40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 5: Vortices, Topology & Transport

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Chapter 5: Vortices, Topology & Transport

I. Physical Picture & Objectives
Magnetic flux in unconventional superconductors organizes into quantized vortices. The core scale ~ ξ, the screening scale ~ λ_L, and the tension landscape T_fil(r)/grad T_fil(r) jointly determine spectra, interactions, lattice orientation, and pinning maps, ultimately shaping dc/ac transport (including Hall and Nernst). This chapter provides a minimal, executable set of vortex–transport equations and workflows that close the loop tension landscape → vortex structure → macroscopic transport.

II. Basic Postulates & Quantization (P50-/S50-)

• P50-1 (Flux quantization)
  In multiply connected superconducting domains, circulation conservation and single-valued phase enforce flux quantization in units Φ0_ref = h/(2e).
• S50-1 (Circulation relation)
  ∮ (m* v_s + q A) · dℓ = n h, hence Φ = ∮ A · dℓ = n Φ0_ref.
• P50-2 (Vortex energy partition)
  Single-vortex energy decomposes as line energy E_line, core energy E_core, and a tension correction ΔE_T:
  E_vortex = E_line + E_core + ΔE_T[T_fil, grad T_fil].
• S50-2 (Line-energy approximation)
  In principal-axis anisotropy,
  E_line/L ≈ (Φ0_ref^2 / 4π μ0_ref) ⟨λ_L^{-2}⟩ · ln(λ_{L,⊥}/ξ_⊥) + O(η_ij),
  where η_ij[T_fil, grad T_fil] is the tension-induced anisotropy correction.
• P50-3 (Orientation locking)
  Coupling between grad T_fil and irreducible representations of the crystal point group locks the vortex-lattice principal axes to the crystal at a preferred angle.

III. Interactions, Lattices & Pinning

• S50-3 (Vortex–vortex interaction)
  Effective 2D pair potential V(R) ≈ (Φ0_ref^2 / 2π μ0_ref) K_0(R / λ_{L,eff}), with modified Bessel K_0; the effective λ_{L,eff} is set jointly by κ_ij and T_fil.
• S50-4 (Pinning potential & force)
  U_p(r) = U_0(r; δ, ε, x, ρ_int) + U_T(r; T_fil), with F_p = - grad U_p; U_T captures tension-landscape–induced pinning.
• S50-5 (Lattice orientation & transitions)
  The orientation φ_0 minimizes E_vortex + Σ V(R) + Σ U_p. Varying |grad T_fil| or external controls H, T_temp can trigger orientation “jumps” and melting–recrystallization transitions.
• M5-1 (Orientation & pinning-map reconstruction)
  Combine scanning magnetometry/flux imaging with SANS/μSR to invert φ_0(H,T_temp) and U_p(r); register sensitivities in the Chapter 8 measurement matrix.

IV. Nonequilibrium Dynamics & TDGL Coarse-Graining

• S50-6 (TDGL form)
  τ (∂ψ/∂t) = - δF/δψ* + ζ(r,t), and E + v_L × B = ρ_ff J / B_c2 + ...,
  with vortex drift v_L, flux-flow resistivity ρ_ff, and noise ζ (metrology definitions in Chapter 2).
• S50-7 (Flux-flow resistivity)
  ρ_ff ≈ ρ_n · B / H_c2(ê) (anisotropy enters via ê). In the presence of J_i^{(T)} (Chapter 4), additional phase/amplitude couplings appear in the microwave/THz band.
• S50-8 (Hall & Nernst)
  Vortex-driven electric field E = B × v_L gives
  σ_xy^v ≍ (n_v Φ0_ref / B) · S_v, ν_v ≍ (S_v / e n_s B),
  where S_v is the single-vortex entropy; the tension correction ΔS_v[T_fil] shifts Hall sign and Nernst peak positions.
• M5-2 (Transport-decomposition pipeline)
  From joint measurements {ρ_xx(B,T), ρ_xy(B,T), ν(B,T), microwave/THz T_arr}, decompose and invert {ρ_n, H_c2(ê), S_v, J^{(T)}}; feed outputs to Chapter 10 (inference) and Chapter 8 (measurement matrix).

