Disordered Solids Without Well-Defined Transverse Phonons: The Nature of   Hard Sphere Glasses
  We probe the Ioffe-Regel limits of glasses with repulsions near the zero-temperature jamming transition by measuring the dynamical structure factors. At zero temperature, the transverse Ioffe-Regel frequency vanishes at the jamming transition with a diverging length, but the longitudinal one does not, which excludes the existence of a diverging length associated with the longitudinal excitations. At low temperatures, the transverse and longitudinal Ioffe-Regel frequencies approach zero at the jamming-like transition and glass transition, respectively. As a consequence, glasses between the glass transition and jamming-like transition, which are hard sphere glasses in the low temperature limit, can only carry well-defined longitudinal phonons and have an opposite pressure dependence of the ratio of the shear modulus to the bulk modulus from glasses beyond the jamming-like transition. 
Xipeng Wang, Wen Zheng, Lijin Wang, and Ning Xu
cond-mat.soft