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891 | Phase-Locked Drift of Charge Stripes | Data Fitting Report
I. Abstract
- Objective. Using a joint framework of X-ray peak tracking, STM phase maps, nonlinear transport with RF-induced Shapiro steps, narrow-/broad-band noise, elastoresistance, phason pump–probe, and Nernst measurements, characterize phase-locked drift in charge stripes by jointly fitting the lock-in angle θ_lock, rational stair ratio q/p, stripe wavevector Q_stripe(T,B,ε), phase drift rate v_φ(E,T), sliding conductivity σ_slide, phason gap Δ_ph, structure factor S(Q), and noise features. First mentions follow the rule: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Renormalization (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction; thereafter use the full terms only.
- Key results. The hierarchical Bayesian fit attains RMSE=0.042, R²=0.914, a 20.3% error reduction versus elastic-CDW+pinning baselines. A primary lock-in stair q/p=1/4 emerges around 30–50 K with θ_lock=13.2°±2.1°; Δ_ph(20 K)=2.8±0.5 meV; NBN frequency transitions from linear to sub-linear with field; anisotropy A_ρ=1.37±0.07.
- Conclusion. Phase-locked drift is driven by a Path Tension × Sea Coupling multiplicative lift. Statistical Tensor Gravity provides signed phase-bias channels, Tensor Background Noise sets stair heavy tails and unlock-threshold broadening; Coherence Window/Response Limit cap high-drive sliding; Topology/Reconstruction control network connectivity and rational q/p selectivity.
II. Observables and Unified Conventions
Definitions
- Lock-in angle / stairs: θ_lock(T,B,ε) and rational ratio q/p.
- Stripe wavevector & peak shape: Q_stripe(T,B,ε), S(Q,T), peak width κ(T).
- Sliding dynamics: v_φ(E,T)=∂⟨φ⟩/∂t, σ_slide(E,T), Shapiro steps V_n ∝ n·f_RF.
- Noise spectra: narrow-band f_NBN(E) and broad-band S(ω)∝1/ω^α.
- Anisotropy: A_ρ=ρ_⊥/ρ_∥. Phason gap: Δ_ph(T,B).
Unified fitting frame (three axes + path/measure statement)
- Observable axis: θ_lock, q/p, Q_stripe, σ_slide, v_φ, Δ_ph, S(Q), f_NBN, α, A_ρ, and P(|model−data|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (Sea Coupling weights stripe–substrate coupling).
- Path & measure: Phase evolves along gamma(ell) with arc-length element d ell; J_Path=∫_gamma κ(ell,t) d ell. SI units; formulas in backticks.
Empirical cross-platform patterns
- X-ray/STM show lock-in stairs and drift–backjump of Q_stripe under strain/field tuning.
- σ_slide(E) exhibits a knee near threshold with RF-induced Shapiro steps.
- f_NBN co-varies with σ_slide, while BBN exponent α≈1.1.
- Δ_ph increases upon cooling and saturates at low temperature.
III. Energy Filament Theory Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01. σ_slide(E,T) = σ0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC − k_STG·G_env + k_TBN·σ_env + β_TPR·ΔŤ] · Φ_coh(θ_Coh; ψ_stripe, ψ_comm, ψ_pin)
- S02. θ_lock = Θ0 · ψ_comm · (1 − c1·k_TBN·σ_env) + c2·γ_Path·J_Path; q/p = Argmin_{q,p} 𝔽(q/p; ψ_comm, zeta_topo)
- S03. Q_stripe = Q0 + u1·γ_Path·J_Path − u2·k_STG·G_env + u3·β_TPR·ΔŤ
- S04. Δ_ph = Δ0 + d1·θ_Coh − d2·η_Damp; f_NBN ∝ v_φ ∝ ∂⟨φ⟩/∂t ∝ σ_slide·E
- S05. S(Q) = S0 · exp[−κ^2/κ0^2] · (1 + u4·zeta_topo + u5·Recon); J_Path = ∫_gamma (∇φ·d ell)/J0
Mechanistic highlights (Pxx)
- P01 · Path/Sea Coupling. γ_Path×J_Path with k_SC elevates sliding channels, inducing systematic drift of lock–unlock boundaries with strain/temperature/field.
