Persistent homology in two-dimensional atomic networks

David Ormrod Morley, Philip Salmon, Mark Wilson

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The topology of two-dimensional network materials is investigated by persistent homology analysis. The constraint of two dimensions allows for a direct comparison of key persistent homology metrics (persistence diagrams, cycles, and Betti numbers) with more traditional metrics such as the ring-size distributions. Two different types of networks are employed in which the topology is manipulated systematically. In the first, comparatively rigid networks are generated for a triangle-raft model, which are representative of materials such as silica bilayers. In the second, more flexible networks are generated using a bond-switching algorithm, which are representative of materials such as graphene. Bands are identified in the persistence diagrams by reference to the length scales associated with distorted polygons. The triangle-raft models with the largest ordering allow specific bands Bn (n = 1, 2, 3, …) to be allocated to configurations of atoms separated by n bonds. The persistence diagrams for the more disordered network models also display bands albeit less pronounced. The persistent homology method thereby provides information on n-body correlations that is not accessible from structure factors or radial distribution functions. An analysis of the persistent cycles gives the primitive ring statistics, provided the level of disorder is not too large. The method also gives information on the regularity of rings that is unavailable from a ring-statistics analysis. The utility of the persistent homology method is demonstrated by its application to experimentally-obtained configurations of silica bilayers and graphene.
Original languageEnglish
Article number124109
Number of pages19
JournalJournal of Chemical Physics
Issue number12
Early online date23 Mar 2021
Publication statusPublished - 28 Mar 2021

Bibliographical note

Funding Information:
D.O.M. and M.W. acknowledge support from the EPSRC Centre for Doctoral Training in Theory and Modeling in Chemical Sciences (TMCS) under Grant No. EP/L015722/1. P.S.S. was supported by the Institute for Mathematical Innovation at the University of Bath (Grant No. IMI/201920/011) and acknowledges helpful discussions with Dr. James Hook. This paper conforms to the RCUK data management requirements.


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