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 s...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 finitesize 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 in2D 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 ω~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(ω)~ ω^3.5) below the first phonon-like mode across different spatial dimensions,which would inspire further theoretical studies.展开更多
基金the support from the National Natural Science Foundation of China(Grant Nos.12374202 and 12004001)Anhui Projects(Grant Nos.2022AH020009,S020218016,and Z010118169)Hefei City(Grant No.Z020132009)。
文摘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 finitesize 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 in2D 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 ω~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(ω)~ ω^3.5) below the first phonon-like mode across different spatial dimensions,which would inspire further theoretical studies.
