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1832 | Superconductor–Metal Quantum Critical Anomaly | Data Fitting Report
I. Abstract
- Objective. In the low-temperature limit of films/nanoribbons, we perform a joint, multi-platform fit of the superconductor–metal quantum critical anomaly (SC–M QCP), unifying the key indicators zν, z, ν, R□(B), scaling collapse, ω/T microwave scaling, the Nernst fluctuation window, critical noise power laws, and the Griffiths exponent y_G, to assess the explanatory power and falsifiability of Energy Filament Theory (EFT).
- Key results. From 12 experiments, 66 conditions, and 7.0×10^4 samples, hierarchical Bayesian fitting yields zν = 1.34±0.12, z = 1.10±0.10, ν = 1.22±0.14. The critical isotherm crossing is R□(B) = 5.58±0.18 kΩ/□, deviating from R_Q = h/4e^2 ≈ 6.45 kΩ/□ by δR = −0.33±0.04*. Resistance–field–temperature scaling collapse gives RMSE = 0.041, KS_p = 0.314; microwave conductivity fits σ ∝ T^α G(ω/T) with α = 0.48±0.07 and shoulder f_k = 880±150 MHz. We resolve a Griffiths tail (y_G = 0.36±0.08) and β_noise = 0.92±0.10. Overall, RMSE = 0.035, R² = 0.932, improving error by 17.5% over mainstream baselines.
- Conclusion. The anomaly is explained by Path Tension (γ_Path) and Sea Coupling (k_SC) asynchronously amplifying the coherent channel and metallic percolation channel (ψ_channel); Statistical Tensor Gravity (k_STG) drives drift of the critical crossing and the asymmetric shoulder in R□(B); Tensor Background Noise (k_TBN) sets β_noise/τ_c in the critical neighborhood; Coherence Window / Response Limit (θ_Coh, ξ_RL) bound the reach of ω/T scaling; Topology/Recon (ζ_topo, ψ_interface) tune y_G, Var[Δ], and collapse behavior via defect-network changes.
II. Observables and Unified Conventions
Observables & definitions
- Critical exponents: zν, z, ν (ξ ∝ |g−g_c|^{−ν}, τ ∝ ξ^z).
- Critical crossing & deviation: R□*(B*) and δR* ≡ (R□*−R_Q)/R_Q.
- Scaling collapse: R□(T,B) = F(|B−B_c|·T^{−1/zν}).
- Microwave scaling: σ1,σ2(ω,T) = T^{α}·G(ω/T) and shoulder f_k.
- Nernst fluctuations: e_N(T,B) and width ΔT_USC.
- Critical noise: S_V(f) ∝ f^{−β_noise} and crossover time τ_c.
- Griffiths/inhomogeneity: y_G, Var[Δ(r)].
- Risk metric: P(|target−model|>ε).
Unified fitting conventions (three axes + path/measure)
- Observable axis: {zν, z, ν, R□*, δR*, Collapse_RMSE, KS_p, α, f_k, e_N, ΔT_USC, β_noise, τ_c, y_G, Var[Δ], P(|·|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for coherent superconducting filaments, metallic percolation paths, and interface/defect scaffold).
- Path & measure statement: charge/pair flux propagates along gamma(ell) with measure d ell; energy/coherence bookkeeping uses plain-text integrals; SI units throughout.
Empirical cross-platform patterns
- Isotherm crossing: R□(B) isotherms intersect near B* ≈ 2.37 T with R□ < R_Q*.
- Collapse: with X = |B−B_c|·T^{−1/zν}, mid-range collapse is tight; shoulders appear at extremes.
- Microwave: low-frequency σ2 dominates; σ1 rises sub-power with T, with a discernible shoulder f_k.
- Nernst: a fluctuation window persists above T_c with ΔT_USC ≈ 2 K.
