40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 3: Tension Landscapes & Pairing Channels

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Chapter 3: Tension Landscapes & Pairing Channels

I. Physical Picture
Macroscopic observables in unconventional superconductors (T_c, H_c1, H_c2, xi, lambda_L, kappa, and transport) are treated as responses to a materials-driven tension landscape T_fil(r) and its gradient grad T_fil(r). T_fil renormalizes pairing kernels and kinetic anisotropy; grad T_fil selects symmetry channels and reshapes gap structure and vortex spectra via nonlocal couplings. This chapter establishes an executable map design variables → tension landscape → pairing channel, stated as principles and minimal statements:

• P30-1 (Tension–coupling principle): Local T_fil(r) and grad T_fil(r) renormalize free-energy density and coupling coefficients, providing first-order control over stability and anisotropy of the order parameter psi.
• P30-2 (Nonlocal response principle): Tension acts through a nonlocal kernel K(R) on psi, setting the coherence-length tensor and the width of critical windows.
• P30-3 (Layer/interface decomposition): In layered/interface systems, T_fil splits into in-plane and interlayer parts; the sign and amplitude of grad T_fil · e_c control parity mixing thresholds.
• P30-4 (Critical-window principle): Slow spatial variations of T_fil translate or shrink critical/coherence windows under fixed external controls; the sensitivity in the measurement matrix is monotone in the relevant knob.

II. Design Variables → Tension Landscape (Constitutive Forms)

1. Design set: theta = { ε_ij(r), x(r), s(r), p(r), δ(r), H(r), T_temp(r), ρ_int(r) } for strain, doping, interlayer spacing/coupling, hydrostatic/chemical pressure, disorder strength, magnetic field, temperature field, and interface roughness.
2. Linearized & nonlocal forms (minimal usable):
   • S30-1 (Local approximation):
     T_fil(r) = T0 + A:ε(r) + α_x x(r) + α_s s^{-1}(r) + α_p p(r) + α_δ δ(r) + α_H |B(r)| + ...
     where A is a fourth-rank tensor and “:`” denotes double contraction; ellipsis covers higher-order and cross terms.
   • S30-2 (Nonlocal kernels):
     T_fil(r) = T_loc(r) + ∫ K_T(r−r') · ζ(r') dV',
     grad T_fil(r) = grad T_loc(r) + ∫ K_G(r−r') · ζ(r') dV',
     with ζ collecting {ε, x, s^{-1}, p, δ, |B|, ...}; the measure dV' is explicit.
3. Derived anisotropy metrics:
   S30-3: kappa_ij = kappa_0 δ_ij + Λ_ij[T_fil, grad T_fil]; principal axes and magnitudes of xi_i, lambda_{L,i} follow from Λ_ij.
4. Calibration workflow (placeholder):
   M3-1: Multi-knob scans over {ε, x, s, p} with joint microwave/THz/DC transport to fit {A, α_*, K_*}, feeding Chapter 8 (measurement matrix).

III. Pairing Symmetry & Gap Anisotropy (Selection Rules)

1. Free-energy extension:
   • S30-4 (GL–tension coupling):
     F = ∫ [ a|psi|^2 + b|psi|^4 + κ_ij (D_i psi)^* (D_j psi) + λ_1 T_fil |psi|^2 + λ_2 (∂_i T_fil) (psi^* D_i psi + c.c.) ] dV,
     with D_i = (∂_i - i q A_i) and λ_1, λ_2 the tension–order-parameter couplings.
   • S30-5 (Anisotropy induction): κ_ij = κ_0 δ_ij + η_ij[T_fil, grad T_fil].
2. Channel selection:
   • P30-5 (Symmetry selection): If grad T_fil transforms as a vector under the crystal point group, it couples linearly to the corresponding irreducible components of psi, shifting free energy and re-weighting d/p channels.
   • P30-6 (Parity mixing condition): When grad T_fil · e_c ≠ 0 with inversion broken at an interface, odd/even parity mixing is allowed with amplitude monotone in |grad T_fil| and interface coupling strength.
3. Observable links: Anisotropy of H_c2, rotation of lambda_L principal axes, Josephson phase relations, and vortex-lattice orientation are read out via Chapters 5–8.

IV. Testable Predictions (from this Chapter)

• Principal-axis locking: The direction of grad T_fil sets principal axes of xi_i/lambda_{L,i}; rotating-strain experiments reveal H_c2(φ) extrema drifting by the same angle (small-angle linear regime).
• Node rearrangement: When η_ij[T_fil, grad T_fil] crosses threshold, d-wave node directions rotate measurably vs. lattice axes; within the same flake, node angles drift monotonically under {ε, p} scans.
• Parity-mixing switch: Heterostructures with sizeable grad T_fil · e_c show enhanced odd harmonics and anomalous phase shifts in Josephson junctions; mixing strength is quasi-linear in |grad T_fil| at low amplitudes.
• Thickness scaling: Nonlocal kernels K_T, K_G in thin films induce detectable deviations in lambda_L(d) and H_c2(d) from local models, enabling kernel-scale estimation.
• Vortex orientation & pinning remap: grad T_fil locks vortex-lattice orientation and reshapes the pinning map; under weak fields, small-angle field rotation triggers orientation “jumps.”
• Critical-window shift/shrinkage: At fixed {x, δ}, uniform ε or p co-shifts T_c and H_c2; superposed slow grad T_fil narrows coherence windows.
• Band-dependent propagation effects: Co-variation of T_fil and n_eff yields coupled drifts of arrival time and amplitude in microwave/THz transmission (see Chapter 7 for T_arr conventions and data contracts).

V. Cross-Volume References & Chapter Anchors

1. Cross-volume citation (fixed style examples): see “EFT.WP.Core.Tension v1.0” S72-; “EFT.WP.Core.Density v1.0” S92-; “EFT.WP.Core.Threads v1.0” P10-/S10-. Citation style follows volume + version + anchor.
2. Anchors in this chapter (P/S/M):
   • Principles: P30-1, P30-2, P30-3, P30-4, P30-5, P30-6.
   • Minimal statements: S30-1 (local constitutive), S30-2 (nonlocal kernels), S30-3 (anisotropy metrics), S30-4 (GL–tension free energy), S30-5 (anisotropy induction).
   • Methods: M3-1 (parameter calibration & kernel-scale estimation; integrated with Chapter 8).
     Formatting, inline symbol policy, and numbering conform to the Full Template v0.1 (Roman-numeral bold heads; “- ” bullets; “1. ” lists; formulas/symbols/definitions in English).

VI. Summary
This chapter provides computable mappings from design variables to the tension landscape, establishes selection rules for pairing symmetry and gap anisotropy, and delivers testable predictions and a calibration workflow. Chapters 4–8 align these P/S/M items with measurement matrices, critical windows, and low-temperature EM measurements to close the loop design variables → tension landscape → pairing channel → observables.