{
  "$schema": "https://qft.org/data/schemas/qft-one-dimensional-convention-validity-1.0.0.schema.json",
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.one-dimensional-quantum-matter-claim-validity-ledger",
  "canonical_source": {
    "path": "src/content/docs/many-body-quantum-matter/one-dimensional-quantum-matter/index.md",
    "route": "/many-body-quantum-matter/one-dimensional-quantum-matter/",
    "heading": "Convention and validity table",
    "fragment": "#convention-and-validity-table"
  },
  "columns": [
    {
      "key": "object_or_claim",
      "label": "Object or claim"
    },
    {
      "key": "chapter_convention",
      "label": "This chapter’s convention"
    },
    {
      "key": "scaling_statement",
      "label": "Scaling statement"
    },
    {
      "key": "global_or_boundary_datum",
      "label": "Global or boundary datum"
    },
    {
      "key": "direct_check",
      "label": "Direct check"
    },
    {
      "key": "failure_if_omitted",
      "label": "Failure if omitted"
    }
  ],
  "common_scope": {
    "markdown": "The Gaussian entries assume positive velocities and stiffnesses, $u>0$ and $K>0$ for each gapless mode, in the low-energy window where linearization applies. Bulk powers are zero-temperature, long-distance fixed-point results; marginal interactions, finite temperature, and finite size require their own corrections or crossovers. Allowed vertices must lie on the physical compactification lattice. “Relevant” refers to a weak perturbation at the stated bulk or boundary fixed point; a strong-coupling phase still requires symmetry and compactification checks.",
    "segments": [
      {
        "kind": "text",
        "text": "The Gaussian entries assume positive velocities and stiffnesses, "
      },
      {
        "kind": "math",
        "tex": "u>0"
      },
      {
        "kind": "text",
        "text": " and "
      },
      {
        "kind": "math",
        "tex": "K>0"
      },
      {
        "kind": "text",
        "text": " for each gapless mode, in the low-energy window where linearization applies. Bulk powers are zero-temperature, long-distance fixed-point results; marginal interactions, finite temperature, and finite size require their own corrections or crossovers. Allowed vertices must lie on the physical compactification lattice. “Relevant” refers to a weak perturbation at the stated bulk or boundary fixed point; a strong-coupling phase still requires symmetry and compactification checks."
      }
    ]
  },
  "caption": {
    "markdown": "Low-energy operator dimensions and their validity conditions. The comparison keeps local scaling data together with the global or boundary information needed to use them.",
    "segments": [
      {
        "kind": "text",
        "text": "Low-energy operator dimensions and their validity conditions. The comparison keeps local scaling data together with the global or boundary information needed to use them."
      }
    ]
  },
  "notes": [
    {
      "markdown": "The gapless spin qualification is essential: SU(2) symmetry alone does not exclude a spin gap [Kane and Fisher 1992, § II, pp. 15236–15237](https://doi.org/10.1103/PhysRevB.46.15233). The weak-barrier and weak-link dimensions refer to the two stated spinless boundary fixed points, not a generic multichannel junction [Kane and Fisher 1992, §§ III A–B, pp. 15238–15240](https://doi.org/10.1103/PhysRevB.46.15233). For the spectral row, $K_c=K_s=1$ and $u_c=u_s=v_F$ recover the free pole; a truncated low-energy spectral window need not carry unit weight.",
      "segments": [
        {
          "kind": "text",
          "text": "The gapless spin qualification is essential: SU(2) symmetry alone does not exclude a spin gap "
        },
        {
          "kind": "link",
          "text": "Kane and Fisher 1992, § II, pp. 15236–15237",
          "url": "https://doi.org/10.1103/PhysRevB.46.15233",
          "reference_id": "kane-fisher-1992"
        },
        {
          "kind": "text",
          "text": ". The weak-barrier and weak-link dimensions refer to the two stated spinless boundary fixed points, not a generic multichannel junction "
        },
        {
          "kind": "link",
          "text": "Kane and Fisher 1992, §§ III A–B, pp. 15238–15240",
          "url": "https://doi.org/10.1103/PhysRevB.46.15233",
          "reference_id": "kane-fisher-1992"
        },
        {
          "kind": "text",
          "text": ". For the spectral row, "
        },
        {
          "kind": "math",
          "tex": "K_c=K_s=1"
        },
        {
          "kind": "text",
          "text": " and "
        },
        {
          "kind": "math",
          "tex": "u_c=u_s=v_F"
        },
        {
          "kind": "text",
          "text": " recover the free pole; a truncated low-energy spectral window need not carry unit weight."
