Decoding conformal field theories: From supervised to unsupervised learning

We use machine learning to classify rational two-dimensional conformal field theories (CFTs). We first use the energy spectra of these minimal models to train a supervised learning algorithm. In contrast to conventional methods that are typically qualitative and also involve system size scaling, our method quantifies the similarity of the spectrum of a system at a fixed size to candidate CFTs. Such an approach allows us to correctly predict the nature and the value of critical points of several strongly correlated spin models using only their energy spectra. Our results are also relevant for the ground-state entanglement Hamiltonian of certain topological phases of matter, described by CFTs. Remarkably, we achieve high prediction accuracy by only using the lowest few Réyni entropies as the input. Finally, using autoencoders, an unsupervised learning algorithm, we find a hidden variable that has a direct correlation with the central charge and discuss prospects for using machine learning to investigate other conformal field theories, including higher-dimensional ones.