We propose a structural diagnostic for thermal phase transitions based on the product of two correlation-derived quantities: the nearest-neighbor mutual information I_nn and the magnetic susceptibility χ. Key Result The thermal cliff diagnostic: κ_T = Ĩ_nn × χ̃ exhibits a sharp peak (cliff) near the critical temperature in systems possessing genuine phase transitions. Findings System Result 2D Ising Sharp cliff near T_c ≈ 2.269 1D Ising No cliff (null test confirmed) Finite-size scaling T_peak(L) → T_c as L → ∞ Contents paper/ — Full manuscript (LaTeX and Markdown) figures/ — Publication-quality figures (PDF and PNG) data/ — Numerical results (CSV format) code/ — Python simulation scripts (Numba-accelerated Monte Carlo) Reproducibility pip install numpy numba matplotlib python code/thermal_cliff_scaling.py
We propose a structural diagnostic for thermal phase transitions based on the product of two correlation-derived quantities: the nearest-neighbor mutual information I_nn and the magnetic susceptibility χ. Key Result The thermal cliff diagnostic: κ_T = Ĩ_nn × χ̃ exhibits a sharp peak (cliff) near the critical temperature in systems possessing genuine phase transitions. Findings System Result 2D Ising Sharp cliff near T_c ≈ 2.269 1D Ising No cliff (null test confirmed) Finite-size scaling T_peak(L) → T_c as L → ∞ Contents paper/ — Full manuscript (LaTeX and Markdown) figures/ — Publication-quality figures (PDF and PNG) data/ — Numerical results (CSV format) code/ — Python simulation scripts (Numba-accelerated Monte Carlo) Reproducibility pip install numpy numba matplotlib python code/thermal_cliff_scaling.py
Critical phenomena are traditionally diagnosed via global order parameters and their cumulants. These methods fail when the order parameter is unknown, undefined, or trivial—as in the Kosterlitz-Thouless transition, magnetization-conserved dynamics, or geometric percolation. We introduce information purity P, a local observable constructed from the absolute nearest-neighbor correlation Δ=⟨|Xi|⟩. The quantity requires only O(Nz) operations per measurement, needs no global magnetization accumulation, and remains well defined in systems with strictly zero global order.Finite-size scaling of equilibrium Monte Carlo data yields dimension-dependent critical values Pc(2D)=0.517±0.007 (2D Ising) and Pc(3D)≈0.36–0.38 (3D Ising), confirming that Pc is not super-universal but serves as a universality-class fingerprint. In the 2D XY model, where the Binder cumulant remains locked at 2/3 across the critical window, P(T) exhibits a measurable slope enhancement near the Kosterlitz-Thouless temperature. For the q-state Potts model, Pc(q) tracks the known critical correlation uc(q). An operational protocol is provided for percolation, a system without a temperature parameter, demonstrating that the local-correlation logic underlying P transfers to non-thermal phase transitions.All calculations employ standard statistical mechanics methods: vectorized Metropolis and Wolff dynamics, binning error analysis, integrated autocorrelation time estimation, and finite-size scaling extrapolation. The results establish P as a computationally inexpensive, locally defined complement to global cumulants, extending the standard critical diagnostics toolkit into regimes where traditional order-parameter-based measures are uninformative or undefined.
On the L=64 lattice of the 3D Ising model—an NP-complete problem rigorously proven to have no analytical solution—the six functional dimensions of the Testability Norms were applied, achieving the detection and classification of the ferromagnetic-paramagnetic phase transition with a statistical confidence of p = 3.33×10⁻⁷⁰. Core findings: The Bai-Perron double-break blind test automatically locked T₁ = 4.39 and T₂ = 4.49 (F = 285.7, significantly outperforming the single-break F = 238.3), precisely aligned with the M_sp switch (4.40, F = 1524) and the U₄ switch (4.50, F = 2335), with deviations of 0.01 and 0.01 respectively. This reveals the Double-Peak Splitting of second-order phase transitions on finite lattices—between two sequential regime switches lies an independent Transition Regime (width approximately 0.1), a region commonly dismissed in standard finite-size scaling (FSS) analysis practice as "rounded crossover requiring extrapolation-based elimination," whose boundaries are algorithmically identifiable. The structure was cross-size replicated on the 2D Potts q=4 model: double-break models significantly outperformed single-break models at L = 16, 32, and 64, with the peak separation shrinking from 0.260 at L = 64 to 0.180 at L = 128, evolving according to scaling laws. The traditional gold standard for phase transition detection, magnetic susceptibility χ, fell completely silent under the Chow test (F = 4.8), yet the 1/χ transformation activated it to 324—revealing that the Chow test requires alignment with the correct mathematical structure to capture physical signals. The regime switch temperature in the M_sp dimension remained stable across all lattice sizes (L = 16, 32, 64, 128) at 4.40–4.46, showing no convergence toward the thermodynamic limit Tc = 4.5115 as L increases—pointing to a Precursor Regime Boundary independent of the thermodynamic limit, cross-model replicated in the M_sp dimension of Potts q=4. The first-order transition (Potts q=3) was confirmed as the direction-inverting type (intercept-shift dominated), and the KT transition (XY model) as single-stage continuous crossover—together with the Double-Peak Splitting of second-order transitions, forming a three-category classification framework of the Factor Hierarchy Law. A three-segment piecewise control function constructed from the Bai-Perron double breaks provides, for this NP-complete system, a quantitative description refined directly from data that is functionally equivalent to the Onsager exact solution for the 2D Ising model—requiring no free energy function. The two inflection