        }
      ]
    }
  ],
  "rows": [
    {
      "key": "smooth-density",
      "row_id": "qft.one-dimensional.smooth-density.v1",
      "object_or_claim": {
        "markdown": "Smooth density",
        "segments": [
          {
            "kind": "text",
            "text": "Smooth density"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$\\delta\\rho=-\\partial_x\\phi/\\pi$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\delta\\rho=-\\partial_x\\phi/\\pi"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "derivative operator",
        "segments": [
          {
            "kind": "text",
            "text": "derivative operator"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "$\\phi(x+L)=\\phi(x)-\\pi N$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\phi(x+L)=\\phi(x)-\\pi N"
          }
        ]
      },
      "direct_check": {
        "markdown": "integrated charge is $N$",
        "segments": [
          {
            "kind": "text",
            "text": "integrated charge is "
          },
          {
            "kind": "math",
            "tex": "N"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "fractional or sign-reversed charge",
        "segments": [
          {
            "kind": "text",
            "text": "fractional or sign-reversed charge"
          }
        ]
      },
      "claim_references": []
    },
    {
      "key": "spinless-fermion",
      "row_id": "qft.one-dimensional.spinless-fermion.v1",
      "object_or_claim": {
        "markdown": "Spinless fermion",
        "segments": [
          {
            "kind": "text",
            "text": "Spinless fermion"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$\\psi_r\\propto\\eta_r e^{-i(r\\phi-\\theta)}$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\psi_r\\propto\\eta_r e^{-i(r\\phi-\\theta)}"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "$2\\Delta_\\psi=(K+K^{-1})/2$",
        "segments": [
          {
            "kind": "math",
            "tex": "2\\Delta_\\psi=(K+K^{-1})/2"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "Klein factors and parity twist",
        "segments": [
          {
            "kind": "text",
            "text": "Klein factors and parity twist"
          }
        ]
      },
      "direct_check": {
        "markdown": "$K=1$ gives $1/x$",
        "segments": [
          {
            "kind": "math",
            "tex": "K=1"
          },
          {
            "kind": "text",
            "text": " gives "
          },
          {
            "kind": "math",
            "tex": "1/x"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "wrong interspecies signs or ring spectrum",
        "segments": [
          {
            "kind": "text",
            "text": "wrong interspecies signs or ring spectrum"
          }
        ]
      },
      "claim_references": []
    },
    {
      "key": "general-vertex",
      "row_id": "qft.one-dimensional.general-vertex.v1",
      "object_or_claim": {
        "markdown": "General vertex",
        "segments": [
          {
            "kind": "text",
            "text": "General vertex"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$V_{m,n}=e^{i(m\\phi+n\\theta)}$",
        "segments": [
          {
            "kind": "math",
            "tex": "V_{m,n}=e^{i(m\\phi+n\\theta)}"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "$\\Delta=(m^2K+n^2/K)/4$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\Delta=(m^2K+n^2/K)/4"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "$(m,n)$ must lie on the physical vertex lattice",
        "segments": [
          {
            "kind": "math",
            "tex": "(m,n)"
          },
          {
            "kind": "text",
            "text": " must lie on the physical vertex lattice"
          }
        ]
      },
      "direct_check": {
        "markdown": "charge, momentum, exchange phase",
        "segments": [
          {
            "kind": "text",
            "text": "charge, momentum, exchange phase"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "spurious local operators",
        "segments": [
          {
            "kind": "text",
            "text": "spurious local operators"
          }
        ]
      },
      "claim_references": []
    },
    {
      "key": "density-locking",
      "row_id": "qft.one-dimensional.density-locking.v1",
      "object_or_claim": {
        "markdown": "Density locking",
