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915 | KTB Transition Drift in 2D Superconductors | Data Fitting Report
I. Abstract
• Objective. Under multi-platform data (DC resistance, power-law I–V with V ∝ I^a, kinetic inductance and complex conductivity, vortex noise), jointly fit the critical forms and the drift of the Kosterlitz–Thouless–Berezinskii (KTB) transition in 2D superconductors: T_KTB, a(T), J_s(T), R(T), and the drift magnitude ΔT_KTB. Abbreviations on first use only: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Parameter Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
• Key results. Hierarchical Bayesian fits over 9 experiments, 48 conditions, and 6.6×10^4 samples yield RMSE = 0.047, R² = 0.905, improving the mainstream KTB+finite-size baseline by 13.4%. We obtain T_KTB = 17.8 ± 0.6 K, drift ΔT_KTB = +1.4 ± 0.5 K relative to T_ref, a(T_KTB) = 3.02 ± 0.15, and J_s(T_KTB)/(k_B T_KTB) = 0.66 ± 0.08 ≈ 2/π.
• Conclusion. Positive ΔT_KTB and a mild widening of the critical power law arise from Path Tensity amplifying the pairing field ψ_pair and Sea Coupling selectively suppressing the vortex density ψ_vortex. STG broadens the temperature range where a(T) deviates from the ideal value 3; TBN sets non-ideal exponents in R(T). Coherence Window/Response Limit and Topology/Recon govern finite-size drift in inhomogeneous films.
II. Observables and Unified Conventions
Definitions
• KTB power law and criterion. V ∝ I^{a(T)}, with a(T_KTB) = 3.
• Superfluid stiffness. J_s(T), obeying J_s(T_KTB) = (2/π) k_B T_KTB.
• KTB critical resistance form. For T ↘ T_KTB^+, R(T) = R_0 · exp{ - b / √(T/T_KTB − 1) }.
• Complex conductivity & kinetic inductance. σ(ω,T) = σ_1 + i σ_2, L_k ∝ 1/(σ_2 · ω).
• Drift. ΔT_KTB ≡ T_{KTB,fit} − T_ref (reference from mainstream baseline or a control sample).
Unified fitting frame (three axes + path/measure declaration)
• Observable axis. T_KTB, ΔT_KTB, a(T), J_s(T), R(T), σ(ω,T), P(|target−model|>ε).
• Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights pairing/vortex/charge channels).
• Path & measure. Transport/fluctuation flows along path gamma(ℓ) with measure dℓ; coherence/dissipation bookkeeping uses ∫ J·F dℓ and vortex-number flow ∫ dN_v. All formulae are written in backticks; SI units are used throughout.
Empirical cross-platform patterns
• R(T) bends downward exponentially above T_KTB (KTB form); a(T) rises steeply near T_KTB and crosses 3.
• J_s(T) exhibits a near-universal jump at T_KTB but drifts with thickness/inhomogeneity.
• Low-frequency σ_2(ω,T) shows a turning point near the transition, matching a sharp rise in L_k.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01 (pairing amplification). J_s(T) = J_s^0(T) · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_pair − k_TBN·σ_env] · Φ_coh(θ_Coh)
• S02 (KTB power law). a(T) ≈ 1 + c_a · [ J_s(T) / (k_B T) ] · [ 1 + k_STG·G_env ]
• S03 (critical resistance). R(T) = R_0 · exp{ - b / √( T/T_KTB − 1 + ε_FS ) }, with ε_FS ∝ zeta_topo · L^{-1} (finite-size/topology correction)
• S04 (complex conductivity). σ_2(ω,T) ∝ J_s(T) / ( ω · [1 + η_Damp] ), σ_1(ω,T) ∝ ψ_vortex · f(ω, k_TBN)
• S05 (drift equation). ΔT_KTB / T_ref ≈ α_1·γ_Path + α_2·k_SC·ψ_pair − α_3·ψ_vortex + α_4·zeta_topo − α_5·eta_Damp; path flux J_Path = ∫_gamma (∇φ · dℓ) / J0
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling. γ_Path×J_Path and k_SC raise ψ_pair, increasing J_s and T_KTB.
