Low-frequency vibrational spectrum of mean-field disordered systems

We study a recently introduced and exactly solvable mean-field model for the density of vibrational states $\mathcal{D}(\ensuremath{\omega})$ structurally disordered system. The is formulated as collection anharmonic oscillators, with random stiffness $\ensuremath{\kappa}$ drawn from distribution $p(\ensuremath{\kappa})$, subjected to constant field $h$ interacting bilinearly coupling strength $J$. investigate properties its ground state at zero temperature. When $p(\ensuremath{\kappa})$ gapped, emergent also small Upon increasing $J$, gap vanishes on critical line in $(h,J)$ phase diagram, whereupon replica symmetry broken. At $h$, form this pseudogap quadratic, $\mathcal{D}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2}$, modes are delocalized, expected previously investigated spin glass models. However, we determine that large enough quartic $\mathcal{D}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{4}$, populated by localized modes, emerges, two regimes being separated special point line. thus uncover systems can generically display both quadratic-delocalized quartic-localized spectrum transition.