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 $D(\omega )\text{\hspace{0.17em}⁣}\sim \text{\hspace{0.17em}⁣}{\omega }^4$ 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 $L\text{\hspace{0.17em}⁣}<\text{\hspace{0.17em}⁣}{L}_D$ is Weibullian, with scaling exponents in excellent agreement with the ${\omega }^4$ law. We further show that the lowest frequency modes are spatially quasi-localized, 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 lengthscales are briefly discussed.