Modeling Quantum Geometry for Fractional Chern Insulators with unsupervised learning

Abstract

Fractional Chern insulators (FCIs) in moire materials present a unique platform for exploring strongly correlated topological phases beyond the paradigm of ideal quantum geometry. While analytical approaches to FCIs and fractional quantum Hall states (FQHS) often rely on idealized Bloch wavefunctions, realistic moire models lack direct tunability of quantum metric and Berry curvature, limiting theoretical and numerical exploration. Here, we introduce an unsupervised machine learning framework to model interacting Hamiltonians directly through the distribution of single-particle form factors. Using a variational autoencoder (VAE), we show that unsupervised learning can not only distinguish FCI and non-FCI states, but also generate new form factors with distinct topological character, not present in the training set. This latent space enables the generation and interpolation of form factors for topological flatbands with Chern number $|C|=1$, enabling the discovery of unobserved many-body states such as charge density waves. Principal component analysis (PCA) further reveals that the dominant patterns in the form factors-reflecting correlations across the Brillouin zone-can be decomposed into components with approximately quantized Chern numbers, providing new insights into the global and topological structure of quantum geometry. Our results highlight the ability of machine learning to generalize and model topological quantum systems, paving the way for the inverse design of form factors with tailored quantum geometry and many-body phases in flatband materials.