{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.hubbard-band-spectral-weight-transfer",
  "title": "Hubbard-band spectral-weight counting, dynamical transfer, and quasiparticle-residue distinction",
  "classification": "schematic constrained by exact spectral-weight identities",
  "scope": {
    "model": "repulsive one-band Hubbard model",
    "state": "zero-temperature atomic counting and a hole-doped strong-coupling regime with separated Hubbard bands",
    "strong_coupling_window": "t << Omega_L << U separates the low-energy addition sector from the upper Hubbard band",
    "strong_coupling_order": "the finite-hopping transfer is retained through first order in t/U, with omitted terms of order t^2/U^2",
    "projected_kinetic_energy_convention": "epsilon_kinetic^P = <Psi_P|H_t|Psi_P>/L for a normalized no-doublon state P|Psi_P> = |Psi_P>; it is negative in the usual hopping convention, so alpha = -2 epsilon_kinetic^P/U is positive through the displayed order",
    "filling": "electron density n = 1 - x with 0 < x < 1 in the doped panels",
    "normalization": "spin-summed single-particle spectral weight per site integrates to 2",
    "frequency": "omega is measured relative to the declared chemical potential",
    "units": "hbar = k_B = 1"
  },
  "status_by_panel": {
    "panel_a": "The half-filled atomic pole locations and spin-summed integrated weights are exact. Stick heights are not a plotted spectral density and are schematic.",
    "panel_b": "The atomic weights and total-weight identities are exact. Alpha is a symbolic, cutoff-dependent dynamical transfer in a separated-band strong-coupling description; equal box widths do not encode numerical weights.",
    "panel_c": "The definitions and non-equivalence of W_low(Omega) and Z_kF are exact. The displayed curve and pole height are explanatory shapes, not solver data or a universal transition profile."
  },
  "panels": [
    {
      "id": "atomic-half-filling",
      "conditions": [
        "t = 0",
        "T = 0",
        "paramagnetic half filling",
        "mu = U/2"
      ],
      "relations": [
        "The spin-summed removal pole is at omega = -U/2 with integrated weight 1.",
        "The spin-summed addition pole is at omega = +U/2 with integrated weight 1.",
        "Removal plus addition weight equals 2."
      ]
    },
    {
      "id": "hole-doping-transfer",
      "conditions": [
        "0 < x < 1",
        "n = 1 - x",
        "T = 0 and t/U << 1",
        "t << Omega_L << U keeps upper and low-energy addition sectors separable",
        "the creation operator and Hamiltonian are transformed consistently into the no-doublon low-energy subspace"
      ],
      "atomic_weights": {
        "removal": "1 - x",
        "low_energy_addition": "2 x",
        "upper_hubbard_band_addition": "1 - x",
        "total_addition": "1 + x",
        "total_removal_plus_addition": "2"
      },
      "finite_hopping_bookkeeping": {
        "total_removal": "1 - x",
        "low_energy_addition": "2 x + alpha",
        "upper_hubbard_band_addition": "1 - x - alpha",
        "total_addition": "1 + x",
        "total_removal_plus_addition": "2",
        "alpha_leading_order": "alpha = -2 epsilon_kinetic^P/U + O(t^2/U^2), where epsilon_kinetic^P = <Psi_P|H_t|Psi_P>/L and P|Psi_P> = |Psi_P>",
        "alpha": "cutoff- and model-dependent dynamical transfer from the upper-band addition sector into the low-energy addition sector"
      },
      "nonnegativity_condition": "For the displayed separated-band bookkeeping, 0 <= alpha <= 1 - x. The common hole-doped strong-coupling case has downward transfer alpha > 0, but the figure does not assign a numerical value."
    },
    {
      "id": "integrated-weight-versus-residue",
      "relations": [
        "W_low(Omega) is a spin-summed local spectral integral over the chosen interval [-Omega, Omega].",
        "W_low(Omega) contains every coherent and incoherent contribution inside that window.",
        "Z_kF is the residue of one momentum-resolved Fermi-level pole when a Fermi-liquid expansion exists.",
        "W_low(Omega) is not equal to Z_kF, 2 x, or 2 x + alpha."
