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arXiv:2602.21468

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

Brandon Yee, Wilson Collins, Maximilian Rutkowski

cond-mat.str-elcond-mat.dis-nncs.LGquant-phSubmitted Feb 25, 2026• Announced Feb 26, 2026• 0 citations• Revised Feb 28, 2026 (v2)
v1 · Feb 25, 2026v2 · Feb 28, 2026v3 · Mar 11, 2026v4 · Apr 17, 2026v5 · Apr 24, 2026
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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 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 $S(π,π)$ and $S(π,0)$ as the dominant order parameters discovered by the VAE, with correlations exceeding $|r| > 0.97$. The RDM-VAE approach successfully captures the Néel-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.

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Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half ($s=\frac{1}{2}$) ${J}_{1}$-${J}_{2}$ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength ${J}_{1}$ and two next-nearest-neighbor (diagonal) bonds of strength ${J}_{2}$. The ${J}_{2}$ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no ${J}_{2}$ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when ${J}_{1}$ and ${J}_{2}$ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum $s=\frac{1}{2}$ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the $s=\frac{1}{2}$ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the $s=\frac{1}{2}$ model are of continuous deconfined type.

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Unsupervised Discovery of Intermediate Phase Order in the Frustrated J1J_1J1​-J2J_2J2​ Heisenberg Model via Prometheus Framework
The spin-1/21/21/2 J1J_1J1​-J2J_2J2​ 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 J1J_1J1​-J2J_2J2​ phase diagram using a multi-scale approach. For L=4L=4L=4, we employ exact diagonalization with full wavefunction analysis via quantum-aware VAE. For larger systems (L=6,8L=6, 8L=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 J2/J1∈[0,1]J_2/J_1 \in [0, 1]J2​/J1​∈[0,1] and comprehensive latent space analysis, we identify the structure factor S(π,π)S(π,π)S(π,π) and S(π,0)S(π,0)S(π,0) as the dominant order parameters discovered by the VAE, with correlations exceeding ∣r∣>0.97|r| > 0.97∣r∣>0.97. The RDM-VAE approach successfully captures the Néel-to-stripe crossover near J2/J1≈0.5J_2/J_1 \approx 0.5J2​/J1​≈0.5--0.60.60.6, 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.
Abstract We extend the Prometheus framework from classical to quantum phase transitions, demonstrat- ing unsupervised discovery of quantum critical phenomena in the disordered transverse field Ising model (DTFIM). Building upon our previous work on classical spin systems [1, 2], we develop a quantum-aware variational autoencoder (Q-VAE) architecture that operates directly on quantum ground state wavefunctions obtained via exact diagonalization. The Q-VAE learns latent represen- tations that capture the essential physics of the quantum phase transition between paramagnetic and ferromagnetic phases, achieving detection of the quantum critical point hc = 1.00 ± 0.02 in the clean limit, consistent with the theoretical value hc/J = 1. We introduce disorder through random transverse fields h i ∼ Uniform[h−W, h+W ] and investigate the disorder-driven infinite-randomness fixed point characteristic of the random TFIM. Our framework successfully identifies the shift in critical behavior with increasing disorder strength and extracts effective critical exponents through finite-size scaling analysis. We formulate and test the hypothesis that the Q-VAE latent space structure reflects the activated dynamical scaling ln ξ ∼ |h − hc| −ψ with ψ ≈ 0.5 expected at the infinite-randomness fixed point. The Prometheus quantum discovery pipeline provides a systematic approach to exploring quantum phase diagrams in systems where analytical solutions are unavail- able, with potential applications to quantum materials, cold atom systems, and quantum computing platforms.
Abstract 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 ≥70%\geq 70\%≥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.
We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3), suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3) in the model. From the crossing of the first derivative of the energies with J2{J}_{2}J2​ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at J2/J1=0.610(5){J}_{2}/{J}_{1}=0.610(5)J2​/J1​=0.610(5).
Investigation of the J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice: A study with projected entangled-pair states
The nature of the quantum spin liquid (QSL) phase in the frustrated J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice remains an open and actively debated problem. In this work, we employ the infinite projected entangled-pair state (PEPS) to systematically investigate the model under different symmetry constraints. Our simulations reveal a direct transition from the 120∘120^\circ120∘ Néel state to a putative QSL at J2/J1≈0.08J_2/J_1\approx 0.08J2​/J1​≈0.08, signaled by the collapse of magnetic order. We further show that, through either an appropriate unitary rotation or spontaneous spin long-range order, the stripe antiferromagnetic phase can also be accurately captured within the infinite PEPS framework. A central focus of our study is the role played by the PEPS symmetry in approximating the QSL ground-state sandwiched between the two magnetic phases. We first found that a fully-symmetric topological Z2\mathbb{Z}_2Z2​ Resonating Valence Bond state, which can be written as a simple PEPS with bond dimension D=3D=3D=3, exhibits a reasonably good variational energy. Motivated by this finding, we have further constructed generic Z2\mathbb{Z}_2Z2​-symmetric PEPS of larger bond dimension (up to D=7D=7D=7). We found that, under wavefunction optimization, spinons condense and, simultaneously, topological vison excitations get confined, hence precluding Z2\mathbb{Z}_2Z2​ topological order. This strongly indicates the gapless (or critical) nature of the QSL phase, which is most naturally consistent with a U(1) Dirac spin liquid scenario.
Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (s=12s=\frac{1}{2}s=21​) J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength J1{J}_{1}J1​ and two next-nearest-neighbor (diagonal) bonds of strength J2{J}_{2}J2​. The J2{J}_{2}J2​ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no J2{J}_{2}J2​ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when J1{J}_{1}J1​ and J2{J}_{2}J2​ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum s=12s=\frac{1}{2}s=21​ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the s=12s=\frac{1}{2}s=21​ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the s=12s=\frac{1}{2}s=21​ model are of continuous deconfined type.
We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1{J}_{1}J1​-J2{J}_{2}J2​ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with LLL === 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different J2/J1{J}_{2}/{J}_{1}J2​/J1​ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high J2/J1{J}_{2}/{J}_{1}J2​/J1​, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.