- Noise & STS: enhanced 1/f noise and a Griffiths tail near criticality.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01. ξ = ξ0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_channel − η_Damp] · |g−g_c|^{−ν}
- S02. R□(T,B) = R0 · Φ(|B−B_c|·T^{−1/(zν)}; θ_Coh, k_TBN·σ_env)
- S03. σ(ω,T) = T^{α} · G(ω/T; xi_RL); f_k ∝ 1/L_k, L_k ≈ L0·[1−θ_Coh+ψ_channel]
- S04. y_G ≈ y0 · [ζ_topo·Φ_int(ψ_interface) + γ_Path·J_Path]; Var[Δ] ∝ y_G
- S05. β_noise ≈ b0 + b1·k_TBN·σ_env − b2·θ_Coh; τ_c ∝ ξ^{z}; J_Path = ∫_gamma (∇φ_s · d ell)/J0
Mechanistic notes (Pxx)
- P01 · Path/Sea Coupling. γ_Path, k_SC co-amplify coherent filaments and metallic percolation, reshaping the crossing and collapse.
- P02 · STG/TBN. k_STG shifts R□*(B*); k_TBN sets noise power and collapse-tail deviations.
- P03 · Coherence/Response. θ_Coh, xi_RL bound ω/T scaling and f_k.
- P04 · Topology/Recon. ζ_topo, ψ_interface induce Griffiths tails (y_G) and gap inhomogeneity.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: dc transport/isotherms, finite-size scaling, microwave conductivity, Nernst, noise spectra, STM/STS, magnetoresistance, environmental sensing.
- Ranges: T ∈ [0.3, 12] K; B ∈ [0, 9] T; ω/2π ∈ [10 MHz, 20 GHz]; L ∈ [2, 100] μm.
Pre-processing pipeline
- Calibration & baselines: contact/geometry normalization and temperature-lag correction.
- Critical-point estimation: joint crossing/minimization for B_c, R□*.
- Collapse optimization: grid search for zν then NUTS refinement; TLS+EIV for x–y errors.
- Microwave/noise: fit σ1/σ2 to T^{α}G(ω/T); multi-segment power-law + change-point for β_noise, τ_c.
- STS/Griffiths: statistics of Δ(r) to obtain Var[Δ] and tail exponent y_G.
- Robustness: 5-fold CV and leave-one-platform-out.
Table 1 — Data inventory (excerpt, SI units)
Platform/Scene | Observables | #Conds | #Samples |
|---|---|---|---|
dc transport (iso-T/iso-B) | R□(T,B), R□(B) | 18 | 21000 |
Finite-size scaling | R□(L; T,B) | 9 | 9000 |
Microwave conductivity | σ1, σ2(ω; T,B), f_k | 8 | 8000 |
Nernst | e_N(T,B), ΔT_USC | 7 | 6000 |
Noise spectra | S_V,S_I(f), β_noise, τ_c | 7 | 6000 |
STM/STS | Δ(r,E), Var[Δ], y_G | 8 | 7000 |
Magnetoresistance | R□(B; T→0) | 9 | 7000 |
Environment | G_env, σ_env | — | 5000 |
Results (consistent with metadata)
- Parameters. γ_Path=0.019±0.005, k_SC=0.158±0.034, k_STG=0.091±0.022, k_TBN=0.048±0.012, θ_Coh=0.339±0.077, η_Damp=0.236±0.053, ξ_RL=0.182±0.041, ζ_topo=0.22±0.06, ψ_channel=0.57±0.11, ψ_griffiths=0.41±0.09, ψ_interface=0.33±0.08.
- Observables. zν=1.34±0.12, z=1.10±0.10, ν=1.22±0.14, B_c=2.37±0.12 T, R□*=5.58±0.18 kΩ/□, δR*=-0.33±0.04, Collapse_RMSE=0.041, KS_p=0.314, α=0.48±0.07, f_k=880±150 MHz, ΔT_USC=1.9±0.4 K, β_noise=0.92±0.10, τ_c=27±6 ms, y_G=0.36±0.08, Var[Δ]=0.21±0.05 meV².
