Instabilities govern the low-frequency vibrational spectrum of amorphous solids

Amorphous solids exhibit an excess of low-frequency modes in their vibrational density of states (VDoS) beyond the Debye prediction, contributing to their anomalous mechanical and thermal properties. Recently a power-law behavior has been observed in the low frequency regime of their VDoS; however, a precise exponent remains a subject of debate. In this study, we demonstrate that boundary-condition-induced instabilities play a key role in the variability of this exponent. We identify two distinct types of elastic branches that differ in the nature of their energy landscape: Fictitious branches, where shear minima cannot be reached through elastic deformation alone and undergo plastic instabilities, and True branches, where elastic deformation can access these minima. Configurations on Fictitious branches display a VDoS scaling as D(ω) ∼ ω3, while those on True elastic branches under simple and pure shear deformations exhibit a scaling of D(ω) ∼ ω5.5. In simulations where this distinction is not made, ensemble averaging leads to a weighted combination of spectra from configurations with different mechanical stabilities, resulting in an exponent close to 4, as commonly reported in the literature. Furthermore, when solids are relaxed to their shear minima, eliminating residual stress, a scaling of D(ω) ∼ ω6.5 emerges in both two and three dimensions. Importantly, our simulations show that the prevalence of solids on Fictitious branches increases with increasing system size. Our findings therefore suggest two possible limiting behaviors for amorphous solids: increasing system size without addressing instabilities may result in a low-frequency VDoS scaling as D(ω) ∼ ω3, while removing residual stresses in each configuration results in a D(ω) ∼ ω6.5 behavior.