{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.cluster-dmft-embedding-geometry",
  "title": "CDMFT and DCA embedding geometries",
  "classification": "schematic comparison of cluster embeddings",
  "panels": [
    {
      "id": "A",
      "title": "CDMFT real-space embedding",
      "purpose": "Show an open real-space cluster embedded through boundary coupling in a self-consistent square-lattice environment.",
      "object_ids": [
        "cdmft_lattice_environment",
        "cdmft_open_cluster",
        "cdmft_cluster_self_energy",
        "cdmft_boundary_coupling",
        "cdmft_translation_property"
      ]
    },
    {
      "id": "B",
      "title": "DCA momentum-space embedding",
      "purpose": "Show a periodic cluster and the corresponding decomposition of a square Brillouin zone into four equal momentum patches.",
      "object_ids": [
        "dca_brillouin_zone",
        "dca_patch_1",
        "dca_patch_2",
        "dca_patch_3",
        "dca_patch_4",
        "dca_representative_momentum",
        "dca_periodic_cluster",
        "dca_patch_self_energy",
        "dca_momentum_resolution",
        "dca_translation_property"
      ]
    }
  ],
  "objects": [
    {
      "id": "cdmft_lattice_environment",
      "panel_id": "A",
      "kind": "lattice environment",
      "content": "A finite drawn fragment represents the self-consistent environment of an otherwise thermodynamic-limit square lattice; the fragment is not a finite-size lattice calculation."
    },
    {
      "id": "cdmft_open_cluster",
      "panel_id": "A",
      "kind": "real-space cluster",
      "content": "Four sites arranged as an open 2 by 2 cluster, with site indices a,b = 1,...,4 and no wrap-around bonds."
    },
    {
      "id": "cdmft_cluster_self_energy",
      "panel_id": "A",
      "kind": "site-matrix self-energy",
      "content": "Sigma_ab(z) acts on the open cluster site indices."
    },
    {
      "id": "cdmft_boundary_coupling",
      "panel_id": "A",
      "kind": "cluster-to-medium coupling",
      "content": "Delta_ab(z) denotes the dynamical boundary coupling between the open cluster and the self-consistent environment."
    },
    {
      "id": "cdmft_translation_property",
      "panel_id": "A",
      "kind": "finite-cluster symmetry statement",
      "content": "The finite open cluster breaks lattice translations."
    },
    {
      "id": "dca_brillouin_zone",
      "panel_id": "B",
      "kind": "momentum domain",
      "content": "A square Brillouin zone divided into a 2 by 2 array of four equal patches."
    },
    {
      "id": "dca_patch_1",
      "panel_id": "B",
      "kind": "momentum patch",
      "content": "Upper-left patch centered at cluster momentum K_1."
    },
    {
      "id": "dca_patch_2",
      "panel_id": "B",
      "kind": "momentum patch",
      "content": "Upper-right patch centered at cluster momentum K_2 and containing the representative momentum k."
    },
    {
      "id": "dca_patch_3",
      "panel_id": "B",
      "kind": "momentum patch",
      "content": "Lower-left patch centered at cluster momentum K_3."
    },
    {
      "id": "dca_patch_4",
      "panel_id": "B",
      "kind": "momentum patch",
      "content": "Lower-right patch centered at cluster momentum K_4."
    },
    {
      "id": "dca_representative_momentum",
      "panel_id": "B",
      "kind": "momentum decomposition",
      "content": "The representative point obeys k = K_2 + k_tilde, with k_tilde measured from the patch center."
    },
    {
      "id": "dca_periodic_cluster",
      "panel_id": "B",
      "kind": "periodic cluster",
      "content": "A 2 by 2 cluster with wrap links in both lattice directions represents periodic cluster boundary conditions."
    },
    {
      "id": "dca_patch_self_energy",
      "panel_id": "B",
      "kind": "cluster-momentum self-energy",
      "content": "Sigma_K(z) labels the cluster self-energy in the cluster-momentum basis."
