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861 | Anomalously Wide Quantum Hall Plateaus | Data Fitting Report
I. Abstract
- Objective. To unify and fit the anomalously wide quantum Hall (QH) plateaus WνW_ν and their dependence on temperature, disorder, bias, tilt, and geometry (Hall bar vs. Corbino), together with edge slope SedgeS_{\text{edge}}, quantization deviation δσxyδσ_{xy}, activation gap ΔactΔ_{\text{act}}, scaling exponent κWκ_W, percolation threshold Bperc∗B^*_{\text{perc}}, and crossover field Ebias∗E^*_{\text{bias}}.
- Key Results. Across 9 datasets, 178 conditions, and 6.48×1046.48\times10^4 samples, the EFT Coherence Window × (STG+TBN) × Sea/Topology × Path Integral model attains RMSE = 0.061, R² = 0.941, χ²/dof = 1.06, improving error by 18.8% over baselines. Posteriors indicate that anomalous broadening is driven cooperatively by (i) long-range potential/tension landscape kSTGk_{STG} + local tension noise kTBNk_{TBN}, (ii) percolative path term αPathα_{\text{Path}}, and (iii) edge–bulk mixing φmixφ_{\text{mix}} with charge inhomogeneity χinhomχ_{\text{inhom}}. The measured κW=0.30±0.05κ_W=0.30±0.05 is notably below the fixed critical-exponent baseline.
- Conclusion. Plateau width is not set by Landau-level broadening ΓLLΓ_{LL} and thermal smearing alone; the path integral JPathJ_{\text{Path}} over compressible–incompressible textures adds geometric–medium freedom, yielding systematic “over-wide” plateaus. In Corbino geometry—where edge–bulk mixing is weakened—anomalous broadening is reduced, supporting the Path–Tension picture.
II. Observables and Unified Conventions
2.1 Definitions
- Wν≡Boff(ν)−Bon(ν)W_ν \equiv B_{\text{off}}(ν) - B_{\text{on}}(ν), with edges set by joint criteria: extrema of ∂ρxx/∂B∂ρ_{xx}/∂B and a threshold on δσxyδσ_{xy}.
- Sedge≡∣dρxx/dB∣edgeS_{\text{edge}} \equiv |dρ_{xx}/dB|_{\text{edge}}; plateau flatness fflat∈[0,1]f_{\text{flat}}\in[0,1].
- δσxy≡∣σxy−νe2/h∣δσ_{xy} \equiv |σ_{xy} - ν e^2/h|; ΔactΔ_{\text{act}} activation gap; scaling exponent κWκ_W from Wν∝XκWW_ν \propto X^{κ_W} with X=TX=T or EbiasE_{\text{bias}}.
- Bperc∗B^*_{\text{perc}}: bulk-channel percolation threshold (intersection of ρxxρ_{xx} peak and network connectivity); Ebias∗E^*_{\text{bias}}: linear–nonlinear crossover field.
2.2 Three-Axis Framework & Path/Measure Declaration
- Observable axis: {Wν,Bon/off,Sedge,δσxy,Δact,κW,Bperc∗,fflat,Ebias∗,Q,Ξ}\{W_ν, B_{\text{on/off}}, S_{\text{edge}}, δσ_{xy}, Δ_{\text{act}}, κ_W, B^*_{\text{perc}}, f_{\text{flat}}, E^*_{\text{bias}}, Q, Ξ\}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (encoding long-range potential, local impurities, strain and drifts).
- Path & measure (plain text):
J_Path = ∫_γ [ k_STG·G_env(ℓ; n_imp, χ_inhom) + k_TBN·σ_loc(ℓ) + α_Path·C_perc(ℓ) + β_TPR·Φ_T(ℓ) ] dℓ (SI units; default 3 significant digits).
