This dataset and code repository accompanies the article of the same title (version 5, May 2026) and provides all simulation data, source code, and raw terminal output necessary to reproduce and verify the results. Study summary We investigate the critical dynamics of the two-dimensional Ising model at the exact critical point (Tc ≈ 2.269185) using the local Metropolis Monte Carlo algorithm. The central focus is the characterization and comparison of two widely used correlation time estimators: the exponential decay criterion τ₁/e and the integrated autocorrelation time τ_int, following the convention of Sokal (1997). Simulations were performed for seven system sizes L ∈ {16, 24, 32, 48, 64, 96, 128} with up to 50 independent replicas per system size, using both Python (NumPy/PCG64) and C++ (std::mt19937, compiled with -O2) implementations. The two implementations were developed independently, enabling cross-validation of results across languages. Main findings The coefficient of variation (CV = std/mean) of τ_int converges to a physical floor of approximately 0.44–0.52 regardless of the number of replicas (8 or 50) or programming language, confirming that this floor is determined by the physics of the estimator under local Metropolis dynamics at criticality — not by statistical sampling limitations. The distribution of τ₁/e is genuinely bimodal at the critical point: individual replicas alternate between low (~200–500 MC steps) and high (~3000–5000 MC steps) correlation times for the same system size, with mean-to-median ratios of up to 2.6× and bimodal ratios (mean of upper half / mean of lower half) of up to 23.5×. This bimodality does not disappear with more replicas, confirming it is a physical property of the estimator. No detectable power-law scaling τ ~ L^z is found for L ≤ 128 in any of the six independent runs (R² < 0.14 in all log-log fits using all 7 system sizes). The accepted asymptotic value z ≈ 2.17 (Li & Sokal, 1989) implies an expected τ_int of approximately 73,600 MC steps at L = 128 — 76× larger than the observed value of 972. This quantifies the dominance of finite-size corrections in the range studied, and justifies treating the effective dynamic exponent z_eff as a continuously varying quantity rather than a fixed constant. The C++ implementation achieves speedups of 98× to 782× relative to Python depending on system size, enabling the full 50-replica run across all seven system sizes to complete in approximately 33 minutes. Cross-validation caveat Cross-validation between Python and C++ is fully valid for τ₁/e. For τ_int, the Python scripts used a fixed equilibration of n_equil = 3000 steps for all L — documentably insufficient for L ≥ 24 where τ_int > 600 (n_equil/τ < 3–5τ). This generates systematic underestimation of up to 86% at L = 64. The valid cross-validation for τ_int is between the C++-8 and C++-50 runs, where CV agreement (~0.52 in both) confirms the physical floor is robust. Connection to the EIE/ZIE framework This study provides the empirical foundation for the EIE/ZIE adaptive meta-control framework (Patent BR 10 2025 024459 4), justifying three architectural decisions: (1) τ_int as the preferred estimator of the correlation time τ_corr; (2) the effective dynamic exponent z_eff treated as a continuously estimated dynamic variable rather than a fixed constant; and (3) the dynamic freezing time τ_freeze introduced as an observable independent of τ_corr, motivated by the bimodal behavior of τ₁/e which reveals that the system alternates between regimes of high and low correlation persistence — a phenomenon invisible to conventional autocorrelation-based diagnostics. Repository contents Cpp/ — C++ source code (3 scripts) and simulation data (6 CSV files + raw terminal output) Python/ — Python source code (2 scripts) and simulation data (4 CSV files + raw screen output) Paper/ — Article in English (PDF) and Portuguese (PDF) README.md — Full repository documentation including file descriptions, column definitions, reproduction instructions, and physical parameters LICENSE.txt — CC BY 4.0 for data; MIT for code; patent notice Joemerson da Silva Lima — Independent Researcher, Cuiabá, Mato Grosso, Brazil joemerson93lima@gmail.com Patent BR 10 2025 024459 4
