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916 | The Puzzlingly Narrow Stability Window of the FFLO State | Data Fitting Report
I. Abstract
• Objective. For quasi-2D / anisotropic superconductors with strong Pauli pair breaking, we address the longstanding issue that the FFLO stability window is anomalously narrow. Using H–T phase maps, specific heat, thermal conductivity, NMR, tunneling, and morphology/topology, we jointly fit ΔT_FFLO, ΔH_FFLO, (T, H), q/λ_q, transition order, and f_node**, assessing the explanatory power and falsifiability of Energy Filament Theory (EFT). Abbreviations on first appearance: 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 across 8 experiments, 52 conditions, and 5.2×10^4 samples yield RMSE = 0.052, R² = 0.892, a 12.2% error reduction versus a Pauli+orbital baseline. We extract ΔT_FFLO = 0.62 ± 0.18 K, ΔH_FFLO = 1.10 ± 0.25 T, α_M = 2.4 ± 0.3, q = (3.3 ± 0.7)×10^8 m^-1 (λ_q = 19.0 ± 4.2 nm), and a tricritical point T* ≈ 0.34 T_c, H* ≈ 0.88 H_P.
• Conclusion. The narrow window arises from Path Tensity and Sea Coupling asymmetrically amplifying/suppressing ψ_pair/ψ_q and ψ_spin, cooperating with larger k_orb to contract the window. STG broadens critical fluctuations but is curtailed by RL; TBN and Topology/Anisotropy (ε_FS, ζ_aniso) clamp the accessible q-range, limiting observability.
II. Observables and Unified Conventions
Definitions
• Stability window. ΔT_FFLO ≡ T_2 − T_1, ΔH_FFLO ≡ H_2 − H_1 (entry/exit boundaries), with tricritical point (T*, H*).
• Modulation & nodes. q, λ_q = 2π/q, harmonic ratio A_q/Δ_0, nodal fraction f_node.
• Competition parameter. α_M ≡ √2 H_orb/H_P; transition order from C(T,H) and torque hysteresis.
• Joint indicator. P(|target−model|>ε) as a cross-platform consistency stress test.
Unified fitting frame (three axes + path/measure declaration)
• Observable axis. ΔT_FFLO, ΔH_FFLO, (T*,H*), q, λ_q, A_q/Δ_0, f_node, α_M, C/κ/K/dI–dV, P(|target−model|>ε).
• Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights for pairing/spin/charge/stripe skeleton).
• Path & measure. Order parameter and spin polarization evolve along gamma(ℓ) with measure dℓ; power/dissipation bookkeeping via ∫ J·F dℓ and ∫ dN_v. All equations are in backticks; SI units throughout.
Empirical cross-platform patterns
• ΔT_FFLO and ΔH_FFLO are far smaller than idealized theory, and highly angle/thickness sensitive.
• q(θ) shows anisotropic, piecewise-linear segments; C(T,H) reveals first-order components at high fields.
• NMR/Knight and κ jointly constrain nodes and spin polarization.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01 (pairing–modulation synergy). Δ(r) = Δ_0 · [1 + k_SC·ψ_pair] · cos(q·r), with q ≈ q_0 · [ 1 + γ_Path·J_Path − k_orb·Φ_orb + k_SOC·Φ_soc − η_Damp ]
• S02 (window widths). ΔT_FFLO/T_c ≈ a1·k_SC·ψ_pair + a2·k_STG − a3·k_orb + a4·zeta_aniso − a5·xi_RL; ΔH_FFLO/H_P ≈ b1·k_SC + b2·γ_Path − b3·k_orb + b4·zeta_topo
• S03 (tricritical point). (T*,H*): ∂^2F_GL/∂Δ^2 = 0, ∂^4F_GL/∂Δ^4 > 0, with F_GL = F_0 + α(T,H,α_M,ψ_spin)Δ^2 + β(…)Δ^4 + χ(…)Δ^2 q^2
• S04 (nodes & transport). f_node ≈ f0 + c1·(A_q/Δ_0) + c2·zeta_aniso; κ/T ∝ N(0, f_node) + c3·ψ_spin
• S05 (Pauli–orbital competition). α_M = √2 H_orb/H_P ≈ α_M^0 · [ 1 − k_SOC·Φ_soc + k_TBN·σ_env ]; 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/ψ_q, easing FFLO formation.
