Dynamic criticality at the jamming transition

We characterize vibrational motion occurring at low temperatures in dense suspensions of soft repulsive spheres over a broad range of volume fractions encompassing the jamming transition at (T = 0, φ = φJ). We find that characteristic time and length scales of thermal vibrations obey critical scaling in the vicinity of the jamming transition. We show in particular that the amplitude and the time scale of dynamic fluctuations diverge symmetrically on both sides of the transition, and directly reveal a diverging correlation length. The critical region near φJ is divided in three different regimes separated by a characteristic temperature scale T⋆(φ) that vanishes quadratically with the distance to φJ. While two of them, (T < T⋆(φ), φ > φJ) and (T < T⋆(φ), φ < φJ), are described by harmonic theories developed in the zero temperature limit, the third one for T > T⋆(φ) is inherently anharmonic and displays new critical properties. We find that the quadratic scaling of T⋆(φ) is due to nonperturbative anharmonic contributions, its amplitude being orders of magnitude smaller than the perturbative prediction based on the expansion to quartic order in the interactions. Our results show that thermal vibrations in colloidal assemblies directly reveal the critical nature of the jamming transition. The critical region, however, is very narrow and has not yet been attained experimentally, even in recent specifically-dedicated experiments.