{
  "$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.quantum-magnetism-frustration-claim-validity-ledger",
  "table_id": "quantum-magnetism-frustration-claim-validity-ledger",
  "canonical_source": {
    "route": "/many-body-quantum-matter/quantum-magnetism-frustration/",
    "heading": "Phase, approximation, and evidence table",
    "anchor": "phase-approximation-and-evidence-table"
  },
  "caption": "Symmetry, excitation, control, and evidence across magnetic regimes.",
  "common_scope": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "qualifications": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "columns": [
    {
      "key": "regime",
      "label": "Regime"
    },
    {
      "key": "symmetry_or_order",
      "label": "Symmetry or order"
    },
    {
      "key": "berry_or_topological_datum",
      "label": "Berry or topological datum"
    },
    {
      "key": "low_energy_excitation",
      "label": "Low-energy excitation"
    },
    {
      "key": "structure_factor_signature",
      "label": "Structure-factor signature"
    },
    {
      "key": "control",
      "label": "Control"
    },
    {
      "key": "distinguishing_check",
      "label": "Distinguishing check"
    }
  ],
  "rows": [
    {
      "key": "heisenberg-ferromagnet",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--heisenberg-ferromagnet",
      "cells": {
        "regime": {
          "markdown": "Heisenberg ferromagnet",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "uniform dipole order",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "summed first-order Berry phase",
          "math": [],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "one quadratic magnon",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "transverse pole near $\\mathbf q=0$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\mathbf q=0"
            }
          ],
          "links": []
        },
        "control": {
          "markdown": "dilute magnons, $1/S$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "1/S"
            }
          ],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "stiffness and absolute moment agree",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "bipartite-antiferromagnet",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--bipartite-antiferromagnet",
      "cells": {
        "regime": {
          "markdown": "Bipartite antiferromagnet",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "staggered dipole order",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "pairwise Berry cancellation plus residual events",
          "math": [],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "linear magnons",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "poles near ordering vector, Bragg weight",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "gradients and $1/S$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "1/S"
            }
          ],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "finite order reduction and sum-rule closure",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "integer-spin-chain",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--integer-spin-chain",
      "cells": {
        "regime": {
          "markdown": "Integer-spin chain",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "no bulk dipole order",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "$\\theta=0$ modulo $2\\pi$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\theta=0"
            },
            {
              "kind": "inlineMath",
              "tex": "2\\pi"
            }
          ],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "massive triplet in the simplest chain",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "gapped one- and multi-particle weight",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "continuum matching plus numerics",
          "math": [],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "gap persists with size; symmetry class stated",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "half-integer-chain",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--half-integer-chain",
      "cells": {
        "regime": {
          "markdown": "Half-integer chain",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "translation and spin symmetry constrain the ground state",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "$\\theta=\\pi$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "\\theta=\\pi"
            }
          ],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "critical spinons or a degenerate gapped alternative",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "continuum or symmetry-breaking tower",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "anomaly/theorem plus model solution",
          "math": [],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "no unique symmetric gapped state",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "valence-bond-solid",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--valence-bond-solid",
      "cells": {
        "regime": {
          "markdown": "Valence-bond solid",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "broken lattice symmetry",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "monopole Berry phases may select pattern",
          "math": [],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "singlets, triplons, domain defects",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "dimer Bragg order; spin gap possible",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "finite-size symmetry sectors",
          "math": [],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "dimer order extrapolates nonzero",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "symmetric-quantum-paramagnet",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--symmetric-quantum-paramagnet",
      "cells": {
        "regime": {
          "markdown": "Symmetric quantum paramagnet",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "no conventional order",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "SPT or intrinsic topological data must be specified",
          "math": [],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "edge degrees of freedom, anyons, or conventional modes depending on phase",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "absence of Bragg order is insufficient",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "representation per cell and topology",
          "math": [],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "projective edge, topology, or trivial deformation",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "itinerant-paramagnon-regime",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--itinerant-paramagnon-regime",
      "cells": {
        "regime": {
          "markdown": "Itinerant paramagnon regime",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "Fermi-surface spin channel",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "no fixed-spin Berry reduction",
          "math": [],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "damped collective spin response",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "broad weight tied to particle–hole continuum",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "weak coupling/RPA or declared extension",
          "math": [],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "band-resolved bubble and total spectral weight agree",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "fractionalization-candidate",
      "row_id": "quantum-magnetism-frustration-claim-validity-ledger--fractionalization-candidate",
      "cells": {
        "regime": {
          "markdown": "Fractionalization candidate",
          "math": [],
          "links": []
        },
        "symmetry_or_order": {
          "markdown": "no required dipole order",
          "math": [],
          "links": []
        },
        "berry_or_topological_datum": {
          "markdown": "emergent gauge/topological sector",
          "math": [],
          "links": []
        },
        "low_energy_excitation": {
          "markdown": "fractional quasiparticles",
          "math": [],
          "links": []
        },
        "structure_factor_signature": {
          "markdown": "structured continuum",
          "math": [],
          "links": []
        },
        "control": {
          "markdown": "controlled model or converged numerics",
          "math": [],
          "links": []
        },
        "distinguishing_check": {
          "markdown": "exclude magnon decay, disorder, and phonons",
          "math": [],
          "links": []
        }
      }
    }
  ],
  "references": [
    {
      "id": "ref-ed7780929bc43e7c",
      "markdown": "Holstein, T., and H. Primakoff. “Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet.” *Physical Review* 58 (1940): 1098–1113. [DOI](https://doi.org/10.1103/PhysRev.58.1098).",
      "urls": [
        "https://doi.org/10.1103/PhysRev.58.1098"
      ]
    }
  ],
  "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."
}