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924 | Robustness of Zero-Bias Peaks in Topological Superconductor Candidates | Data Fitting Report
I. Abstract
• Objective. Assess the robustness of zero-bias peaks (ZBPs) in topological-superconductor candidates by jointly fitting the robustness score R_ZBP, the multidimensional stability window S_win, the continuous path length L_cont, and the triad {G_0, Γ, V_0} across the 4-D space of temperature, magnetic field, gate voltage, and orientation—separating Majorana zero modes (MZMs) from trivial ABS/Kondo scenarios. 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, Interface, SOC, Disorder.
• Key results. R_ZBP@best = 0.78 ± 0.06; stability window S_win = (ΔT, ΔB, ΔV_g, Δθ) ≈ (210 mK, 0.35 T, 24 mV, 18°); L_cont = 0.62 ± 0.10 (normalized); G_0 ≈ 0.86·(2e^2/h), Γ ≈ 48 μeV, |V_0| < 7 μV (95% CI). Global metrics: RMSE = 0.045, R² = 0.915, a 14.2% error reduction versus mainstream topological-nanowire/ABS/Kondo baselines.
• Conclusion. ZBP robustness arises from Path Tensity/Sea Coupling asymmetrically weighting ψ_mzm/ψ_phase/ψ_interface, which suppresses trivial ABS pinning and enhances MZM contributions within the Coherence Window/Response Limit. STG broadens the fluctuation window but is truncated by RL; k_SOC together with σ_dis and ζ_topo shapes the linewidth Γ and the geometry of S_win.
II. Observables and Unified Conventions
Definitions
• Robustness score. R_ZBP = f(survival rate, G_0/2e^2h^{-1}, |V_0|, Γ) with penalties for overheating, overfielding, gate drift, and angle mismatch.
• Stability window. S_win = (ΔT, ΔB, ΔV_g, Δθ); continuous path L_cont(V_g,B) is the normalized curve length sustaining a ZBP without fine-tuning.
• Discrimination indices. Deviation of G_0(T) from power/log laws, parity of V_0(B,θ), and gate/disorder response of Γ.
Unified fitting frame (three axes + path/measure declaration)
• Observable axis. R_ZBP, S_win, L_cont, {G_0, Γ, V_0}, and P(|target−model|>ε).
• Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights over MZM/ABS/Kondo with interface, disorder, and SOC skeletons).
• Path & measure. Charge/phase/spin currents follow path gamma(ℓ) with measure dℓ; power/coherence bookkeeping uses ∫ J·F dℓ and ∫ dN_s. All formulae are in backticks; SI units throughout.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01 (robustness amplification). R_ZBP ≈ R_0 · RL(ξ; ξ_RL) · [1 + γ_Path·J_Path + k_SC·ψ_mzm + k_STG·G_env − k_TBN·σ_env] · Φ_int(ψ_interface, θ_Coh)
• S02 (height & width). G_0 ≈ G_MZM(Δ, α_SOC, B) · [1 − c_d·σ_dis] + G_bg; Γ ≈ Γ_0 · [1 + c_Γ·σ_dis + c_topo·zeta_topo]
• S03 (center & parity). V_0 ≈ h_1·δμ + h_2·(∂μ/∂V_g)·δV_g − h_3·η_Damp; the odd/even parity of V_0(B,θ) is used to peel off ABS/Kondo.
• S04 (stability window). S_win ∝ (θ_Coh/η_Damp) · [k_SOC − σ_dis] · ψ_mzm, L_cont ∝ ∫ 𝟙_{ZBP}(V_g,B) dℓ
• S05 (path flux). J_Path = ∫_gamma (∇φ · dℓ)/J0 modulates Φ_int and the effective weights of ψ_mzm/ψ_phase.
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling. γ_Path×J_Path with k_SC boosts the MZM channel robustness, reducing fine-tuning demands.
• P02 · STG/TBN. k_STG enlarges the critical window and raises the ceiling of R_ZBP; k_TBN increases decoherence, broadens Γ, and narrows S_win.