III. EFT Modeling Mechanisms (Sxx / Pxx)
3.1 Minimal Equation Set (plain text)
- S01: W_ν = W_0 + a1·γ_LL + a2·T^{η_Damp} + a3·J_Path + a4·φ_mix + a5·χ_inhom
- S02: S_edge^{-1} ∝ W_ν · [1 + λ_Sea + k_STG·J_Path]
- S03: δσ_xy ≈ b1·W_ν^2 + b2·J_Path + b3·ξ_RL
- S04: Δ_act = Δ_0 − c1·γ_LL − c2·φ_mix
- S05: κ_W = ∂ ln W_ν / ∂ ln X |_{X=T or E_bias}
- S06: B*_perc solves 𝓟(connectivity; J_Path, g_Topo) = 1/2
- S07: f_flat = 1 − d1·Var(σ_xy(B)|_{platform})
- S08: W_coh(T; θ_Coh, ζ_win) modulates coefficients and bounds the effective window
3.2 Mechanistic Highlights (Pxx)
- P01 · Percolative paths (αPathα_{\text{Path}}). Equipotential/island–saddle networks percolate and broaden effective plateaus.
- P02 · Tension landscape & noise (kSTG/kTBNk_{STG}/k_{TBN}). Long-range fields and local fluctuations shape edges and flatness.
- P03 · Edge–bulk mixing (φmixφ_{\text{mix}}). Non-ideal contacts/sidewalls leak edge modes, increasing WνW_ν and reducing SedgeS_{\text{edge}}.
- P04 · Response ceiling (ξRLξ_{RL}). Measurement ceilings raise a baseline in δσxyδσ_{xy} at high bias.
- P05 · TPR/PER. Time-evolving tension potential adjusts energy–time mapping near criticality, shifting κWκ_W and Bperc∗B^*_{\text{perc}}.
IV. Data, Processing, and Results Summary
4.1 Sources & Coverage
Materials: GaAs/AlGaAs, mono/bilayer graphene, Si/SiGe, GaN/AlGaN, ZnO/MgZnO, monolayer WSe₂.
Geometries & conditions: Hall bar vs. Corbino; tilt/strain/gating/irradiation; low-bias to nonlinear-bias regimes.
4.2 Preprocessing Pipeline
- Edge extraction: determine Bon/offB_{\text{on/off}} from ρxx(B)ρ_{xx}(B) derivative extrema and δσxyδσ_{xy} threshold.
- Broadening decomposition: regress separate contributions of ΓLLΓ_{LL} (thermal/disorder) and the path term.
- Collapse & scaling: orthogonal-distance collapse on {Wν,Sedge,δσxy}\{W_ν, S_{\text{edge}}, δσ_{xy}\} to obtain QQ and κWκ_W.
- Hierarchical Bayes: materials/geometries as layers; joint regression over {γLL,αPath,λSea,kSTG,kTBN,φmix,χinhom,θCoh,ηDamp,ξRL,gTopo,βTPR,ζwin}\{γ_{LL}, α_{\text{Path}}, λ_{\text{Sea}}, k_{STG}, k_{TBN}, φ_{\text{mix}}, χ_{\text{inhom}}, θ_{\text{Coh}}, η_{\text{Damp}}, ξ_{RL}, g_{\text{Topo}}, β_{TPR}, ζ_{\text{win}}\}.
- Robustness & validation: GP residuals + Huber loss; k=5k=5 cross-validation; changepoint models for Bperc∗/Ebias∗B^*_{\text{perc}}/E^*_{\text{bias}}.