      ]
    }
  ],
  "visual_encoding": {
    "removal_sector": "solid outline with pale gray fill and a direct text label",
    "low_energy_addition_sector": "heavier solid outline with white fill and a direct text label",
    "upper_band_addition_sector": "dashed gray outline with very pale fill and a direct text label",
    "dynamical_transfer": "solid curved arrow labeled alpha from upper-band to low-energy addition",
    "local_integration_window": "pale vertical band bounded by -Omega and +Omega",
    "quasiparticle_pole": "separate solid vertical stick with an open endpoint and a direct Z_kF label",
    "color_dependency": "none; category and status remain legible through text, fill, weight, and dash pattern"
  },
  "reader_takeaway": "The exact total spectral weight is conserved while doping and finite hopping repartition it among removal and addition sectors; the integrated low-energy area remains a different object from the quasiparticle residue.",
  "intended_alt_text": "Three stacked panels show equal half-filled atomic removal and addition poles whose spin-summed weights total two; hole doping gives removal weight 1 minus x, low-energy addition 2x, and upper-band addition 1 minus x, while finite hopping transfers a symbolic alpha from upper- to low-energy addition without changing the total; a final comparison separates a local low-energy integration window from the residue of one momentum-resolved quasiparticle pole.",
  "intended_caption": "Spectral-weight bookkeeping for the repulsive one-band Hubbard model. Exact atomic counting fixes total spin-summed weight 2 and, at hole density x, addition weight 1+x. In a separated-band strong-coupling description through first order in t/U, finite hopping transfers a cutoff-dependent amount alpha = -2 epsilon_kinetic^P/U from upper-band to low-energy addition, with epsilon_kinetic^P < 0 in the usual convention. The local integrated weight W_low(Omega) includes coherent and incoherent contributions and is not the momentum-resolved pole residue Z_kF. Pole sticks, boxes, curves, and alpha are schematic and not a universal transition profile.",
  "scientific_checks": [
    "At x = 0 and alpha = 0, both half-filled atomic sectors have spin-summed weight 1.",
    "Atomic hole counting gives (1 - x) + 2 x + (1 - x) = 2.",
    "Finite-hopping bookkeeping gives (1 - x) + (2 x + alpha) + (1 - x - alpha) = 2.",
    "The addition-sector sum remains 1 + x before and after the alpha transfer.",
    "Through first order in t/U, alpha = -2 epsilon_kinetic^P/U is positive when the projected kinetic energy is negative in the declared convention.",
    "At x = 0 in the no-hole projected state, P H_t P cannot move an electron, so epsilon_kinetic^P and the leading alpha vanish.",
    "The upper-band addition weight remains nonnegative when alpha <= 1 - x.",
    "No visual area or box width is asserted to equal Z_kF."
  ],
  "exclusions": [
    "A quantitative DMFT, exact-diagonalization, or experimental spectrum.",
    "A universal line shape, critical interaction, phase boundary, or coherent-peak trajectory.",
    "A definition of a Mott state from two peaks alone.",
    "An alpha value without a Hamiltonian, temperature, band-separating cutoff, and controlled calculation.",
    "Electron doping above one electron per site without a particle-hole reformulation."
  ],
  "sources": [
    {
      "citation": "A. Brooks Harris and Robert V. Lange, Single-Particle Excitations in Narrow Energy Bands, Physical Review 157 (1967) 295-314",
      "url": "https://doi.org/10.1103/PhysRev.157.295",
      "use": "strong-coupling Hubbard-band moments and transfer"
    },
    {
      "citation": "H. Eskes, M. B. J. Meinders, and G. A. Sawatzky, Anomalous Transfer of Spectral Weight in Doped Strongly Correlated Systems, Physical Review Letters 67 (1991) 1035-1038",
      "url": "https://doi.org/10.1103/PhysRevLett.67.1035",
      "use": "atomic 2x hole-counting rule and dynamical spectral-weight transfer"
    },
    {
      "citation": "M. B. J. Meinders, H. Eskes, and G. A. Sawatzky, Spectral-Weight Transfer: Breakdown of Low-Energy-Scale Sum Rules in Correlated Systems, Physical Review B 48 (1993) 3916-3926",
      "url": "https://doi.org/10.1103/PhysRevB.48.3916",
      "use": "redistribution between low- and high-energy spectral sectors at finite hopping"
    },
    {
      "citation": "Mohit Randeria, Rajdeep Sensarma, Nandini Trivedi, and Fu-Chun Zhang, Particle-Hole Asymmetry in Doped Mott Insulators: Implications for Tunneling and Photoemission Spectroscopies, Physical Review Letters 95 (2005) 137001",
      "url": "https://doi.org/10.1103/PhysRevLett.95.137001",
      "use": "matched low-energy electron operator and alpha = -2 epsilon_kinetic^P/U through first order in t/U"
    },
    {
      "citation": "Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, Metal-Insulator Transitions, Reviews of Modern Physics 70 (1998) 1039-1263",
      "url": "https://doi.org/10.1103/RevModPhys.70.1039",
      "use": "coherent versus incoherent spectral weight and Mott-diagnostic limitations"
    }
  ],
  "consumer": {
    "page_id": "qft.topic.correlated-electrons.mott-transition-spectral-weight",
    "route": "/many-body-quantum-matter/correlated-lattice-fermions-mott-physics/mott-transition-spectral-weight/",
    "placement": "after the matched-operator alpha derivation and before the reflowing spectral-weight table",
    "integration_status": "embedded with full-size SVG access, a relationship-oriented alt text and caption, adjacent derivation, an equivalent table, and a public semantic-JSON link"
  }
}