# Continuum limit of the vibrational properties of amorphous solids

## TL;DR

This study investigates the low-frequency vibrational modes of amorphous solids, revealing deviations from mean-field predictions and showing a mixture of phonon and localized modes in the continuum limit.

## Contribution

It provides the first large-scale analysis demonstrating the breakdown of mean-field scaling and the coexistence of phonon and localized vibrational modes in amorphous solids.

## Key findings

- Mean-field theory scaling law is violated at low frequency.
- Vibrational modes in the continuum limit are a mixture of phonons and localized modes.
- Soft localized modes follow a universal non-Debye scaling law.

## Abstract

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.