Scaling of the Non-Phononic Spectrum of Two-Dimensional Glasses

Low-frequency vibrational harmonic modes of glasses are frequently used to understand their universal low-temperature properties. One well studied feature is the excess low-frequency density of states over the Debye model prediction. Here we examine the system size dependence of the density of states for two-dimensional glasses. For systems of fewer than 100 particles, the density of states scales with the system size as if all the modes were plane-wave-like. However, for systems greater than 100 particles we find a different system-size scaling of the cumulative density of states below the first transverse sound mode frequency, which can be derived from the assumption that these modes are quasi-localized. Moreover, for systems greater than 100 particles, we find that the cumulative density of states scales with frequency as a power law with the exponent that leads to the exponent $\beta =3.5$ for the density of states independent of system size.