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

## TL;DR

This study links the low-frequency vibrational spectrum in glasses to heterogeneous regions, showing how parental temperature influences non-Goldstone modes and their localization, with implications for understanding glass dynamics.

## Contribution

It reveals how parental temperature affects the ratio of Goldstone to non-Goldstone modes and introduces a method to estimate heterogeneity size from vibrational spectra.

## Key findings

- Parental temperature modifies the GM/NGM ratio in vibrational spectra.
- The low-frequency spectrum follows a power law with an exponent dependent on temperature.
- Heterogeneity size diverges mildly as temperature approaches the dynamical crossover.

## Abstract

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 $T^*$ modifies the GM/NGM ratio in $D(\omega)$. In particular, the phonon attenuation is reflected in a parental temperature dependency of the exponent $s(T^*)$ in the low-frequency power law $D(\omega) \sim \omega^{s(T^*)}$, with $2 \leq s(T^*) \leq 4 $. Secondly, by comparing $s(T^*)$ with $s(p)$, i. e., the same quantity obtained by pinning \mttp{a} $p$ particle fraction, we suggest that $s(T^*)$ reflects the presence of dynamical heterogeneous regions of size $\xi^3 \propto p$. Finally, we provide an estimate of $\xi$ as a function of $T^*$, finding a mild power law divergence, $\xi \sim (T^* - T_d)^{-\alpha/3}$, with $T_d$ the dynamical crossover temperature and $\alpha$ falling in the range $\alpha \in [0.8,1.0]$.