Modes in Structural Glasses

Low-frequency vibrational modes play a central role in determining various basic properties of glasses, yet their statistical and mechanical properties are not fully understood. Using extensive numerical simulations of several model glasses in three dimensions, we show that in systems of linear size $L$ sufficiently smaller than a crossover size $L_D$, the low-frequency tail of the density of states follows $D(\omega )\sim {\omega }^4$ up to the vicinity of the lowest Goldstone mode frequency. We find that the sample-to-sample statistics of the minimal vibrational frequency in systems of size $L<L_D$ is Weibullian, with scaling exponents in excellent agreement with the ${\omega }^4$ law. We further show that the lowest-frequency modes are spatially quasilocalized and that their localization and associated quartic anharmonicity are largely frequency independent. The effect of preparation protocols on the low-frequency modes is elucidated, and a number of glassy length scales are briefly discussed.

Figure 1

(a) Sample-to-sample mean MVF $⟨{\omega }_{\min }⟩$ rescaled by $2\pi \sqrt{\mu /\rho }/a_0$ vs sample length $L$, for the 3DIPL, KABLJ, and HARM models. The dashed line represents the expectation for the lowest-frequency Goldstone modes $2\pi \sqrt{\mu /\rho }/L$. (b) Sample-to-sample distributions of the minimal vibrational frequency $P({\omega }_{\min })$ for the 3DIPL system. (c) The same distributions plotted as a function of the rescaled frequency $\omega L^{3/5}$. The continuous magenta line represents the Weibull distribution $W(y)\propto y^4{e}^{-(y/y_0{)}^5}$, with $y_0\approx 4$.

Figure 2

Left column: density of vibrational modes $D(\omega )$ measured in (a) the 3DIPL system and (c) the KABLJ and HARM systems. The continuous magenta lines all represent the scaling $D(\omega )\sim {\omega }^4$. Right column: same distributions plotted vs the rescaled frequencies $\omega L/(2\pi \sqrt{\mu /\rho })$. The vertical dashed lines represent the lowest Goldstone mode frequency expectation. Distributions were shifted vertically for visibility.

Figure 3

Scatter plots of the product $Ne$ (a) and anharmonicity $\chi$ (b) vs the rescaled MVF. Median participation ratio $e$ and anharmonicity $\chi$ vs system size are shown in the insets of panels (a) and (b), respectively.

Figure 4

(a) Spatial decay profile (see Ref. [32] for exact definition) of the lowest-frequency mode amongst our entire ensemble of minimal frequency modes with $N=10^6$. (b) The ensemble lowest-frequency mode (only components larger than a tenth of the mode’s maximal component are shown). On the $x\text{−}y$, $x\text{−}z$, and $y\text{−}z$ planes, the respective projections of the mode are shown, allowing for a visual estimation of its spatial scale.

Figure 5

(a) The mean rescaled MVFs depend on system size as $L^{-3/5}$, both for rapidly and slowly quenched samples, suggesting the robustness of the ${\omega }^4$ law to different preparation protocols. (b)–(d) Distributions of (b) MVFs ${\omega }_{\min }$, (c) athermal shear modulus $\mu$, and (d) participation ratio $e$ measured in our slowly and rapidly quenched solids as indicated by the legend. See text for discussion.