Persistent homology in two-dimensional atomic networks

What is it about?

We use the emerging method of persistence homology in topological analysis to characterise the structure of 2D network-forming materials, and apply the method to experimentally obtained configurations of silica bilayers and graphene.

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Photo by Alina Grubnyak on Unsplash

Photo by Alina Grubnyak on Unsplash

Why is it important?

Persistence homology is an emerging method for analysing the topology of network-forming amorphous materials. To date, the method provides largely qualitative descriptors, with some doubt as to the “added value” compared to more traditional measures of network structures. Here, we address this issue by investigating two-dimensional atomic networks where the ring structure is well defined and straightforward to visualise. In particular, the level of disorder is controlled systematically via the choice of model (triangle raft versus bond-switch) and the choice of temperature within a given class of model. In the persistence diagrams used to visualise the output from persistent homology, a band structure Bn is observed, which is unambiguously shown to originate from atoms separated by n bonds. The method thereby provides information on two and higher body correlations that is not accessible from structure factors or radial distribution functions. The persistent homology method also gives the primitive ring statistics, provided the level of disorder is not too large, together with information on the regularity of rings, which is unavailable from the ring statistics. Finally, the method is applied to experimentally obtained configurations of silica bilayers and graphene, and the potential utility of persistence homology as an analytic tool for materials characterisation is examined.

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