40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 11: Case Library — Cuprates / Fe-based / Heavy Fermion / 2D

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Chapter 11: Case Library — Cuprates / Fe-based / Heavy Fermion / 2D

I. Chapter Goals & Structure
This chapter presents end-to-end exemplars for four representative systems—Cuprates, Fe-based, Heavy-Fermion, and 2D: data acquisition & cleaning → two T_arr conventions & de-embedding → measurement matrix y = M(θ) → inversion & model comparison → falsification & design recommendations. All symbols/formulas/definitions are in English. Data contracts follow Chapters 6–8 and 10. Anchors in this chapter use S110-* / M11-* / I11-*.

II. Common Data Contract & Pipeline (All Four Families)

• S110-1 (Case dataset card — required)
• case_id: "cuprates|fe-based|heavy-fermion|2d"
• specimen: {batch: "...", growth: "...", geometry: "...", thickness_d: "..."}
• measurement:
• bands: ["microwave","THz","optical"]
• convention: "pulled_const|integrand" # choose one T_arr convention
• delta_form: "c_ref^-1 * ∫ n_eff dℓ" # or "∫ (n_eff/c_ref) dℓ"
• gamma: "piecewise: free|fixture|substrate|film|sample"
• d_ell: "m"
• deembed: {scheme: "OSLT|TRL", refs: ["blank","substrate-only"]}
• unwrap: {method: "phase_unwrap_v2"}
• observables: ["T_arr(ω)","Δf/f","1/Q","|S21|","arg S21","M(H)","R(H,T,θ,φ)"]
• priors: {...} # see Chapter 10
• references: ["Ch.6","Ch.7","Ch.8","Ch.10"]
• S110-2 (Cleaning & artifact control)
  Handle phase jumps, multipath, instrument drift, hotspots; include piecewise unwrapping and de-embedding uncertainties inside Σ_y.
• S110-3 (Measurement matrix & weights)
  Set w(ω) ∝ L_coh(ω) from W_coh(ω); build block-diagonal J = ∂y/∂θ and Σ_y.
• S110-4 (Inversion & comparison)
  Prefer NUTS; enable SMC/NS for multi-modality or evidence; model family {𝓜0, 𝓜1, 𝓜2, 𝓜3} (see Chapter 10).
• S110-5 (Falsification lines)
  Residual whiteness/spectral slope; co-variation of H_c2(θ) with principal axes of λ_L; same-sense drift of ΔT_arr and amplitude; common slope deviation in thickness scalings.
• M11-1 (Card → cleaning)
  Import case card → OSLT/TRL de-embedding → phase unwrapping → outlier masking → estimate Σ_y, w(ω).
• M11-2 (Cleaning → measurement matrix)
  Generate J, B, Σ_y; persist under the measurement_matrix section.
• M11-3 (Matrix → inversion)
  Run NUTS/VI/SMC with priors; output {posterior, logZ}.
• M11-4 (Posterior → falsification)
  PPC/LOO, SBC coverage; on failure, propose model upgrade (enable λ_2 or {K_T, K_G}).
• M11-5 (Posterior → design recommendations)
  Feed θ̂ into Chapter 9 objective Φ(θ); output design-variable adjustments Δθ_design and process-recipe draft.
• I11-1 load_case_card(path) -> {data, meta}
• I11-2 clean_and_deembed(data, scheme) -> {clean, Σ_y, weights}
• I11-3 build_J(clean, config) -> {J, B, Σ_y}
• I11-4 infer_theta(J, Σ_y, priors, algo) -> {posterior, logZ, diag}
• I11-5 falsify_and_ppc(posterior, clean) -> {tests, passrate}
• I11-6 recommend_design(posterior, objectives) -> {Δθ_design, recipe}

III. Case A: Cuprates

• S110-6 (Physics focus)
  Strong anisotropy with potential d-wave nodes; grad T_fil may rotate node angles and induce mild parity mixing.
• M11-A1 (Data & cleaning)
  TDTS T_arr(ω) + μSR/resonator λ_L(T) + angle-resolved H_c2(θ); execute M11-1.
• S110-7 (Model & priors)
  Start with 𝓜0; if H_c2(θ) and λ_L principal axes misalign, activate λ_2 (𝓜1).
• M11-A2 (Inference)
  Target θ = {λ_{L,i}(T), ξ_i(T), κ_ij, σ1, σ2}; report γ(T)=ξ_∥/ξ_⊥ and axis angle.
• S110-8 (Falsification)
  Node-shift: if η_ij[T_fil, grad T_fil] crosses threshold, H_c2(θ) extrema drift relative to lattice axes; PPC requires white residuals.
• Design recommendations
  If γ too large and T_c limited, use strain knob to reduce in-plane anisotropy (A:ε < 0 direction) or tune Γ_if to suppress excessive parity mixing (see Chapter 9).
• Risks
  Phase separation/defects raise δ, co-degrading λ_L and ξ (S9F-2).

