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

Glasses are known to possess low-frequency excess modes beyond the Debye prediction. For decades, it has been assumed that evolution of low-frequency density of excess modes D(ω) with frequency ω follows a power-law scaling: D(ω) ∼ ω^γ. However, it remains debated on the value of γ at low frequencies below the first phonon-like mode in finite-size glasses. Early simulation studies reported γ = 4 at low frequencies in two- (2D), three- (3D), and four-dimensional (4D) glasses, whereas recent observations in 2D and 3D glasses suggested γ = 3.5 in a lower-frequency regime. It is uncertain whether the low-frequency scaling of D(ω) ∼ ω^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. It is found that the system size dependence of D(ω) below the first phonon-like mode varies with spatial dimensions: D(ω) increases in 2D glasses but decreases in 3D and 4D glasses as the system size increases. Furthermore, we demonstrate that the ω^3.5 scaling, rather than the ω⁴ 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(ω) ∼ ω^3.5 below the first phonon-like mode across different spatial dimensions, which would inspire further theoretical studies.