Mean-field model of interacting quasilocalized excitations in glasses

Structural glasses feature quasilocalized excitations whose frequencies ωω follow a universal density of states D(ω)∼ω4D(ω)∼ω4. Yet, the underlying physics behind this universality is not yet 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 κ0κ0) in the absence of interactions, interact among themselves through random couplings (characterized by strength JJ) and with the surrounding elastic medium (an interaction characterized by a constant force hh). We first show that the model gives rise to a gapless density of states D(ω)=Agω4D(ω)=Agω4 for a broad range of model parameters, expressed in terms of the strength of 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 Ag(h,J,κ0)Ag(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 Ag(h,J,κ0)Ag(h,J,κ0) is a non-monotonic function of JJ for a fixed hh, varying predominantly exponentially with −(κ0h2/3/J2)−(κ0h2/3/J2) in the weak interactions (small JJ) regime --- reminiscent of recent observations in computer glasses --- and predominantly decays as a power-law for larger JJ, in a regime where hh 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.