{
  "schema_version": "qft-overview-figure/v1",
  "artifact_id": "qft.artifact.many-body-quantum-matter.disorder-localization-glasses-structure-and-dictionary",
  "title": "Disorder, localization and glasses: structure and dictionary",
  "schematic": true,
  "not_to_scale": true,
  "evidence_cutoff": null,
  "description_url": "/many-body-quantum-matter/disorder-localization-glasses/quenched-ensembles-averaging/#nonvisual-description-of-the-disorder-structure-figure",
  "nodes": {
    "ensemble": {
      "role": "stage",
      "content_tex": "Quenched disorder ensemble\\\\ $P[V]$, correlation length, symmetry, and order of limits"
    },
    "annealed": {
      "role": "note",
      "content_tex": "Distinguish quenched and annealed averages; self-averaging alone is insufficient"
    },
    "normalization": {
      "role": "stage",
      "content_tex": "Normalized disorder observables\\\\ replicas $n\\to0$ or boson--fermion supersymmetry"
    },
    "limits": {
      "role": "note",
      "content_tex": "Check convergence, analytic continuation, and rare-sample tails"
    },
    "soft": {
      "role": "stage",
      "content_tex": "Retarded--advanced soft modes\\\\ diffuson and, when allowed, cooperon: $(Dq^2-i\\omega)^{-1}$"
    },
    "sigma": {
      "role": "stage",
      "content_tex": "Symmetry-class nonlinear sigma model\\\\ $Q^2=1$, gradients, frequency, topology, interactions"
    },
    "interference": {
      "role": "branch",
      "content_tex": "Coherent interference\\\\ weak localization or antilocalization"
    },
    "interaction": {
      "role": "branch",
      "content_tex": "Diffusive interactions\\\\ Altshuler--Aronov and Finkel'stein flow"
    },
    "class": {
      "role": "branch",
      "content_tex": "Class and topology\\\\ target space and topological term"
    },
    "scaling": {
      "role": "result",
      "content_tex": "Conductance scaling\\\\ $\\beta(g)=\\mathrm d\\ln g/\\mathrm d\\ln L$"
    },
    "boundary": {
      "role": "result",
      "content_tex": "Localization statement\\\\ class, dimension, energy, and boundary fixed"
    },
    "evidence": {
      "role": "result",
      "content_tex": "Finite-size conclusion: fit relevant and irrelevant drift, distributions, boundary conditions, and dephasing cutoffs"
    }
  },
  "edges": {
    "ensemble--construction--normalization": {
      "from": "ensemble",
      "to": "normalization",
      "relation": "construction",
      "label_tex": ""
    },
    "annealed--condition--ensemble": {
      "from": "annealed",
      "to": "ensemble",
      "relation": "condition",
      "label_tex": ""
    },
    "limits--condition--normalization": {
      "from": "limits",
      "to": "normalization",
      "relation": "condition",
      "label_tex": ""
    },
    "normalization--construction--soft": {
      "from": "normalization",
      "to": "soft",
      "relation": "construction",
      "label_tex": "weak static disorder"
    },
    "soft--construction--sigma": {
      "from": "soft",
      "to": "sigma",
      "relation": "construction",
      "label_tex": "$q\\ell,|\\omega|\\tau\\ll1$"
    },
    "sigma--construction--interference": {
      "from": "sigma",
      "to": "interference",
      "relation": "construction",
      "label_tex": ""
    },
    "sigma--construction--interaction": {
      "from": "sigma",
      "to": "interaction",
      "relation": "construction",
      "label_tex": ""
    },
    "sigma--construction--class": {
      "from": "sigma",
      "to": "class",
      "relation": "construction",
      "label_tex": ""
    },
    "interference--construction--scaling": {
      "from": "interference",
      "to": "scaling",
      "relation": "construction",
      "label_tex": ""
    },
    "interaction--construction--scaling": {
      "from": "interaction",
      "to": "scaling",
      "relation": "construction",
      "label_tex": ""
    },
    "interaction--construction--boundary": {
      "from": "interaction",
      "to": "boundary",
      "relation": "construction",
      "label_tex": ""
    },
    "class--construction--boundary": {
      "from": "class",
      "to": "boundary",
      "relation": "construction",
      "label_tex": ""
    },
    "scaling--construction--evidence": {
      "from": "scaling",
      "to": "evidence",
      "relation": "construction",
      "label_tex": ""
    },
    "boundary--construction--evidence": {
      "from": "boundary",
      "to": "evidence",
      "relation": "construction",
      "label_tex": ""
    }
  },
  "context": [],
  "linear_equivalent": {
    "heading": "Nonvisual description of the disorder structure figure",
    "columns": [
      "Figure element or arrow",
      "Relation encoded",
      "Condition or resulting statement"
    ],
    "rows": {
      "row-d9f05f0d79c1f6bd": {
        "key": "Quenched ensemble",
        "cells": [
          "Quenched ensemble",
          "The construction begins with $P[V]$, its correlation length and symmetries, and the order of thermodynamic and disorder limits.",
          "The dashed side condition distinguishes $\\overline{\\ln Z}$ from $\\ln\\overline Z$; self-averaging alone does not justify equality."
