Universal non-phononic density of states in 2D, 3D and 4D glasses

It is now well established that structural glasses possess disorder- and frustration-induced soft quasilocalized excitations, which play key roles in various glassy phenomena. Recent work has established that in model glass-formers in three dimensions, these non-phononic soft excitations may assume the form of quasilocalized, harmonic vibrational modes whose frequency follows a universal density of states $D(\omega)\!\sim\!\omega^4$, independently of microscopic details, and for a broad range of glass preparation protocols. Here we further establish the universality of the non-phononic density of vibrational modes by direct measurements in model structural glasses in two dimensions and four dimensions. We also investigate their degree of localization, which is generally weaker in lower spatial dimensions, giving rise to a pronounced system-size dependence of the non-phononic density of states in two dimensions, but not in higher dimensions. Finally, we identify a fundamental glassy frequency scale $\omega_c$ above which the universal $\omega^4$ law breaks down.