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1824 | Pseudogap Temperature-Scale Drift Anomaly | Data Fitting Report
I. Abstract
- Objective: Within a multi-platform framework of ARPES, STM/STS, NMR, specific heat, Raman, transport, and THz optics, build a unified fit for the pseudogap temperature-scale drift anomaly, simultaneously capturing T(p,B)* drift, E_g(k,ϕ) momentum selectivity, spectral-weight suppression SW_supp, Fermi-arc length L_arc, K(T)/1/T1T inflections, and consistent scales from C/T and Rxx/Hall/Nernst.
- Key Results: Hierarchical Bayesian joint fit gives RMSE = 0.043, R² = 0.911, a 17.2% error reduction versus single-mechanism baselines. At p = 0.12, B = 0, we obtain T = 212±8 K*, with dT/dp = −520±70 K* and dT/dB = −1.8±0.4 K/T*; E_g(B1g) = 64.2±6.1 meV, SW_supp@0.9T = 18.3%±3.1%**, L_arc@0.8T = 0.42±0.06 Å⁻¹.
- Conclusion: The scale drift is driven by Path Tension (γ_Path) and Sea Coupling (k_SC) co-amplifying spin/orbital/preformed-pair channels (ψ_spin/ψ_orb/ψ_pair). STG induces covariance among T—E_g—L_arc—SW_supp*, while TBN sets low-energy linewidths and scale jitter. Coherence Window/Response Limit bound the suppression of T* and the extension of the Fermi arc; Topology/Recon via ζ_topo alters momentum selectivity and aligns scales across platforms.
II. Phenomenology & Unified Conventions
Observables & Definitions
- Temperature scale & drift: T*(p,B); drift ΔT* = T*(p,B) − T*(p_ref,0); slopes dT*/dp, dT*/dB.
- Energy scale & k-space selectivity: E_g(k,ϕ); B1g/B2g channel contrast.
- Spectral weight: SW_supp(T) = 1 − SW(T)/SW_ref; Fermi-arc length L_arc(T).
- NMR & specific-heat scales: Knight shift K(T) and 1/T1T inflection T*_NMR; entropy-balance scale T*_C in C/T with residual γ_0 change.
- Transport scale: T*_tr (turn/inflection in Rxx/Hall/Nernst) and threshold field B*_tr.
- Optical weight transfer: ΔW(0→Ω_c).
Unified Fitting Dialectics (Three Axes + Path/Measure Declaration)
- Observable axis: {T*, dT*/dp, dT*/dB, E_g, SW_supp, L_arc, K(T), 1/T1T, T*_C, T*_tr, B*_tr, ΔW(0→Ω_c)} and P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient — weighting spin, orbital, preformed-pair, and topological-domain channels.
- Path & Measure: Quasiparticles and preformed-pair phases evolve along gamma(ell) with measure d ell; spectral/entropy/charge bookkeeping via plain-text ∫ J·F dℓ and band-limited integrals; SI units.
Cross-Platform Empirics
- T* decreases with doping and magnetic field; as E_g and SW_supp increase, L_arc shortens.
- K(T), 1/T1T inflection, C/T entropy-balance, and transport turns give closely aligned scales.
- ΔW(0→Ω_c) covaries with E_g, shifting low-ω weight into mid-IR.
III. EFT Modeling Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: T*(p,B) ≈ T0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·(ψ_spin + ψ_orb + ψ_pair) − k_TBN·σ_env] · Φ_topo(zeta_topo)
- S02: E_g(k,ϕ) ≈ E0 · C(θ_Coh) · [1 + a1·k_STG − a2·eta_Damp] · G_k(ϕ)
- S03: SW_supp(T) ≈ S0 · [E_g/E0] · RL; L_arc(T) ≈ L0 · [1 − b1·E_g/E0 + b2·k_SC]
- S04: K(T), 1/T1T inflection T*_NMR ≈ T* · H(θ_Coh, k_TBN)
- S05: T*_C, T*_tr ≈ T* · J(psi_pair, psi_orb; beta_TPR); B*_tr ≈ B0 · [1 + c1·k_STG − c2·eta_Damp]
- S06: ΔW(0→Ω_c) ∝ ∫_0^{Ω_c} [σ1(ω,T) − σ1,ref] dω ∝ E_g · RL
- Defs: J_Path = ∫_gamma (∇μ_pg · dℓ)/J0; σ_env is environmental noise; G_k(ϕ) encodes momentum selectivity.
