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.

Glasses exhibit spatially inhomogeneous elastic properties, which can be investigated by measuring their elastic moduli at a local scale. Various methods to evaluate the local elastic modulus have been proposed in the literature. A first possibility is to measure the local stress-local strain curve and to obtain the local elastic modulus from the slope of the curve, or equivalently to use a local fluctuation formula. Another possible route is to assume an affine strain and to use the applied global strain instead of the local strain for the calculation of the local modulus. Most recently a third technique has been introduced, which is easy to be implemented and has the advantage of low computational cost. In this contribution, we compare these three approaches by using the same model glass and reveal the differences among them caused by the non-affine deformations.