This dataset and code repository accompanies the article of the same title (version 5, May 2026) and provides all simulation data, source code, and raw terminal output necessary to reproduce and verify the results. Study summary We investigate the critical dynamics of the two-dimensional Ising model at the exact critical point (Tc ≈ 2.269185) using the local Metropolis Monte Carlo algorithm. The central focus is the characterization and comparison of two widely used correlation time estimators: the exponential decay criterion τ₁/e and the integrated autocorrelation time τ_int, following the convention of Sokal (1997). Simulations were performed for seven system sizes L ∈ {16, 24, 32, 48, 64, 96, 128} with up to 50 independent replicas per system size, using both Python (NumPy/PCG64) and C++ (std::mt19937, compiled with -O2) implementations. The two implementations were developed independently, enabling cross-validation of results across languages. Main findings The coefficient of variation (CV = std/mean) of τ_int converges to a physical floor of approximately 0.44–0.52 regardless of the number of replicas (8 or 50) or programming language, confirming that this floor is determined by the physics of the estimator under local Metropolis dynamics at criticality — not by statistical sampling limitations. The distribution of τ₁/e is genuinely bimodal at the critical point: individual replicas alternate between low (~200–500 MC steps) and high (~3000–5000 MC steps) correlation times for the same system size, with mean-to-median ratios of up to 2.6× and bimodal ratios (mean of upper half / mean of lower half) of up to 23.5×. This bimodality does not disappear with more replicas, confirming it is a physical property of the estimator. No detectable power-law scaling τ ~ L^z is found for L ≤ 128 in any of the six independent runs (R² < 0.14 in all log-log fits using all 7 system sizes). The accepted asymptotic value z ≈ 2.17 (Li & Sokal, 1989) implies an expected τ_int of approximately 73,600 MC steps at L = 128 — 76× larger than the observed value of 972. This quantifies the dominance of finite-size corrections in the range studied, and justifies treating the effective dynamic exponent z_eff as a continuously varying quantity rather than a fixed constant. The C++ implementation achieves speedups of 98× to 782× relative to Python depending on system size, enabling the full 50-replica run across all seven system sizes to complete in approximately 33 minutes. Cross-validation caveat Cross-validation between Python and C++ is fully valid for τ₁/e. For τ_int, the Python scripts used a fixed equilibration of n_equil = 3000 steps for all L — documentably insufficient for L ≥ 24 where τ_int > 600 (n_equil/τ < 3–5τ). This generates systematic underestimation of up to 86% at L = 64. The valid cross-validation for τ_int is between the C++-8 and C++-50 runs, where CV agreement (~0.52 in both) confirms the physical floor is robust. Connection to the EIE/ZIE framework This study provides the empirical foundation for the EIE/ZIE adaptive meta-control framework (Patent BR 10 2025 024459 4), justifying three architectural decisions: (1) τ_int as the preferred estimator of the correlation time τ_corr; (2) the effective dynamic exponent z_eff treated as a continuously estimated dynamic variable rather than a fixed constant; and (3) the dynamic freezing time τ_freeze introduced as an observable independent of τ_corr, motivated by the bimodal behavior of τ₁/e which reveals that the system alternates between regimes of high and low correlation persistence — a phenomenon invisible to conventional autocorrelation-based diagnostics. Repository contents Cpp/ — C++ source code (3 scripts) and simulation data (6 CSV files + raw terminal output) Python/ — Python source code (2 scripts) and simulation data (4 CSV files + raw screen output) Paper/ — Article in English (PDF) and Portuguese (PDF) README.md — Full repository documentation including file descriptions, column definitions, reproduction instructions, and physical parameters LICENSE.txt — CC BY 4.0 for data; MIT for code; patent notice Joemerson da Silva Lima — Independent Researcher, Cuiabá, Mato Grosso, Brazil joemerson93lima@gmail.com Patent BR 10 2025 024459 4
