{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.bcs-spectrum-coherence-factors",
  "title": "BCS quasiparticle spectrum and coherence factors",
  "classification": "exact dimensionless curves for a uniform BCS mean-field saddle",
  "reader_question": "How does a uniform pairing saddle open a quasiparticle gap at the normal-state crossing, and how do the particle and hole weights change across the Fermi surface?",
  "dominant_point": "The BdG eigenvalues are particle-hole partners separated by 2|Delta| at xi=0, while the minimum positive quasiparticle excitation is |Delta|; on the positive-energy branch, u^2 and v^2 exchange continuously and are equal at the Fermi surface.",
  "scope": {
    "system": "uniform, spin-balanced, two-component BCS mean-field saddle in one reduced Nambu block",
    "dispersion_assumption": "The accessible normal-state dispersion crosses the chemical potential on a regular Fermi surface, so xi=0 occurs and the displayed minimum at xi=0 is physically available.",
    "pairing": "momentum-independent s-wave pairing amplitude over the displayed local dispersion window",
    "hamiltonian": "H_BdG = xi tau_3 + Delta tau_1",
    "basis_convention": "The uniform gap is chosen real and positive, Delta > 0, by a Nambu-basis phase convention; the plotted energies and squared weights are phase independent.",
    "branch_convention": "Panel (b) shows the normalized positive-energy eigenvector in the basis (c_(k up), c^dagger_(-k down))^T.",
    "temperature": "The normalized curves apply for any fixed nonzero mean-field |Delta(T)|; they do not determine Delta(T).",
    "units": "All horizontal and energy variables are divided by |Delta|; the weights are dimensionless."
  },
  "data": {
    "authoring_path": "figures-src/many-body-quantum-matter/bcs-spectrum-coherence-factors.csv",
    "public_path": "public/figures/many-body-quantum-matter/bcs-spectrum-coherence-factors.csv",
    "generator": "figures-src/many-body-quantum-matter/bcs-spectrum-coherence-factors.mjs",
    "domain": {
      "xi_over_abs_delta_min": -4,
      "xi_over_abs_delta_max": 4,
      "step": 0.02,
      "data_rows": 401
    },
    "columns": [
      "xi_over_abs_delta",
      "eplus_over_abs_delta",
      "eminus_over_abs_delta",
      "normal_particle_over_abs_delta",
      "normal_hole_over_abs_delta",
      "u_squared",
      "v_squared",
      "u_times_v"
    ],
    "numeric_format": "The abscissa is fixed to two decimal places and every derived value to twelve decimal places; the file uses UTF-8, comma separators, LF line endings, and a final newline."
  },
  "panels": [
    {
      "id": "spectrum",
      "label": "(a) Gap opening and particle-hole partners",
      "horizontal_axis": "x = xi/|Delta|",
      "vertical_axis": "E/|Delta|",
      "curves": [
        {
          "id": "positive_bdg_branch",
          "formula": "+sqrt(x^2 + 1)",
          "style": "black solid"
        },
        {
          "id": "negative_bdg_branch",
          "formula": "-sqrt(x^2 + 1)",
          "style": "black solid"
        },
        {
          "id": "normal_particle_branch",
          "formula": "+x",
          "style": "gray dashed"
        },
        {
          "id": "normal_hole_branch",
          "formula": "-x",
          "style": "gray dashed"
        }
      ],
      "landmarks": [
        {
          "x": 0,
          "positive_energy": 1,
          "negative_energy": -1,
          "meaning": "The minimum positive excitation is |Delta| and the partner-branch separation is 2|Delta|."
        }
      ],
      "meaning": "Pairing removes the normal Nambu crossing and creates particle-hole-related branches; the negative branch is the doubled partner of the positive branch, not a second independent orbital."
    },
    {
      "id": "coherence_weights",
      "label": "(b) Weights on the positive-energy branch",
      "horizontal_axis": "x = xi/|Delta|",
      "vertical_axis": "dimensionless weight",
      "curves": [
        {
          "id": "u_squared",
          "formula": "(1 + x/sqrt(x^2 + 1))/2",
          "style": "black solid"
        },
        {
          "id": "v_squared",
          "formula": "(1 - x/sqrt(x^2 + 1))/2",
          "style": "gray dashed"
        }
      ],
      "identities": [
        "u^2 + v^2 = 1",
        "u^2 - v^2 = xi/E",
        "u v = |Delta|/(2E) for the displayed positive real basis convention"
      ],
      "landmarks": [
        {
          "x": 0,
          "u_squared": 0.5,
          "v_squared": 0.5,
          "u_times_v": 0.5,
          "meaning": "The positive-energy quasiparticle is an equal particle-hole mixture at the normal-state Fermi surface."
        }
      ],
      "meaning": "Far below the Fermi surface the positive-energy branch is hole-like; far above it the same branch is particle-like, with a continuous interchange across xi=0."
