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Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.

This paper extends the key concept of persistence within Topological Data Analysis (TDA) in a new direction. TDA quantifies topological shapes hidden in unorganized data such as clouds of unordered points. In the 0-dimensional case the distance-based persistence is determined by a single-linkage (SL) clustering of a finite set in a metric space. Equivalently, the 0D persistence captures only edge-lengths of a Minimum Spanning Tree (MST). Both SL dendrogram and MST are unstable under perturbations of points. We define the new stable-under-noise mergegram, which outperforms previous isometry invariants on a classification of point clouds by PersLay.