IV. Case B: Fe-based

• S110-9 (Physics focus)
  Multi-band with potential s±; strong interlayer coupling and interface effects.
• M11-B1 (Data & cleaning)
  Multi-band S21 & T_arr, thickness scans {λ_L(d), H_c2(d)}, magnetoresistance R(H,T,θ); execute M11-1.
• S110-10 (Model & priors)
  Prefer 𝓜1 (includes λ_2); when films approach kernel scales, switch to 𝓜2 (enable {K_T, K_G}).
• M11-B2 (Inference)
  Joint thickness-scaling inversion for {K_T,K_G} and λ_{L,i}, ξ_i; report posterior intervals of kernel radii.
• S110-11 (Falsification)
  If log–log {λ_L(d), H_c2(d)} show no common-slope deviation, 𝓜2 is unnecessary; Bayes factors favor 𝓜1.
• Design recommendations
  Control s^{-1} and grad s via intercalation/strain to optimize in-plane/out-of-plane coupling, targeting shorter ξ_∥ and higher H_c2.
• Risks
  Excess Γ_if drives interfacial reconstruction (S9F-3), spoiling target grad T_fil · e_c.

V. Case C: Heavy-Fermion

• S110-12 (Physics focus)
  Strong correlations at low energy scales; possible odd-parity pairing and anisotropic vortex dynamics.
• M11-C1 (Data & cleaning)
  Ultra-low-T cavity/resonator Δf/f, 1/Q, small-field M(H), T_arr(ω); strict drift modeling.
• S110-13 (Model & priors)
  Start with 𝓜1; if vortex-transport anomalies dominate, couple Chapter 5 S50-* via S_v correction into M(·).
• M11-C2 (Inference)
  Target θ = {λ_{L,i}, ξ_i, κ_ij, σ1, σ2, S_v}; report low-T extrapolations of H_c1,H_c2.
• S110-14 (Falsification)
  Hall sign window and Nernst-peak drift must match posterior ΔS_v[T_fil]; PPC must reproduce phase/group-delay co-variation.
• Design recommendations
  Gate ∂T_fil/∂p, ∂T_fil/∂ε via pressure/strain to tune anisotropy without degrading ρ_s.
• Risks
  Thermal budgets may induce phase reconstruction, harming reproducibility (shorten life_window in dataset card).

VI. Case D: 2D (KTB)

• S110-15 (Physics focus)
  KTB transition with phase-stiffness threshold; nonlocal kernels dominate for d ~ ξ.
• M11-D1 (Data & cleaning)
  I–V scaling, T_arr(ω), thickness scan λ_L(d); execute M11-1.
• S110-16 (Model & priors)
  Use 𝓜2 (nonlocal kernels) with T_BKT prior; adopt smoothness prior (GP) for σ2(ω).
• M11-D2 (Inference)
  Jointly fit T_BKT, Gi_2D, {K_T,K_G}, and λ_L, ξ; output the gap T_c − T_BKT.
• S110-17 (Falsification)
  If α(T_BKT) ≠ 3 or ρ_s(T_BKT) ≠ 2k_B T_BKT/π, reject KTB or flag insufficient data.
• Design recommendations
  Use patterning/strain maps to constrain grad T_fil, controlling T_BKT and coherence-window breadth.
• Risks
  Ultra-thin films may lose percolation, causing mismatches between |S21| and T_arr (geometry corrections required).

VII. Unified Outputs & Report Template

• S110-18 (Report fields)
• case_report:
• summary: {...} # key findings & falsification results
• posterior: {θ_hat: "...", ci: "...", η_hat: "..."}
• evidence: {logZ: "...", model_rank: ["𝓜k", "..."]}
• falsification: {tests: ["node_shift","thickness_scaling","parity_gate"], pass: "..."}
• design: {Δθ_design: "...", recipe: "...", risk: "..."}
• reproducibility: {scripts: "...", env: "...", anchors: ["S110-*","M11-*","I11-*"]}
• M11-6 (Reproducibility bundle)
  Export cleaning scripts, inference configs, environment digest, and anchor list; validate check_dim and citation style.

VIII. Cross-Chapter Links & Anchors (This Chapter)

• Internal links (fixed style): This volume Ch. 3 (tension landscape & pairing), Ch. 4 (free energy & field equations), Ch. 5 (vortex/topology/transport), Ch. 6 (critical/coherence windows), Ch. 7 (arrival time & de-embedding), Ch. 8 (measurement matrix), Ch. 10 (inversion & model comparison). Methods/data specs: Methods.Cleaning v1.0, Data.DatasetCards v1.0, Data.Pipeline v1.0, Methods.Repro v1.0.
• Anchors (S/M/I):
  S110-1—S110-18; M11-1—M11-6; I11-1—I11-6.

IX. Summary
Under a unified contract and measurement matrix, this chapter completes closed-loop exemplars for four systems—from data → inversion → falsification → design recommendations. When evidence favors model upgrades, enable λ_2 and/or {K_T, K_G} per rules, and write back posteriors to Chapter 9 for recipe optimization and to Chapter 12 for implementation bindings.