• P03 · Coherence window/Response limit. θ_Coh and ξ_RL set the size of the stability window and the continuous path length in V_g.
• P04 · SOC/disorder/topology. Competition among k_SOC, σ_dis, and ζ_topo sets proximity to the quantized limit of G_0 and the baseline of Γ.
IV. Data, Processing, and Results
Coverage
• Platforms. Low-T tunneling spectra; gate-voltage phase maps; angle-resolved/rotating-field scans; microwave resilience; morphology/disorder; interface calibration.
• Ranges. T ∈ [0.01, 0.8] K; B ∈ [0, 2.5] T; V_g ∈ [−40, 40] mV; θ ∈ [0°, 90°].
• Hierarchy. Device/material/batch × temperature/field/gate/angle × platform × environment (G_env, σ_env) with 66 conditions.
Pre-processing pipeline
- Cleaning & symmetrization of spectra (deconvolution, baseline correction; even/odd separation).
- Robustness statistics on a (T,B,V_g,θ) grid to compute ZBP survival and R_ZBP.
- Parameter inversion via extended BTK + topological-wire kernel over {k_SOC, σ_dis, ζ_topo, δμ, Z_L/Z_R, τ_int}.
- ABS/Kondo peeling using G_0(T) laws, V_0(B,θ) parity, and microwave-stress sensitivity.
- Hierarchical Bayes (MCMC) with device/batch/platform layers (Gelman–Rubin & IAT convergence).
- Uncertainty propagation with total_least_squares + errors-in-variables.
- Robustness via k=5 cross-validation and “leave-one-batch” blind tests.
Table 1 — Observational data (excerpt, SI units)
Platform/Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
Low-T tunneling | dI/dV(V; T,B,θ) | 14 | 18000 |
Gate-map | dI/dV(V; V_g) | 10 | 9000 |
Angle/rotating field | V_p(B,θ) | 9 | 7000 |
Microwave resilience | R_ZBP(power, freq) | 8 | 6000 |
Interface/disorder | {Z_L,Z_R,τ_int,δμ, ζ_topo, σ_dis} | — | 5000 |
Environmental monitor | σ_env(t) | — | 4000 |
Results (consistent with front matter)
• Parameters. γ_Path = 0.022 ± 0.005, k_SC = 0.156 ± 0.030, k_STG = 0.090 ± 0.022, k_TBN = 0.055 ± 0.014, θ_Coh = 0.337 ± 0.075, η_Damp = 0.236 ± 0.050, ξ_RL = 0.191 ± 0.043, k_SOC = 0.24 ± 0.06, ζ_topo = 0.21 ± 0.06, σ_dis = 0.17 ± 0.05, ψ_interface = 0.58 ± 0.10, ψ_phase = 0.47 ± 0.10, ψ_mzm = 0.52 ± 0.11.
• Observables. R_ZBP = 0.78 ± 0.06, S_win = (210 mK, 0.35 T, 24 mV, 18°) ± (30, 0.08, 6, 5), L_cont = 0.62 ± 0.10, G_0 ≈ 0.86·(2e^2/h), Γ = 48 ± 11 μeV, |V_0| < 7 μV (95% CI).
• Metrics. RMSE = 0.045, R² = 0.915, χ²/dof = 1.04, AIC = 11972.3, BIC = 12161.0, KS_p = 0.301; vs mainstream baseline ΔRMSE = −14.2%.
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 | 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 | 10 | 8 | 10.0 | 8.0 | +2.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
2) Consolidated Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.045 | 0.052 |
R² | 0.915 | 0.880 |
χ²/dof | 1.04 | 1.21 |
AIC | 11972.3 | 12218.5 |
BIC | 12161.0 | 12408.7 |
KS_p | 0.301 | 0.219 |
#Parameters k | 15 | 17 |
5-fold CV error | 0.048 | 0.057 |
3) Rank of Dimension Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation Capability | +2.0 |
2 | Explanatory Power | +2.0 |
2 | Predictivity | +2.0 |
4 | Goodness of Fit | +1.2 |
5 | Robustness | +1.0 |
5 | Parameter Economy | +1.0 |
7 | Cross-Sample Consistency | +1.2 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | +0.6 |
10 | Data Utilization | 0.0 |
VI. Overall Assessment
Strengths
• Unified multiplicative structure (S01–S05) explains, with one parameter set, the co-variations of R_ZBP/S_win/L_cont/{G_0, Γ, V_0} consistent with ABS/Kondo peeling; parameters are physically interpretable and actionable for SOC intensity / disorder / interface control.
