  This article proposes a topological method that extracts hierarchical
structures of various amorphous solids. The method is based on the persistence
diagram (PD), a mathematical tool for capturing shapes of multiscale data. The
input to the PDs is given by an atomic configuration and the output is
expressed as 2D histograms. Then, specific distributions such as curves and
islands in the PDs identify meaningful shape characteristics of the atomic
configuration. Although the method can be applied to a wide variety of
disordered systems, it is applied here to silica glass, the Lennard-Jones
system, and Cu-Zr metallic glass as standard examples of continuous random
network and random packing structures. In silica glass, the method classified
the atomic rings as short-range and medium-range orders and unveiled
hierarchical ring structures among them. These detailed geometric
characterizations clarified a real space origin of the first sharp diffraction
peak and also indicated that PDs contain information on elastic response. Even
in the Lennard-Jones system and Cu-Zr metallic glass, the hierarchical
structures in the atomic configurations were derived in a similar way using
PDs, although the glass structures and properties substantially differ from
silica glass. These results suggest that the PDs provide a unified method that
extracts greater depth of geometric information in amorphous solids than
conventional methods.
