Relation between Heterogeneous Frozen Regions in Supercooled Liquids and Non-Debye Spectrum in the Corresponding Glasses

Recent numerical studies on glassy systems provide evidence for a population of non-Goldstone modes (NGMs) in the low-frequency spectrum of the vibrational density of states 𝐷⁡(𝜔). Similarly to Goldstone modes (GMs), i.e., phonons in solids, NGMs are soft low-energy excitations. However, differently from GMs, NGMs are localized excitations. Here we first show that the parental temperature 𝑇* modifies the GM/NGM ratio in 𝐷⁡(𝜔). In particular, the phonon attenuation is reflected in a parental temperature dependency of the exponent 𝑠⁡(𝑇*) in the low-frequency power law 𝐷⁡(𝜔)∼𝜔𝑠⁡(𝑇*), with 2≤𝑠⁡(𝑇*)≤4. Second, by comparing 𝑠⁡(𝑇*) with 𝑠⁡(𝑝), i.e., the same quantity obtained by pinning a 𝑝 particle fraction, we suggest that 𝑠⁡(𝑇*) reflects the presence of dynamical heterogeneous regions of size 𝜉3∝𝑝. Finally, we provide an estimate of 𝜉 as a function of 𝑇*, finding a mild power law divergence, 𝜉∼(𝑇*−𝑇𝑑)−𝛼/3, with 𝑇𝑑 the dynamical crossover temperature and 𝛼 falling in the range 𝛼∈[0.8,1.0].