  The low-frequency vibrational and low-temperature thermal properties of
amorphous solids are markedly different from those of crystalline solids. This
situation is counter-intuitive because any solid material is expected to behave
as a homogeneous elastic body in the continuum limit, in which vibrational
modes are phonons following the Debye law. A number of phenomenological
explanations have been proposed, which assume elastic heterogeneities, soft
localized vibrations, and so on. Recently, the microscopic mean-field theories
have been developed to predict the universal non-Debye scaling law. Considering
these theoretical arguments, it is absolutely necessary to directly observe the
nature of the low-frequency vibrations of amorphous solids and determine the
laws that such vibrations obey. Here, we perform an extremely large-scale
vibrational mode analysis of a model amorphous solid. We find that the scaling
law predicted by the mean-field theory is violated at low frequency, and in the
continuum limit, the vibrational modes converge to a mixture of phonon modes
following the Debye law and soft localized modes following another universal
non-Debye scaling law.