Este conjunto de dados e repositório de código acompanha o artigo de mesmo título (versão 5, maio de 2026) e fornece todos os dados de simulação, código-fonte e saída bruta do terminal necessários para reproduzir e verificar os resultados, estão no repositorio https://doi.org/10.5281/zenodo.20180520 Resumo do estudo Investigamos a dinâmica crítica do modelo de Ising bidimensional no ponto crítico exato (Tc ≈ 2,269185) usando o algoritmo de Monte Carlo de Metropolis local. O foco central é a caracterização e comparação de dois estimadores de tempo de correlação amplamente utilizados: o critério de decaimento exponencial τ₁/e e o tempo de autocorrelação integrado τ_int, seguindo a convenção de Sokal (1997). As simulações foram realizadas para sete tamanhos de sistema L ∈ {16, 24, 32, 48, 64, 96, 128} com até 50 réplicas independentes por tamanho de sistema, usando implementações em Python (NumPy/PCG64) e C++ (std::mt19937, compilado com -O2). As duas implementações foram desenvolvidas independentemente, permitindo a validação cruzada dos resultados entre as linguagens. Principais conclusões O coeficiente de variação (CV = desvio padrão/média) de τ_int converge para um limite inferior físico de aproximadamente 0,44–0,52, independentemente do número de réplicas (8 ou 50) ou da linguagem de programação, confirmando que esse limite inferior é determinado pela física do estimador sob a dinâmica local de Metropolis na criticidade — e não por limitações de amostragem estatística. A distribuição de τ₁/e é genuinamente bimodal no ponto crítico: réplicas individuais alternam entre tempos de correlação baixos (~200–500 passos de Monte Carlo) e altos (~3000–5000 passos de Monte Carlo) para o mesmo tamanho de sistema, com razões média-mediana de até 2,6× e razões bimodais (média da metade superior / média da metade inferior) de até 23,5×. Essa bimodalidade não desaparece com mais réplicas, confirmando que se trata de uma propriedade física do estimador. Não foi detectada nenhuma lei de potência τ ~ L^z para L ≤ 128 em nenhuma das seis simulações independentes (R² < 0,14 em todos os ajustes log-log usando todos os 7 tamanhos de sistema). O valor assintótico aceito z ≈ 2,17 (Li & Sokal, 1989) implica um τ_int esperado de aproximadamente 73.600 passos de Monte Carlo em L = 128 — 76 vezes maior que o valor observado de 972. Isso quantifica a dominância das correções de tamanho finito na faixa estudada e justifica tratar o expoente dinâmico efetivo z_eff como uma quantidade que varia continuamente, em vez de uma constante fixa. A implementação em C++ alcança ganhos de velocidade de 98× a 782× em relação ao Python, dependendo do tamanho do sistema, permitindo que a execução completa com 50 réplicas em todos os sete tamanhos de sistema seja concluída em aproximadamente 33 minutos. Ressalva sobre validação cruzada A validação cruzada entre Python e C++ é totalmente válida para τ₁/e. Para τ_int, os scripts em Python usaram uma equilibração fixa de n_equil = 3000 passos para todos os valores de L — comprovadamente insuficiente para L ≥ 24, onde τ_int > 600 (n_equil/τ < 3–5τ). Isso gera uma subestimação sistemática de até 86% em L = 64. A validação cruzada válida para τ_int ocorre entre as execuções em C++-8 e C++-50, onde a concordância da validação cruzada (~0,52 em ambas) confirma que o limite físico é robusto. Conexão com a estrutura EIE/ZIE Este estudo fornece a base empírica para a estrutura de meta-controle adaptativo EIE/ZIE (Patente BR 10 2025 024459 4), justificando três decisões arquitetônicas: (1) τ_int como o estimador preferido do tempo de correlação τ_corr; (2) o expoente dinâmico efetivo z_eff tratado como uma variável dinâmica estimada continuamente em vez de uma constante fixa; e (3) o tempo de congelamento dinâmico τ_freeze introduzido como um observável independente de τ_corr, motivado pelo comportamento bimodal de τ₁/e que revela que o sistema alterna entre regimes de alta e baixa persistência de correlação — um fenômeno invisível para diagnósticos convencionais baseados em autocorrelação. O conteúdo