  Characterization of medium-range order in amorphous materials and its
relation to short-range order is discussed. A new topological approach is
presented here to extract a hierarchical structure of amorphous materials,
which is robust against small perturbations and allows us to distinguish it
from periodic or random configurations. The method is called the persistence
diagram (PD) and it introduces scales into many-body atomic structures in order
to characterize the size and shape. We first illustrate how perfect crystalline
and random structures are represented in the PDs. Then, the medium-range order
in the amorphous silica is characterized by using the PD. The PD approach
reduces the size of the data tremendously to much smaller geometrical summaries
and has a huge potential to be applied to broader areas including complex
molecular liquid, granular materials, and metallic glasses.