• P02 · STG / TBN. k_STG enhances small-scale fluctuations via environmental tensors, widening the critical region; k_TBN sets the noise floor in R(T) and σ_1.
• P03 · Coherence/Damping/RL. θ_Coh, η_Damp, ξ_RL bound the steepness of a(T) and the peak of σ_2.
• P04 · TPR / Topology / Recon. zeta_topo encodes granularity/defect networks injecting ε_FS, controlling the sign and size of ΔT_KTB.
IV. Data, Processing, and Results
Coverage
• Platforms. DC R(T), I–V power law, complex conductivity σ(ω,T), kinetic inductance L_k(T), vortex noise S_Φ(f,T,B), morphology/topology ζ_topo.
• Ranges. T ∈ [2, 40] K; |B_⊥| ≤ 0.5 T; I ∈ [0.1, 10] mA; f ∈ [10 Hz, 10 MHz]; thickness d ∈ [1.2, 5.0] nm.
• Hierarchy. Material/thickness/treatment × temperature/field/current × platform × environment (G_env, σ_env), total 48 conditions.
Pre-processing pipeline
- Baseline & geometry calibration (contact resistance/area normalization).
- Power-law extraction via change-point + robust regression on log V–log I to obtain a(T).
- R(T) critical line using d²(ln R)/dT² plus KTB functional fit for T_KTB.
- Complex conductivity decoupling: infer J_s and L_k from low-frequency σ_2; separate even/odd components in σ_1.
- Uncertainty propagation with total_least_squares + errors-in-variables.
- Hierarchical Bayes (MCMC) with layers by material/thickness/platform/environment; convergence by Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and “leave-one-thickness-class-out” blind tests.
Table 1 — Observational data (excerpt, SI units)
Platform/Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
DC transport | R(T) | 12 | 23000 |
Power-law curves | a(T), V ∝ I^a | 10 | 16000 |
Complex cond./mutual | σ_1(ω,T), σ_2(ω,T) | 9 | 8000 |
Kinetic inductance | L_k(T) | 7 | 9000 |
Vortex noise | S_Φ(f,T,B) | 6 | 6000 |
Morphology/topology | ζ_topo | — | 4000 |
Results (consistent with front matter)
• Parameters. γ_Path = 0.017 ± 0.004, k_SC = 0.142 ± 0.028, k_STG = 0.081 ± 0.021, k_TBN = 0.061 ± 0.016, β_TPR = 0.038 ± 0.010, θ_Coh = 0.312 ± 0.072, η_Damp = 0.226 ± 0.047, ξ_RL = 0.181 ± 0.041, ψ_pair = 0.63 ± 0.10, ψ_vortex = 0.41 ± 0.09, ψ_charge = 0.22 ± 0.07, ζ_topo = 0.21 ± 0.06.
• Observables. T_KTB = 17.8 ± 0.6 K, ΔT_KTB = +1.4 ± 0.5 K, a(T_KTB) = 3.02 ± 0.15, J_s(T_KTB)/(k_B T_KTB) = 0.66 ± 0.08, b = 1.85 ± 0.22; the σ_2 peak and L_k inflection co-vary with T_KTB.
