Vibrations and Diverging Length Scales Near the Unjamming Transition

We numerically study the vibrations of jammed packings of particles interacting with finite-range, repulsive potentials at zero temperature. As the packing fraction $\varphi$ is lowered towards the onset of unjamming at ${\varphi }_c$, the density of vibrational states approaches a nonzero value in the limit of zero frequency. For $\varphi >{\varphi }_c$, there is a crossover frequency, ${\omega }^{*}$ below which the density of states drops towards zero. This crossover frequency obeys power-law scaling with $\varphi \mathrm{\text{−}}{\varphi }_c$. Characteristic length scales, determined from the dominant wave vector contributing to the eigenmode at ${\omega }^{*}$, diverge as power laws at the unjamming transition.