    }
  ],
  "selected_anchors": [
    {
      "x": -4,
      "energy": 4.123105625618,
      "u_squared": 0.014928749927,
      "v_squared": 0.985071250073,
      "u_times_v": 0.121267812518
    },
    {
      "x": -1,
      "energy": 1.414213562373,
      "u_squared": 0.146446609407,
      "v_squared": 0.853553390593,
      "u_times_v": 0.353553390593
    },
    {
      "x": 0,
      "energy": 1,
      "u_squared": 0.5,
      "v_squared": 0.5,
      "u_times_v": 0.5
    },
    {
      "x": 1,
      "energy": 1.414213562373,
      "u_squared": 0.853553390593,
      "v_squared": 0.146446609407,
      "u_times_v": 0.353553390593
    },
    {
      "x": 4,
      "energy": 4.123105625618,
      "u_squared": 0.985071250073,
      "v_squared": 0.014928749927,
      "u_times_v": 0.121267812518
    }
  ],
  "visual_encoding": {
    "paired_branches_and_u_squared": "black solid curves with direct labels",
    "normal_branches_and_v_squared": "gray dashed curves with direct labels",
    "exact_xi_zero_landmarks": "circular markers",
    "canvas": "explicit white",
    "color_dependency": "none",
    "reading_order": "spectrum panel first, coherence-weight panel second, normalization identities last",
    "phone_layout": "one narrow vertical stack designed to fit the 320-pixel content column without local horizontal panning",
    "minimum_label_size": "LaTeX small"
  },
  "nonclaims": [
    "The curves do not solve the gap or number equation, determine |Delta|, or predict T_c.",
    "The negative-energy branch is a particle-hole-redundant BdG partner and is not counted as a second physical orbital.",
    "The excitation threshold |Delta| and the 2|Delta| separation of the displayed BdG branches are distinct quantities; neither should be silently substituted for the spectroscopic threshold of a specific probe.",
    "If the allowed single-particle states do not include xi=0, as can occur when the chemical potential lies below a band minimum, the global positive-energy minimum need not equal |Delta|; that regime is outside this regular-Fermi-surface plot.",
    "The figure does not establish phase stiffness, Meissner response, a microscopic pairing mediator, a material gap value, or validity beyond the uniform mean-field saddle.",
    "Anisotropic, multiband, spin-orbit-coupled, imbalanced, retarded, disordered, and strong-coupling systems require enlarged kernels or modified self-consistency and are outside the plotted model.",
    "Line gray values, plot dimensions, and the finite domain |xi| <= 4|Delta| carry no uncertainty or material-specific meaning."
  ],
  "scientific_checks": [
    "Diagonalizing xi tau_3 + Delta tau_1 gives eigenvalues plus and minus sqrt(xi^2 + Delta^2).",
    "At xi=0, the positive excitation is |Delta| and the separation between the partner branches is 2|Delta|.",
    "Every generated row satisfies E^2 = xi^2 + |Delta|^2, u^2 + v^2 = 1, and 2uv = |Delta|/E within the recorded numerical tolerance.",
    "The generated data obey E(-x) = E(x) and u^2(-x) = v^2(x).",
    "As x tends to positive infinity, u^2 tends to one and v^2 tends to zero; the limits exchange as x tends to negative infinity.",
    "The TeX source reads the exact governed CSV columns named in this record; every rendered curve and landmark has a matching semantic entry."
  ],
  "sources": [
    {
      "citation": "John Bardeen, Leon N. Cooper, and J. Robert Schrieffer, Theory of Superconductivity, Physical Review 108 (1957) 1175-1204",
      "doi": "10.1103/PhysRev.108.1175",
      "use": "mean-field quasiparticle spectrum and coherence factors"
    },
    {
      "citation": "Yoichiro Nambu, Quasi-Particles and Gauge Invariance in the Theory of Superconductivity, Physical Review 117 (1960) 648-663",
      "doi": "10.1103/PhysRev.117.648",
      "use": "particle-hole doubled representation and basis-covariant interpretation"
    }
  ],
  "alt_text": "Two vertically stacked plots. In the first, solid particle-hole partner branches bend away from the dashed normal-state crossing, leaving minimum positive energy |Delta| and a 2|Delta| separation at xi=0. In the second, u squared rises from nearly zero to nearly one while v squared falls from nearly one to nearly zero; the two weights cross at one half when xi=0.",
  "caption": "For a uniform BCS saddle whose normal-state dispersion crosses xi=0, the Nambu branches plus and minus sqrt(xi squared plus |Delta| squared) avoid the normal crossing. The minimum positive excitation is |Delta|, whereas the separation of the particle-hole partner branches at xi=0 is 2|Delta|. On the positive branch, particle and hole weights interchange through u squared equals v squared equals one half while their sum remains one. The curves are exact dimensionless BCS formulas, not material data.",
  "rights": {
    "basis": "Original QFT.org quantitative figure generated from exact equations; no third-party figure, geometry, data, or artwork was copied, traced, or restyled.",
    "creator": "OpenAI Codex, for QFT.org",
    "date": "2026-08-31"
  }
}