Ordered ground state configurations of the asymmetric Wigner bilayer system—Revisited with unsupervised learning

We refer to Ref. 1 to a review on various machine learning tools applied to physical systems.

At this point, it should be mentioned that other related techniques are also available, which may even directly operate on the structural data (i.e., on a data set of coordinates of the particles)—see, for instance, Refs. 64 and 65.

See Ref. 2 for an in-depth discussion on different, problem specific similarity measures in data science problems. Methods from unsupervised machine learning¹ can be used to analyze a data set of feature vectors (or of order parameters in our case) for certain similarity measures in the features that may permit us to algorithmically organize the elements of the data set into an initially unknown set of categories.² 

Depending on the particularly applied clustering algorithm the number of clusters may be a preset parameter to the algorithm or may even be identified by the algorithm during execution.

The total number of possible compositions at each value of η is $Ntot=1+12∑n=2N(n−nmod2)$ given that $0<N2≤N12$ for N₁ > 1, and only counting the monolayer structure with N₂ = 0 and N₁ = 1 once. For N = 40, we thus have N_tot = 401.

For a discussion about alternative dimensional reduction and clustering tools and a justification for opting for PCA and k-means clustering in this manuscript, we refer to the supplementary material, Sec. III.

The “elbow” in a linear scree plot,⁶⁶ such as shown in the main panel of Fig. 3, is defined as the point where the contribution of the significant eigenvalues (here realized via the PEVs) seems to level off.⁴ Notably, this is a subjective measure as multiple elbows can occur in a scree plot. In this contribution, we opted for the second elbow (i.e., for i-values larger than i = 9) in Fig. 3—as opposed to the first elbow (at i = 4)—in an effort to include more information and to be on the safe side in our further analysis (cf., the supplementary material, Figs. 7 and 8, for the significance of PC_i>4 in the state diagram of the asymmetric Wigner bilayer system).

It should be noted that we do not claim that the first three PCs are sufficient to fully describe the structural diversity of the asymmetric Wigner bilayer system. We have decided to use PC₁ through PC3 in Fig. 5 simply for illustrative purposes in an effort to motivate the need for a more refined investigation of the diagram of states. Using other permutations of PC₁ through PC₉ will lead to similar plots that differ, however, in significant details, which, in turn, allows for a more systematic classification of structural phases in the system.

Nowadays, it is easily possible to train a neural network in a supervised way with the objective of performing classification tasks.¹ For our purposes, such a task would be to classify structural data into a number of K different categories (identified, for instance, by unsupervised clustering), which would allow us to directly classify a structure from its geometric, structural data, e.g., via coordinates and lattice vectors.⁶⁷ The output of the classifier would then be the probability of a structure falling into any of the K clusters or families (when using “softmax” activation in the output layer of the neural network and “categorical cross-entropy loss” during training¹), which may give additional insight when comparing competing structures.

We report at this occasion that for all K-values both k- and k^∗-means clustering results nicely correlate when evaluating the adjusted mutual information scores between the k- and k^∗-means clustering, i.e., $IK(ki,kj*)$ as defined in the  Appendix E; for more details and graphical representations, we refer to Ref. 40.

Although most of the suggested ground state candidates identified by the evolutionary algorithm in Refs. 11–13 are very likely to represent the true ground state configurations of the asymmetric Wigner bilayer system at a given state point, there is no rigorous proof that they are, indeed, the ground states. Thus, whenever we use the term “ground state solutions,” we refer to “ground state candidate solutions” of the asymmetric Wigner bilayer system.

Also, for the k^∗-means clustering, we rely on the set of analytically labeled data, w^(sym), of the entire data set, X^(asym), which correspond to the ground state solutions of the symmetric case, A = 1, in the evaluation of $S(ki*,kj*|w(sym))$⁠, given by Eq. (D1): we, respectively, compare in $S(ki*,kj*|w(sym))$ the labels w^(sym) with $ki(sym)$ and $kj(sym)$⁠, i.e., the fraction of the samples $ki*$ and $kj*$⁠, which, respectively, corresponds to the known ground state structures of the symmetric Wigner bilayer system.