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(?). 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(?). In particular, the phonon attenuation is reflected in a parental temperature dependency of the exponent s(T*) in the low-frequency power law D(?)~?s(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 ?3 p. Finally, we provide an estimate of ? as a function of T*, finding a mild power law divergence, ?~(T*-Td)-?/3, with Td the dynamical crossover temperature and ? falling in the range ?[0.8,1.0].