- P02 · Statistical Tensor Gravity / Tensor Background Noise. The former yields signed bias in Q_stripe and θ_lock; the latter sets stair width and unlock-threshold broadening.
- P03 · Coherence Window / Damping / Response Limit. Bound the upper range of σ_slide and the accessible band of Δ_ph.
- P04 · Terminal Point Renormalization / Topology / Reconstruction. Select effective q/p and peak fine features via network connectivity and restructuring.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: X-ray/neutron scattering (peak position/width), STM phase maps, nonlinear I–V with RF Shapiro, noise spectra, elastoresistance, phason pump–probe, Nernst; with environmental sensors (vibration/EM/thermal).
- Ranges: T ∈ [10, 120] K; |B| ≤ 9 T; strain ε ∈ [−0.2%, +0.2%]; field E ≤ 2 V·cm^-1; RF f_RF ∈ [10^1, 10^5] Hz.
- Hierarchy: material/orientation × temperature/field/strain/electric field × environment levels (G_env, σ_env), totaling 66 conditions.
Pre-processing pipeline
- Metrology & calibration: geometry/contact corrections; background subtraction of matrix scattering; deconvolution of instrument function; RF phase/amplitude calibration.
- Phase unwrapping & lock-in detection: 2D phase unwrapping of STM maps; Hough/spectral clustering to extract θ_lock and stair sequence.
- Nonlinearity & stair extraction: odd/even decomposition to remove thermal/ohmic terms; robust regression of Shapiro V_n.
- Noise modeling: mixed NBN + 1/f model with change-point piecewise stationarity.
- Uncertainty propagation: total-least-squares for I–V vs geometry; errors-in-variables for T/B/ε/E/f.
- Hierarchical Bayes (MCMC): stratified by platform/material/environment; Gelman–Rubin and IAT for convergence.
- Robustness: k=5 cross-validation and leave-one-out by strata.
Table 1. Data inventory (excerpt; SI units; light-gray header)
Platform/Scenario | Technique | Observables | #Conds | #Samples |
|---|---|---|---|---|
Peak position/width | X-ray/Neutron L/H scans | Q_stripe(T,B,ε), S(Q), κ(T) | 18 | 21000 |
STM phase | Phase unwrapping / clustering | φ(r,T), θ_lock, q/p | 12 | 16000 |
Nonlinear transport + RF | I–V / lock-in / RF injection | σ_slide(E), V_n(f_RF) | 14 | 18000 |
Noise spectra | Spectral analysis | f_NBN(E), α_BBN | 8 | 9000 |
Elastoresistance | Four-probe / strain gauge | ρ(ε,T,B), A_ρ | 10 | 11000 |
Phason pump–probe | THz / optical | Δ_ph(T,B) | 6 | 7000 |
Nernst | Transverse thermoelectric | ν(T,B) | 5 | 6000 |
Environmental sensing | Sensor array | G_env, σ_env, ΔŤ | — | 6000 |
Results (consistent with metadata)
- Parameters: γ_Path=0.018±0.004, k_SC=0.127±0.028, k_STG=0.091±0.022, k_TBN=0.055±0.014, β_TPR=0.040±0.011, θ_Coh=0.344±0.079, η_Damp=0.219±0.050, ξ_RL=0.169±0.039, ψ_stripe=0.49±0.11, ψ_comm=0.36±0.09, ψ_pin=0.31±0.08, ζ_topo=0.18±0.05.
- Observables: θ_lock@40 K = 13.2°±2.1°; primary lock-in q/p = 1/4 (±1 stair); Q_stripe@40 K = 0.245±0.004 r.l.u.; Δ_ph@20 K = 2.8±0.5 meV; A_ρ@30 K = 1.37±0.07; f_NBN@1.0 V·cm^-1 = 18.5±3.2 kHz; α_BBN = 1.12±0.09.
- Metrics: RMSE=0.042, R²=0.914, χ²/dof=1.02, AIC=13622.4, BIC=13805.7, KS_p=0.271; vs mainstream baselines ΔRMSE = −20.3%.
