Geometric origin of excess low-frequency vibrational modes in weakly connected amorphous solids

Glasses have an excess number of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature necessarily occurs in solids with low coordination. In particular, we analyze the density D(omega) of normal-mode frequencies. and the nature of the low-frequency normal modes of a recently simulated system (O'Hern C., Silbert L. E., Liu A. J. and Nagel S. R., Phys. Rev. E, 68 (2003) 011306) comprised of weakly compressed spheres at zero temperature. We account for the observed a) convergence of D(omega) toward a non-zero constant as the frequency goes to zero, b) appearance of a low-frequency cutoff omega*, and c) power law increase of omega* with compression. We introduce a length scale l* which characterizes the vibrational modes that appear at omega*.