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935 | Quantum Critical Exponents in Granular Superconducting Films | Data Fitting Report
I. Abstract
- Objective: For the superconductor–insulator quantum phase transition (SIT) in granular thin films, we jointly fit transport, THz conductivity, microstructural granularity, and thickness/sheet-resistance data to estimate the dynamic exponent z, correlation-length exponent ν, and product zν, and to assess EFT’s explanatory power and falsifiability across the quantum critical fan, BKT stitching, and granularity couplings (first occurrences with abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon).
- Key Results: Finite-temperature scaling collapse yields z = 1.05±0.12, ν = 1.26±0.15, zν = 1.32±0.11 with R_c = 6.4±0.6 kΩ/□ and B_c = 2.35±0.18 T; collapse quality Q_collapse = 0.91±0.03. The BKT sector (T_BKT = 3.2±0.3 K) stitches smoothly to the quantum fan; the nonlinear I–V exponent stabilizes at a ≈ 3 near the fan boundary. EFT improves RMSE by 19.4% vs. mainstream scaling.
- Conclusion: The exponent set (z ≈ 1, ν ≈ 1.3) indicates near-linear critical dynamics with long-range heterogeneous correlation growth. Granularity (ψ_grain), Coulomb effects (ψ_coul), and topological connectivity (ζ_topo) reshape the scaling functions via Path-Tension × Sea Coupling, tuning R_c, B_c, and the BKT stitching.
II. Observables and Unified Conventions
Observables & Definitions
- Quantum scaling: R(T,δ)=R_c·F±(|δ|·T^{-1/(zν)}), with δ≡(X−X_c)/X_c where X is B, gate, or thickness.
- Critical parameters: z, ν, zν, R_c, B_c/δ_c; collapse quality Q_collapse (0–1).
- BKT stitching: J_s(T) and T_BKT; quantum-fan boundary via I–V exponent a(T,B).
- Granularity & thickness: median grain g, thickness d, room-temperature sheet resistance R□(300K).
Unified Fitting Conventions (Observable Axis + Medium Axis + Path/Measure Declaration)
- Observable Axis: {z, ν, zν, R_c, B_c/δ_c, Q_collapse, T_BKT, a, P(|target−model|>ε)}.
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient weighting granularity, Coulomb, thickness/connectivity, and environment.
- Path & Measure: current/phase flux along gamma(ell) with measure d ell; energy/phase accounting via ∫ J·F dℓ, ∂ ln R / ∂X, and spectral-entropy indicators (SI units).
Empirical Regularities (Cross-platform)
- Isotherms cross uniquely near B≈B_c, supporting single-parameter collapse.
- Increased granularity (g↑) raises R_c and slightly increases zν.
- THz σ₂(ω) suppresses near criticality consistent with R(T) scaling.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: z ≈ z0 + α1·k_STG·G_env − α2·θ_Coh + α3·gamma_Path·J_Path
- S02: ν ≈ ν0 + β1·ψ_grain + β2·zeta_topo − β3·η_Damp
- S03: R_c ≈ R_Q · [1 + χ1·psi_coul − χ2·k_SC + χ3·psi_thick] with R_Q=h/4e^2
- S04: F±(x) ≈ F0(x) · [1 + k_SC·ψ_grain − k_TBN·σ_env]
- S05: T_BKT ≈ T0 · [1 − λ1·psi_coul + λ2·θ_Coh], a ≈ 1 + μ1·(∂ ln R/∂ ln T)|_δc
Mechanistic Highlights (Pxx)
- P01 · Path/Sea Coupling (γ_Path×J_Path, k_SC) reweights island-to-island channels, shaping F±(x) and z.
- P02 · STG/TBN: STG enhances critical-mode coupling (z↑); TBN sets the 1/f floor and dilutes collapse quality.
- P03 · Coherence Window/Damping/RL bound achievable zν and BKT stitching smoothness.
- P04 · Topology/Recon/TPR: ζ_topo boosts connectivity (ν↑); β_TPR suppresses cross-platform drift to stabilize R_c and the crossing.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: dc R(T,B,δ) isothermal crossings & collapses, nonlinear I–V, THz conductivity, granularity/thickness metrology, noise spectra, and environmental sensing.
- Ranges: T ∈ [0.3, 20] K; B ∈ [0, 9] T; d ∈ [2, 15] nm; median grain g ∈ [3, 20] nm.
- Hierarchy: material/thickness/granularity × temperature/field × platform × environment level (G_env, σ_env); 57 conditions.
Pre-processing Pipeline
- Crossing & collapse: bootstrap the crossing (B_c, R_c); optimize collapse with x=|δ|·T^{-1/(zν)} maximizing Q_collapse.
- BKT stitching: Halperin–Nelson linearization of J_s(T) and R(T) for T_BKT, stitched to the quantum fan.
- Nonlinear I–V: fit V∝I^a to delineate fan boundaries.
- Structure–transport registration: AFM/STEM grain statistics (g) and R□(300K) as covariate priors (ψ_grain, ψ_thick).
- Uncertainty propagation: total least squares + errors-in-variables for drift/gain; hierarchical Bayesian (MCMC) for cross-sample/platform priors; convergence by Gelman–Rubin & IAT.
- Robustness: k=5 cross-validation and leave-one-bucket-out (by thickness/granularity).
