{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.meissner-screening-response-limit-order",
  "title": "Meissner screening and response-limit order",
  "classification": "original hybrid exact-and-schematic electromagnetic-response figure",
  "reader_question": "What does a positive static transverse kernel do physically, and why does a zero-frequency optical delta not by itself prove Meissner screening?",
  "dominant_point": "The static transverse response produces exponential bulk magnetic screening, whereas the spatially uniform dynamic response reaches a different limit that can remain nonzero in a nonsuperconducting system with protected ballistic current.",
  "scope": {
    "state": "equilibrium linear response",
    "geometry": "isotropic three-dimensional local bulk occupying x > 0 with a planar surface at x = 0",
    "field": "weak static B_z parallel to the surface with B_z(0) = B_0 > 0",
    "units": "SI electromagnetic units",
    "local_condition": "|p| xi_0 much less than 1, with the local London kernel replacing K_T(p,0)",
    "thickness": "bulk thickness large compared with lambda_L; thin-film Pearl electrodynamics is excluded"
  },
  "conventions": {
    "orientation": "x points into the bulk, B = B_z hat z, and the positive screening current is j = j_y hat y",
    "ampere_law": "j_y = -(1/mu_0) d B_z/dx",
    "static_response": "D_s = lim_{|p| -> 0} K_T(p, omega = 0) after the thermodynamic limit and transverse projection",
    "dynamic_response": "For a fixed unit current polarization e, D_opt^(e) = lim_{omega -> 0} K_ee^R(0, omega), where K_ee^R = e_i K_ij^R e_j",
    "line_meanings": "solid black is normalized magnetic field; dashed gray with open markers is normalized screening current; solid and dashed arrows distinguish static-transverse and uniform-dynamic paths"
  },
  "panels": {
    "screening_profile": {
      "classification": "exact normalized solution under the declared local-bulk assumptions",
      "horizontal_axis": "x/lambda_L from 0 to 5",
      "vertical_axis": "B_z/B_0 and j_y/j_0",
      "equations": [
        "lambda_L^{-2} = mu_0 D_s",
        "B_z/B_0 = exp(-x/lambda_L)",
        "j_0 = B_0/(mu_0 lambda_L)",
        "j_y/j_0 = exp(-x/lambda_L)"
      ],
      "boundary_conditions": [
        "B_z(0) = B_0",
        "B_z(infinity) = 0"
      ],
      "exact_anchors": [
        { "x_over_lambda": 0, "profile": 1 },
        { "x_over_lambda": 1, "profile": 0.36787944117144233 },
        { "x_over_lambda": 2, "profile": 0.1353352832366127 },
        { "x_over_lambda": 3, "profile": 0.049787068367863944 },
        { "x_over_lambda": 5, "profile": 0.006737946999085467 }
      ],
      "integrated_current": "integral from 0 to infinity of j_y dx = B_0/mu_0"
    },
    "limit_order": {
      "classification": "schematic limit-path diagram, not to scale",
      "axes": ["|p|", "omega"],
      "meissner_path": "set omega = 0, project transverse, then take |p| -> 0",
      "meissner_observable": "D_s = lim_{|p| -> 0} K_T(p,0)",
      "optical_path": "set p = 0, analytically continue to the retarded kernel, then take omega -> 0",
      "optical_observable": "For fixed polarization e, D_opt^(e) = lim_{omega -> 0} K_ee^R(0,omega)",
      "control": "A nonsuperconducting state can have D_s = 0 and D_opt > 0 when current overlaps an exactly conserved quantity and has no effective relaxation channel."
    }
  },
  "data": {
    "authoring_csv": "figures-src/many-body-quantum-matter/meissner-screening-response-limit-order.csv",
    "public_csv": "public/figures/many-body-quantum-matter/meissner-screening-response-limit-order.csv",
    "domain": "0 <= x/lambda_L <= 5",
    "step": 0.25,
    "columns": ["x_over_lambda", "B_over_B0", "j_over_j0"],
    "generator": "figures-src/many-body-quantum-matter/meissner-screening-response-limit-order.mjs"
  },
  "visual_encoding": {
    "palette": "black, gray, pale gray, and white",
    "canvas": "explicit opaque white",
    "layout": "two vertically stacked phone-first panels",
    "minimum_label_size": "LaTeX small",
    "color_independence": "line styles, markers, arrow styles, and direct labels redundantly encode every distinction"
  },
  "nonclaims": [
    "The exact half-space profile does not apply to Pippard nonlocality, thin-film Pearl electrodynamics, anisotropic surface eigenmodes, nonlinear fields, or material-specific data.",
    "The limit-path panel is schematic and does not assign a numerical scale to |p| or omega.",
    "A clean lattice is not asserted to have D_opt > 0; interactions and Umklapp can relax current.",
    "The superconducting optical delta coefficient is not asserted to equal D_s when ballistic quasiparticle or other protected contributions remain.",
    "The figure does not derive fluxoid quantization or Josephson relations."
  ],
  "scientific_checks": [
    "Substitution of B_z = B_0 exp(-x/lambda_L) satisfies d^2 B_z/dx^2 = B_z/lambda_L^2 and both half-space boundary conditions.",
    "Ampere's law with B = B_z hat z gives j_y = -(1/mu_0) d B_z/dx = B_0 exp(-x/lambda_L)/(mu_0 lambda_L).",
    "The normalized field and current profiles are identical, positive, and strictly decreasing.",
    "The analytic current integral equals B_0/mu_0.",
    "The static path fixes omega = 0 before the transverse |p| -> 0 limit; the dynamic path fixes p = 0 before omega -> 0.",
    "The nonsuperconducting control separates a protected optical delta from zero equilibrium Meissner weight."
  ],
  "sources": [
    {
      "citation": "J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of Superconductivity, Physical Review 108 (1957) 1175-1204",
      "url": "https://doi.org/10.1103/PhysRev.108.1175",
      "use": "gauge-consistent microscopic Meissner response and local-versus-nonlocal screening"
    },
    {
      "citation": "D. J. Scalapino, S. R. White, and S. C. Zhang, Insulator, Metal, or Superconductor: The Criteria, Physical Review B 47 (1993) 7995-8007",
      "url": "https://doi.org/10.1103/PhysRevB.47.7995",
      "use": "distinct static-transverse and uniform-dynamic current-response limits"
    }
  ],
  "alt_text": "In a local London half-space, the magnetic field and parallel screening current decay over the penetration depth. A second panel approaches the response origin along two different paths: the static transverse path defines Meissner weight, while the spatially uniform dynamic path defines optical or Drude weight and can remain finite in a nonsuperconducting state with protected ballistic current.",
  "caption": "A positive local static transverse kernel gives the exact half-space profiles B_z/B_0 = j_y/j_0 = exp(-x/lambda_L), with j_0 = B_0/(mu_0 lambda_L). The solid static-transverse path to the response origin defines D_s and magnetic screening; the dashed uniform-dynamic path defines D_opt and can remain finite without superconductivity when current has a protected conserved overlap. Panel (a) is exact under the stated isotropic bulk London assumptions; panel (b) is schematic and not to scale.",
  "rights": {
    "basis": "Original QFT.org figure derived from the page equations and limit definitions; no third-party figure, data, layout, or artwork was copied or restyled.",
    "creator": "OpenAI Codex, for QFT.org",
    "date": "2026-08-31"
  }
}