Vibrational Properties of Hard and Soft Spheres are Unified at Jamming

Glasses and granular materials are characterized by the appearance of amorphous rigidity at the jamming point. While grains described as soft spheres jam in the limit of zero pressure, in a thermal hard sphere glass jamming is only achieved at infinite pressure when all the available space is filled and particles are in enduring contact with one another. The criticality near jamming has been largely explored from the soft side of the transition both in theory and numerics. The study of the vibrational properties in hard spheres has proven to be challenging to measure numerically because no analytic interaction is defined. Recently, an effective interaction between hard spheres at zero temperature has been proposed, of which minimization is able to produce typical configurations of low temperature colloidal glasses. This protocol ensures a proper definition of the vibrational spectrum near and at jamming. We observe a variety of low frequency modes which, for a certain range of density, agree with those found in jammed soft spheres. Therefore, the vibrational properties of hard and soft spheres near jamming are fully governed by the geometry of the system and the critical point represents a smooth joining between the two descriptions.