Density of states in random lattices with translational invariance

We propose a random matrix approach to describe vibrations in disordered systems. The dynamical matrix $M$ is taken in the form $M=AA^T$ where $A$ is a real random matrix. It guaranties that $M$ is a positive definite matrix. This is necessary for mechanical stability of the system. We built matrix $A$ on a simple cubic lattice with translational invariance and interaction between nearest neighbors. It was found that for a certain type of disorder acoustical phonons cannot propagate through the lattice and the density of states $g(\omega)$ is not zero at $\omega=0$. The reason is a breakdown of affine assumptions and inapplicability of the macroscopic elasticity theory. Young modulus goes to zero in the thermodynamic limit. It reminds of some properties of a granular matter at the jamming transition point. Most of the vibrations are delocalized and similar to diffusons introduced by Allen, Feldman et al., Phil. Mag. B 79, 1715 (1999). We show how one can gradually return rigidity and phonons back to the system increasing the width of the so-called phonon gap (the region where $g(\omega)\propto\omega^2$). Above the gap the reduced density of states $g(\omega)/\omega^2$ shows a well-defined Boson peak which is a typical feature of glasses. Phonons cease to exist above the Boson peak and diffusons are dominating. It is in excellent agreement with recent theoretical and experimental data.