Density of excess modes below the first phonon mode in four-dimensional glasses

Abstract

Glasses are known to possess low-frequency excess modes beyond the Debye prediction. For decades, it has been assumed the evolution of the low-frequency density of excess modes $D(\omega)$ with the frequency $\omega$ follows a power-law scaling: $D(\omega)\sim \omega^{\gamma}$. However, it remains debated on the value of $\gamma$ at low frequencies below the first phonon-like mode in finite-size glasses. Early simulation studies reported $\gamma=4$ at low frequencies in two- (2D), three- (3D) and four-dimensional (4D) glasses, whereas recent observations in 2D and 3D glasses suggested $\gamma=3.5$ in a lower-frequency regime. It's uncertain whether the low-frequency scaling of $D(\omega)\sim \omega^{3.5}$ could be generalized to 4D glasses. Here, we conduct numerical simulation studies of excess modes at frequencies below the first phonon-like mode in 4D model glasses. We find the system size dependence of $D(\omega)$ below the first phonon-like mode varies with spatial dimensions: $D(\omega)$ increases in 2D glasses but decreases in 3D and 4D glasses as the system size increases. Furthermore, we demonstrate that the $\omega^{3.5}$ scaling, rather than the $\omega^{4}$ scaling, works in the lowest-frequency regime accessed in 4D glasses, regardless of interaction potentials and system sizes examined. Therefore, our findings in 4D glasses, combined with previous results in 2D and 3D glasses, suggest a common low-frequency scaling of $D(\omega)\sim \omega^{3.5}$ below the first phonon-like mode across different spatial dimensions, which would inspire further theoretical studies.