do repositório está disponível na versão de publicação v1, https://doi.org/10.5281/zenodo.20180520 Cpp/ — Código-fonte em C++ (3 scripts) e dados de simulação (6 arquivos CSV + saída bruta do terminal) Python/ — Código-fonte em Python (2 scripts) e dados de simulação (4 arquivos CSV + saída bruta da tela) Artigo em inglês (PDF) e português (PDF) README.md — Documentação completa do repositório, incluindo descrições de arquivos, definições de colunas, instruções de reprodução e parâmetros físicos. LICENSE.txt — CC BY 4.0 para dados; MIT para código; aviso de patente Joemerson da Silva Lima — Pesquisador Independente, Cuiabá, Mato Grosso, Brasil joemerson93lima@gmail.com Patente BR 10 2025 024459 4
I simulated the two-dimensional Ising model at its exact critical temperature with two Markov chain Monte Carlo methods, the single-spin Metropolis algorithm and the Wolff cluster algorithm, on periodic L×L lattices from L = 8 to L = 64. For each method I measured the integrated autocorrelation time of the energy and of the absolute magnetization, using four independent runs per lattice size. Both methods reproduce the exact finite-lattice energy within statistical error. For Metropolis, the autocorrelation time of |M| grows as L^z with z = 2.11(3) for L ≥ 16 and 2.16(5) for L ≥ 24, in line with the accepted value z ≈ 2.17 for local dynamics. For Wolff, measured in sweep-equivalent units, I get z = 0.29(1) from the energy and 0.15(1) from |M| for L ≥ 24, within the range of published values. In CPU time per statistically independent sample of |M|, Wolff was 5 times faster at L = 8 and 275 times faster at L = 64, and the gap grows roughly as L^2. All code and raw results are available, and the full study runs in under an hour on two CPU cores.
In the present work, we present an extensive Monte Carlo simulation study on the dynamical properties of the two-dimensional random-bond Ising model. The correlation time τ of the Swendsen-Wang and Wolff cluster algorithms is calculated at the critical point. The dynamic critical exponent z of both algorithms is also measured by using the numerical data for several lattice sizes up to L=512. It is found for both algorithms that the autocorrelation time decreases considerably and the critical slowing-down effect reduces upon the introduction of bond disorder. Additionally, simulations with the Metropolis algorithm are performed, and the critical slowing-down effect is observed to be more pronounced in the presence of disorder, confirming the previous findings in the literature. Moreover, the existence of the non-self-averaging property of the model is demonstrated by calculating the scaled form of the standard deviation of autocorrelation times. Finally, the critical exponent ratio of the magnetic susceptibility is estimated by using the average cluster size of the Wolff algorithm.
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
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
We present a high-precision finite-size scaling (FSS) analysis of the two-dimensional Ising model using single-spin Metropolis Monte Carlo simulations on lattice sizes L=12L = 12L=12–128128128. Critical behaviour is characterised through multiple independent observables, including susceptibility peaks, Binder cumulants, and scaling collapse, yielding a consistent estimate of the critical temperature Tc=2.2690(8)T_c = 2.2690(8)Tc=2.2690(8) with excellent agreement to established benchmarks. Temperature-resolved integrated autocorrelation times τint(T) au_{mathrm{int}}(T)τint(T) are computed to account for statistical correlations, demonstrating classical critical slowing down with scaling τint∝Lz au_{mathrm{int}} propto L^zτint∝Lz, where z≈2.1z approx 2.1z≈2.1. All uncertainties are corrected using correlation-aware error propagation. Building on these results, we introduce a Scale–Coherence Junction (SCJ) interpretation, defining a thermodynamic coherence energy scale: E_coh = k_B * T_c This provides a phenomenological bridge between critical phenomena and coherence thresholds, suggesting that maximal system-wide correlations at criticality correspond to enhanced sensitivity to environmental interactions.