{
  "$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.fractionalization-emergent-gauge-fields-claim-validity-ledger",
  "table_id": "fractionalization-claim-validity-table",
  "canonical_source": {
    "route": "/many-body-quantum-matter/fractionalization-emergent-gauge-fields/",
    "heading": "Claim-validity table",
    "anchor": "claim-validity-table"
  },
  "caption": "Gauge structure, physical diagnostics, and limits of fractionalization claims.",
  "common_scope": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "qualifications": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "columns": [
    {
      "key": "claim",
      "label": "Claim"
    },
    {
      "key": "gauge_structure_and_compactness",
      "label": "Gauge structure and compactness"
    },
    {
      "key": "matter_and_constraint",
      "label": "Matter and constraint"
    },
    {
      "key": "symmetry_data",
      "label": "Symmetry data"
    },
    {
      "key": "positive_gauge_invariant_diagnostic",
      "label": "Positive gauge-invariant diagnostic"
    },
    {
      "key": "leading_confinement_or_alternative_mechanism",
      "label": "Leading confinement or alternative mechanism"
    },
    {
      "key": "evidence_ceiling",
      "label": "Supported conclusion"
    }
  ],
  "rows": [
    {
      "key": "gapped-z2-spin-liquid",
      "row_id": "fractionalization-claim-validity-table--gapped-z2-spin-liquid",
      "cells": {
        "claim": {
          "markdown": "Gapped Z2 spin liquid",
          "math": [],
          "links": []
        },
        "gauge_structure_and_compactness": {
          "markdown": "Compact Z2",
          "math": [],
          "links": []
        },
        "matter_and_constraint": {
          "markdown": "Spinons plus visons; local parity Gauss law",
          "math": [],
          "links": []
        },
        "symmetry_data": {
          "markdown": "Fractionalization class for each anyon",
          "math": [],
          "links": []
        },
        "positive_gauge_invariant_diagnostic": {
          "markdown": "Topological sectors, mutual $\\pi$ braiding, vison/spinon gaps, long-range entanglement",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\pi"
            }
          ],
          "links": []
        },
        "leading_confinement_or_alternative_mechanism": {
          "markdown": "Spinon condensation, vison condensation, valence-bond or magnetic order",
          "math": [],
          "links": []
        },
        "evidence_ceiling": {
          "markdown": "Absence of order is insufficient; sector and excitation data must converge",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "u-1-coulomb-spin-liquid",
      "row_id": "fractionalization-claim-validity-table--u-1-coulomb-spin-liquid",
      "cells": {
        "claim": {
          "markdown": "U(1) Coulomb spin liquid",
          "math": [],
          "links": []
        },
        "gauge_structure_and_compactness": {
          "markdown": "Compact U(1), normally $3+1$D",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "3+1"
            }
          ],
          "links": []
        },
        "matter_and_constraint": {
          "markdown": "Gapped charges satisfying Gauss law",
          "math": [],
          "links": []
        },
        "symmetry_data": {
          "markdown": "Monopole and charge quantum numbers stated",
          "math": [],
          "links": []
        },
        "positive_gauge_invariant_diagnostic": {
          "markdown": "Photon pinch-point structure plus charge/monopole spectrum and Wilson response",
          "math": [],
          "links": []
        },
        "leading_confinement_or_alternative_mechanism": {
          "markdown": "Monopole proliferation, Higgsing, disorder, or finite-temperature crossover",
          "math": [],
          "links": []
        },
        "evidence_ceiling": {
          "markdown": "Photon-like scattering alone can have non-gauge alternatives",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "dirac-spin-liquid",
      "row_id": "fractionalization-claim-validity-table--dirac-spin-liquid",
      "cells": {
        "claim": {
          "markdown": "Dirac spin liquid",
          "math": [],
          "links": []
        },
        "gauge_structure_and_compactness": {
          "markdown": "Compact U(1) in $2+1$D with gapless fermions",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "2+1"
            }
          ],
          "links": []
        },
        "matter_and_constraint": {
          "markdown": "Projected Dirac spinons, flavor number declared",
          "math": [],
          "links": []
        },
        "symmetry_data": {
          "markdown": "Projective symmetry group and monopole quantum numbers",
          "math": [],
          "links": []
        },
        "positive_gauge_invariant_diagnostic": {
          "markdown": "Correlation exponents/operator spectrum consistent across physical channels",
          "math": [],
