Finite-size effects in the nonphononic density of states in computer glasses

The universal form of the density of nonphononic, quasilocalized vibrational modes of frequency $\omega$ in structural glasses, ${\cal D}(\omega)$, was predicted theoretically decades ago, but only recently revealed in numerical simulations. In particular, it has been recently established that, in generic computer glasses, ${\cal D}(\omega)$ increases from zero frequency as $\omega^4$, independent of spatial dimension and of microscopic details. However, in [E. Lerner, and E. Bouchbinder, Phys. Rev. E 96, 020104(R) (2017)] it was shown that the preparation protocol employed to create glassy samples may affect the form of their resulting ${\cal D}(\omega)$: glassy samples rapidly quenched from high temperature liquid states were shown to feature ${\cal D}(\omega)\!\sim\!\omega^\beta$ with $\beta\!<\!4$, presumably limiting the degree of universality of the $\omega^4$ law. Here we show that exponents $\beta\!<\!4$ are only seen in small glassy samples quenched from high-temperatue liquid states --- whose sizes are comparable to or smaller than the size of the disordered core of soft quasilocalized vibrations --- while larger glassy samples made with the same protocol feature the universal $\omega^4$ law. Our results demonstrate that observations of $\beta\!<\!4$ in the nonphononic density of states stem from finite-size effects, and we thus conclude that the $\omega^4$ law should be featured by any sufficiently large glass quenched from a melt.