Geometry of hopping processes and local excitations in glasses

In amorphous materials, groups of particles can rearrange locally into a new stable configuration. Such elementary excitations are key as they determine the response to external stresses, as well as to thermal and quantum fluctuations. Yet, understanding what controls their geometry remains a challenge. Here we build a scaling description of their geometry and energy in terms of the distance to an instability, corresponding to a vanishing frequency $\omega_c$ in their vibrational spectrum, as predicted for instance at the dynamical transition in mean field approaches of supercooled liquids. We successfully test our predictions in materials with a gapped spectrum and in regular ultrastable glasses. Our analysis explains why string-like rearrangements are observed to relax liquids only at low temperatures, as otherwise excitations display too small displacements to probe the granularity of the material.