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.
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.
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
We perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
Abstract For systems above the marginal dimension d*, where mean field theory starts to become valid, such as Ising models in d = 5 for which d* = 4, hyperscaling is invalid and hence it was suggested that finite size scaling is not ruled by the correlation length ξ (∝ |t| −1/2 in Landau theory, t being the distance from the critical point) but by a “thermodynamic length” l (∝ |t| −2/d). Early simulation work by Binder et al. using nearest neighbor hypercubic L5 lattices with L ⩽ 7 yielded some evidence for this prediction, but the renormalized coupling constant gL = −3 + 〈M4〉/〈M2〉2 at Tc was gL ≈ −1.0 instead of the prediction of Brézin and Zinn‐Justin, gL(Tc) = −3 + Γ4(1/4)/(8 π2) ≈ −0.812. In the present work, we try to shed light on this controversy obtaining much more precise Monte Carlo data using multihistogram techniques and lattices as large as L = 17 (i.e., 1419857 Ising spins). While our value of Tc agrees nicely with recent high temperature series work (KBTc/J ≈ 8.7774 ± 0.0035), the coupling at Tc {gL(Tc) ≈ −0.958 ± 0.050} confirms the work for small lattices and disagrees with the analytical prediction. Hence the data are better consistent with a shift of the effective critical temperature Tc(L) ‐ Tc(∝) ∝ L−d/2 rather than with L−(d−2) according to the analytical theory. If the latter behavior is correct, it can hence be seen for extremely large systems only.
The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature Tc despite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating Tc of the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles C(T). Different approaches were compared, including scaling models for the peak temperature Tmax(N) and an envelope construction across the set of C(T)-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for Tmax with a single model parameter following an N/(N+1)-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation.
It is shown by the Monte Carlo method that finite-size scaling (FSS) holds in the two-dimensional randomly coupled Ising ferromagnet. It is also demonstrated that the form of the universal FSS function constructed via a FSS scheme depends on the strength of the random coupling in the case of strongly disordered systems. Monte Carlo measurements of thermodynamic (infinite-volume-limit) data of the correlation length $(\ensuremath{\xi})$ up to $\ensuremath{\xi}\ensuremath{\simeq}200$ along with measurements of the fourth-order cumulant ratio (Binder cumulant ratio) at criticality are analyzed in light of two competing scenarios of weak universality and the multiplicative logarithmic correction. It is demonstrated that the data are likely to be more consistent with the former scenario than the latter which is the conventional scenario.