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

TL;DR

This paper uses a novel machine learning framework called Prometheus to unsupervisedly identify phases in the frustrated $J_1$-$J_2$ Heisenberg model, revealing the Ne9el-to-stripe transition through scalable quantum correlation analysis.

Contribution

It introduces a scalable RDM-based VAE approach for unsupervised phase discovery in frustrated quantum systems beyond full wavefunction analysis.

Findings

Successfully identified dominant order parameters with high correlation.

Captured the Ne9el-to-stripe crossover near $J_2/J_1 \u2248 0.5$--$0.6$.

Demonstrated that local quantum correlations suffice for phase identification.

Abstract

The spin-$1/2$ $J_1$-$J_2$ Heisenberg model on the square lattice exhibits a debated intermediate phase between N\'eel 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 applied to classical (2D, 3D Ising) and quantum (disordered transverse field Ising) phase transitions -- to systematically explore the $J_1$-$J_2$ phase diagram using a multi-scale approach. For $L=4$, we employ exact diagonalization with full wavefunction analysis via quantum-aware VAE. For larger systems ($L=6, 8$), we introduce a reduced density matrix (RDM) based methodology using DMRG ground states, enabling scaling beyond the exponential barrier of full Hilbert space representation. Through dense parameter scans of $J_2/J_1 \in [0, 1]$ and comprehensive latent space analysis, we identify the structure factor $S(\pi,\pi)$ and $S(\pi,0)$ as the dominant order parameters discovered by the VAE, with correlations exceeding $|r| > 0.97$. The RDM-VAE approach successfully captures the N\'eel-to-stripe crossover near $J_2/J_1 \approx 0.5$--$0.6$, demonstrating that local quantum correlations encoded in reduced density matrices contain sufficient information for unsupervised phase discovery. This work establishes a scalable pathway for applying machine learning to frustrated quantum systems where full wavefunction access is computationally prohibitive.