Relation between heterogeneous frozen regions in supercooled liquids and non-Debye spectrum in the corresponding glasses

Recent numerical studies on glassy systems provide evidences for a population of non-Goldstone modes (NGMs) in the low-frequency spectrum of the vibrational density of states $D(\omega )$. 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 $T^{\ast }$ modifies the GM/NGM ratio in $D(\omega )$. In particular, the phonon attenuation is reflected in a parental temperature dependency of the exponent $s(T^{\ast })$ in the low-frequency power law $D(\omega) \sim \omega^{s(T^)}$,with$2 \leq s(T^) \leq 4 $\mathrm{.}Secondly,bycomparing$s(T^*)$with$s(p)$,i\mathrm{.}e\mathrm{.},thesamequantityobtainedbypinning\text{mttp}a$p$ particle fraction, we suggest that $s(T^{\ast })$ reflects the presence of dynamical heterogeneous regions of size ${\xi }^3\propto p$. Finally, we provide an estimate of $\xi$ as a function of $T^{\ast }$, finding a mild power law divergence, $\xi \sim (T^{\ast }-T_d{)}^{-\alpha \mathrm{/}3}$, with $T_d$ the dynamical crossover temperature and $\alpha$ falling in the range $\alpha \in [0.8,1.0]$.