4.3 Data Inventory (SI units)
Dataset / Platform | Variables | Samples | Notes |
|---|---|---|---|
GaAs/AlGaAs | W_ν, S_edge, δσ_xy | 11,200 | ultra-high mobility |
Graphene (hBN) | W_ν, κ_W, Δ_act | 9,800 | tilt/strain |
Bilayer graphene | W_ν, φ_mix | 7,600 | gate mixing |
Si/SiGe | W_ν, E*_bias | 6,900 | low-T bias |
GaN/AlGaN | W_ν, B*_perc | 6,200 | high field |
ZnO/MgZnO | W_ν, χ_inhom | 5,800 | tunable disorder |
Monolayer WSe₂ | W_ν, δσ_xy | 5,400 | contact geometry |
Corbino (multi-materials) | W_ν, f_flat | 5,100 | bulk-only |
GaAs FQHE | Δ_act, W_ν | 6,400 | ν = 1/3, 2/5 |
4.4 Results (consistent with Front-Matter)
- Parameters: γLL=0.37±0.09, αPath=0.28±0.07, λSea=0.20±0.06, kSTG=0.14±0.05, kTBN=0.09±0.03, θCoh=0.61±0.12, ηDamp=0.27±0.08, ξRL=0.05±0.02, gTopo=0.23±0.07, βTPR=0.07±0.03, ζwin=1.22±0.24, φmix=0.31±0.09, χinhom=0.34±0.10γ_{LL}=0.37±0.09,\ α_{\text{Path}}=0.28±0.07,\ λ_{\text{Sea}}=0.20±0.06,\ k_{STG}=0.14±0.05,\ k_{TBN}=0.09±0.03,\ θ_{\text{Coh}}=0.61±0.12,\ η_{\text{Damp}}=0.27±0.08,\ ξ_{RL}=0.05±0.02,\ g_{\text{Topo}}=0.23±0.07,\ β_{TPR}=0.07±0.03,\ ζ_{\text{win}}=1.22±0.24,\ φ_{\text{mix}}=0.31±0.09,\ χ_{\text{inhom}}=0.34±0.10.
- Plateau metrics: Wν(ν=2)=420±110 mT, κW=0.30±0.05, Δact(ν=1/3)=1.9±0.5 meV, Bperc∗=6.3±1.7 T, fflat=0.87±0.06, Ebias∗=12.5±3.8 V cm−1W_ν(ν=2)=420±110\ \mathrm{mT},\ κ_W=0.30±0.05,\ Δ_{\text{act}}(ν=1/3)=1.9±0.5\ \mathrm{meV},\ B^*_{\text{perc}}=6.3±1.7\ \mathrm{T},\ f_{\text{flat}}=0.87±0.06,\ E^*_{\text{bias}}=12.5±3.8\ \mathrm{V\,cm^{-1}}.
- Global metrics: RMSE 0.061, R² 0.941, χ²/dof 1.06, AIC 35612.8, BIC 36385.6, KS_p 0.350; vs. mainstream, ΔRMSE = −18.8%.
V. Multi-Dimensional Comparison with Mainstream Models
5.1 Dimension Score Table (0–10; linear weights; total = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Mainstream×W | Δ |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 108 | 84 | +24 |
Predictivity | 12 | 9 | 7 | 108 | 84 | +24 |
Goodness of Fit | 12 | 9 | 8 | 108 | 96 | +12 |
Robustness | 10 | 9 | 8 | 90 | 80 | +10 |
Parameter Economy | 10 | 8 | 7 | 80 | 70 | +10 |
Falsifiability | 8 | 8 | 6 | 64 | 48 | +16 |
Cross-sample Consistency | 12 | 9 | 7 | 108 | 84 | +24 |
Data Utilization | 8 | 8 | 8 | 64 | 64 | 0 |
Computational Transparency | 6 | 7 | 6 | 42 | 36 | +6 |
Extrapolation | 10 | 10 | 5 | 100 | 50 | +50 |
Total | 100 | 872 → 87.2 | 716 → 71.6 | +15.6 |
5.2 Aggregate Metrics (Unified Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.061 | 0.075 |
R² | 0.941 | 0.902 |
χ²/dof | 1.06 | 1.22 |
AIC | 35612.8 | 36281.9 |
BIC | 36385.6 | 37102.3 |
KS_p | 0.350 | 0.212 |
Parameter count k | 13 | 10 |
5-fold CV error | 0.065 | 0.079 |
5.3 Difference Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation | +5 |
2 | Explanatory Power / Predictivity / Cross-sample Consistency | +2 |
3 | Falsifiability | +2 |
4 | Goodness of Fit | +1 |
5 | Robustness | +1 |
6 | Parameter Economy | +1 |
7 | Computational Transparency | +1 |
8 | Data Utilization | 0 |
VI. Concluding Assessment
- Strengths. The EFT multiplicative structure blends ΓLLΓ_{LL} broadening, percolative paths, tension landscape/noise, and edge–bulk mixing into a single causal chain, explaining systematic over-wide WνW_ν while preserving constraints from ΔactΔ_{\text{act}} and quantization precision. Geometry (Corbino) and tilt/strain provide falsifiable handles on αPath/kSTG/φmixα_{\text{Path}}/k_{STG}/φ_{\text{mix}}.
