{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.cooper-instability-kinematics-flow",
  "title": "Cooper-shell kinematics and signed eigenchannel flow",
  "classification": "original schematic with an exact representative one-loop flow",
  "reader_question": "Why does a zero-total-momentum pair retain enough low-energy phase space to produce the Cooper logarithm, and what does the resulting one-loop flow say for attractive and repulsive eigenchannels?",
  "dominant_point": "Antipodal partners can both remain in a shrinking shell while the full Fermi-surface label survives; after channel projection, repulsion flows toward zero whereas attraction reaches a finite logarithmic scale where the normal-state expansion fails, without by itself determining the transition temperature or ordered state.",
  "scope": {
    "dimension": "d >= 2; panel (a) draws a two-dimensional cross-section",
    "state": "normal Fermi liquid with well-defined quasiparticles over the retained shell",
    "fermi_surface": "one smooth inversion-symmetric Fermi surface with xi(-k) = xi(k); the time-reversed internal or Kramers label is suppressed",
    "interaction": "weak, nonsingular, antisymmetrized four-fermion interaction projected into independent pairing eigenchannels",
    "pair_momentum": "zero total momentum in panel (a)",
    "approximation": "constant density of states over the shell and leading-logarithmic one-loop flow",
    "units": "hbar = k_B = 1"
  },
  "conventions": [
    "N(0) is the density of states per paired species in the normalization used on the owner page.",
    "The signed dimensionless eigenvalue is g = N(0) V: g < 0 is attractive and g > 0 is repulsive.",
    "The logarithmic flow variable is ell = log(Lambda_0/Lambda).",
    "E denotes the active infrared cutoff, such as temperature, finite total pair momentum, imbalance, pair breaking, or a competing instability scale.",
    "Filled markers and a solid diameter denote the incoming antipodal pair; open markers and a dashed diameter denote the outgoing antipodal pair.",
    "Black solid and gray dashed curves, together with direct labels, distinguish attraction from repulsion; color carries no information."
  ],
  "layout": {
    "reading_order": [
      "panel (a): retained shell, incoming and outgoing antipodal pairs, full Fermi-surface label, and logarithmic radial measure",
      "panel (b): beta function, repulsive solution, attractive solution, and the finite failure scale"
    ],
    "orientation": "single vertical stack",
    "phone_first": true,
    "horizontal_panning_required": false,
    "minimum_label_size": "LaTeX small"
  },
  "panels": [
    {
      "id": "antipodal-shell-kinematics",
      "label": "(a) Antipodal shell kinematics",
      "scale_status": "schematic; shell thickness, angles, and marker positions are not to scale",
      "objects": [
        "smooth Fermi surface",
        "retained shell |xi| < Lambda",
        "incoming filled pair (k,-k)",
        "outgoing open pair (k_prime,-k_prime)",
        "unscaled full-surface label n_hat"
      ],
      "relations": [
        "k + (-k) = 0 and k_prime + (-k_prime) = 0",
        "xi(-k) = xi(k) in the declared inversion-symmetric illustration",
        "G(k,i omega) G(-k,-i omega) = 1/(omega^2 + xi^2)",
        "Pi_pp(E) approximately equals N(0) integral_E^Lambda d|xi|/|xi| = N(0) log(Lambda/E)"
      ],
      "meaning": "The radial energy integral shrinks, but the pair can rotate through the full Fermi surface while both propagators remain low energy; that surviving phase space produces the logarithm."
    },
    {
      "id": "signed-eigenchannel-flow",
      "label": "(b) Each eigenchannel flows separately",
      "scale_status": "the curves are exact solutions of the displayed one-loop equation for the stated dimensionless initial values; they are not material fits",
      "beta_function": "d g/d ell = -g^2",
      "solution": "g(ell) = g_0/(1 + g_0 ell)",
      "examples": [
        {
          "id": "repulsive",
          "initial_value": 0.2,
          "sign": "repulsive",
          "domain": "ell >= 0",
          "behavior": "g(ell) remains positive and approaches 0 from above"
        },
        {
          "id": "attractive",
          "initial_value": -0.2,
          "sign": "attractive",
          "domain": "0 <= ell < 5",
          "pole_scale": 5,
          "energy_scale": "E_* = Lambda_0 exp(-5)",
          "behavior": "g(ell) becomes increasingly negative and the normal-state one-loop solution terminates at ell_* = 5"
        }
      ],
      "meaning": "The finite attractive pole is a breakdown scale of perturbation theory around the normal state. It is not, without a paired saddle and fluctuation analysis, a calculation of T_c."
