
VOLUME 93  ISSUE 10 
PAGE 660

Density of states in random lattices with translational invariance
Y. M. Beltukov, D. A. Parshin
Saint Petersburg State Polytechnical University, 195251 Saint Petersburg, Russia
Abstract
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(ω) is not zero at ω=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 socalled
phonon gap (the region where ). Above the gap
the reduced density of states g(ω)/ω^{2} shows a welldefined 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.

