Hierarchical Jamming

The free energy landscape of mean field marginal glasses is ultrametric. We demonstrate that this feature persists in finite three dimensional systems which are out of equilibrium by finding sets of minima which are nearby in configuration space. By calculating the distance between these nearby minima, we produce a small region of the distance metric. This metric exhibits a clear hierarchical structure and shows the signature of an ultrametric space. That such a hierarchy exists for the jamming energy landscape provides direct evidence for the existence of a marginal phase along the zero temperature jamming line.