Detecting the 3D Ising model phase transition with a ground-state-trained autoencoder

Abstract

We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the 3D Ising model. A 3D convolutional neural network autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature $T_c=4.5128(58)$ and the correlation-length critical exponent $ν=0.63(27)$, consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.

Figures (4)

• Figure 1: MSE-derived quantities coming from the CAE. The vertical, gray band indicates the literature average $T_{\rm c}=4.511535(31)$. Left: Ensemble average of the MSE for our smallest ($L=50$) and largest ($L=130$) lattices from two runs of the CAE. Solid lines, drawn to guide the eye, indicate $L=50$ while dashed lines indicate $L=130$. Right: $\chi_{\rm MSE}$ for $L=50$ from two runs of the CAE. Vertical bands in the inset indicate the extracted $T_{\rm c}(50)$ for each run.
• Figure 2: Left: Dependence of $\chi_{\rm MSE}$ vs $T$ on $L$ for CAE1. The vertical, gray band indicates $T_{\rm c}$. Right: Correlation of $1-\ev{{\rm MSE}}$ with the order parameter for CAE1. Statistical uncertainties are suppressed for visibility.
• Figure 3: Extraction of $T_{\rm c}$ and $\nu$ for the 3D Ising model using the MSE. The horizontal, gray band indicates the literature average for $T_{\rm c}$.
• Figure 4: Architecture of the 3D CAE used in this work.