Density of states below the first sound mode in 3D glasses

Glasses feature universally low-frequency excess vibrational modes beyond Debye prediction, which could help rationalize, e.g., the glasses' unusual temperature dependence of thermal properties compared to crystalline solids. The way the density of states of these low-frequency excess modes D(ω) depends on the frequency ω has been debated for decades. Recent simulation studies of 3D glasses suggest that D(ω) scales universally with ω⁴ in a low-frequency regime below the first sound mode. However, no simulation study has ever probed as low frequencies as possible to test directly whether this quartic law could work all the way to extremely low frequencies. Here, we calculated D(ω) below the first sound mode in 3D glasses over a wide range of frequencies. We find D(ω) scales with ω^β with β < 4 at very low frequencies examined, while the ω⁴ law works only in a limited intermediate-frequency regime in some glasses. Moreover, our further analysis suggests our observation does not depend on glass models or glass stabilities examined. The ω⁴ law of D(ω) below the first sound mode is dominant in current simulation studies of 3D glasses, and our direct observation of the breakdown of the quartic law at very low frequencies thus leaves an open but important question that may attract more future numerical and theoretical studies.