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 $\phi$ is lowered towards the onset of unjamming at $\phi_{c}$, the density of vibrational states approaches a non-zero value in the limit of zero frequency. For $\phi>\phi_{c}$, there is a crossover frequency, $\omega^{*}$ below which the density of states drops towards zero. This crossover frequency obeys power-law scaling with $\phi-\phi_{c}$. Characteristic length scales, determined from the dominant wavevector contributing to the eigenmode at $\omega^{*}$, diverge as power-laws at the unjamming transition.