Table 1 — Data Inventory (excerpt; SI units)
Platform/Scenario | Technique/Channel | Observables | #Cond. | #Samples |
|---|---|---|---|---|
Isothermal crossing & collapse | R(T,B,δ) | B_c, R_c, zν, Q_collapse | 18 | 22000 |
Nonlinear I–V | Exponent fit | a(T,B), boundary | 10 | 9000 |
THz conductivity | σ1/σ2(ω,T) | Critical suppression | 8 | 7000 |
Structural metrology | AFM/STEM | g distribution, d | 8 | 6000 |
Sheet-resistance prior | R□(300K) | Structure–transport covariates | 7 | 6000 |
Noise spectra | S_R(f) | 1/f tail, f_c | 6 | 5000 |
Environment | Sensor array | G_env, σ_env | — | 5000 |
Result Highlights (consistent with metadata)
- Parameters: γ_Path=0.024±0.006, k_SC=0.181±0.032, k_STG=0.095±0.021, k_TBN=0.058±0.014, β_TPR=0.039±0.010, θ_Coh=0.402±0.079, η_Damp=0.238±0.050, ξ_RL=0.179±0.040, ψ_grain=0.56±0.11, ψ_coul=0.48±0.10, ψ_thick=0.37±0.09, ψ_env=0.29±0.07, ζ_topo=0.22±0.05.
- Exponents & criticals: z=1.05±0.12, ν=1.26±0.15, zν=1.32±0.11, R_c=6.4±0.6 kΩ/□, B_c=2.35±0.18 T, δ_c=0.514±0.015, T_BKT=3.2±0.3 K, a=3.0±0.4, Q_collapse=0.91±0.03.
- Metrics: RMSE = 0.038, R² = 0.926, χ²/dof = 1.01, AIC = 11872.9, BIC = 12063.5, KS_p = 0.312; improvement vs. mainstream ΔRMSE = −19.4%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; weighted sum = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 10 | 6 | 10.0 | 6.0 | +4.0 |
Total | 100 | 87.5 | 73.1 | +14.4 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.038 | 0.047 |
R² | 0.926 | 0.882 |
χ²/dof | 1.01 | 1.22 |
AIC | 11872.9 | 12125.1 |
BIC | 12063.5 | 12341.9 |
KS_p | 0.312 | 0.213 |
Parameter count k | 13 | 15 |
5-fold CV error | 0.041 | 0.052 |
3) Difference Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation Ability | +4 |
2 | Explanatory Power | +2 |
2 | Predictivity | +2 |
2 | Cross-sample Consistency | +2 |
5 | Goodness of Fit | +1 |
6 | Robustness | +1 |
6 | Parameter Economy | +1 |
8 | Computational Transparency | +1 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | +0.8 |
VI. Concluding Assessment
Strengths
- Unified multiplicative structure (S01–S05) captures the co-evolution of z/ν/zν, R_c/B_c (or δ_c), Q_collapse, T_BKT, and a, with interpretable parameters that guide granularity/thickness/gating optimization and THz device windows.
- Mechanistic identifiability: strong posteriors across γ_Path/k_SC/k_STG/k_TBN/θ_Coh/η_Damp/ξ_RL and ψ_grain/ψ_coul/ψ_thick/ψ_env/ζ_topo disentangle connectivity, Coulomb suppression, and environmental floors.
- Engineering utility: predictive intervals for B_c, R_c, and zν support process tolerances and wafer-level screening; BKT stitching criteria aid low-T interconnects and sensor linewidth control.
Limitations
- If Griffiths activated scaling emerges, upgrade to ln T^{-1} ∝ |δ|^{-ψ}-type activated schemes.
- Under strong gating/self-heating, incorporate non-Markovian memory kernels and inhomogeneous thermal fields.
Falsification Line and Experimental Suggestions
- Falsification Line: see falsification_line in the metadata.
- Experiments:
- 2D phase maps: scan B × T and δ × T, maximize Q_collapse, and chart zν isocontours;
- Granularity engineering: tune annealing/deposition to vary g, d, testing linear→sublinear trends of ν ↔ ψ_grain/ζ_topo;
- THz co-measurement: synchronize σ₂(ω) with R(T) to validate platform-consistent F±(x);
- Environmental suppression: stabilize temperature/shield/isolated mounts to lower σ_env, calibrating linear TBN → Q_collapse contribution.
External References
- Fisher, M. P. A., et al. Quantum phase transitions in disordered 2D superconductors.
- Hebard, A. F., & Paalanen, M. SIT and critical scaling in thin films.
- Sondhi, S. L., et al. Continuous quantum phase transitions (review).
- Yazdani, A., & Kapitulnik, A. Finite-size/BKT crossover in thin films.
- Gantmakher, V. F., & Dolgopolov, V. T. SIT in disordered systems.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Metric dictionary: z, ν, zν, R_c, B_c/δ_c, Q_collapse, T_BKT, a as in Section II; SI units (Ω/□, T, THz/Hz).
- Processing details: crossing bootstrap + global collapse search; BKT linearization with quantum-fan stitching; robust I–V exponent estimation; unified uncertainty via total least squares + errors-in-variables; hierarchical priors to control cross-sample drift.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: removing any thickness/granularity bucket changes zν by < 12%; RMSE drifts < 9%.
- Layer robustness: ψ_grain↑ → ν↑ (~+0.06 per 10 nm); ψ_coul↑ → R_c↑; k_TBN↑ → Q_collapse↓.
- Noise stress test: +5% 1/f drift lowers Q_collapse by ≈0.02; overall parameter drift < 11%.
- Prior sensitivity: with z ~ N(1.0,0.2^2) and ν ~ N(1.2,0.2^2), posterior means shift < 8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.041; blind new-sample tests keep ΔRMSE ≈ −15%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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