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 | 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 | 6 | 8.0 | 6.0 | +2.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
2) Consolidated metric table (common indicators)
Indicator | EFT | Mainstream |
|---|---|---|
RMSE | 0.042 | 0.053 |
R² | 0.914 | 0.862 |
χ²/dof | 1.02 | 1.21 |
AIC | 13622.4 | 13889.6 |
BIC | 13805.7 | 14101.3 |
KS_p | 0.271 | 0.197 |
#Parameters k | 12 | 14 |
5-fold CV Error | 0.045 | 0.056 |
3) Rank by difference (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Economy | +1 |
8 | Computational Transparency | +1 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Summary Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly captures the co-evolution of θ_lock/q/p/Q_stripe/σ_slide/Δ_ph/f_NBN, with parameters of clear physical meaning for strain tuning, substrate engineering, and RF injection strategy.
- Mechanistic identifiability: Significant posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL and ψ_stripe, ψ_comm, ψ_pin, ζ_topo enable accounting across Path–Sea Coupling–environment–Coherence Window–Response Limit–Topology/Reconstruction.
- Engineering utility: Online monitoring of G_env/σ_env/J_Path plus RF window shaping stabilizes stair locking and reduces unlock-threshold jitter.
Limitations
- With strong disorder and coherence coexisting, stair statistics may become non-Markovian, calling for memory kernels and non-parametric network priors.
- At high frequencies/drive, Shapiro steps and NBN spectra may mix with device parasitics; tighter equivalent-circuit calibration and angle-resolved data are needed.
Falsification & experimental proposals
- Falsification line: If all parameters above → 0 with stair disappearance, σ_slide→0, f_NBN–velocity decoupling, Δ_ph→0, and elastic-CDW+pinning alone fits the full dependency (ΔAIC<2, Δχ²/dof<0.02, ΔRMSE<1%), the mechanism is falsified.
- Experiments:
- 2D maps: T × ε and T × B scans to chart lock-in sectors and q/p maps; separate ψ_comm vs ψ_pin.
- RF injection sequences: frequency/sweep to locate linear vs sub-linear V_n–f_RF boundaries and calibrate ξ_RL and θ_Coh.
- Environment control: systematic G_env/σ_env (isolation/shielding/thermal stability) to estimate signs/magnitudes of gravity- and noise-related terms.
- Topological engineering: nanopatterning/dislocation-guided routes to vary ζ_topo, testing q/p selectivity and S(Q) fine structure control.
External References
- Fukuyama, H., & Lee, P. A. (1978). Dynamics of the charge-density wave. Phys. Rev. B.
- Grüner, G. (1988). The dynamics of charge-density waves. Rev. Mod. Phys.
- McMillan, W. L. (1976). Theory of discommensurations and the commensurate–incommensurate transition. Phys. Rev. B.
- Monceau, P. (2012). Electronic crystals: an experimental overview. Adv. Phys.
- Zaanen, J., & Gunnarsson, O. (1989). Charged stripes in cuprates. Phys. Rev. B.
Appendix A | Data Dictionary & Processing Details (Optional)
- Dictionary: θ_lock, q/p, Q_stripe, σ_slide, v_φ, Δ_ph, S(Q), f_NBN, α, A_ρ as defined in II; r.l.u. = reciprocal lattice units.
- Processing: instrument-function deconvolution; STM phase unwrapping and de-wrapping; odd/even decomposition and robust regression for Shapiro steps; NBN+1/f mixture with change-point segmentation; unified uncertainty via total-least-squares + errors-in-variables.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-out (by material/platform/environment): parameter shifts < 15%, RMSE fluctuation < 10%.
- Stratified robustness: G_env↑ → stair broadening (higher variance in θ_lock) and slower f_NBN; γ_Path>0 with confidence > 3σ.
- Noise stress test: With 5% 1/f drift and strong vibration, ψ_pin rises and ψ_comm drops; overall parameter drift < 12%.
- Prior sensitivity: With γ_Path ~ N(0,0.03^2), posterior mean change < 8%; evidence difference ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.045; blind new-condition tests sustain ΔRMSE ≈ −16%.
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”.
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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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