Probing the Debye spectrum in glasses using small system sizes

We investigate the low-frequency spectrum in a three-dimensional model of structural glass focusing on small system sizes, and using different observables, i.e., the density of states D(?), the cumulative of the density of states F(?), and the dynamical structure factor S(q,?) in the harmonic approximation. When the glass is obtained by an instantaneous quench from high temperatures, we show that extended "phonon-like" modes always populate the low-energy spectrum. Looking at the properties of the dynamical structure factor S(q,?), we observe that in agreement with early studies of Lennard-Jones glasses [V. Mazzacurati, G. Ruocco, and M. Sampoli, Europhys. Lett. 34, 681 (1996)], there are still extended modes below the lowest resonant peak. These modes give rise to a plateau in the S(q,?) for ?->0. This result indicates that the low-energy spectrum of extended modes in glasses can be probed using small system sizes and performing instantaneous quench from high parental temperatures. As we recently observed [M. Paoluzzi, L. Angelani, G. Parisi, and G. Ruocco, Phys. Rev. Lett. 123, 155502 (2019)], the situation changes when the glassy configuration is obtained by an instantaneous quench from lower temperatures. The former protocol suppresses extended modes below the lowest resonant peak emphasizing the localized modes with D(?)~?4.