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

Brandon Yee; Wilson Collins; Pairie Koh; Maximilian Rutkowski

cond-mat.dis-nnFeb 16, 2026

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

Authors: Brandon Yee, Wilson Collins, Pairie Koh, Maximilian Rutkowski

Organizations: Physics Lab, Yee Collins Research Group

Abstract

We extend the Prometheus framework for unsupervised phase transition discovery from two-dimensional classical systems to three-dimensional classical systems and quantum many-body systems. Building upon preliminary observations from a 2D Ising model student abstract [Yee et al., 2026], we address two fundamental questions: (1) Does the framework scale to higher dimensions where exact solutions are unavailable? (2) Can it generalize to quantum phase transitions driven by quantum fluctuations rather than thermal fluctuations? For the 3D Ising model on lattices up to $L{=}32$ , we achieve critical temperature detection within 0.01% of literature values ( $\Tc/J = 4.511 \pm 0.005$ ) and extract critical exponents with ${\geq}70\%$ accuracy, with statistical analysis correctly identifying the 3D Ising universality class ( $p = 0.72$ ). For quantum systems, we develop quantum-aware VAE (Q-VAE) architectures operating on complex-valued wavefunctions with fidelity-based loss functions, achieving 2% accuracy in quantum critical point detection for the transverse field Ising model. For the disordered TFIM, we perform a consistency check of activated dynamical scaling $\ln ξ\sim |h - \hc|^{-ψ}$ , extracting tunneling exponent $ψ= 0.48 \pm 0.08$ consistent with theoretical predictions ( $ψ= 0.5$ , $Δχ^2 = 12.3$ , $p < 0.001$ ). This demonstrates that unsupervised learning can identify qualitatively different types of critical behavior, serving as a consistency check on known IRFP physics.

Figures & tables

Method	$T_{c}/J$	Uncertainty	Confidence
Latent susceptibility peak	$4.512$	$\pm 0.008$	95%
Binder cumulant crossing	$4.511$	$\pm 0.006$	97%
Gradient maximum	$4.510$	$\pm 0.010$	92%
Reconstruction error peak	$4.513$	$\pm 0.009$	94%
Ensemble weighted average	$\mathbf{4.511}$	$\pm 0.005$	98%
Literature ( Ferrenberg et al., 2018 )	$4.511528$	$\pm 0.000006$	—

Table 1: Critical temperature detection methods for 3D Ising model (

L=32

Exponent	Extracted	Literature ( Pelissetto and Vicari, 2002 )	Abs. Error	Rel. Accuracy
$\beta$	$0.328\pm 0.015$	$0.3265(3)$	$0.002$	72%
$\gamma$	$1.24\pm 0.06$	$1.2372(5)$	$0.003$	70%
$\nu$	$0.632\pm 0.025$	$0.6301(4)$	$0.002$	75%
$\eta$	$0.034\pm 0.008$	$0.0364(5)$	$0.002$	70%
Average				72%

Table 2: Critical exponents for the 3D Ising model.

Universality Class	$\chi^{2}$	DOF	$p$ -value	Consistent?
3D Ising	2.1	3	0.72	Yes
Mean-field ( $d>4$ )	45.3	3	$<0.001$	No
2D Ising	38.7	3	$<0.001$	No
3D XY	12.8	3	0.005	No
3D Heisenberg	18.4	3	$<0.001$	No

Table 3: Universality class discrimination via

\chi^{2}

analysis.

System Size $L$	Pearson $r$	$p$ -value
8	$0.96\pm 0.02$	$<10^{-20}$
10	$0.97\pm 0.01$	$<10^{-25}$
12	$0.97\pm 0.01$	$<10^{-30}$
14	$0.97\pm 0.01$	$<10^{-30}$

Table 4: Quantum order parameter correlation across system sizes.

Method	$h_{c}/J$	Confidence
Latent variance peak	$1.02\pm 0.03$	92%
Reconstruction error	$0.98\pm 0.04$	88%
Fidelity susceptibility	$1.00\pm 0.02$	95%
Binder cumulant	$1.01\pm 0.03$	90%
Ensemble weighted average	$\mathbf{1.00\pm 0.02}$	94%
Exact value ( Pfeuty, 1970 )	$1.0$	—

Table 5: Quantum critical point detection for clean TFIM.

Exponent	Extracted	Exact	Deviation	Within $1\sigma$ ?
$\nu$	$1.05\pm 0.10$	$1.0$	$0.05$	Yes ( $0.5\sigma$ )
$z$	$0.95\pm 0.15$	$1.0$	$0.05$	Yes ( $0.3\sigma$ )
$\beta$	$0.13\pm 0.03$	$0.125$	$0.005$	Yes ( $0.2\sigma$ )

Table 6: Quantum critical exponents for clean TFIM.

Disorder $W/J$	$h_{c}/J$	Uncertainty
0.0	$1.00\pm 0.02$	—
0.2	$1.02\pm 0.03$	$\pm 0.03$
0.5	$1.04\pm 0.04$	$\pm 0.04$
1.0	$1.06\pm 0.05$	$\pm 0.05$
2.0	$1.08\pm 0.06$	$\pm 0.06$

Table 7: Quantum critical point vs. disorder strength.

