{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.fermi-surface-shell-sector-and-channels",
  "title": "Whole-surface shell scaling, local curved sectors, and exceptional interaction channels",
  "scope": {
    "system": "normal fermions with a smooth, strictly convex two-dimensional Fermi surface",
    "state": "zero or low temperature with a sharp quasiparticle pole over the retained shell",
    "interaction": "weak, nonsingular, short-range four-fermion interaction",
    "geometry": "one connected inversion-symmetric Fermi curve; lattice anisotropy is allowed only while smoothness and nonzero curvature remain",
    "units": "hbar = k_B = 1"
  },
  "scale_status": {
    "overall": "schematic",
    "panel_a": "Shell thickness, patch arc length, and the two highlighted antipodal regions are not drawn to scale.",
    "panel_b": "The parabolic curvature term and cutoff inequality are equation-checked; displayed curvature and widths are schematic.",
    "panel_c": "Marker positions illustrate the limiting forward and zero-total-momentum Cooper families rather than a numerical scattering event."
  },
  "scaling_schemes": [
    {
      "id": "whole-surface-shell",
      "coordinates": "k = k_F(n_hat) + k_perp n_hat",
      "scaled": [
        "omega -> s omega",
        "xi -> s xi",
        "k_perp -> s k_perp"
      ],
      "unscaled": "the Fermi-surface point n_hat and its surface measure",
      "momentum_frequency_field_scaling": "psi -> s^(-3/2) psi",
      "use": "global Wilsonian classification of the angular coupling functions"
    },
    {
      "id": "curvature-resolved-local-sector",
      "dispersion": "xi = v_F k_perp + k_parallel^2/(2 m_c) + higher-order terms",
      "scaled": [
        "omega -> s omega",
        "k_perp -> s k_perp",
        "k_parallel -> s^(1/2) k_parallel"
      ],
      "cutoff_assumption": "Lambda_parallel tracks Lambda_perp^(1/2) up to a scale-independent factor so that curvature remains leading; another mesh exponent defines another rescaling and field dimension.",
      "momentum_frequency_field_scaling_in_d_dimensions": "psi -> s^(-(d+5)/4) psi",
      "two_dimensional_specialization": "psi -> s^(-7/4) psi",
      "use": "sector counting, curvature control, and local patch theories"
    }
  ],
  "density_of_states": {
    "total": "N(0) = g integral_FS dS_k / [(2 pi)^d v_F(k)]",
    "two_spin_pair_measure": "nu_pair(0) = N(0)/2",
    "normalized_surface_measure": "d mu(n_hat) = dS_k / [(2 pi)^d v_F(k) nu_pair(0)], with integral_FS d mu = 1",
    "cooper_eigenproblem": "integral_FS d mu(n_hat_prime) V(n_hat,n_hat_prime) phi_a(n_hat_prime) = V_a phi_a(n_hat)",
    "dimensionless_eigenvalue": "lambda_a = nu_pair(0) V_a"
  },
  "cutoff_relations": {
    "energy_cutoff": "E_Lambda = v_F Lambda_perp",
    "curvature_condition": "Lambda_parallel^2/(2 m_c) <= E_Lambda",
    "maximal_natural_tangential_width": "Lambda_parallel approximately sqrt(2 m_c v_F Lambda_perp)",
    "circular_quadratic_band": "m_c v_F = k_F, so Lambda_parallel approximately sqrt(2 k_F Lambda_perp)",
    "patch_count_in_two_dimensions": "N_patch approximately L_F/(2 Lambda_parallel) for a nonoverlapping equal-width partition; the square-root count belongs to curvature-tracking sectors, while a narrower whole-surface cell decomposition is different bookkeeping rather than the same anisotropic fixed-point scaling",
    "double_counting_control": "Use a disjoint partition or a partition of unity whose weights sum to one on the retained shell."
  },
  "channel_families": [
    {
      "id": "forward",
      "kinematics": "outgoing momenta approach the incoming patches as transfer q -> 0",
      "low_energy_data": "Landau forward-scattering function",
      "exchange_note": "The exchanged forward configuration is related by fermionic antisymmetry and is not a separately countable coupling."
    },
    {
      "id": "cooper",
      "kinematics": "(k,-k) -> (k_prime,-k_prime) at zero total momentum",
      "low_energy_data": "angular or point-group pairing kernel",
      "one_loop_status": "with V_a defined using the normalized per-species surface measure and lambda_a = nu_pair(0) V_a, each dimensionless eigenvalue obeys d lambda_a/dt = -lambda_a^2 at leading logarithmic order"
    },
    {
      "id": "generic-fixed-angle-transfer",
      "kinematics": "a generic fixed-angle transfer cannot keep the full internal phase space inside a shrinking regular shell",
      "low_energy_status": "not logarithmically enhanced and suppressed relative to the exceptional families in the stated regular regime",
      "local_power_counting_qualification": "A smooth contact vertex written with all four fields in one common anisotropic patch frame scales as s^((d-1)/2). Distinct patch normals do not share one homogeneous local momentum delta; the global classification retains and sums the unscaled patch labels."
    }
  ],
  "visual_encoding": {
    "retained_shell": "pale annulus or band bounded by dashed curves",
    "fermi_surface": "solid black curve",
    "highlighted_patches": "heavy black arcs",
    "incoming_momenta": "filled markers and a solid Cooper diameter",
    "outgoing_momenta": "open markers and a dashed Cooper diameter",
    "cutoffs": "directly labeled braces or double arrows"
  },
  "exclusions": [
    "One spatial dimension, where the Fermi surface is two points and generic four-fermion interactions are marginal.",
    "Nested segments, van Hove points, and inflection points with vanishing or anomalously small curvature.",
    "Singular long-range gauge or Coulomb kernels and fermions coupled to additional gapless bosons.",
    "A proof of an ordered phase below a Cooper strong-coupling stopping scale.",
    "A unique patch number independent of regulator and partition choice."
  ],
  "sources": [
    {
      "citation": "Joseph Polchinski, Effective Field Theory and the Fermi Surface, TASI 1992 lectures, arXiv manuscript pp. 13-23",
      "url": "https://arxiv.org/pdf/hep-th/9210046",
      "use": "fixed Fermi-surface label, normal scaling, special four-fermion kinematics, and Cooper flow"
    },
    {
      "citation": "Ramamurti Shankar, Renormalization-Group Approach to Interacting Fermions, Reviews of Modern Physics 66 (1994) 129-192, sections V-VII",
      "url": "https://doi.org/10.1103/RevModPhys.66.129",
      "use": "whole-surface Wilson RG, forward and Cooper angular functions, one-loop channel flows, and patch-number control"
    },
    {
      "citation": "Giuseppe Benfatto, Alessandro Giuliani, and Vieri Mastropietro, Fermi Liquid Behavior in the 2D Hubbard Model at Low Temperatures, Annales Henri Poincare 7 (2006) 809-898",
      "url": "https://doi.org/10.1007/s00023-006-0270-z",
      "use": "curvature-resolved angular sectors and the bounded finite-temperature weak-coupling application"
    }
  ],
  "semantic_equivalent": {
    "page_route": "/many-body-quantum-matter/fermi-liquids-beyond/patch-rg-instabilities/",
    "sections": [
      "One surface supports two useful scalings",
      "Momentum conservation selects exceptional channels"
    ],
    "note": "The comparison table and adjacent derivations preserve every scaling rule, cutoff relation, channel family, and validity qualification without relying on the SVG."
  }
}