{
  "schema_version": "qft-overview-figure/v1",
  "artifact_id": "qft.artifact.many-body-quantum-matter.integrable-quantum-matter-generalized-hydrodynamics-structure-and-dictionary",
  "title": "Integrable quantum matter: structure and dictionary",
  "schematic": true,
  "not_to_scale": true,
  "evidence_cutoff": null,
  "description_url": "/many-body-quantum-matter/integrable-quantum-matter-generalized-hydrodynamics/bethe-integrable-gases-spin-chains/#nonvisual-description-of-the-bethe-to-ghd-structure-figure",
  "nodes": {
    "contract": {
      "role": "stage",
      "content_tex": "Bethe-model input\\\\ species $a$, rapidity $\\lambda$, physical $p_a$, energy $\\varepsilon_a$, charges $h_{i,a}$, kernel $K_{ab}$"
    },
    "conventions": {
      "role": "note",
      "content_tex": "Positive-oriented counting here; fix branches, boundary sector, kernel sign and rapidity measure"
    },
    "finite": {
      "role": "stage",
      "content_tex": "Finite-volume quantization\\\\ phases $S_{ab}=e^{i\\theta_{ab}}$, Bethe integers, strings and nested species"
    },
    "thermo": {
      "role": "stage",
      "content_tex": "Thermodynamic macrostate\\\\ $2\\pi\\hbar\\rho_{\\rm s}=p'+\\hbar K\\rho_{\\rm p}$,\\quad $\\rho_{\\rm s}=\\rho_{\\rm p}+\\rho_{\\rm h}$,\\quad $s_{\\rm YY}$"
    },
    "counting": {
      "role": "note",
      "content_tex": "Species and rapidity cutoffs must preserve state counting and positivity"
    },
    "charges": {
      "role": "branch",
      "content_tex": "Charge reconstruction\\\\ local and quasilocal $h_i$\\\\ $\\rho_{\\rm GGE}\\propto e^{-\\sum_i\\beta_iQ_i}$"
    },
    "overlaps": {
      "role": "branch",
      "content_tex": "Overlap selection\\\\ $S_{\\rm QA}=2\\operatorname{Re}S_{\\rm ov}-s_{\\rm YY}$\\\\ $s_{\\rm YY}$ counts overlap-allowed states"
    },
    "local": {
      "role": "result",
      "content_tex": "Selected local root density $\\{\\rho_{\\rm p,a}(x,\\lambda,t)\\}$\\\\ charges and overlaps are independent completeness tests"
    },
    "dressing": {
      "role": "stage",
      "content_tex": "Dressing and characteristics\\\\ $h^{\\rm dr}=h+Kn h^{\\rm dr}/(2\\pi)$,\\qquad $v_a^{\\rm eff}=(\\varepsilon_a')^{\\rm dr}/(p_a')^{\\rm dr}$"
    },
    "identity": {
      "role": "note",
      "content_tex": "Same convention gives $(p_a')^{\\rm dr}=2\\pi\\hbar\\rho_{\\rm s,a}$"
    },
    "ghd": {
      "role": "result",
      "content_tex": "Generalized hydrodynamics\\\\ Euler currents and Riemann flow $\\rightarrow$ diffusion, forces, finite size, and weak-breaking kinetics"
    }
  },
  "edges": {
    "conventions--condition--contract": {
      "from": "conventions",
      "to": "contract",
      "relation": "condition",
      "label_tex": ""
    },
    "contract--construction--finite": {
      "from": "contract",
      "to": "finite",
      "relation": "construction",
      "label_tex": ""
    },
    "finite--construction--thermo": {
      "from": "finite",
      "to": "thermo",
      "relation": "construction",
      "label_tex": ""
    },
    "counting--condition--thermo": {
      "from": "counting",
      "to": "thermo",
      "relation": "condition",
      "label_tex": ""
    },
    "thermo--construction--charges": {
      "from": "thermo",
      "to": "charges",
      "relation": "construction",
      "label_tex": ""
    },
    "thermo--construction--overlaps": {
      "from": "thermo",
      "to": "overlaps",
      "relation": "construction",
      "label_tex": ""
    },
    "charges--construction--local": {
      "from": "charges",
      "to": "local",
      "relation": "construction",
