From Classical to Quantum: Extending Prometheus for Unsupervised Discovery of Phase Transitions in Three Dimensions and Quantum Systems

TL;DR

The paper generalizes Prometheus from 2D classical systems to 3D classical and quantum many-body systems, combining a 3D convolutional VAE and a quantum-aware VAE (Q-VAE) with fidelity-based losses to achieve unsupervised discovery of phase transitions. It delivers precise Tc detection (0.01% for the 3D Ising model) and robust critical exponents (∼70–75% accuracy) with proper universality-class identification, all without analytical guidance. In quantum regimes, it achieves 2% accuracy for the clean transverse-field Ising model critical point and uncovers activated scaling at the infinite-randomness fixed point in the disordered TFIM, extracting a tunneling exponent ψ ≈ 0.48 that agrees with theory ψ = 0.5. The work demonstrates that a single, architecture-aware unsupervised framework can reveal order parameters, critical points, and exotic critical behavior across classical and quantum domains, offering a scalable, automated pipeline for exploring phase diagrams where traditional methods struggle. Together with the 2D foundation, this establishes a systematic validation pathway from exact-solvable 2D systems to complex 3D and quantum settings, underscoring the potential of ML as a genuine discovery engine in physics.

Abstract

We extend the Prometheus framework for unsupervised phase transition discovery from 2D classical systems to 3D classical and quantum many-body systems, addressing scalability in higher dimensions and generalization to quantum fluctuations. For the 3D Ising model ($L \leq 32$), the framework detects the critical temperature within 0.01\% of literature values ($T_c/J = 4.511 \pm 0.005$) and extracts critical exponents with $\geq 70\%$ accuracy ($β= 0.328 \pm 0.015$, $γ= 1.24 \pm 0.06$, $ν= 0.632 \pm 0.025$), correctly identifying the 3D Ising universality class via $χ^2$ comparison ($p = 0.72$) without analytical guidance. For quantum systems, we developed quantum-aware VAE (Q-VAE) architectures using complex-valued wavefunctions and fidelity-based loss. Applied to the transverse field Ising model, we achieve 2\% accuracy in quantum critical point detection ($h_c/J = 1.00 \pm 0.02$) and successfully discover ground state magnetization as the order parameter ($r = 0.97$). Notably, for the disordered transverse field Ising model, we detect exotic infinite-randomness criticality characterized by activated dynamical scaling $\ln ξ\sim |h - h_c|^{-ψ}$, extracting a tunneling exponent $ψ= 0.48 \pm 0.08$ consistent with theoretical predictions ($ψ= 0.5$). This demonstrates that unsupervised learning can identify qualitatively different types of critical behavior, not just locate critical points. Our systematic validation across classical thermal transitions ($T = 0$ to $T > 0$) and quantum phase transitions ($T = 0$, varying $h$) establishes that VAE-based discovery generalizes across fundamentally different physical domains, providing robust tools for exploring phase diagrams where analytical solutions are unavailable.

Figures (18)

• Figure 1: VAE latent space for 3D Ising model ($L = 32$). Points represent individual spin configurations colored by temperature: blue (low $T$, ferromagnetic), red (high $T$, paramagnetic). Clear phase separation emerges along the first latent dimension, which correlates with magnetization ($r = 0.997$). The critical region ($T \approx T_c$) shows increased scatter reflecting enhanced fluctuations.
• Figure 2: Correlation between discovered latent order parameter $\phi_1$ and physical magnetization $m$ for $L = 32$. Linear relationship ($r = 0.997$) confirms successful unsupervised discovery. Error bars show standard error across configurations at each temperature.
• Figure 3: Latent susceptibility $\chi_\phi$ vs temperature for system sizes $L = 8$ to $32$. Peaks sharpen and shift toward $T_c = 4.5115$ as $L$ increases, consistent with finite-size scaling theory. Extrapolation to $L \to \infty$ yields $T_c/J = 4.511 \pm 0.005$.
• Figure 4: Finite-size scaling collapse for 3D Ising order parameter. Rescaled data for $L = 8$ to $32$ collapse onto a universal curve with optimized parameters $T_c/J = 4.510$, $\nu = 0.632$, $\beta/\nu = 0.518$. Collapse quality: 0.92 indicating excellent agreement with finite-size scaling theory.
• Figure 5: Bootstrap uncertainty distributions for critical exponents from 1000 resampling iterations. Histograms show approximately Gaussian distributions centered on extracted values (red lines) with literature values (blue dashed) falling within the spread. This validates the bootstrap uncertainty estimates.