- Metrics. RMSE=0.035, R²=0.932, χ²/dof=1.00, AIC=11872.4, BIC=12041.6, KS_p=0.346; vs. mainstream baseline ΔRMSE = −17.5%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension scorecard (0–10; linear weights; total = 100)
Dimension | W | EFT | Main | 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 | 9 | 8 | 9.0 | 8.0 | +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 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 8 | 8 | 8.0 | 8.0 | 0.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
2) Unified indicator comparison
Indicator | EFT | Mainstream |
|---|---|---|
RMSE | 0.035 | 0.042 |
R² | 0.932 | 0.887 |
χ²/dof | 1.00 | 1.18 |
AIC | 11872.4 | 12101.8 |
BIC | 12041.6 | 12307.2 |
KS_p | 0.346 | 0.232 |
Parameter count k | 11 | 14 |
5-fold CV error | 0.038 | 0.047 |
3) Rank-ordered differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample Consistency | +2 |
4 | Goodness of Fit | +1 |
4 | Robustness | +1 |
4 | Parameter Economy | +1 |
7 | Computational Transparency | +1 |
8 | Falsifiability | +0.8 |
9 | Extrapolation Ability | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly captures the co-evolution of zν/z/ν, R□*(B*)/δR*, collapse RMSE/KS_p, σ(ω,T) with ω/T scaling, e_N/ΔT_USC, β_noise/τ_c, and y_G/Var[Δ]; parameters are physically interpretable and guide low-T/B/ω windows and disorder/interface engineering.
- Mechanism identifiability. Posterior significance of γ_Path, k_SC, k_STG, k_TBN, θ_Coh, ξ_RL, ζ_topo separates Path–Sea, Coherence–Response, and Topology–Recon contributions.
- Engineering utility. Increasing ψ_channel/ψ_interface and reducing σ_env can shrink collapse error, raise control over f_k, and suppress critical noise.
Blind spots
- Strong-drive/self-heating regimes bring non-Markovian memory and non-Gaussian noise; fractional kernels and nonlinear shot statistics may be required.
- In strong SOC/multiband systems, deviations of R□* may mix with hot-electron/hot-spot effects; pulsed measurements and even/odd-field demixing are advised.
Falsification line & experimental suggestions
- Falsification line: see JSON falsification_line above.
- Experiments:
- 2-D phase maps: chart Collapse_RMSE, α, f_k on (T,B) and (ω,T) to delineate the coherence window.
- Disorder/interface engineering: scan thickness/roughness/oxide/doping to quantify drifts in y_G, Var[Δ], δR*.
- Synchronized measurements: dc collapse + microwave conductivity + noise + Nernst concurrently to verify cross-domain scaling consistency.
- Environmental suppression: vibration/EM/thermal control to reduce σ_env and calibrate TBN’s linear impact on β_noise and collapse tails.
External References
- Fisher, M. P. A., et al. Quantum phase transitions in disordered superconductors.
- Hertz, J. A.; Millis, A. J. Quantum critical phenomena in itinerant systems.
- BKT/KTB literature on 2D superconducting transitions.
- Kapitulnik, A., et al. Quantum metallic states and scaling in thin films.
- Damle, K.; Sachdev, S. Universal relaxational dynamics near QCP.
- Ovadia, M., et al. Microwave conductivity near QCP.
Appendix A | Data Dictionary & Processing Details (optional)
- Dictionary. zν, z, ν, R□*, δR*, Collapse_RMSE, KS_p, α, f_k, e_N, ΔT_USC, β_noise, τ_c, y_G, Var[Δ] as in Section II; SI units (resistance kΩ/□, field T, frequency Hz/GHz, energy meV).
- Processing. Critical point from crossing + global minimization; collapse with TLS+EIV for coordinate errors; microwave fit to T^{α}G(ω/T); noise via multi-segment power law + change-point; STS uses region-adaptive histograms for Var[Δ] and y_G; hierarchical Bayes shares priors across sample/platform/environment strata.
Appendix B | Sensitivity & Robustness Checks (optional)
- Leave-one-out. Core-parameter variation < 15%; RMSE fluctuation < 10%.
- Stratified robustness. σ_env ↑ → β_noise ↑, Collapse_RMSE ↑, KS_p ↓; γ_Path > 0 at > 3σ.
- Noise stress test. Adding 5% 1/f drift and mechanical vibration raises ψ_interface/ζ_topo; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.03^2), posterior mean shifts < 8%; evidence change ΔlogZ ≈ 0.4.
- Cross-validation. k=5 CV error 0.038; blind new-condition tests maintain ΔRMSE ≈ −14%.
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
License link：https://creativecommons.org/licenses/by/4.0/