Vibrational Properties of Hard and Soft Spheres Are Unified at Jamming

The unconventional thermal properties of jammed amorphous solids are directly related to their density of vibrational states. While the vibrational spectrum of jammed soft sphere solids has been fully described, the vibrational spectrum of hard spheres, a model glass former often related to physical colloidal glasses, is still unknown due to the difficulty of treating the nonanalytic interaction potential. We bypass this difficulty using the recently described effective interaction potential for the free energy of thermal hard spheres. By minimizing this effective free energy, we mimic the rapid compression of hard spheres and produce typical configurations of the thermal system. We measure the resulting vibrational spectrum and characterize its evolution toward the jamming point where configurations of hard and soft spheres are trivially unified. For densities approaching jamming from below, we observe low-frequency modes which agree with those found in numerical simulations of jammed soft spheres. Our measurements of the vibrational structure demonstrate that the jamming universality extends away from jamming: hard sphere thermal systems below jamming exhibit the same vibrational spectra as thermal and athermal soft sphere systems above the transition.