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.
This paper introduces persistent homology, which is a powerful tool to characterize the shape of data using the mathematical concept of topology. We explain the fundamental idea of persistent homology from scratch using some examples. We also review some applications of persistent homology to materials researches and software for persistent homology data analysis. HomCloud, one of persistent homology software, is especially featured in this paper.
This paper addresses a cyclone identification algorithm with the superlevel set filtration of the persistent homology together with the merge-tree reconstruction of data. Based on the information of peaks and saddles of the scaler field, the newly developed algorithm divides the analysis area into several homology classes, each of which satisfies the peak-to-saddle difference larger than a criterion that should be set in advance. Applied to the 850-hPa relative vorticity in the western North Pacific at 1200 UTC on 2 March 2013, 3 homology classes were found with the criterion of 100 × 10−6 s−1 and 17 homology classes were found with the criterion of 50 × 10−6 s−1. The merge-tree restructuring clarified the neighbour relation among homology classes. The result suggests that the weak criterion detected too much homology classes, some of which are small peaks inside of a single cyclone. The climatology feature density provides the Pacific storm track with the strict criterion. Finally, a possible way to extend toward cyclone tracking with the persistent homology is discussed.
Topological data analysis is an emerging mathematical concept for characterizing shapes in multi-scale data. In this field, persistence diagrams are widely used as a descriptor of the input data, and can distinguish robust and noisy topological properties. Nowadays, it is highly desired to develop a statistical framework on persistence diagrams to deal with practical data. This paper proposes a kernel method on persistence diagrams. A theoretical contribution of our method is that the proposed kernel allows one to control the effect of persistence, and, if necessary, noisy topological properties can be discounted in data analysis. Furthermore, the method provides a fast approximation technique. The method is applied into several problems including practical data in physics, and the results show the advantage compared to the existing kernel method on persistence diagrams.
Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.
Understanding disordered structure is difficult due to insufficient information in experimental data. Here, we overcome this issue by using a combination of diffraction and simulation to investigate oxygen packing and network topology in glassy (g-) and liquid (l-) MgO–SiO2 based on a comparison with the crystalline topology. We find that packing of oxygen atoms in Mg2SiO4 is larger than that in MgSiO3, and that of the glasses is larger than that of the liquids. Moreover, topological analysis suggests that topological similarity between crystalline (c)- and g-(l-) Mg2SiO4 is the signature of low glass-forming ability (GFA), and high GFA g-(l-) MgSiO3 shows a unique glass topology, which is different from c-MgSiO3. We also find that the lowest unoccupied molecular orbital (LUMO) is a free electron-like state at a void site of magnesium atom arising from decreased oxygen coordination, which is far away from crystalline oxides in which LUMO is occupied by oxygen’s 3s orbital state in g- and l-MgO–SiO2, suggesting that electronic structure does not play an important role to determine GFA. We finally concluded the GFA of MgO–SiO2 binary is dominated by the atomic structure in terms of network topology.
Abstract Silicate glasses have evolved from basic structural materials to enabling materials for advanced applications. In this article, we unravel the origin of the mixed alkali effect for alkali silicate 22.7R 2 O–77.3SiO 2 glasses (R = Na and/or K) by identifying the variation in the alkali ion location around the non-bridging oxygen atoms. To do so, we constructed a state-of-the art structural model, which reproduces both diffraction and NMR data with a particular focus on the behavior of the alkali ions. A novel topological analysis using persistent homology found that sodium-potassium silicate glass shows a significant reduction in large cavities as a result of the mixed alkali effect. Furthermore, a highly correlated pair arrangement between sodium and potassium ions around non-bridging oxygen atoms was identified. The potassium ions can be trapped in K–O polyhedra due to the increased bridging oxygen coordination; therefore, the correlated pair arrangement is likely the intrinsic origin of the mixed alkali effect.
This paper introduces a new percolation model motivated from polymer materials. The mathematical model is defined over a random cubical set in the $d$-dimensional space $\mathbb{R}^d$ and focuses on generations and percolations of $(d-1)$-dimensional holes as higher dimensional topological objects. Here, the random cubical set is constructed by the union of unit faces in dimension $d-1$ which appear randomly and independently with probability $p$, and holes are formulated by the homology generators. Under this model, the upper and lower estimates of the critical probability $p_c^{\rm hole}$ of the hole percolation are shown in this paper, implying the existence of the phase transition. The uniqueness of infinite hole cluster is also proven. This result shows that, when $p > p_c^{\rm hole}$, the probability $P_p(x^*\overset{\rm hole}{\longleftrightarrow} y^*)$ that two points in the dual lattice $(\mathbb{Z}^d)^*$ belong to the same hole cluster is uniformly greater than 0.
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.
