Moiré patterns appear when two atomically thin layers are rotated a little relative to each other. Some rotation angles produce an exact larger repeating cell, called commensurate angles. These angles matter because they let researchers use a finite periodic model to predict electronic behavior. The new paper introduces a way to spot those commensurate angles directly from atomic positions, even when the lattice is disturbed or only partially sampled.

The authors treat atomic coordinates as point-cloud data and use a tool from topological data analysis (TDA) called persistent homology. Persistent homology watches how simple geometric features—like groups of nearby atoms or loop-like gaps—appear and disappear as you zoom out. The output, called a persistence diagram (PD), is a compact summary of structure across scales. The team adds a “small-neighborhood separation” filter to remove many repeated small motifs that would otherwise drown out the larger-scale pattern.

To decide whether a twisted configuration matches a reference (for example an untwisted or ideal cell), the researchers compute a Wasserstein distance between persistence diagrams. Sharp minima in that distance indicate commensurate angles. They benchmarked this topological approach against geometric and Fourier-based (spectral) similarity measures and found the topological descriptors are more stable when atom positions are perturbed. The paper reports robustness under a range of positional disorder in bond length, from weak to strong perturbations.

The authors also show how to transfer these topological fingerprints to sparse datasets. They fit a constrained Gaussian process (a statistical surrogate model) so the descriptors can be predicted from only a few sampled configurations. As a concrete material example, they applied the method to sparsely sampled, density-functional-theory (DFT)–relaxed structures of a twisted MoTe2 dichalcogenide bilayer, and to fully relaxed twisted MoTe bilayers in their tests.