Фононы, диффузоны и бозонный пик в двумерных решетках со случайными связями

Within the model of stable random matrices possessing translational invariance, a two-dimensional (on a square lattice) disordered oscillatory system with random strongly fluctuating bonds is considered. By a numerical analysis of the dynamic structure factor S ( q , ω), it is shown that vibrations with frequencies below the Ioffe-Regel frequency ω_IR are ordinary phonons with a linear dispersion law ω( q ) ∝ q and a reciprocal lifetime б ~ q ^3. Vibrations with frequencies above ω_IR, although being delocalized, cannot be described by plane waves with a definite dispersion law ω( q ). They are characterized by a diffusion structure factor with a reciprocal lifetime б ~ q ^2, which is typical of a diffusion process. In the literature, they are often referred to as diffusons. It is shown that, as in the three-dimensional model, the boson peak at the frequency ωb in the reduced density of vibrational states g (ω)/ω is on the order of the frequency ω_IR. It is located in the transition region between phonons and diffusons and is proportional to the Young’s modulus of the lattice, ω_ b ≃ E .