    }
  ],
  "relations": [
    {
      "from": "antipodal shell phase space",
      "to": "Cooper logarithm",
      "meaning": "The full surface label and radial d|xi|/|xi| measure survive at zero total pair momentum."
    },
    {
      "from": "Cooper logarithm",
      "to": "independent channel flows",
      "meaning": "Diagonalizing the antisymmetrized pairing kernel isolates signed eigenvalues whose leading logarithms resum separately."
    },
    {
      "from": "attractive pole",
      "to": "normal-state breakdown",
      "meaning": "The pole licenses an instability claim, not a completed phase, transition temperature, phase stiffness, or microscopic mechanism."
    }
  ],
  "nonclaims": [
    "The circular drawing does not assert rotational invariance for the general result; a smooth lattice Fermi surface uses its own surface measure and point-group eigenfunctions.",
    "The figure does not cover a vanishing density of states, a van Hove singularity, nesting, one dimension, singular long-range interactions, or non-Fermi-liquid propagators.",
    "Finite pair momentum, population imbalance, orbital mismatch, pair-breaking fields, retardation, and competing channels can cut off or reorganize the flow and are not drawn.",
    "The attractive pole does not by itself identify a mediator, select the nonlinear combination within a degenerate representation, prove phase coherence, or establish superconductivity.",
    "The shell width, angular positions, curve colors, and arrow lengths encode no material parameter, probability, or uncertainty."
  ],
  "scientific_checks": [
    "With xi(-k) = xi(k), multiplying the two normal-state propagators gives 1/(omega^2 + xi^2) with a positive denominator.",
    "The Matsubara sum gives tanh(xi/2T)/(2 xi); integrating both signs of xi with N(0) per paired species yields the leading N(0) log(Lambda/E) term.",
    "The displayed solution differentiates to d g/d ell = -g^2 and satisfies g(0) = g_0.",
    "For g_0 = -0.20, 1 + g_0 ell vanishes at ell_* = 5; for g_0 = +0.20, the denominator grows and g approaches zero from above.",
    "The corresponding scale E_* = Lambda_0 exp(-5) is identified only as the normal-state breakdown scale, not as T_c.",
    "Every distinction remains available in monochrome through marker fill, line style, direct labels, and panel order."
  ],
  "sources": [
    {
      "citation": "L. N. Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Physical Review 104 (1956) 1189-1190",
      "doi": "10.1103/PhysRev.104.1189",
      "use": "zero-total-momentum pair instability"
    },
    {
      "citation": "Joseph Polchinski, Effective Field Theory and the Fermi Surface, TASI 1992 lectures",
      "url": "https://arxiv.org/abs/hep-th/9210046",
      "use": "surviving Fermi-surface label, exceptional Cooper kinematics, and leading-logarithmic flow"
    },
    {
      "citation": "Ramamurti Shankar, Renormalization-Group Approach to Interacting Fermions, Reviews of Modern Physics 66 (1994) 129-192",
      "doi": "10.1103/RevModPhys.66.129",
      "use": "channel decomposition, signed beta function, solution, and validity assumptions"
    }
  ],
  "alt_text": "A thin shell surrounds a Fermi surface, with filled incoming momenta k and minus k and open outgoing momenta k-prime and minus k-prime at antipodal points; their zero total momentum preserves the full surface label and produces the radial logarithm. Below, the exact one-loop flow sends g0 equals plus 0.20 toward zero, while g0 equals minus 0.20 runs to a pole at ell equals 5, where the normal-state expansion fails rather than directly defining T_c.",
  "caption": "Zero-total-momentum partners can stay antipodal inside the shrinking shell while the full Fermi-surface label remains, leaving the radial measure d|xi|/|xi| and its Cooper logarithm. After symmetry projection, the representative repulsive channel g0 = +0.20 flows toward zero, whereas the attractive channel g0 = -0.20 reaches ell_* = 5. That pole marks failure of the normal-state expansion, not by itself the transition temperature, ordered state, or pairing mechanism. Panel (a) is schematic; panel (b) plots the displayed one-loop solution.",
  "rights": {
    "basis": "Original QFT.org diagram derived from independently checked equations and relations; no third-party figure, geometry, data, or artwork was copied or restyled.",
    "creator": "OpenAI Codex, for QFT.org",
    "date": "2026-08-31"
  }
}