        ]
      },
      "row-8b80a7dbd0547668": {
        "key": "Ensemble → normalized observables",
        "cells": [
          "Ensemble → normalized observables",
          "Disorder observables are normalized by replicas with $n\\to0$ or by boson–fermion supersymmetry.",
          "The representation is fixed before any diffusive or localization approximation."
        ]
      },
      "row-c277838bc5f07567": {
        "key": "Normalization side condition",
        "cells": [
          "Normalization side condition",
          "Convergence, analytic continuation, and rare-sample tails constrain the normalization step.",
          "Failure preserves only the directly defined disorder average, not a controlled replica or supersymmetric continuation."
        ]
      },
      "row-e82efb6261f9a294": {
        "key": "Normalized observables → retarded–advanced soft modes",
        "cells": [
          "Normalized observables → retarded–advanced soft modes",
          "Weak static disorder produces diffusons and, when symmetry permits, cooperons with pole $(Dq^2-i\\omega)^{-1}$.",
          "This arrow assumes the weak-disorder regime."
        ]
      },
      "row-4d90c3107a84afbd": {
        "key": "Soft modes → nonlinear sigma model",
        "cells": [
          "Soft modes → nonlinear sigma model",
          "The long-wavelength modes are organized into a symmetry-class field $Q$ with $Q^2=1$ and gradient, frequency, topological, and interaction terms.",
          "The displayed expansion requires $q\\ell\\ll1$ and $\\lvert\\omega\\rvert\\tau\\ll1$."
        ]
      },
      "row-690c8dcd41e5569c": {
        "key": "Sigma model → interference branch",
        "cells": [
          "Sigma model → interference branch",
          "Cooperon structure produces weak localization or antilocalization.",
          "The sign and channel count depend on symmetry, dephasing, magnetic field, and spin–orbit scales."
        ]
      },
      "row-08a3a1dda1ac26b7": {
        "key": "Sigma model → interaction branch",
        "cells": [
          "Sigma model → interaction branch",
          "Diffusive density and spin modes produce Altshuler–Aronov corrections and Finkel’stein flow.",
          "Interaction amplitudes and frequency renormalization must use the same normalization as the diffusive action."
        ]
      },
      "row-291a28f230429c40": {
        "key": "Sigma model → class and topology branch",
        "cells": [
          "Sigma model → class and topology branch",
          "The target space and allowed topological term follow from the symmetry class.",
          "The conclusion remains dimension-, boundary-, and energy-specific."
        ]
      },
      "row-ab489ca9f5a03df5": {
        "key": "Interference and interaction branches → conductance scaling",
        "cells": [
          "Interference and interaction branches → conductance scaling",
          "Their controlled corrections feed $\\beta(g)=\\mathrm d\\ln g/\\mathrm d\\ln L$.",
          "A perturbative logarithm is not itself a critical point."
        ]
      },
      "row-2378de00263f280d": {
        "key": "Interaction plus class/topology branches → localization statement",
        "cells": [
          "Interaction plus class/topology branches → localization statement",
          "These branches determine which class-, dimension-, energy-, and boundary-fixed localization statement is admissible.",
          "Conductance scaling and localization statements jointly require finite-size fits of relevant and irrelevant drift, distributions, boundary conditions, and dephasing cutoffs."
        ]
      }
    }
  }
}