V. Dimensionality & Geometry: Thin Films/2D and KTB

• P50-4 (2D topological transition)
  In effectively 2D geometries, a Kosterlitz–Thouless–Berezinskii (KTB) unbinding transition occurs at a phase-stiffness threshold.
• S50-9 (KTB criteria)
  ρ_s(T_BKT) = (2/π) k_B T_BKT, with I–V scaling V ∝ I^{α(T)} and α(T_BKT)=3.
• S50-10 (Thickness scaling)
  When d ~ ξ or approaches kernel scales (Chapter 3 K_T, K_G), systematic deviations emerge in λ_L(d), H_c2(d), ρ_ff(d), and ν(d) from local models, enabling kernel-scale estimation.
• M5-3 (KTB & thickness pipeline)
  Extract joint scalings for {T_BKT, ρ_s(T), α(T)} and {λ_L(d), H_c2(d)}; invert admissible ranges of {K_T, K_G} and back-propagate to Chapters 3–4 coefficients.

VI. Coupling to Tension Landscapes & Testable Predictions

1. P50-5 (Tension–vortex coupling)
   grad T_fil selectively raises/lowers orientation-dependent line energy, reorganizing the lattice and the pinning map.
2. Predictions
   • Orientation locking & jumps: Under small-angle field rotation, φ_0 exhibits discrete jumps at a threshold |grad T_fil|; jump angle scales ~linearly with |grad T_fil|.
   • Hall sign window: ΔS_v[T_fil] rewrites the sign domain of σ_xy^v, enabling strain/pressure-gated Hall sign reversals; the threshold grows with |grad T_fil|.
   • Nernst peak drift: The ν(B,T) peak drifts monotonically under {ε, p} scans, co-rotating with the principal axes of {λ_L, ξ}.
   • Thickness-scaling deviations: Log-plots of {λ_L(d), H_c2(d)} show a common slope shift, estimating nonlocal kernel scales.
   • Pinning-map reconfiguration: After thermal/field cycling, pinning maxima migrate along grad T_fil; the shift correlates with the cycle amplitude (tension-landscape rewrite).

VII. Metrology & Data Contracts (Links to Chs. 7/8/12)

• M5-4 (Imaging–transport composite matrix)
  Merge imaging {φ_0, n_v(r), U_p(r)} and transport {ρ_xx, ρ_xy, ν} into a measurement vector y; expand sensitivities of y = M(θ) over {κ_ij, ξ, λ_L, ΔS_v, U_T} and register condition numbers and identifiability in Chapter 8.
• M5-5 (Dataset/Pipeline cards)
  Unified fields: B, T_temp, ε, p, d, geometry, freq; units and u(x)/U per the metrology baseline; references/see must carry volume + version + anchor explicitly.
• I5-* (Implementation bindings, placeholders)
  infer_pinning_map(dataset) -> {U_p(r), F_p(r)}
  fit_flux_flow(dataset) -> {ρ_n, H_c2(ê), σ_xy^v, ν}
  estimate_Kkernels(thickness_scan) -> {K_T, K_G}
  Final prototypes and returns are fixed in Chapter 12.

VIII. Cross-Volume References & Chapter Anchors

• Cross-volume (fixed style): EFT.WP.Core.Tension v1.0 S72-; EFT.WP.Core.Density v1.0 S92-; EFT.WP.Core.Sea v1.0 P20-/S20-; EFT.WP.Methods.Imaging v1.0 M2-; EFT.WP.Methods.Cleaning v1.0 M3-; metrology per EFT.WP.Core.Metrology v1.0 Ch.1–3,5.
• Anchors (this chapter, P/S/M/I):
  Principles: P50-1, P50-2, P50-3, P50-4, P50-5
  Statements: S50-1—S50-10
  Methods: M5-1—M5-5
  Implementations: I5-* (see Chapter 12)

IX. Summary
This chapter defines a minimal, executable framework for vortex spectra, lattice orientation, and pinning maps under tension landscapes, and connects TDGL coarse-graining and 2D KTB physics to macroscopic transport (ρ_xx, ρ_xy, ν). Together with Chapter 7’s arrival-time conventions and Chapter 8’s measurement-matrix/identifiability analysis, Chapter 10 can directly perform parameter inversion and model comparison.