    },
    {
      "id": "dca_momentum_resolution",
      "panel_id": "B",
      "kind": "resolution scale",
      "content": "For cluster linear extent L_c, the characteristic patch resolution is Delta k approximately 2 pi / L_c."
    },
    {
      "id": "dca_translation_property",
      "panel_id": "B",
      "kind": "finite-cluster symmetry statement",
      "content": "Periodic boundary conditions preserve translations on the DCA cluster while lattice momentum is coarse grained into patches."
    }
  ],
  "relations": [
    {
      "from": "cdmft_open_cluster",
      "to": "cdmft_lattice_environment",
      "type": "embedded_in",
      "meaning": "The finite open cluster is coupled self-consistently to the thermodynamic-limit lattice environment rather than treated as an isolated finite lattice."
    },
    {
      "from": "cdmft_boundary_coupling",
      "to": "cdmft_open_cluster",
      "type": "couples_boundary_sites",
      "meaning": "Dashed links connect every edge of the drawn 2 by 2 cluster to neighboring environment sites."
    },
    {
      "from": "cdmft_cluster_self_energy",
      "to": "cdmft_open_cluster",
      "type": "defined_on",
      "meaning": "Sigma_ab(z) is a matrix on the open-cluster site labels."
    },
    {
      "from": "cdmft_open_cluster",
      "to": "cdmft_translation_property",
      "type": "breaks_at_finite_cluster",
      "meaning": "The open finite cluster makes its internal sites inequivalent and breaks lattice translations at finite N_c."
    },
    {
      "from": "dca_brillouin_zone",
      "to": "dca_patch_1",
      "type": "partitioned_into",
      "meaning": "Patch 1 is one of four equal cells in the square Brillouin zone."
    },
    {
      "from": "dca_brillouin_zone",
      "to": "dca_patch_2",
      "type": "partitioned_into",
      "meaning": "Patch 2 is one of four equal cells in the square Brillouin zone."
    },
    {
      "from": "dca_brillouin_zone",
      "to": "dca_patch_3",
      "type": "partitioned_into",
      "meaning": "Patch 3 is one of four equal cells in the square Brillouin zone."
    },
    {
      "from": "dca_brillouin_zone",
      "to": "dca_patch_4",
      "type": "partitioned_into",
      "meaning": "Patch 4 is one of four equal cells in the square Brillouin zone."
    },
    {
      "from": "dca_representative_momentum",
      "to": "dca_patch_2",
      "type": "lies_in",
      "meaning": "The representative k lies inside the patch centered at K_2 and is decomposed as k = K_2 + k_tilde."
    },
    {
      "from": "dca_patch_self_energy",
      "to": "dca_brillouin_zone",
      "type": "approximates_within_patch",
      "meaning": "Across all four cells, DCA uses Sigma(k,z) approximately equal to Sigma_K(z) for momenta in the patch centered at K."
    },
    {
      "from": "dca_periodic_cluster",
      "to": "dca_translation_property",
      "type": "preserves",
      "meaning": "Periodic cluster boundary conditions preserve translations on the finite DCA cluster."
    },
    {
      "from": "dca_momentum_resolution",
      "to": "dca_brillouin_zone",
      "type": "sets_patch_scale",
      "meaning": "A regular cluster of linear extent L_c has characteristic momentum resolution Delta k approximately 2 pi / L_c."
    }
  ],
  "assumptions": [
    "The displayed example is a two-dimensional square lattice with a regular N_c = 4, L_c = 2 cluster and four equal DCA patches.",
    "z denotes a complex frequency argument; no real-frequency or Matsubara-axis choice is needed for the geometric comparison.",
    "The environment grid in panel A is only a representative fragment of the self-consistent thermodynamic-limit lattice.",
    "Both constructions solve an auxiliary finite cluster coupled to a self-consistent medium and approximate the thermodynamic-limit lattice self-energy.",
    "The diagram is schematic, not to scale, and contains no numerical data."
