  Glasses possess more low-frequency vibrational modes than predicted by Debye
theory. These excess modes are crucial for the understanding the low
temperature thermal and mechanical properties of glasses, which differ from
those of crystalline solids. Recent simulational studies suggest that the
density of the excess modes scales with their frequency $\omega$ as $\omega^4$
in two and higher dimensions. Here, we present extensive numerical studies of
two-dimensional model glass formers over a large range of glass stabilities. We
find that the density of the excess modes follows $D_\text{exc}(\omega)\sim
\omega^2 $ up to around the boson peak, regardless of the glass stability. The
stability dependence of the overall scale of $D_\text{exc}(\omega)$ correlates
with the stability dependence of low-frequency sound attenuation. However, we
also find that in small systems, where the first sound mode is pushed to higher
frequencies, at frequencies below the first sound mode there are excess modes
with a system size independent density of states that scales as $\omega^3$.
