Unsupervised Latent-Space Discovery of Quantum Phases in the Bose–Hubbard Model

We demonstrate that a variational autoencoder (VAE) can autonomously learn the quantum phase structure of the Bose–Hubbard model using only local observables. Based on data generated from the Gutzwiller mean-field approximation, the VAE organizes the system into a low-dimensional latent space that cleanly separates superfluid and Mott-insulating regimes and reconstructs the conventional (U/t, μ/t) phase diagram without supervision. Extending the input to finite temperature, the latent geometry captures the thermal suppression of superfluidity and the smooth deformation of phase boundaries. These results establish latent-space learning as a scalable, prior-free framework for identifying phase structure in high-dimensional quantum systems.