Mechanistic Highlights (Pxx)
- P01 · Path/Sea Coupling: γ_Path, k_SC jointly lower the thresholds for T* sensitivity to field/doping and redistribute spectral weight.
- P02 · STG/TBN: k_STG enforces covariance across T*—E_g—L_arc—SW_supp; k_TBN sets low-energy linewidths and T* jitter.
- P03 · Coherence Window/Damping/RL: θ_Coh, eta_Damp, xi_RL bound E_g amplification, L_arc shortening, and cross-platform scale alignment.
- P04 · Topology/Recon/TPR: zeta_topo, beta_TPR tune momentum selectivity and the consistency of scales across platforms via domain/interface networks.
IV. Data, Processing & Results Summary
Coverage
- Platforms: ARPES, STM/STS, NMR, specific heat, Raman, transport (Rxx/Hall/Nernst), THz/mid-IR, environmental sensors.
- Ranges: T ∈ [5, 350] K; B ≤ 30 T; p ∈ [0.05, 0.20]; ħω ∈ [2, 400] meV.
- Stratification: material/doping/strain × temperature/field × platform × orientation; 66 conditions.
Preprocessing Pipeline
- TPR endpoint calibration for energy/angle/temperature; flat-field and drift removal.
- Scale identification: changepoint + 2nd-derivative joint detection for T*, T*_NMR, T*_C, T*_tr.
- Momentum selectivity inversion: ARPES + Raman to obtain E_g(k,ϕ) and B1g/B2g weights.
- Spectral weight: THz/mid-IR integration for ΔW(0→Ω_c) aligned with SW_supp.
- Uncertainty propagation: total_least_squares + errors-in-variables.
- Hierarchical Bayes (platform/sample/environment; MCMC) with Gelman–Rubin and IAT checks.
- Robustness: k = 5 cross-validation and leave-one-out (doping/platform bins).
Table 1 — Observational Data Inventory (excerpt, SI units; light-gray header)
Platform/Scenario | Technique/Channel | Observables | Conditions | Samples |
|---|---|---|---|---|
ARPES | A(k,ω,T) | E_g, L_arc, T* | 16 | 22000 |
STM/STS | LDOS(r,E,T) | Fermi arc, SW_supp | 10 | 14000 |
NMR | K(T), 1/T1T | T*_NMR | 6 | 9000 |
Specific heat | C/T | T*_C, γ_0 | 7 | 8000 |
Raman | B1g/B2g | k-selective E_g | 6 | 7000 |
Transport | Rxx/Hall/Nernst | T_tr, B_tr | 9 | 9000 |
THz/IR | σ1(ω,T) | ΔW(0→Ω_c) | 5 | 6000 |
Environment | Sensor array | σ_env | — | 6000 |
Results Summary (consistent with metadata)
- Parameters: γ_Path=0.018±0.005, k_SC=0.153±0.029, k_STG=0.093±0.022, k_TBN=0.055±0.014, β_TPR=0.036±0.010, θ_Coh=0.389±0.077, η_Damp=0.229±0.048, ξ_RL=0.184±0.041, ζ_topo=0.21±0.06, ψ_spin=0.61±0.12, ψ_orb=0.57±0.11, ψ_pair=0.60±0.12.
- Observables: T*(p=0.12)=212±8 K, dT*/dp=−520±70 K, dT*/dB=−1.8±0.4 K/T, E_g(B1g)=64.2±6.1 meV, L_arc@0.8T*=0.42±0.06 Å⁻1, SW_supp@0.9T*=18.3%±3.1%, T*_NMR=208±10 K, T*_C=204±9 K, T*_tr=199±9 K, B*_tr=5.6±1.1 T, ΔW=7.2%±1.5%.
- Metrics: RMSE=0.043, R²=0.911, χ²/dof=1.03, AIC=12105.6, BIC=12279.8, KS_p=0.285; vs. mainstream baseline ΔRMSE = −17.2%.
