Mean-field model of interacting quasilocalized excitations in glasses

Structural glasses feature quasilocalized excitations whose frequencies ω follow a universal density of states D(ω)∼ω⁴. Yet, the underlying physics behind this universality is not fully understood. Here we study a mean-field model of quasilocalized excitations in glasses, viewed as groups of particles embedded inside an elastic medium and described collectively as anharmonic oscillators. The oscillators, whose harmonic stiffness is taken from a rather featureless probability distribution (of upper cutoff κ₀) in the absence of interactions, interact among themselves through random couplings (characterized by a strength J) and with the surrounding elastic medium (an interaction characterized by a constant force h). We first show that the model gives rise to a gapless density of states D(ω) = A_g ω⁴ for a broad range of model parameters, expressed in terms of the strength of the oscillators’ stabilizing anharmonicity, which plays a decisive role in the model. Then — using scaling theory and numerical simulations — we provide a complete understanding of the non-universal prefactor A_g(h, J,κ₀), of the oscillators’ interaction-induced mean square displacement and of an emerging characteristic frequency, all in terms of properly identified dimensionless quantities. In particular, we show that A_g(h, J,κ₀) is a non-monotonic function of J for a fixed h, varying predominantly exponentially with -(κ₀h²/³/J²) in the weak interactions (small J) regime — reminiscent of recent observations in computer glasses — and predominantly decays as a power-law for larger J, in a regime where h plays no role. We discuss the physical interpretation of the model and its possible relations to available observations in structural glasses, along with delineating some future research directions.