# Vibrational spectrum derived from the local mechanical response in disordered solids

## TL;DR

This paper demonstrates that quasilocalized vibrations in disordered solids can be derived from elasticity theory with quenched disorder, revealing a new elastic instability related to local dipolar forces that explains their origin.

## Contribution

The study introduces a novel elastic instability mechanism for quasilocalized vibrations, expanding the theoretical understanding of vibrational modes in disordered solids.

## Key findings

- Identified a new elastic instability related to local dipolar forces.
- Derived the quartic law for quasilocalized vibrational density of states.
- Mode structure and energetics are consistent with QLVs.

## Abstract

The low-frequency vibrations of glasses are markedly different from those of crystals. These vibrations have recently been categorized into two types: spatially extended vibrations, whose vibrational density of states (vDOS) follows a non-Debye quadratic law, and quasilocalized vibrations (QLVs), whose vDOS follows a quartic law. The former are explained by elasticity theory with quenched disorder and microscopic replica theory as being a consequence of elastic instability, but the origin of the latter is still debated. Here, we show that the latter can also be directly derived from elasticity theory with quenched disorder. We find another elastic instability that the theory encompasses but that has been overlooked so far, namely, the instability of the system against a local dipolar force. This instability gives rise to an additional contribution to the vDOS, and the spatial structure and energetics of the mode originating from this instability are consistent with those of the QLVs. Finally, we construct a model in which the additional contribution to the vDOS follows a quartic law.