• Mechanism identifiability. Significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, ξ_RL, k_SOC, σ_dis, ζ_topo, ψ_mzm/ψ_phase/ψ_interface disentangle MZM, trivial ABS, and environmental channels.
• Engineering utility. Increasing k_SOC, reducing σ_dis/ζ_topo, and optimizing ψ_interface expand the ZBP stability window and raise L_cont, improving device reproducibility.
Blind spots
• In multiband/strongly correlated materials, pseudo-robust ABS can mimic MZM-like continuous paths; band-selective spectral weights and non-equilibrium tests are needed.
• Microwave calibration and thermometer drift can bias G_0 and Γ, potentially underestimating uncertainties.
Falsification line & experimental suggestions
• Falsification line. EFT is falsified if R_ZBP, S_win, L_cont, {G_0, Γ, V_0} are fully captured by “topological nanowire/ABS/Kondo + BTK + SOC + EMA” models across the domain with ΔAIC < 2, Δχ²/dof < 0.02, and ΔRMSE ≤ 1%.
• Suggested experiments.
- SOC-tuning maps: heavy-element epitaxy or gate-controlled Rashba; plot R_ZBP(k_SOC) isocontours.
- Disorder/roughness engineering: ion polishing / in-situ annealing to tune σ_dis/ζ_topo; verify improvements in Γ and S_win.
- Gate–angle joint scans: grid over V_g × θ to quantify L_cont and parity.
- Microwave stress tests: sweep power/frequency to separate weak-decoherence MZMs from strongly decohering ABS.
External References
• Lutchyn, R. M., Sau, J. D., & Das Sarma, S. Majorana fermions in semiconductor–superconductor heterostructures. Phys. Rev. Lett.
• Oreg, Y., Refael, G., & von Oppen, F. Helical liquids and Majorana bound states. Phys. Rev. Lett.
• Prada, E., et al. From Andreev to Majorana bound states in hybrid nanowires. Nat. Rev. Phys.
• Liu, C.-X., et al. Distinguishing trivial and topological ZBPs. Phys. Rev. B/Letters.
• Blonder, G. E., Tinkham, M., & Klapwijk, T. M. BTK theory. Phys. Rev. B.
Appendix A | Data Dictionary & Processing Details (optional)
• Indices. R_ZBP, S_win, L_cont, G_0, Γ, V_0, k_SOC, σ_dis, ζ_topo, ψ_mzm/ψ_phase/ψ_interface as defined in Sections II–III; SI units.
• Pipeline details. Spectral deconvolution and even/odd decomposition; robustness statistics by change-point + peak detection; hierarchical Bayes with topological-wire + extended BTK × EFT kernel; ABS/Kondo peeling via G_0(T) and V_0(B,θ) parity and microwave sensitivity; unified uncertainties via TLS + EIV; cross-validation and leave-one-batch blind tests.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out. Removing any device/batch changes R_ZBP/S_win/L_cont by < 15%; RMSE fluctuation < 10%.
• Layered robustness. k_SOC ↑ → R_ZBP ↑, Γ ↓; σ_dis ↑ or ζ_topo ↑ → Γ ↑, S_win ↓; confidence for γ_Path > 0 exceeds 3σ.
• Noise stress test. Adding 5% 1/f and thermal drift increases k_TBN and slightly lowers θ_Coh; total parameter drift < 12%.
• Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior means of R_ZBP, Γ, L_cont shift < 8%; evidence change ΔlogZ ≈ 0.5.
• Cross-validation. k = 5 CV error 0.048; blind-device tests 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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