Prometheus: Unsupervised Discovery of Phase Transitions in Three-Dimensional Spin Systems Using Variational Autoencoders

We present Prometheus, an unsupervised machine learning framework for discovering phase transitions and extracting critical exponents in three-dimensional spin systems. Building upon our previous work on the two-dimensional Ising model, we extend the variational autoencoder (VAE) approach to three-dimensional systems, demonstrating that the learned latent representations capture the essential physics of the ferromagnetic-paramagnetic phase transition. Our 3D convolutional VAE architecture achieves $\geq 70\%$ accuracy in critical exponent extraction for the 3D Ising model, with the critical temperature $\Tc = 4.511 \pm 0.005$ in excellent agreement with the theoretical value $\Tc/J = 4.5115$. The framework incorporates a novel quantum discovery mechanism that enables systematic exploration of parameter spaces and automated detection of novel phase transition phenomena. We validate our approach through extensive finite-size scaling analysis, bootstrap confidence intervals, and comparison with established Monte Carlo results. The Prometheus framework provides a powerful tool for unsupervised exploration of phase diagrams in complex many-body systems where analytical solutions are unavailable.