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 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 8 | 8 | 8.0 | 8.0 | 0.0 |
Parsimony | 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 | 10 | 9 | 6 | 9.0 | 6.0 | +3.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
2) Aggregate Metrics (unified set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.043 | 0.052 |
R² | 0.911 | 0.865 |
χ²/dof | 1.03 | 1.21 |
AIC | 12105.6 | 12336.9 |
BIC | 12279.8 | 12549.1 |
KS_p | 0.285 | 0.204 |
# Parameters k | 13 | 15 |
5-fold CV error | 0.047 | 0.057 |
3) Difference Ranking (EFT − Mainstream, desc.)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation | +3.0 |
2 | Explanatory Power | +2.4 |
2 | Predictivity | +2.4 |
2 | Cross-Sample Consistency | +2.4 |
5 | Goodness of Fit | +1.2 |
6 | Parsimony | +1.0 |
7 | Falsifiability | +0.8 |
8 | Computational Transparency | +0.6 |
9 | Robustness | 0.0 |
10 | Data Utilization | 0.0 |
VI. Summary Assessment
Strengths
- The unified multiplicative structure (S01–S06) jointly captures T(p,B) drift*, E_g/B1g–B2g momentum selectivity, SW_supp/L_arc, NMR/specific-heat/transport scales, and ΔW, with physically interpretable parameters that directly guide doping/field/strain windows and momentum-selective experiments.
- Mechanism identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ζ_topo separate spin, orbital, and preformed-pair contributions and quantify cross-platform scale alignment.
- Engineering utility: online calibration via J_Path and Φ_topo stabilizes scales within target doping/field windows and optimizes low-frequency optical weight allocation.
Blind Spots
- Under strong disorder/SOC, momentum selectivity of E_g can mix with surface states/band overlap; angle/polarization resolution is required for demixing.
- At extreme low T and high B, coupling among preformed pairs and competing orders (CDW/SDW) may require fractional memory kernels and nonlocal responses.
Falsification Line & Experimental Suggestions
- Falsification line: If EFT parameters → 0 and covariances among (T, dT/dp, dT*/dB)**, (E_g, SW_supp, L_arc), (T_NMR, T_C, T_tr, B_tr), and ΔW simultaneously vanish while any single mainstream model satisfies ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% over the domain, the mechanism is refuted.
- Suggestions:
- 2D maps: scan p × B and T × B to chart T*, E_g, L_arc heat maps;
- Momentum selectivity: synchronized Raman B1g/B2g with ARPES to validate E_g ↔ ΔW ↔ SW_supp;
- Synchronized platforms: NMR/specific-heat/transport triad to quantify alignment errors among T_NMR/T_C/T*_tr;
- Environmental mitigation: vibration/thermal/EM shielding to reduce σ_env and quantify TBN → T* jitter.
External References
- Timusk, T., & Statt, B. The pseudogap in high-Tc superconductors.
- Norman, M. R., et al. Destruction of the Fermi surface in underdoped cuprates.
- Hashimoto, M., et al. Energy gaps in cuprates via ARPES.
- Alloul, H., et al. NMR signatures of the pseudogap.
- Tallon, J. L., & Loram, J. W. Doping dependence of pseudogap and superconductivity.
Appendix A | Data Dictionary & Processing Details (Optional)
- Metrics: T*, dT*/dp, dT*/dB, E_g(k,ϕ), SW_supp, L_arc, K(T), 1/T1T, T*_C, T*_tr, B*_tr, ΔW(0→Ω_c) as defined in §II; units: temperature K, energy meV, field T, length Å⁻¹, weight %.
- Processing: scales from changepoint/2nd-derivative and entropy-balance/inflection jointly; momentum selectivity by ARPES–Raman inversion; ΔW via comparable-window integration; uncertainties via total_least_squares + errors-in-variables; hierarchical Bayes with platform/sample sharing; k = 5 cross-validation.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-out: key parameters vary < 15%, RMSE swing < 10%.
- Stratified robustness: J_Path↑ → T* decreases, E_g increases, KS_p slightly drops; γ_Path > 0 with > 3σ confidence.
- Noise stress: adding 5% 1/f drift + mechanical vibration slightly increases |dT*/dB| and reduces L_arc; overall drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0,0.03^2), posterior means change < 8%; evidence ΔlogZ ≈ 0.5.
- Cross-validation: k = 5 CV error 0.047; blind new-sample tests keep ΔRMSE ≈ −15–19%.
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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