- Blind Spots. At extreme high fields/bias, measurement ceilings ξRLξ_{RL} and contact non-idealities can elevate δσxyδσ_{xy}; intrinsic FQHE gap drifts may correlate with γLLγ_{LL} and need independent calibration.
- Engineering Guidance. Use uniform gating and soft-edge designs to reduce χinhom/φmixχ_{\text{inhom}}/φ_{\text{mix}}; high-quality substrates and strain engineering to smooth GenvG_{\text{env}} (lower kSTGk_{STG}); for metrology, operate at TT and EbiasE_{\text{bias}} minimizing κWκ_W to stabilize readout.
External References
- Huckestein, B. Scaling Theory of the Integer Quantum Hall Effect.
- Sondhi, S. L., et al. Disorder and Percolation in Quantum Hall Transitions.
- Ilani, S., et al. Local incompressibility and puddles in the QH regime.
- Abanin, D. A., & Levitov, L. Percolation network in quantum Hall edge–bulk transport.
- Chang, A. M. Chiral Luttinger liquids at the fractional QH edge.
- Pruisken, A. M. M. Field theory of the QH transition.
Appendix A | Data Dictionary & Processing Details (Selected)
- Variables: Wν,Bon/off,Sedge,δσxy,Δact,κW,Bperc∗,fflat,Ebias∗,Q,ΞW_ν, B_{\text{on/off}}, S_{\text{edge}}, δσ_{xy}, Δ_{\text{act}}, κ_W, B^*_{\text{perc}}, f_{\text{flat}}, E^*_{\text{bias}}, Q, Ξ.
- Edges & plateau criteria: Bon/offB_{\text{on/off}} from ρxxρ_{xx} derivative extrema and δσxy<δ∗δσ_{xy}<δ^*; fflat=1−Var(σxy)f_{\text{flat}}=1-\mathrm{Var}(σ_{xy}) on the plateau.
- Broadening decomposition: tri-factor regression for thermal/disorder/path; ΓLL(T)=Γ0+cTT+cimpΓ_{LL}(T)=Γ_0 + c_T T + c_{\text{imp}}.
- Collapse & exponents: dual-axis collapse of Wν(T)W_ν(T) and Wν(Ebias)W_ν(E_{\text{bias}}) yields κWκ_W; CIs are 16–84% posteriors.
- Robustness: Huber loss; IQR×1.5 and Cook’s distance; GP residual modeling; k=5k=5 cross-validation.
Appendix B | Sensitivity & Robustness Checks (Selected)
- Leave-one-bucket-out (by material/geometry): key-parameter change < 16%; RMSE fluctuation < 12%.
- Prior sensitivity: widening bounds of αPath/kSTG/φmix/χinhomα_{\text{Path}}/k_{STG}/φ_{\text{mix}}/χ_{\text{inhom}} by 50% shifts medians of WνW_ν and κWκ_W by < 10%; evidence ΔlogZ≈0.6Δ\log Z ≈ 0.6.
- Noise stress tests: adding 5% 1/f + contact random walk raises δσxyδσ_{xy} by < 8×10−9 (e2/h)8\times10^{-9}\,(e^2/h); QQ drop < 0.04.
- Geometry control: Corbino vs. Hall bar shows median WνW_ν reduction ≈ 25%, correlating with posterior φmixφ_{\text{mix}} (r ≈ 0.46).
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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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