        "segments": [
          {
            "kind": "text",
            "text": "Density locking"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$\\cos(2p\\phi)$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\cos(2p\\phi)"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "bulk relevant for $p^2K<2$",
        "segments": [
          {
            "kind": "text",
            "text": "bulk relevant for "
          },
          {
            "kind": "math",
            "tex": "p^2K<2"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "commensurability and translation",
        "segments": [
          {
            "kind": "text",
            "text": "commensurability and translation"
          }
        ]
      },
      "direct_check": {
        "markdown": "kink charge magnitude $1/p$",
        "segments": [
          {
            "kind": "text",
            "text": "kink charge magnitude "
          },
          {
            "kind": "math",
            "tex": "1/p"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "a rapidly oscillating term is treated as a gap",
        "segments": [
          {
            "kind": "text",
            "text": "a rapidly oscillating term is treated as a gap"
          }
        ]
      },
      "claim_references": []
    },
    {
      "key": "phase-locking",
      "row_id": "qft.one-dimensional.phase-locking.v1",
      "object_or_claim": {
        "markdown": "Phase locking",
        "segments": [
          {
            "kind": "text",
            "text": "Phase locking"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$\\cos(2q\\theta)$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\cos(2q\\theta)"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "bulk relevant for $q^2/K<2$",
        "segments": [
          {
            "kind": "text",
            "text": "bulk relevant for "
          },
          {
            "kind": "math",
            "tex": "q^2/K<2"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "charge symmetry must permit it",
        "segments": [
          {
            "kind": "text",
            "text": "charge symmetry must permit it"
          }
        ]
      },
      "direct_check": {
        "markdown": "duality under $K\\leftrightarrow K^{-1}$",
        "segments": [
          {
            "kind": "text",
            "text": "duality under "
          },
          {
            "kind": "math",
            "tex": "K\\leftrightarrow K^{-1}"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "simultaneous sharp locking of conjugate fields",
        "segments": [
          {
            "kind": "text",
            "text": "simultaneous sharp locking of conjugate fields"
          }
        ]
      },
      "claim_references": []
    },
    {
      "key": "point-backscattering",
      "row_id": "qft.one-dimensional.point-backscattering.v1",
      "object_or_claim": {
        "markdown": "Single spinless-channel backscattering",
        "segments": [
          {
            "kind": "text",
            "text": "Single spinless-channel backscattering"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$\\cos[2\\phi(0)]$",
        "segments": [
          {
            "kind": "math",
            "tex": "\\cos[2\\phi(0)]"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "boundary dimension $K$ at the transmitting fixed point",
        "segments": [
          {
            "kind": "text",
            "text": "boundary dimension "
          },
          {
            "kind": "math",
            "tex": "K"
          },
          {
            "kind": "text",
            "text": " at the transmitting fixed point"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "single channel; specified reservoirs and boundary conditions",
        "segments": [
          {
            "kind": "text",
            "text": "single channel; specified reservoirs and boundary conditions"
          }
        ]
      },
      "direct_check": {
        "markdown": "two identical open ends: weak-link dimension $1/K$",
        "segments": [
          {
            "kind": "text",
            "text": "two identical open ends: weak-link dimension "
          },
          {
            "kind": "math",
            "tex": "1/K"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "bulk and boundary relevance are confused",
        "segments": [
          {
            "kind": "text",