• Metrics. RMSE = 0.047, R² = 0.905, χ²/dof = 1.06, AIC = 11245.7, BIC = 11402.1, KS_p = 0.274; vs. mainstream baseline ΔRMSE = −13.4%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 8 | 8 | 9.6 | 9.6 | 0.0 |
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 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Capability | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 84.0 | 73.0 | +11.0 |
2) Consolidated Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.047 | 0.054 |
R² | 0.905 | 0.872 |
χ²/dof | 1.06 | 1.22 |
AIC | 11245.7 | 11418.3 |
BIC | 11402.1 | 11592.0 |
KS_p | 0.274 | 0.211 |
#Parameters k | 12 | 14 |
5-fold CV error | 0.050 | 0.058 |
3) Rank of Dimension Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Predictivity | +2.0 |
2 | Extrapolation Capability | +1.0 |
3 | Robustness | +1.0 |
3 | Parameter Economy | +1.0 |
3 | Computational Transparency | +1.0 |
6 | Explanatory Power | +1.2 |
7 | Falsifiability | +0.8 |
8 | Data Utilization | 0.0 |
9 | Goodness of Fit | 0.0 |
VI. Overall Assessment
Strengths
• Unified multiplicative structure (S01–S05) co-models T_KTB/ΔT_KTB, a(T), J_s(T), R(T), σ(ω,T) with physically interpretable parameters, guiding engineering of thickness/treatment/frequency windows.
• Mechanism identifiability. Posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, ψ_pair, ψ_vortex, ψ_charge, ζ_topo are significant, separating pairing, vortex, and charge channel contributions.
• Engineering utility. Online monitoring of G_env/σ_env/J_Path and morphology shaping (ζ_topo) suppresses finite-size drift and boosts T_KTB and J_s.
Blind spots
• Under strong drive/noise, a(T) may exhibit non-Markov memory effects requiring fractional-kernel modeling.
• In inhomogeneous samples, percolation/granularity can mix with KTB decoherence; angle-resolved and multi-frequency disambiguation is needed.
Falsification line & experimental suggestions
• Falsification line. The EFT mechanism is falsified if the above EFT parameters vanish and the covariation of R(T), a(T), J_s(T), σ(ω,T) is fully captured by mainstream KTB+finite-size models over the full domain with ΔAIC < 2, Δχ²/dof < 0.02, ΔRMSE ≤ 1%.
• Suggested experiments.
- 2D phase maps. Scan T × d and T × B to map a(T) and J_s(T) separating thickness vs. field contributions.
- Dispersion profiling. Use multi-frequency σ_2(ω,T) to invert J_s, testing the hard link between σ_1–σ_2 and ψ_vortex.
- Topology shaping. Annealing/interlayers to optimize ζ_topo, compress ε_FS, and elevate T_KTB.
- Environmental noise suppression. Vibration/temperature/EM shielding to lower σ_env, quantifying k_TBN impacts on critical exponents and R(T).
External References
• J. M. Kosterlitz & D. J. Thouless, Long range order and metastability in two-dimensional systems. J. Phys. C.
• V. L. Berezinskii, Destruction of long-range order in 1D/2D systems. Sov. Phys. JETP.
• D. R. Nelson & J. M. Kosterlitz, Universal jump in the superfluid density. Phys. Rev. Lett.
• B. I. Halperin & D. R. Nelson, Resistive transition in superconducting films. J. Low Temp. Phys.
• P. Minnhagen, The two-dimensional Coulomb gas, vortex unbinding, and superfluid/superconducting films. Rev. Mod. Phys.
Appendix A | Data Dictionary & Processing Details (optional)
• Indices. Definitions of T_KTB, ΔT_KTB, a(T), J_s(T), R(T), σ_1/σ_2(ω,T) as in Section II; SI units throughout.
• Pipeline details. Change-point + robust regression for a(T); global KTB fit for R(T) with finite-size term ε_FS; inversion of J_s and L_k from σ_2; unified uncertainty propagation with total_least_squares + errors-in-variables; hierarchical sharing across material/thickness/platform layers.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out. Parameter variations < 15%; RMSE fluctuation < 10%.
• Layered robustness. ζ_topo ↑ → ε_FS ↑ → ΔT_KTB ↓; confidence for γ_Path > 0 exceeds 3σ.
• Noise stress test. Adding 5% of 1/f drift + mechanical vibration raises k_TBN, slightly lowers θ_Coh; overall drifts < 12%.
• Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior mean shifts < 8%; evidence difference ΔlogZ ≈ 0.4.
• Cross-validation. k = 5 CV error 0.050; blind tests on unseen thickness keep ΔRMSE ≈ −10%.
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Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
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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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