points of this function were automatically identified by the Bai-Perron algorithm, without relying on any a priori definition of free energy or order parameter, directly demonstrating the capacity of the Testability Norms to extract physical laws from data. Methodological contribution and ultimate asset: This paper distills the entire body of verification into the Multi-Regime State Decoupling Algorithm (MRSD) based on the Factor Hierarchy Law. Its core mathematical logic proceeds in three steps: (1) input any complex high-dimensional dataset; (2) execute the structural optimization operator \left\{T_k\right\}=arg\min_{}{\sum\nolimits^{k+1}_{j=1}{}}\mathrm{RSS}_j(T)+λ⋅k; (3) output a dynamic regime allocation matrix, informing the user at which point the system logic undergoes "rule resetting" and at which point the structure enters "rigid locking." The algorithm has been cross-model verified on the 3D Ising, 2D Potts q=4, and 2D XY models, requiring no free energy function or a priori order parameter definition—only that the data be objective. It redefines finite-size effects from "errors requiring correction" to independent statistical regimes with rigorous boundaries that are algorithmically identifiable, completing the key leap of the information-theoretic duality between the Factor Hierarchy Law and the Ehrenfest classification from analytically solvable systems to systems without analytical solutions. The Testability Norms achieved full verification closure on an NP-complete system across all six functional dimensions. Leave-one-out cross-validation (Tc range [4.47, 4.48], standard deviation 0.0021), Bootstrap threshold regression (95% CI [4.44, 4.48]), and the shrinking external field convergence verification (Tc monotonically shifted from 4.48 to 4.50) jointly confirm the high robustness of the conclusions. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.
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
The critical behavior at the phase transition of the ferromagnetic two-dimensional anisotropic Ising model with next-nearest neighbor (NNN) couplings in the presence of the field is determined using mainly Monte Carlo (MC) method. This method is used to investigate the phase diagram of the model and to verify the existence of a divergence at null temperature which often appears in two-dimensional systems. We analyze also the influence of the report of the NNN interactions [Formula: see text] and the magnetic field [Formula: see text] on the critical temperature of the system, and we show that the critical temperature depends on the magnetic field for positive values of the interaction. Finally, we have investigated other thermodynamical qualities such as the magnetic susceptibility [Formula: see text]. It has been shown that their thermal behavior depends qualitatively and quantitatively on the strength of NNN interactions and the magnetic field.
We investigate phase transitions in the two-dimensional Ising model with competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions, where the NNN coupling generates a degenerate superantiferromagnetic (SAF) ground state, by combining machine-learning methods with Monte Carlo simulations and finite-size scaling (FSS). Going beyond earlier studies, we show that unsupervised learning can recover order-parameter and Binder cumulant analogs that obey FSS. Principal component analysis (PCA) reveals the two-component structure of the SAF order parameter, and k-means clustering is used to visualize qualitative differences between ordered and disordered configurations in the principal-component space. A fully connected neural network (FCNN), trained without prior knowledge of the order parameter or the critical temperature, learns the strength of SAF stripe ordering and uses it as the internal feature for phase classification. FSS of thermodynamic observables, the PCA-derived quantity, and the FCNN output yields consistent estimates of the critical temperature and correlation-length exponent. These results demonstrate that machine learning can decode ordering patterns, extract meaningful order-parameter analogs, and reproduce critical behavior in frustrated Ising systems.
What defines the operational boundary between quantum information and classical magnetism? We present a model-independent methodology for detecting characteristic scales in quantum systems through generalized susceptibility analysis χ(σ)=|d〈O〉/dσ|, where σ represents observation scales across spatial, temporal, and decoherence dimensions. Through comprehensive experiments on a Rigetti Ankaa-3 quantum processor, we identify an operational threshold at γc=0.6737±0.036 under tested noise profiles, exhibiting exceptional peak clarity κ=8.58—the sharpest signal observed across all phenomena. This pronounced transition, validated through bootstrap analysis and multiple susceptibility metrics, suggests critical-like behavior in the quantum-to-classical crossover, though we carefully refrain from claiming a thermodynamic phase transition. The framework successfully extracts characteristic scales without prior theoretical knowledge: correlation length ξ=8.00 qubits, distinct ordering timescales for ferromagnetic (tc=0.36) versus antiferromagnetic (tc=0.91) systems, and the quantum critical field hc=1.821 in the transverse-field Ising model. Robustness analysis confirms stability across smoothing parameters, methods, and metric definitions. Beyond providing actionable guidance for noisy intermediate-scale quantum-era quantum computing, our findings suggest that magnetic correlations exhibit characteristic transitions at specific decoherence scales—an operational perspective connecting quantum information theory and condensed matter physics. While framing this as methodological advance rather than fundamental physics, the identification of critical-like information-to-physics transitions provides practical diagnostics for quantum device characterization and optimization.