Universal disorder-induced broadening of phonon bands: from disordered lattices to glasses

The translational symmetry of solids, either ordered or disordered, gives rise to the existence of low-frequency phonons. In ordered systems, either crystalline solids or isotropic homogeneous continua, some phonons characterized by different wavevectors are degenerate, i.e. they share the exact same frequency ω; in finite-size systems, phonons form a discrete set of bands with nq(ω)-fold degeneracy. Here we focus on understanding how this degeneracy is lifted in the presence of disorder, and its physical implications. Using standard degenerate perturbation theory and simple statistical considerations, we predict the dependence of the disorder-induced frequency width of phonon bands to be , where σ is the strength of disorder and N is the total number of particles. This theoretical prediction is supported by extensive numerical calculations for disordered lattices - characterized by topological, mass, stiffness and positional disorder - and for computer glasses, where disorder is self-generated, thus establishing its universal nature. The predicted scaling of phonon band widths leads to the identification of a crossover frequency ω† ~ L-2/(d+2) in systems of linear size L in đ > 2 dimensions, where the disorder-induced width of phonon bands becomes comparable to the frequency gap between neighboring bands. Consequently, phonons continuously cover the frequency range ω > ω†, where the notion of discrete phonon bands becomes ill-defined. Two basic applications of the theory are presented; first, we show that the phonon scattering lifetime is proportional to (Δω)-1 for ω < ω†. Second, the theory is applied to the basic physics of glasses, allowing us to determine the range of frequencies in which the recently established universal density of states of non-phononic excitations can be directly probed for different system sizes