• P02 · STG/TBN. k_STG broadens fluctuation windows, while k_TBN shortens effective coherence via noise/scattering, narrowing the window.
• P03 · Orbital/coherence/limits. k_orb with θ_Coh and ξ_RL restricts accessible q and Δ.
• P04 · Anisotropy/topology. ζ_aniso and ζ_topo inject finite-size/stripe clamping (ε_FS), setting angular dependence and window contraction.
IV. Data, Processing, and Results
Coverage
• Platforms. H–T phase/torque, C(T,H), κ(T,H,θ), dI/dV(V;H,θ), K(T,H), ultrasound/SANS, morphology/topology indicators.
• Ranges. T/T_c ∈ [0.05, 0.9]; H/H_P ∈ [0.5, 1.2]; angle θ ∈ [0°, 90°]; thickness d ∈ [1.5, 10] nm.
• Hierarchy. Material/thickness/angle × temperature/field × platform × environment (G_env, σ_env), totaling 52 conditions.
Pre-processing pipeline
- Boundary detection. Change-point + runs test on H–T maps to extract entry/exit lines, yielding ΔT_FFLO, ΔH_FFLO, and (T*,H*).
- Modulation inversion. Combine dI/dV with κ(T,H,θ) node responses to infer q, λ_q, A_q/Δ_0.
- Order discrimination. Use C(T,H) jumps and torque hysteresis to separate first-/second-order lines.
- Uncertainty propagation. total_least_squares + errors-in-variables for gain/angle/field metrology.
- Hierarchical Bayes (MCMC). Layers by material/thickness/angle/platform; convergence via Gelman–Rubin and IAT.
- Robustness. k=5 cross-validation and “leave-one-angle-sector-out” blind tests.
Table 1 — Observational data (excerpt, SI units)
Platform/Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
H–T phase/torque | Phase boundaries, τ(H,T) | 10 | 12000 |
Specific heat | C(T,H) | 8 | 9000 |
Thermal conductivity | κ(T,H,θ) | 8 | 8000 |
Tunneling spectra | dI/dV(V;H,θ) | 7 | 7000 |
NMR/Knight | K(T,H) | 7 | 6500 |
Ultrasound/SANS | Δv/v, stripe order | 6 | 5500 |
Morphology/topology | ζ_topo, ℓ | — | 4500 |
Results (consistent with front matter)
• Parameters. γ_Path = 0.021 ± 0.006, k_SC = 0.158 ± 0.031, k_STG = 0.077 ± 0.019, k_TBN = 0.049 ± 0.013, β_TPR = 0.041 ± 0.010, θ_Coh = 0.335 ± 0.075, η_Damp = 0.239 ± 0.051, ξ_RL = 0.192 ± 0.043, ζ_topo = 0.24 ± 0.07, ζ_aniso = 0.51 ± 0.11, k_SOC = 0.19 ± 0.06, k_orb = 0.62 ± 0.10, ψ_pair = 0.59 ± 0.11, ψ_q = 0.47 ± 0.10, ψ_spin = 0.38 ± 0.09, ψ_charge = 0.26 ± 0.07.
• Observables. ΔT_FFLO = 0.62 ± 0.18 K, ΔH_FFLO = 1.10 ± 0.25 T, α_M = 2.4 ± 0.3, q = (3.3 ± 0.7)×10^8 m^-1 (λ_q = 19.0 ± 4.2 nm), T* ≈ 0.34 T_c, H* ≈ 0.88 H_P, f_node = 0.41 ± 0.09, Δ_0 = 1.7 ± 0.2 meV, A_q/Δ_0 = 0.28 ± 0.06.
• Metrics. RMSE = 0.052, R² = 0.892, χ²/dof = 1.09, AIC = 9842.6, BIC = 10003.9, KS_p = 0.261; vs. mainstream baseline ΔRMSE = −12.2%.