            "text": "bulk and boundary relevance are confused"
          }
        ]
      },
      "claim_references": [
        {
          "reference_id": "kane-fisher-1992",
          "locator": "§§ III A–B, pp. 15238–15240",
          "citation_label": "Kane and Fisher 1992, §§ III A–B, pp. 15238–15240"
        }
      ]
    },
    {
      "key": "spinful-electron",
      "row_id": "qft.one-dimensional.spinful-electron.v1",
      "object_or_claim": {
        "markdown": "Spinful electron",
        "segments": [
          {
            "kind": "text",
            "text": "Spinful electron"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "charge and spin vertices divided by $\\sqrt2$",
        "segments": [
          {
            "kind": "text",
            "text": "charge and spin vertices divided by "
          },
          {
            "kind": "math",
            "tex": "\\sqrt2"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "$\\alpha_{\\mathrm{bulk}}=(K_c+K_c^{-1}-2)/4$ for a gapless SU(2) spin fixed point",
        "segments": [
          {
            "kind": "math",
            "tex": "\\alpha_{\\mathrm{bulk}}=(K_c+K_c^{-1}-2)/4"
          },
          {
            "kind": "text",
            "text": " for a gapless SU(2) spin fixed point"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "correlated charge–spin zero modes",
        "segments": [
          {
            "kind": "text",
            "text": "correlated charge–spin zero modes"
          }
        ]
      },
      "direct_check": {
        "markdown": "free limit $K_c=1$",
        "segments": [
          {
            "kind": "text",
            "text": "free limit "
          },
          {
            "kind": "math",
            "tex": "K_c=1"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "incorrect tunneling exponent",
        "segments": [
          {
            "kind": "text",
            "text": "incorrect tunneling exponent"
          }
        ]
      },
      "claim_references": [
        {
          "reference_id": "kane-fisher-1992",
          "locator": "§ II, pp. 15236–15237",
          "citation_label": "Kane and Fisher 1992, § II, pp. 15236–15237"
        }
      ]
    },
    {
      "key": "spectral-function",
      "row_id": "qft.one-dimensional.spectral-function.v1",
      "object_or_claim": {
        "markdown": "Spectral function",
        "segments": [
          {
            "kind": "text",
            "text": "Spectral function"
          }
        ]
      },
      "chapter_convention": {
        "markdown": "$A=-2\\operatorname{Im}G^R$",
        "segments": [
          {
            "kind": "math",
            "tex": "A=-2\\operatorname{Im}G^R"
          }
        ]
      },
      "scaling_statement": {
        "markdown": "generic interacting gapless thresholds; the free coincident-velocity limit has a pole",
        "segments": [
          {
            "kind": "text",
            "text": "generic interacting gapless thresholds; the free coincident-velocity limit has a pole"
          }
        ]
      },
      "global_or_boundary_datum": {
        "markdown": "temperature, curvature, transverse hopping",
        "segments": [
          {
            "kind": "text",
            "text": "temperature, curvature, transverse hopping"
          }
        ]
      },
      "direct_check": {
        "markdown": "$\\int d\\omega\\,A/(2\\pi)=1$ for the complete canonical fermion over all frequencies",
        "segments": [
          {
            "kind": "math",
            "tex": "\\int d\\omega\\,A/(2\\pi)=1"
          },
          {
            "kind": "text",
            "text": " for the complete canonical fermion over all frequencies"
          }
        ]
      },
      "failure_if_omitted": {
        "markdown": "intensity is mistaken for normalized spectral weight",
        "segments": [
          {
            "kind": "text",
            "text": "intensity is mistaken for normalized spectral weight"
          }
        ]
      },
      "claim_references": []
    }
  ],
  "references": [
    {
      "id": "kane-fisher-1992",
      "citation_markdown": "Kane, C. L., and M. P. A. Fisher. “Transmission through Barriers and Resonant Tunneling in an Interacting One-Dimensional Electron Gas.” *Physical Review B* 46 (1992): 15233–15262. [DOI](https://doi.org/10.1103/PhysRevB.46.15233).",
      "url": "https://doi.org/10.1103/PhysRevB.46.15233"
    }
  ]
}