  ],
  "limits": [
    {
      "condition": "finite N_c in CDMFT",
      "statement": "The open cluster breaks lattice translations and introduces boundary and cluster-shape dependence."
    },
    {
      "condition": "finite N_c in DCA",
      "statement": "The periodic cluster preserves cluster translations but replaces full momentum dependence by patchwise coarse graining with Delta k approximately 2 pi / L_c."
    },
    {
      "condition": "N_c = 1",
      "statement": "Both CDMFT and DCA reduce to single-site DMFT."
    },
    {
      "condition": "systematic N_c -> infinity refinement",
      "statement": "A controlled sequence of enlarging open clusters or refining periodic momentum patches approaches the exact lattice."
    }
  ],
  "nonclaims": [
    "Neither finite auxiliary cluster is the thermodynamic lattice itself.",
    "The N_c = 4 drawings are not exact solutions and do not establish convergence for an observable.",
    "The diagram does not claim that CDMFT or DCA is universally superior or more accurate for every observable.",
    "The diagram does not assign a universal error bar, convergence rate, phase boundary, solver accuracy, or computational cost.",
    "The equal drawn patch areas and link lengths carry no quantitative weights beyond the stated four-patch geometry."
  ],
  "visual_encodings": {
    "canvas": "explicit white background for light, dark, monochrome, and print contexts",
    "panel_order": "A above B gives a phone-first vertical reading order",
    "environment": "thin gray square-lattice bonds and small filled gray sites",
    "cdmft_cluster": "pale rectangular field, heavy solid intracluster bonds, and large open sites",
    "cdmft_boundary_coupling": "heavy dashed gray bonds from cluster boundary sites to environment sites",
    "dca_brillouin_zone": "heavy solid square outline",
    "dca_patch_boundaries": "dashed gray vertical and horizontal bisectors",
    "dca_patch_centers": "filled black points directly labeled K_1 through K_4",
    "representative_momentum": "open point k connected to K_2 by an arrow labeled k_tilde",
    "periodic_cluster": "open sites and heavy solid bonds with curved arrowed wrap links",
    "direct_labels": "Sigma_ab(z), Delta_ab(z), Sigma_K(z), k = K + k_tilde, and Delta k approximately 2 pi / L_c appear next to the objects they qualify",
    "color_dependency": "none; every distinction also uses shape, fill, line weight, dash pattern, arrows, and direct text labels"
  },
  "sources": [
    {
      "citation": "Gabriel Kotliar, Sergej Y. Savrasov, Gunnar Palsson, and Giulio Biroli, Cellular Dynamical Mean Field Approach to Strongly Correlated Systems, Physical Review Letters 87 (2001) 186401",
      "doi": "10.1103/PhysRevLett.87.186401",
      "url": "https://doi.org/10.1103/PhysRevLett.87.186401",
      "use": "open real-space cellular embedding, site-matrix self-energy, and boundary-induced translation breaking"
    },
    {
      "citation": "Matthias H. Hettler, M. Mukherjee, Mark Jarrell, and H. R. Krishnamurthy, Dynamical Cluster Approximation: Nonlocal Dynamics of Correlated Electron Systems, Physical Review B 61 (2000) 12739-12756",
      "doi": "10.1103/PhysRevB.61.12739",
      "url": "https://doi.org/10.1103/PhysRevB.61.12739",
      "use": "periodic DCA cluster, cluster momenta, Brillouin-zone patching, and patchwise self-energy"
    },
    {
      "citation": "Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler, Quantum Cluster Theories, Reviews of Modern Physics 77 (2005) 1027-1080",
      "doi": "10.1103/RevModPhys.77.1027",
      "url": "https://doi.org/10.1103/RevModPhys.77.1027",
      "use": "CDMFT-DCA comparison, finite-cluster boundary properties, resolution scale, and single-site and infinite-cluster limits"
    }
  ]
}