Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling

Ali Tozar

Published Jan 1, 2026• 0 citations

Abstract

Replication Data and Code for: "Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling" This repository contains the complete source code, simulation datasets, and analysis scripts required to reproduce all findings presented in the associated manuscript. The study introduces a symmetry-corrected machine learning pipeline (Gauge-Fixed PCA + Logistic Regression) to identify phase transitions and estimate critical temperatures with high precision, validated on the two-dimensional Ising model. Repository Contents: 01_generate_data.py — Generates 2D Ising model spin configurations using the Wolff cluster algorithm. Produces equilibrium datasets for lattice sizes L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} with 10,000 independent configurations per temperature point. 02_analysis_pipeline_v2.py — Core analysis script: performs gauge fixing (σ′=σ⋅sgn(m)\sigma' = \sigma \cdot \mathrm{sgn}(m) σ′=σ⋅sgn(m)), PCA dimensionality reduction (d=16d = 16 d=16, cross-validated), confusion-scheme classification with parabolic sub-grid interpolation, bootstrap uncertainty quantification (Nboot=50N_{\rm boot} = 50 Nboot=50), and pFMp_{\rm FM} pFM computation with isotonic calibration. Outputs summary_ml_v2.json. 03_plot_main_figures.py — Reproduces Figures 1–6 of the main manuscript. 04_plot_supplementary_v2.py — Reproduces Supplementary Figures S1–S5 and Table 1. ablation_preprocessing_S5.py — Standalone preprocessing ablation script: runs raw and ∣m∣|m| ∣m∣-only comparisons and generates Supplementary Figure S5. Does not modify main results. summary_ml_v2.json — Pre-computed results (gauge-fixed pipeline) for immediate reproduction of all main figures without re-running Monte Carlo simulations. summary_ml_v2_raw.json — Pre-computed ablation results: raw (no gauge fixing) preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. summary_ml_v2_absm.json — Pre-computed ablation results: ∣m∣|m| ∣m∣-only preprocessing, L∈{32,64,96}L \in \{32, 64, 96\} L∈{32,64,96}. Table1_Tc_Comparison.tex — LaTeX source for Table 1. requirements.txt — Python dependencies. README.txt — Step-by-step reproduction instructions. Methodology Highlights: System sizes: L∈{32,48,64,96}L \in \{32, 48, 64, 96\} L∈{32,48,64,96} 10,000 independent configurations per temperature point; Wolff cluster algorithm with 10410^4 104 thermalization steps Symmetry correction: global Z2\mathbb{Z}_2 Z2 gauge fixing prior to PCA Confusion scheme: validation cross-entropy minimization over T0∈[2.10,2.45]T_0 \in [2.10, 2.45] T0∈[2.10,2.45] (60 candidate values, band Δ=0.05\Delta = 0.05 Δ=0.05, 5-fold CV) Bootstrap: 50 configuration-level resamples; parabolic sub-grid interpolation for discretization bias reduction Key result: Tc≈2.2717(8)T_c \approx 2.2717(8) Tc≈2.2717(8) for L=96L = 96 L=96 (0.6%0.6\% 0.6% from Onsager); universality class confirmed via FSS data collapse with ν=1\nu = 1 ν=1

DOI: 10.5281/zenodo.18235186

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Capturing the universality class of the 2D ising model via symmetry-corrected machine learning and finite-size scaling

Ali Tozar

Physica A Statistical Mechanics and its ApplicationsMar 22, 20260 citations

Replication Data and Code for: "Capturing the Universality Class of the 2D Ising Model via Symmetry-Corrected Machine Learning and Finite-Size Scaling" This repository contains the complete source code, simulation datasets, and analysis scripts required to reproduce all findings presented in the associated manuscript. The study introduces a symmetry-corrected machine learning pipeline (Gauge-Fixed PCA + Logistic Regression) to identify phase transitions and estimate critical temperatures with high precision, validated on the two-dimensional Ising model. Repository Contents: 01_generate_data.py — Generates 2D Ising model spin configurations using the Wolff cluster algorithm. Produces equilibrium datasets for lattice sizes L∈{32,48,64,96}L \in {32, 48, 64, 96} L∈{32,48,64,96} with 10,000 independent configurations per temperature point. 02_analysis_pipeline_v2.py — Core analysis script: performs gauge fixing (σ′=σ⋅sgn(m)\sigma' = \sigma \cdot \mathrm{sgn}(m) σ′=σ⋅sgn(m)), PCA dimensionality reduction (d=16d = 16 d=16, cross-validated), confusion-scheme classification with parabolic sub-grid interpolation, bootstrap uncertainty quantification (Nboot=50N_{\rm boot} = 50 Nboot=50), and pFMp_{\rm FM} pFM computation with isotonic calibration. Outputs summary_ml_v2.json. 03_plot_main_figures.py — Reproduces Figures 1–6 of the main manuscript. 04_plot_supplementary_v2.py — Reproduces Supplementary Figures S1–S5 and Table 1. ablation_preprocessing_S5.py — Standalone preprocessing ablation script: runs raw and ∣m∣|m| ∣m∣-only comparisons and generates Supplementary Figure S5. Does not modify main results. summary_ml_v2.json — Pre-computed results (gauge-fixed pipeline) for immediate reproduction of all main figures without re-running Monte Carlo simulations. summary_ml_v2_raw.json — Pre-computed ablation results: raw (no gauge fixing) preprocessing, L∈{32,64,96}L \in {32, 64, 96} L∈{32,64,96}. summary_ml_v2_absm.json — Pre-computed ablation results: ∣m∣|m| ∣m∣-only preprocessing, L∈{32,64,96}L \in {32, 64, 96} L∈{32,64,96}. Table1_Tc_Comparison.tex — LaTeX source for Table 1. requirements.txt — Python dependencies. README.txt — Step-by-step reproduction instructions. Methodology Highlights: System sizes: L∈{32,48,64,96}L \in {32, 48, 64, 96} L∈{32,48,64,96} 10,000 independent configurations per temperature point; Wolff cluster algorithm with 10410^4 104 thermalization steps Symmetry correction: global Z2\mathbb{Z}_2 Z2 gauge fixing prior to PCA Confusion scheme: validation cross-entropy minimization over T0∈[2.10,2.45]T_0 \in [2.10, 2.45] T0∈[2.10,2.45] (60 candidate values, band Δ=0.05\Delta = 0.05 Δ=0.05, 5-fold CV) Bootstrap: 50 configuration-level resamples; parabolic sub-grid interpolation for discretization bias reduction Key result: Tc≈2.2717(8)T_c \approx 2.2717(8) Tc≈2.2717(8) for L=96L = 96 L=96 (0.6%0.6% 0.6% from Onsager); universality class confirmed via FSS data collapse with ν=1\nu = 1 ν=1