Phonons, diffusons, and the boson peak in two-dimensional lattices with random bonds

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 $S(\mathbf{q},\omega)$, it is shown that vibrations with frequencies below the Ioffe–Regel frequency $\omega_{\operatorname{IR}}$ are ordinary phonons with a linear dispersion law $\omega(q)\propto q$ and a reciprocal lifetime $\Gamma\sim q^{3}$. Vibrations with frequencies above $\omega_{\operatorname{IR}}$, although being delocalized, cannot be described by plane waves with a definite dispersion law $\omega(q)$. They are characterized by a diffusion structure factor with a reciprocal lifetime $\Gamma\sim q^{2}$, 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 $\omega_b$ in the reduced density of vibrational states $g(\omega)/\omega$ is on the order of the frequency $\omega_{\operatorname{IR}}$. It is located in the transition region between phonons and diffusons and is proportional to the Young's modulus of the lattice, $\omega_b\simeq E$.