A Theory of Jamming and Elastic Instability in Low Temperature Amorphous Solids

Abstract

Despite intense research, no first-principles theory has yet rationalized the rich phenomena and vibrational anomalies in amorphous solids. This monograph presents an analytical study, which in takes the first steps towards an exhaustive microscopic theory. The derived microscopic theory successfully describes the jammed and unjammed phase of disordered systems at zero temperature. Employing the Zwanzig-Mori projection operator formalism, we expand the Mode-Coupling Theory to coherently describe the jammed state. We identify a symmetry constraint in the sequence of local scattering events, compelling us to go beyond the standard self-consistent Born approximation: A planar theory does not recover the characteristic Rayleigh-sound attenuation in amorphous solids. Thus, we include non-planar contributions in the theoretical model. The universal vibrational properties of amorphous solids at low temperatures are recovered in the jammed phase. We identify a diffusive regime of modes, characterized by a plateau in the vibrational Density of States. Here, the energy suffices to resolve the local disorder. Below the disorder-dominated regime, modes can propagate, and the system can be approximated as an elastic medium. The vanishing of the transverse speed of sound heralds the unjamming instability. The diffusive regime extends down to zero frequency directly at the critical point. In the unjammed phase, the theory predicts no viscous flow but the presence of modes with zero restoring forces. As a consequence, injected momentum causes plastic rearrangements of stable sub-clusters whose size diverges at the jamming transition. Utilizing the Euclidean Random Matrix model, we support our theory by deriving an equivalent theory in the harmonic approximation of the potential energy. The quantitative predictions of the theory are in good agreement with Philipp Baumgärtel's numerical solution of the scalar Euclidean Random Matrix model published in (Vogel et al. Phys. Rev. X, vol. 15:011.030, 2025).