$W/J$	$\psi$	$\chi^{2}_{\rm act.}$	$\chi^{2}_{\rm power}$	$\Delta\chi^{2}$
0.0	$0.02\pm 0.05$	15.2	14.8	$-0.4$ (n.s.)
0.2	$0.25\pm 0.10$	12.1	13.5	$+1.4$ (n.s.)
0.5	$0.35\pm 0.09$	10.3	14.7	$+4.4$ ( $p<0.05$ )
1.0	$0.42\pm 0.08$	9.1	17.2	$+8.1$ ( $p<0.01$ )
2.0	$\mathbf{0.48\pm 0.08}$	8.7	21.0	$\mathbf{+12.3}$ ( $\mathbf{p<0.001}$ )
Theory (IRFP)	$0.5$

Table 8: Tunneling exponent

\psi

vs. disorder strength for

L=12

Method	Supervision	System Size	Detects $\psi$ ?
SDRG ( Fisher, 1995 )	Requires RG flow	Large ( $L\sim 10^{3}$ )	Yes
QMC ( Rieger and Young, 1996 )	Manual analysis	Medium ( $L\sim 100$ )	Indirect
Exact diag. + manual	Manual fitting	Small ( $L\lesssim 14$ )	Yes
Prometheus (this work)	None	Small ( $L\lesssim 14$ )	Yes

Table 9: Methods for detecting infinite-randomness fixed point.

System	$\lambda_{c}$ Error	Order param. $r$	Exp. acc.	Key result
2D Ising ( Yee et al., 2026 )	0.04%	0.998	65%	Exact validation
3D Ising (this work)	0.01%	0.997	72%	No exact solution
Clean TFIM (this work)	2%	0.97	65%	Quantum discovery
DTFIM (this work)	8%	0.97	—	Activated $\psi=0.48$

Table 10: Performance summary: Prometheus across physical domains.

Capability	Prometheus	Supervised
Phase classification accuracy	90–95%	$>99\%$
Requires labeled data	No	Yes
Discovers unknown phases	Yes	No
Extracts critical exponents	Yes (65–72%)	Sometimes
Identifies universality classes	Yes	Sometimes
Detects exotic criticality	Yes	No

Table 11: Unsupervised (Prometheus) vs. supervised methods.

Method	$T_{c}$ Accuracy	Manual Effort	System Size
Specialized MC ( Hasenbusch, 2010 )	$10^{-5}$	High	$L\sim 100$
Standard MC ( Ferrenberg and Landau, 1991 )	$10^{-3}$	Medium	$L\sim 50$
Prometheus (3D)	$10^{-4}$	None	$L\sim 32$
Exact diag. + manual	$10^{-10}$	High	$L\lesssim 14$
Prometheus (quantum)	$10^{-2}$	None	$L\lesssim 14$

Table 12: Unsupervised ML vs. traditional numerical methods.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Layer	Type	Input Shape	Output Shape	Kernel	Stride
Input	—	$(1,32,32,32)$	$(1,32,32,32)$	—	—
Conv3D-1	3D Conv + ReLU	$(1,32,32,32)$	$(32,16,16,16)$	3	2
Conv3D-2	3D Conv + ReLU	$(32,16,16,16)$	$(64,8,8,8)$	3	2
Conv3D-3	3D Conv + ReLU	$(64,8,8,8)$	$(128,4,4,4)$	3	2
Flatten	—	$(128,4,4,4)$	$(8192)$	—	—
FC-1	Linear + ReLU	$(8192)$	$(256)$	—	—

Appendix

Table 13: 3D Ising encoder architecture for

L=32

Layer	Type	Input Shape	Output Shape	Kernel	Stride
Input	—	$(8)$	$(8)$	—	—
FC-1	Linear + ReLU	$(8)$	$(256)$	—	—
FC-2	Linear + ReLU	$(256)$	$(8192)$	—	—
Reshape	—	$(8192)$	$(128,4,4,4)$	—	—
ConvT3D-1	3D ConvT + ReLU	$(128,4,4,4)$	$(64,8,8,8)$	3	2
ConvT3D-2	3D ConvT + ReLU	$(64,8,8,8)$	$(32,16,16,16)$	3	2

Appendix

Table 14: 3D Ising decoder architecture.

Hyperparameter	Value
Optimizer	Adam
Learning rate (initial)	$1\times 10^{-3}$
$\beta_{1}$ / $\beta_{2}$	0.9 / 0.999
Weight decay	$1\times 10^{-5}$
Batch size	64
Latent dimension $z_{\text{dim}}$	8

Appendix

Table 15: 3D Ising training hyperparameters.

Layer	Type	Input Dim	Output Dim
Input	Flatten	$2\times 2^{12}$	8192
FC-1	Linear + LayerNorm + ReLU	8192	512
FC-2	Linear + LayerNorm + ReLU	512	256
FC-3	Linear + LayerNorm + ReLU	256	128
FC-4	Linear	128	16
Split	—	16	$\mu(8),\log\sigma^{2}(8)$

Appendix

Table 16: Q-VAE encoder architecture for

L=12

TFIM.

Layer	Type	Input Dim	Output Dim
Input	—	8	8
FC-1	Linear + LayerNorm + ReLU	8	128
FC-2	Linear + LayerNorm + ReLU	128	256
FC-3	Linear + LayerNorm + ReLU	256	512
FC-4	Linear	512	8192
Reshape	—	8192	$2\times 2^{12}$

Appendix

Table 17: Q-VAE decoder architecture.

Hyperparameter	Value
Optimizer	Adam
Learning rate (initial)	$5\times 10^{-4}$
$\beta_{1}$ / $\beta_{2}$	0.9 / 0.999
Weight decay	$1\times 10^{-5}$
Batch size	32
Latent dimension $z_{\text{dim}}$	8

Appendix

Table 18: Q-VAE training hyperparameters.

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

Brandon Yee, Wilson Collins, Maximilian Rutkowski

Physics Lab, Yee Collins Research Group

Phase Transitions Quantum Many-Body Physics

Open paper AskPDF arXiv

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 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(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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Phase Transitions

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