Unsupervised Discovery of Intermediate Phase Order in the Frustrated $J_1$-$J_2$ Heisenberg Model via Prometheus Framework

Abstract

The spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice exhibits a debated intermediate phase between Néel antiferromagnetic and stripe ordered regimes, with competing theories proposing plaquette valence bond, nematic, and quantum spin liquid ground states. We apply the Prometheus variational autoencoder framework -- previously validated on classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the $J_1$-$J_2$ phase diagram via unsupervised analysis of exact diagonalization ground states for a $4 \times 4$ lattice. Through dense parameter scans of $J_2/J_1 \in [0.3, 0.7]$ with step size 0.01 and comprehensive latent space analysis, we investigate the nature of the intermediate regime using unsupervised order parameter discovery and critical point detection via multiple independent methods. This work demonstrates the application of rigorously validated machine learning methods to open questions in frustrated quantum magnetism, where traditional order parameter identification is challenged by competing interactions and limited accessible system sizes.

Figures (6)

• Figure 1: Ground state energy density $e = E/N$ as a function of frustration ratio for $L=4$. The energy exhibits a minimum near $J_2/J_1 \approx 0.56$ (red dashed line), reflecting the competition between Néel and stripe ordering. The detected critical point at $J_2/J_1 = 0.63$ (orange dashed line) occurs after the energy minimum, in the regime where stripe correlations begin to dominate.
• Figure 2: Evolution of physical observables across the frustration parameter range for $L=4$. The staggered magnetization $m_s$ (scaled by 10) and Néel structure factor $S(\pi,\pi)$ decrease monotonically through the intermediate regime, while the stripe structure factor $S(\pi,0)$ increases. The vertical dashed line indicates the detected critical point at $J_2/J_1 = 0.63 \pm 0.004$.
• Figure 3: Latent space trajectory showing the evolution of ground states as $J_2/J_1$ varies from 0.30 to 0.70. Points are colored by frustration ratio. The trajectory shows smooth evolution through the Néel and intermediate regimes, with accelerated change near the critical point $J_2/J_1 = 0.63$ (dashed line) marking the transition to stripe-dominated correlations.
• Figure 4: Correlation matrix between Q-VAE latent dimensions ($z_0$--$z_7$) and physical observables. Strong correlations ($|r| > 0.85$) are highlighted in color (red for positive, blue for negative). The dominant latent dimension $z_0$ shows strong correlations with energy, staggered magnetization, and structure factors, indicating successful unsupervised discovery of the relevant order parameters.
• Figure 5: Fidelity susceptibility $\chi_F$ as a function of frustration ratio $J_2/J_1$. The peak at $J_2/J_1 = 0.63 \pm 0.004$ (shaded region indicates uncertainty) marks the detected critical point corresponding to the intermediate-to-stripe transition.