{
  "$schema": "https://qft.org/data/schemas/qft-many-body-comparison-1.0.0.schema.json",
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.band-geometry-symmetry-protected-matter-claim-validity-ledger",
  "table_id": "band-geometry-claim-validity-table",
  "canonical_source": {
    "route": "/many-body-quantum-matter/band-geometry-symmetry-protected-matter/",
    "heading": "Claim-validity table",
    "anchor": "claim-validity-table"
  },
  "caption": "Claims, diagnostics, and validity limits for band geometry and symmetry-protected matter.",
  "common_scope": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "qualifications": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "columns": [
    {
      "key": "claim",
      "label": "Claim"
    },
    {
      "key": "dimension_and_symmetry",
      "label": "Dimension and symmetry"
    },
    {
      "key": "stable_object",
      "label": "Stable object"
    },
    {
      "key": "bulk_diagnostic",
      "label": "Bulk diagnostic"
    },
    {
      "key": "boundary_or_response_test",
      "label": "Boundary or response test"
    },
    {
      "key": "interaction_disorder_limit",
      "label": "Interaction/disorder limit"
    },
    {
      "key": "negative_test",
      "label": "Negative test"
    }
  ],
  "rows": [
    {
      "key": "localized-occupied-bands",
      "row_id": "band-geometry-claim-validity-table--localized-occupied-bands",
      "cells": {
        "claim": {
          "markdown": "Localized occupied bands",
          "math": [],
          "links": []
        },
        "dimension_and_symmetry": {
          "markdown": "Any $d$; declared space-group action",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "d"
            }
          ],
          "links": []
        },
        "stable_object": {
          "markdown": "Isolated projector up to smooth occupied-frame gauge",
          "math": [],
          "links": []
        },
        "bulk_diagnostic": {
          "markdown": "Exponentially localized symmetry-compatible Wannier basis, when it exists",
          "math": [],
          "links": []
        },
        "boundary_or_response_test": {
          "markdown": "Boundary not required",
          "math": [],
          "links": []
        },
        "interaction_disorder_limit": {
          "markdown": "Disorder needs a real-space projector; interactions need a many-body replacement",
          "math": [],
          "links": []
        },
        "negative_test": {
          "markdown": "Gap closes, bands are entangled without a disentanglement window, or symmetry-compatible Wannier functions are obstructed",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "chern-insulator",
      "row_id": "band-geometry-claim-validity-table--chern-insulator",
      "cells": {
        "claim": {
          "markdown": "Chern insulator",
          "math": [],
          "links": []
        },
        "dimension_and_symmetry": {
          "markdown": "$d=2$, charge conservation; no time reversal",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "d=2"
            }
          ],
          "links": []
        },
        "stable_object": {
          "markdown": "Class-A bundle after adding trivial bands",
          "math": [],
          "links": []
        },
        "bulk_diagnostic": {
          "markdown": "Integer $C=(2\\pi)^{-1}\\int_{\\mathrm{BZ}}\\Omega$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "C=(2\\pi)^{-1}\\int_{\\mathrm{BZ}}\\Omega"
            }
          ],
          "links": []
        },
        "boundary_or_response_test": {
          "markdown": "Net chirality has magnitude $\\lvert\\Delta C\\rvert$ and sign fixed by the interface orientation; quantized Hall response",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\lvert\\Delta C\\rvert"
            }
          ],
          "links": []
        },
        "interaction_disorder_limit": {
          "markdown": "Mobility gap can replace a spectral gap; intrinsic order is outside the band index",
          "math": [],
          "links": []
        },
        "negative_test": {
          "markdown": "Hall sign conflicts after orientation/charge translation, or the Fermi level crosses extended bulk states",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "time-reversal-topological-insulator",
      "row_id": "band-geometry-claim-validity-table--time-reversal-topological-insulator",
      "cells": {
        "claim": {
          "markdown": "Time-reversal topological insulator",
          "math": [],
          "links": []
        },
        "dimension_and_symmetry": {
          "markdown": "$d=2$ nontrivial Z2 or $d=3$ strong phase; charge conservation and $T^2=-1$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "d=2"
            },
            {
              "kind": "inlineMath",
              "tex": "d=3"
            },
            {
              "kind": "inlineMath",
              "tex": "T^2=-1"
            }
          ],
          "links": []
        },
        "stable_object": {
          "markdown": "Quaternionic occupied bundle in the stable limit",
          "math": [],
          "links": []
        },
        "bulk_diagnostic": {
          "markdown": "Z2 Wilson-loop flow or equivalent invariant",
          "math": [],
          "links": []
        },
        "boundary_or_response_test": {
