We look for inherent structural characteristics hidden behind amorphous solid formation by using zero-temperature jammed packings of frictionless particles as models. Differently from previous geometrical approaches, we introduce a microscopic mechanical or vibrational order parameter $\mathrm{\Psi }$, which characterizes the susceptibility of particle motion to infinitesimal thermal excitation. We show that (i) the distribution of $\mathrm{\Psi }$ has a power-law tail toward high $\mathrm{\Psi }$ and (ii) the spatial organization of $\mathrm{\Psi }$ is characterized by a nontrivial scale-free correlation. Both findings (i) and (ii) are regarded as a real-space manifestation of marginal stability due to critical self-organization of jammed packings toward mechanical equilibrium. Furthermore, we find that the power-law exponent of the $\mathrm{\Psi }$ distribution tail shows a critical-like scaling behavior toward the unjamming transition, which unveils an intriguing interplay between jamming criticality and marginal stability. Our microscopic order parameter provides new structural insights into the marginal stability and instability of jammed packings and may shed light on the important common structural feature of amorphous solids.

• Received 26 November 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.215502