Random matrix approach to the boson peak and Ioffe-Regel criterion in amorphous solids

We present a random matrix approach to study the general vibrational properties of stable amorphous solids with translational invariance using the correlated Wishart ensemble. Within this approach, both analytical and numerical methods can be applied. Applying the random matrix theory to the correlated Wishart ensemble, we found the analytical form of the vibrational density of states and the dynamical structure factor. The ratio between the number of bonds and the number of degrees of freedom controls the Ioffe-Regel frequency, which determines the crossover between low-frequency propagating phonons and diffusons at higher frequencies. The reduced vibrational density of states shows the boson peak, whose frequency is close to the Ioffe-Regel crossover. In the isostatic case, we obtain the low-frequency cusplike singularity of the vibrational density of states, which was observed numerically. We also present a simple numerical random matrix model with finite interaction radius, whose properties rapidly converge to the analytical results with increasing the interaction radius. For a finite interaction radius, the numerical model demonstrates the presence of quasilocalized vibrations with a power-law low-frequency density of states.