          "markdown": "Odd Kramers-pair parity on a 2D edge; odd Dirac-cone parity on a 3D strong surface, with the protecting symmetries preserved",
          "math": [],
          "links": []
        },
        "interaction_disorder_limit": {
          "markdown": "When the band description fails, use many-body invariants",
          "math": [],
          "links": []
        },
        "negative_test": {
          "markdown": "Boundary is gapped without breaking symmetry only because extra topological order or another anomalous sector was added",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "crystalline-higher-order-phase",
      "row_id": "band-geometry-claim-validity-table--crystalline-higher-order-phase",
      "cells": {
        "claim": {
          "markdown": "Crystalline/higher-order phase",
          "math": [],
          "links": []
        },
        "dimension_and_symmetry": {
          "markdown": "Declared spatial symmetry and boundary geometry",
          "math": [],
          "links": []
        },
        "stable_object": {
          "markdown": "Symmetry representation or real-space obstruction under specified band additions",
          "math": [],
          "links": []
        },
        "bulk_diagnostic": {
          "markdown": "Symmetry indicator, Wilson loop, nested invariant, or defect response",
          "math": [],
          "links": []
        },
        "boundary_or_response_test": {
          "markdown": "Symmetry-related surface, hinge, corner, or disclination signature",
          "math": [],
          "links": []
        },
        "interaction_disorder_limit": {
          "markdown": "Disorder must preserve symmetry exactly or statistically as claimed",
          "math": [],
          "links": []
        },
        "negative_test": {
          "markdown": "Bulk or defect invariant fails under its stated symmetry and stable-equivalence assumptions; a termination-localized feature alone is not decisive because boundary decorations can change it",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "interacting-spt",
      "row_id": "band-geometry-claim-validity-table--interacting-spt",
      "cells": {
        "claim": {
          "markdown": "Interacting SPT",
          "math": [],
          "links": []
        },
        "dimension_and_symmetry": {
          "markdown": "On-site or crystalline symmetry stated",
          "math": [],
          "links": []
        },
        "stable_object": {
          "markdown": "Short-range-entangled many-body ground state modulo symmetric product states",
          "math": [],
          "links": []
        },
        "bulk_diagnostic": {
          "markdown": "Quantized response, symmetry defect, entanglement invariant, or anomaly",
          "math": [],
          "links": []
        },
        "boundary_or_response_test": {
          "markdown": "Boundary cannot be trivially gapped while preserving symmetry and locality",
          "math": [],
          "links": []
        },
        "interaction_disorder_limit": {
          "markdown": "Free indices are insufficient; intrinsic topological order is excluded by definition",
          "math": [],
          "links": []
        },
        "negative_test": {
          "markdown": "Proposed signal is reproducible by a symmetric trivial state or depends only on single-particle bands",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "weyl-or-nodal-semimetal",
      "row_id": "band-geometry-claim-validity-table--weyl-or-nodal-semimetal",
      "cells": {
        "claim": {
          "markdown": "Weyl or nodal semimetal",
          "math": [],
          "links": []
        },
        "dimension_and_symmetry": {
          "markdown": "Codimension and protecting symmetry stated",
          "math": [],
          "links": []
        },
        "stable_object": {
          "markdown": "Topological charge on a gapped enclosing surface for an isolated point node or a gapped linking loop for a nodal line",
          "math": [],
          "links": []
        },
        "bulk_diagnostic": {
          "markdown": "Chern, winding, or Berry-phase charge on the relevant enclosing object",
          "math": [],
          "links": []
        },
        "boundary_or_response_test": {
          "markdown": "Arc or drumhead structure subject to projected-node geometry and termination",
          "math": [],
          "links": []
        },
        "interaction_disorder_limit": {
          "markdown": "Strong disorder/interactions may broaden, annihilate, or reconstruct nodes",
          "math": [],
          "links": []
        },
        "negative_test": {
          "markdown": "The relevant enclosing surface or linking loop is not gapped, opposite charges coincide, or the protection is broken",
          "math": [],
          "links": []
        }
      }
    }
  ],
  "references": [
    {
      "id": "ref-ab4bc90d79ec3018",
      "markdown": "M. Zahid Hasan and Charles L. Kane, “Colloquium: Topological Insulators,” *Reviews of Modern Physics* **82** (2010) 3045–3067, [doi:10.1103/RevModPhys.82.3045](https://doi.org/10.1103/RevModPhys.82.3045).",
      "urls": [
        "https://doi.org/10.1103/RevModPhys.82.3045"
      ]
    }
  ],
  "reference_scope": "The reference catalog reproduces the canonical chapter bibliography. Cell links are associated with references only when their resolved URL matches; uncited chapter references are not asserted as claim-specific support."
}