Solvable model of interacting quasilocalized excitations in glasses

Structural glasses feature quasilocalized excitations whose frequencies $\omega$ follow a universal density of states (DOS) ${\cal D}(\omega)\!\sim\!\omega^4$. Yet, the underlying physics behind this universality is not fully understood. Here we study a solvable model of quasilocalized excitations in glasses, viewed as groups of particles described collectively as anharmonic oscillators, which are coupled among themselves through random interactions that are characterized by a disorder strength $J$. We show that the model generically gives rise to a gapless ${\cal D}(\omega)\!=\!A_g\omega^4$ density of states, where $A_g$ -- describing the number and the characteristic frequency of soft quasilocalized excitations -- is shown to vary exponentially with $1/J^2$. Finally, for sufficiently large disorder strength $J$, excitations are predicted to become delocalized through a spin-glass (Gardner-like) transition. We discuss the possible relations between our findings and available observations.