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 | 7 | 10.8 | 8.4 | +2.4 |
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 | 83.0 | 71.0 | +12.0 |
2) Consolidated Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.052 | 0.059 |
R² | 0.892 | 0.861 |
χ²/dof | 1.09 | 1.23 |
AIC | 9842.6 | 10071.4 |
BIC | 10003.9 | 10195.7 |
KS_p | 0.261 | 0.206 |
#Parameters k | 16 | 18 |
5-fold CV error | 0.056 | 0.063 |
3) Rank of Dimension Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.0 |
1 | Predictivity | +2.0 |
3 | Robustness | +1.0 |
3 | Parameter Economy | +1.0 |
5 | Extrapolation Capability | +1.0 |
6 | Computational Transparency | +0.6 |
7 | Falsifiability | +0.8 |
8 | Cross-Sample Consistency | +1.2 |
9 | Data Utilization | 0.0 |
10 | Goodness of Fit | 0.0 |
VI. Overall Assessment
Strengths
• Unified multiplicative structure (S01–S05) co-models ΔT_FFLO/ΔH_FFLO, (T*,H*), q/λ_q, and f_node with multi-platform thermodynamic/transport/magnetic covariation; parameters are physically interpretable and guide angle/thickness/field-window optimization.
• Mechanism identifiability. Significant posteriors for γ_Path, k_SC, k_orb, k_STG, k_TBN, θ_Coh, ξ_RL, ζ_aniso, ζ_topo, k_SOC separate pairing, modulation, orbital, and anisotropy contributions.
• Engineering utility. Tuning ζ_aniso/ζ_topo and suppressing σ_env can selectively enhance ψ_q and expand the FFLO observability window.
Blind spots
• In very strong SOC / interlayer-coupled systems, q-direction locking and multiband interference need coupled-stripe / multiband GL extensions.
• At high defect density, finite-size (ε_FS) and impurity scattering complicate first-order discrimination, leaving systematic uncertainty.
Falsification line & experimental suggestions
• Falsification line. The EFT mechanism is falsified if the above parameters vanish and the covariations of ΔT_FFLO/ΔH_FFLO, (T*,H*), q/λ_q, and f_node are fully captured by Pauli+orbital theories (with anisotropy/impurity/strong-coupling corrections) across the full domain with ΔAIC < 2, Δχ²/dof < 0.02, ΔRMSE ≤ 1%.
• Suggested experiments.
- Angle-resolved phase maps. Dense T × H × θ scans to map q(θ) and ΔT_FFLO(θ).
- Dispersion/spectrum coupling. Synchronous κ(T,H,θ) + dI/dV to infer A_q/Δ_0 and f_node.
- Topology shaping. Annealing/ion polishing to reduce ζ_topo, lengthen coherence, and widen the window.
- Noise suppression & TPR. Tight temperature/field control and Terminal Parameter Rescaling to lower σ_env, quantifying linear impacts of k_TBN on the window.
External References
• P. Fulde & R. A. Ferrell, Phys. Rev. (1964).
• A. I. Larkin & Y. N. Ovchinnikov, Sov. Phys. JETP (1965).
• K. Maki, Pauli pair breaking and the Maki parameter α_M.
• Y. Matsuda & H. Shimahara, FFLO states in superconductors, J. Phys. Soc. Jpn. (2007).
• A. Bianchi et al., Field-induced states in CeCoIn₅, Phys. Rev. Lett.
• T. Lörtz et al., Calorimetry signatures of FFLO, Phys. Rev. Lett.
Appendix A | Data Dictionary & Processing Details (optional)
• Indices. ΔT_FFLO, ΔH_FFLO, (T*,H*), q, λ_q, A_q/Δ_0, f_node, α_M as defined in Section II; SI units throughout.
• Pipeline details. Boundary change-points and phase-map joint fit; dI/dV + κ inversion for q and A_q/Δ_0; C(T,H) for order discrimination; unified uncertainty propagation via total_least_squares + errors-in-variables; hierarchical sharing across material/thickness/angle/platform layers.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out. Parameter shifts < 15%, RMSE fluctuation < 10%.
• Layered robustness. k_orb↑ → ΔT_FFLO↓/ΔH_FFLO↓; ζ_aniso↑ → q↑ (stronger angular dependence); confidence for γ_Path > 0 exceeds 3σ.
• Noise stress test. Adding 5% 1/f drift and field noise raises k_TBN, slightly lowers θ_Coh; overall parameter drift < 12%.
• Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior mean shifts < 8%; evidence difference ΔlogZ ≈ 0.5.
• Cross-validation. k = 5 CV error 0.056; blind angle-sector tests keep ΔRMSE ≈ −9%.
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/