  The vibrational density of states of glasses is considerably different from
that of crystals. In particular, there exist spatially localized vibrational
modes in glasses. The density of states of these non-phononic modes has been
observed to follow $g(\omega) \propto \omega^4$, where $\omega$ is the
frequency. However, in two-dimensional systems, the abundance of phonons makes
it difficult to accurately determine this non-phononic density of states
because they are strongly coupled to non-phononic modes and yield strong
system-size and preparation-protocol dependencies. In this article, we utilize
the random pinning method to suppress phonons and disentangle their coupling
with non-phononic modes and successfully calculate their density of states as
$g(\omega) \propto \omega^4$. We also study their localization properties and
confirm that low-frequency non-phononic modes in pinned systems are truly
localized without far-field contributions. We finally discuss the excess
density of states over the Debye value that results from the hybridization of
phonons and non-phononic modes.
