{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.dmft-self-consistency-and-validation-loop",
  "title": "DMFT self-consistency and validation loop",
  "classification": "schematic process diagram",
  "scope": {
    "method": "single-site equilibrium dynamical mean-field theory",
    "interaction": "local Hubbard interaction",
    "state": "homogeneous one-band normal state in the scalar formulas; matrix indices must be restored for broken symmetry or multiple orbitals",
    "control": "exact local mapping on the declared infinite-dimensional or infinite-connectivity lattice sequence with scaled hopping; approximate in finite dimension"
  },
  "conventions": {
    "kinetic_hamiltonian": "H_t = -sum_{ij,sigma} t_ij c_i_sigma^dagger c_j_sigma",
    "thermal_green_function": "G_ij,sigma(tau) = -<T_tau c_i_sigma(tau) c_j_sigma^dagger(0)>",
    "density_of_states": "rho_0(epsilon) = N^(-1) sum_k delta(epsilon-epsilon_k), integral rho_0 d epsilon = 1",
    "weiss_field": "Gcal_0^(-1) = z + mu - Delta",
    "impurity_dyson_equation": "Sigma_imp = Gcal_0^(-1) - G_imp^(-1)"
  },
  "nodes": [
    {
      "order": 1,
      "id": "inputs",
      "label": "Declare inputs and sector",
      "content": "rho_0(epsilon), U, mu, T, symmetry constraints, and a causal trial local self-energy Sigma^(m)(z)"
    },
    {
      "order": 2,
      "id": "lattice_projection",
      "label": "Project through the lattice",
      "content": "G_loc^(m)(z) = integral d epsilon rho_0(epsilon) / [z + mu - epsilon - Sigma^(m)(z)]"
    },
    {
      "order": 3,
      "id": "weiss_bath",
      "label": "Construct the Weiss field and bath",
      "content": "[Gcal_0^(m)]^(-1) = [G_loc^(m)]^(-1) + Sigma^(m); Delta^(m) = z + mu - [Gcal_0^(m)]^(-1)"
    },
    {
      "order": 4,
      "id": "impurity_solver",
      "label": "Solve the interacting impurity",
      "content": "S_imp[Gcal_0^(m),U] returns G_imp^(m) and the unmixed Sigma_imp^(m)"
    },
    {
      "order": 5,
      "id": "fixed_point_test",
      "label": "Compare the unmixed return",
      "content": "Test G_imp^(m) = G_loc^(m) and Sigma_imp^(m) = Sigma^(m) on the declared frequency window"
    },
    {
      "order": 6,
      "id": "independent_controls",
      "label": "Apply independent controls",
      "content": "Solver and bath refinement; causality and analyticity; spectral moments and normalization; branch continuation and symmetry-sector checks"
    },
    {
      "order": 7,
      "id": "bounded_claim",
      "label": "Bounded local-observable claim",
      "content": "Exact for the declared model as coordination tends to infinity; in finite dimension, a single-site-DMFT result until spatial errors are tested"
    }
  ],
  "edges": [
    {"from": "inputs", "to": "lattice_projection", "style": "solid", "meaning": "required update"},
    {"from": "lattice_projection", "to": "weiss_bath", "style": "solid", "meaning": "required update"},
    {"from": "weiss_bath", "to": "impurity_solver", "style": "solid", "meaning": "required update"},
    {"from": "impurity_solver", "to": "fixed_point_test", "style": "solid", "meaning": "required update"},
    {"from": "fixed_point_test", "to": "lattice_projection", "style": "solid feedback", "condition": "residuals do not pass", "meaning": "mix the returned self-energy and repeat"},
    {"from": "fixed_point_test", "to": "independent_controls", "style": "solid", "condition": "residuals pass", "meaning": "fixed-point convergence is necessary but not sufficient"},
    {"from": "independent_controls", "to": "impurity_solver", "style": "dashed feedback", "condition": "a solver or physicality control fails", "meaning": "refine the solver, bath representation, or postprocessing before repeating"},
    {"from": "independent_controls", "to": "bounded_claim", "style": "solid", "condition": "all declared controls pass", "meaning": "permit only the stated evidence-bounded local result"}
  ],
  "nonclaims": [
    "A small fixed-point residual does not certify an impurity solver, causality, analytic continuation, thermodynamic stability, or finite-dimensional locality.",
    "Mixing changes the numerical iteration but not the stationary DMFT equation.",
    "Failure of one iteration is not by itself a physical spinodal.",
    "The diagram contains no quantitative performance, accuracy, phase-boundary, or cost data."
  ],
  "sources": [
    {
      "citation": "Antoine Georges and Gabriel Kotliar, Physical Review B 45 (1992) 6479-6483, especially pp. 6479-6480, Eqs. (2)-(5)",
      "doi": "10.1103/PhysRevB.45.6479",
      "use": "self-consistent impurity mapping"
    },
    {
      "citation": "Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, Reviews of Modern Physics 68 (1996) 13-125, especially Sections II-III",
      "doi": "10.1103/RevModPhys.68.13",
      "use": "locality, cavity construction, Weiss field, and DMFT self-consistency"
    },
    {
      "citation": "Emanuel Gull et al., Reviews of Modern Physics 83 (2011) 349-404, especially Section II.2 and Sections IX-X",
      "doi": "10.1103/RevModPhys.83.349",
      "use": "Monte Carlo sampling errors, estimators, performance, and technical controls"
    }
  ]
}