          "links": []
        },
        "leading_confinement_or_alternative_mechanism": {
          "markdown": "Relevant monopole, mass, pairing/Higgs instability, velocity anisotropy, or confinement",
          "math": [],
          "links": []
        },
        "evidence_ceiling": {
          "markdown": "Mean-field cones and one finite-size spectrum support only a candidate",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "spinon-fermi-surface-state",
      "row_id": "fractionalization-claim-validity-table--spinon-fermi-surface-state",
      "cells": {
        "claim": {
          "markdown": "Spinon Fermi-surface state",
          "math": [],
          "links": []
        },
        "gauge_structure_and_compactness": {
          "markdown": "Effectively noncompact U(1) over a scale window",
          "math": [],
          "links": []
        },
        "matter_and_constraint": {
          "markdown": "Gauge-charged Fermi surface and exact constraint",
          "math": [],
          "links": []
        },
        "symmetry_data": {
          "markdown": "Translation/filling constraints stated",
          "math": [],
          "links": []
        },
        "positive_gauge_invariant_diagnostic": {
          "markdown": "Gauge-invariant continuum response with a probe-specific observable map and stability tests",
          "math": [],
          "links": []
        },
        "leading_confinement_or_alternative_mechanism": {
          "markdown": "Pairing, compactness, disorder, density-wave order, or strong-coupling breakdown",
          "math": [],
          "links": []
        },
        "evidence_ceiling": {
          "markdown": "One-loop exponents depend on the declared patch/control scheme",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "deconfined-critical-point",
      "row_id": "fractionalization-claim-validity-table--deconfined-critical-point",
      "cells": {
        "claim": {
          "markdown": "Deconfined critical point",
          "math": [],
          "links": []
        },
        "gauge_structure_and_compactness": {
          "markdown": "Compact microscopic CP1; monopoles constrained by lattice symmetry",
          "math": [],
          "links": []
        },
        "matter_and_constraint": {
          "markdown": "Spinons coupled to U(1) gauge field",
          "math": [],
          "links": []
        },
        "symmetry_data": {
          "markdown": "Néel, VBS, and monopole operator map stated",
          "math": [],
          "links": []
        },
        "positive_gauge_invariant_diagnostic": {
          "markdown": "Joint scaling, spectral relations, and emergent symmetry with drift tests",
          "math": [],
          "links": []
        },
        "leading_confinement_or_alternative_mechanism": {
          "markdown": "Relevant monopoles, walking flow, or weak first order",
          "math": [],
          "links": []
        },
        "evidence_ceiling": {
          "markdown": "Emergent symmetry over finite sizes is not proof of an asymptotic fixed point; a dangerously irrelevant monopole can generate a second crossover scale even at a continuous transition",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "fracton-like-order",
      "row_id": "fractionalization-claim-validity-table--fracton-like-order",
      "cells": {
        "claim": {
          "markdown": "Fracton-like order",
          "math": [],
          "links": []
        },
        "gauge_structure_and_compactness": {
          "markdown": "Model-specific subsystem or tensor gauge structure",
          "math": [],
          "links": []
        },
        "matter_and_constraint": {
          "markdown": "Restricted-mobility charges obeying local constraints",
          "math": [],
          "links": []
        },
        "symmetry_data": {
          "markdown": "Lattice foliation/subsystem structure stated",
          "math": [],
          "links": []
        },
        "positive_gauge_invariant_diagnostic": {
          "markdown": "Mobility selection rules, operator algebra, and subextensive topology dependence",
          "math": [],
          "links": []
        },
        "leading_confinement_or_alternative_mechanism": {
          "markdown": "Perturbations restoring mobility, boundary sectors, or conventional order",
          "math": [],
          "links": []
        },
        "evidence_ceiling": {
          "markdown": "No single universal classification covers all models called fractonic",
          "math": [],
          "links": []
        }
      }
    }
  ],
  "references": [
    {
      "id": "ref-30442160e5dfc543",
      "markdown": "Lucile Savary and Leon Balents, “Quantum Spin Liquids: A Review,” *Reports on Progress in Physics* **80** (2017) 016502, [doi:10.1088/0034-4885/80/1/016502](https://doi.org/10.1088/0034-4885/80/1/016502).",
      "urls": [
        "https://doi.org/10.1088/0034-4885/80/1/016502"
      ]
    }
  ],
  "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."
}