Phase probabilities in first-order transitions using machine learning

TL;DR

The paper addresses extracting critical energies $e_o$ and $e_d$ and latent heat in systems with first-order transitions using a ternary classifier trained on energy-resolved spin configurations. It combines microcanonical population annealing (MCPA) sampling to generate large, nearly uncorrelated configurations and a CNN that outputs probabilities for OS, CS, DS. Application to the $q$-state Potts model with $q=10$ and $q=20$ shows that the estimated $e_o$, $e_d$, and latent heat $\mathcal{L}=e_d-e_o$ agree with exact values and exhibit limited finite-size effects up to $L=60$. The results reveal that phase-probability fluctuations encode finite-size rounding and droplets in the coexistence region, offering a new ML-based tool for analyzing first-order transitions and potentially generalizing to other three-phase problems.

Abstract

We set out to explore the possibility of investigating the critical behavior of systems with first-order phase transition using deep machine learning. We propose a machine learning protocol with ternary classification of instantaneous spin configurations using known values of disordered phase energy and ordered phase energy. The trained neural network is used to predict whether a given sample belong to one or another phase of matter. This allows us to estimate for the first time the probability that configurations with a certain energy belong to the ordered phase, coexistence phase, and disordered phase. Based on these probabilities, we obtained estimates of the values of the critical energies and latent heat for the Potts model with 10 and 20 components, which undergoes a strong discontinuous transition. We also found that the probabilities may reflect geometric transitions in the coexistence phase.

Figures (10)

• Figure 1: The fraction of $\epsilon$ configurations in the population with energy $e$. The red vertical lines indicate the critical energies $e_o$ and $e_d$. Top figure: PM-10 model, bottom figure: PM-20 model.
• Figure 2: Typical spin configurations for the 10-state Potts model on the $L=30$ lattice at energies from left to right: $e=-1.9$ in the ordered phase, $e=-1.4$ in the coexistence phase, and $e=-0.9$ in the disordered phase. The upper panel is the raw dataset RD and lower panel is the majority/minority MD dataset. The vertical colored bar marks the spin number.
• Figure 3: Average magnetization $<M>$ of the samples at a given energy $e$. The insets show symbols and colors for grid sizes $L=30$ - blue triangles, 40 - orange rhombuses, 50 - green dots and 60 - red squares. Top figure: model PM-10, bottom figure: model PM-20.
• Figure 4: Probabilities of phases $P_{xS}(E)$ for $L=30$ (first row), 40 (second row), 50 (third row) and 60 (last row) for 10-state Potts model, PM-10. Left panel is the training/testing with the raw dataset RD and right panel is the training/testing with the majority/minority dataset MD. Blue dashed lines are ordered phase probabilities $P_{OS}$, orange solid lines are coexistence phase probabilities $P_{CS}$, and green dotted lines are disordered phase probabilities $P_{DS}$. The vertical lines denote the exact values of the critical energies of the ordered and disordered phase.
• Figure 5: Same as in Fig. \ref{['fig4']} for 20-state Potts model, PM-20.