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.T he atomic configurations of amorphous solids are difficult to characterize. Because they have no periodicity as found in crystalline solids, only local structures have been analyzed in detail. Although short-range order (SRO) defined by the nearest neighbor is thoroughly studied, it is not sufficient to fully understand the atomic structures of amorphous solids. Therefore, medium-range order (MRO) has been discussed to properly characterize amorphous solids (1-3). Many experimental and simulation studies (4-7) have suggested signatures of MRO such as a first sharp diffraction peak (FSDP) in the structure factor of the continuous random network structure, and a split second peak in the radial distribution function of the random packing structure. However, in contrast to SRO, the geometric interpretation of MRO and the hierarchical structures among different ranges are not yet clear.Among the available methods, the distributions of bond angle and dihedral angle are often used to identify the geometry beyond the scale of SRO. They cannot, however, provide a complete description of MRO because they only deal with the atomic configuration up to the third nearest neighbors. Alternatively, ring statistics are also applied as a conventional combinatorial topological method (2, 8, 9). However, this method is applicable only for the continuous random network or crystalline structures, and furthermore it cannot describe length scale. Therefore, methodologies that precisely characterize hierarchical structures beyond SRO and are applicable to a wide variety of amorphous solids are highly desired.In recent years, ...

Abstract. 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 manybody 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.

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We study persistence modules defined on commutative ladders. This class of persistence modules frequently appears in topological data analysis, and the theory and algorithm proposed in this paper can be applied to these practical problems. A new algebraic framework deals with persistence modules as representations on associative algebras and the Auslander-Reiten theory is applied to develop the theoretical and algorithmic foundations. In particular, we prove that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander-Reiten quivers. Furthermore, a generalization of persistence diagrams is introduced by using Auslander-Reiten quivers. We provide an algorithm for computing persistence diagrams for the commutative ladders of length 3 by using the structure of Auslander-Reiten quivers.