{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.bdg-vortex-core-spectrum-resolution",
  "title": "Conventional vortex-core BdG spectrum and resolution",
  "classification": "original controlled schematic with exact dimensionless fixture curves",
  "reader_question": "Why can a clean conventional s-wave vortex produce a near-zero local spectral feature without a protected zero mode?",
  "dominant_point": "A conventional vortex supports core-localized, particle-hole-paired half-integer CdGM levels; when resolution is comparable to their small spacing, ordinary levels can merge into one zero-bias-looking feature without becoming topological.",
  "scope": {
    "vortex": "clean, singly quantized, axisymmetric, conventional s-wave vortex",
    "regime": "quasiclassical k_F xi_A much greater than one",
    "radial_coordinate": "x = r / xi_A",
    "andreev_length": "xi_A = v_F / Delta_infinity",
    "prescribed_gap_profile": "|Delta(r)| / Delta_infinity = tanh(x)",
    "phase_winding": "Delta(r,phi) is proportional to exp(i phi), so the phase winds by 2 pi",
    "localization_envelope": "W(x) = exp[-2 integral_0^r |Delta(r')| dr' / v_F] = sech(x)^2",
    "e_f_over_delta_infinity": 25,
    "omega0_over_delta_infinity": 0.04,
    "cdgm_formula": "E_mu approximately mu omega0",
    "angular_momenta": [
      -5.5,
      -4.5,
      -3.5,
      -2.5,
      -1.5,
      -0.5,
      0.5,
      1.5,
      2.5,
      3.5,
      4.5,
      5.5
    ],
    "omega0_scale": "omega0 is of order Delta_infinity^2 / E_F; its order-one coefficient depends on the core profile and microscopic model"
  },
  "panels": [
    {
      "id": "radial_profile",
      "meaning": "The prescribed pair amplitude rises from zero while the Andreev quasiparticle envelope decays away from the core.",
      "curves": [
        {
          "id": "gap",
          "formula": "tanh(r / xi_A)",
          "style": "solid"
        },
        {
          "id": "envelope",
          "formula": "sech(r / xi_A)^2",
          "style": "dashed"
        }
      ]
    },
    {
      "id": "cdgm_ladder",
      "meaning": "Half-integer CdGM levels occur in plus-minus pairs and do not include mu = 0 in the conventional asymptotic ladder.",
      "displayed_mu": [
        -5.5,
        -4.5,
        -3.5,
        -2.5,
        -1.5,
        -0.5,
        0.5,
        1.5,
        2.5,
        3.5,
        4.5,
        5.5
      ]
    },
    {
      "id": "resolution_diagnostic",
      "meaning": "The same six equal-weight spectral sticks are convolved with narrow and broad normalized Lorentzian kernels to isolate the effect of resolution alone.",
      "included_mu": [
        -2.5,
        -1.5,
        -0.5,
        0.5,
        1.5,
        2.5
      ],
      "equal_weight": 0.16666666666666666,
      "dimensionless_energy": "epsilon = E / omega0",
      "dimensionless_width": "eta_tilde = eta / omega0",
      "response_formula": "R_eta(epsilon) = (1/6) sum_mu [eta_tilde / (pi ((epsilon - mu)^2 + eta_tilde^2))]",
      "eta_over_omega0": [
        0.12,
        1
      ]
    }
  ],
  "data_files": [
    "bdg-vortex-core-spectrum-resolution-radial.csv",
    "bdg-vortex-core-spectrum-resolution-levels.csv",
    "bdg-vortex-core-spectrum-resolution-resolution.csv"
  ],
  "diagnostics": {
    "max_envelope_identity_residual": 3.3306690738754696e-16,
    "max_resolved_symmetry_residual": 1.6653345369377348e-16,
    "max_merged_symmetry_residual": 5.551115123125783e-17,
    "resolved_nearest_to_zero_ratio": 8.215679275626774,
    "merged_central_contrast": 0.004743061864673349,
    "merged_maximum_location_over_omega0": 0.39,
    "resolved_mixture_integral_over_abs_x_le_1000": 0.9999236054048647,
    "merged_mixture_integral_over_abs_x_le_1000": 0.9993633785830256
  },
  "nonclaims": [
    "The tanh profile is a declared analytic fixture, not a self-consistent material-specific vortex solution.",
    "The formula for omega0 states its parametric scale; the order-one coefficient is not universal.",
    "Equal spectral-stick weights isolate resolution and are not calculated tunnelling matrix elements or a physical vortex local density of states.",
    "The Lorentzian width is an analysis resolution parameter, not a derived quasiparticle lifetime.",
    "A near-zero or zero-bias-looking feature in this fixture is not a bulk invariant, protected Majorana mode, or experimental platform identification.",
    "Only the low-energy CdGM ladder is displayed; the continuum near |E| = Delta_infinity lies outside the plotted energy window."
  ],
  "scientific_checks": [
    "The gap vanishes at the core and approaches Delta_infinity, while the localization envelope equals one minus tanh(x)^2 and decreases monotonically.",
    "Every displayed level has half-integer mu, a partner at minus mu, and energy E_{-mu} = -E_mu; no mu = 0 row exists.",
    "The fixture has E_F / Delta_infinity = 25 and omega0 / Delta_infinity = 0.04, keeping the displayed levels well below the continuum edge.",
    "Both resolution curves use the identical six sticks and equal weights summing to one.",
    "Each Lorentzian component has unit integral on the full real line; finite-cutoff analytic checks approach one within the recorded tolerances.",
    "The narrow kernel resolves the nearest plus-minus pair, whereas the broad curve is flat to within one percent of its maximum across the central unresolved feature."
  ],
  "sources": [
    {
      "citation": "C. Caroli, P. G. de Gennes, and J. Matricon, Bound Fermion States on a Vortex Line in a Type II Superconductor, Physics Letters 9 (1964) 307-309",
      "doi": "10.1016/0031-9163(64)90375-0",
      "use": "half-integer conventional vortex-core ladder and minigap scale"
    },
    {
      "citation": "F. Gygi and M. Schluter, Self-consistent Electronic Structure of a Vortex Line in a Type-II Superconductor, Physical Review B 43 (1991) 7609-7621",
      "doi": "10.1103/PhysRevB.43.7609",
      "use": "self-consistent vortex-core structure and local spectral interpretation boundary"
    }
  ],
  "alt_text": "A conventional vortex gap rises from zero while a core-state envelope decays; a half-integer CdGM ladder has mirrored positive and negative levels but no zero, and stronger Lorentzian broadening merges the nearest ordinary peaks into one central feature.",
  "caption": "Controlled schematic for a clean, singly quantized conventional s-wave vortex in the quasiclassical regime. The prescribed tanh profile suppresses the pair field and localizes quasiparticle weight in the core. Its asymptotic CdGM levels occur in particle-hole-related pairs E_mu approximately mu omega0, with half-integer mu and omega0 of order Delta_infinity squared over E_F; there is no protected zero in this conventional ladder. The final panel convolves equal-weight spectral sticks only to demonstrate resolution: eta comparable to omega0 can make ordinary levels look like one zero-bias feature. It is not material data or a calculated tunnelling local density of states.",
  "rights": {
    "basis": "Original QFT.org curves and layout generated from the declared analytic fixture; no third-party figure, geometry, data, or artwork was copied or restyled.",
    "creator": "OpenAI Codex, for QFT.org",
    "date": "2026-08-31"
  }
}