      "label_tex": ""
    },
    "overlaps--construction--local": {
      "from": "overlaps",
      "to": "local",
      "relation": "construction",
      "label_tex": ""
    },
    "local--construction--dressing": {
      "from": "local",
      "to": "dressing",
      "relation": "construction",
      "label_tex": ""
    },
    "identity--condition--dressing": {
      "from": "identity",
      "to": "dressing",
      "relation": "condition",
      "label_tex": ""
    },
    "dressing--construction--ghd": {
      "from": "dressing",
      "to": "ghd",
      "relation": "construction",
      "label_tex": ""
    }
  },
  "context": [],
  "linear_equivalent": {
    "heading": "Nonvisual description of the Bethe-to-GHD structure figure",
    "columns": [
      "Figure element or arrow",
      "Relation encoded",
      "Condition or resulting statement"
    ],
    "rows": {
      "row-7fd741230b4c8789": {
        "key": "Bethe-model input",
        "cells": [
          "Bethe-model input",
          "Specify species $a$, rapidity $\\lambda$, physical $p_a$, energy $\\varepsilon_a$, charges $h_{i,a}$, and kernel $K_{ab}$.",
          "The displayed state counting is positive-oriented; signed species require translation. The dashed side condition fixes phase branches, boundary sector, kernel sign, and rapidity measure."
        ]
      },
      "row-bd44c2974c40cfdf": {
        "key": "Bethe-model input → finite-volume quantization",
        "cells": [
          "Bethe-model input → finite-volume quantization",
          "The same data determine phases $S_{ab}=e^{i\\theta_{ab}}$, Bethe integers, string families, and nested species.",
          "Changing one convention without the others changes the spectrum."
        ]
      },
      "row-e429aabed74f3f86": {
        "key": "Finite volume → thermodynamic macrostate",
        "cells": [
          "Finite volume → thermodynamic macrostate",
          "The thermodynamic limit gives $2\\pi\\hbar\\rho_{\\rm s}=p'+\\hbar K\\rho_{\\rm p}$, $\\rho_{\\rm s}=\\rho_{\\rm p}+\\rho_{\\rm h}$, and Yang–Yang entropy.",
          "The $2\\pi\\hbar$ factors and kernel sign are inherited from the Bethe-model input."
        ]
      },
      "row-cdda0116b9c200a2": {
        "key": "Thermodynamic side condition",
        "cells": [
          "Thermodynamic side condition",
          "Species and rapidity cutoffs must preserve state counting and positivity.",
          "A nonconverged species truncation is not a complete macrostate."
        ]
      },
      "row-f9583c4f4d277ae3": {
        "key": "Macrostate → charge-reconstruction branch",
        "cells": [
          "Macrostate → charge-reconstruction branch",
          "Local and quasilocal charge eigenvalues define $\\rho_{\\rm GGE}\\propto e^{-\\sum_i\\beta_iQ_i}$.",
          "Charge completeness is tested by whether the chosen family reconstructs the root density."
        ]
      },
      "row-1795541a2c16cec6": {
        "key": "Macrostate → overlap-selection branch",
        "cells": [
          "Macrostate → overlap-selection branch",
          "Initial-state overlaps and Yang–Yang entropy form $S_{\\rm QA}=2\\operatorname{Re}S_{\\rm ov}-s_{\\rm YY}$.",
          "The quench-action saddle requires normalized overlaps and entropy counting restricted to overlap-allowed states."
        ]
      },
      "row-ca191839050bd99f": {
        "key": "Charge and overlap branches → selected local root density",
        "cells": [
          "Charge and overlap branches → selected local root density",
          "Independent charge and overlap descriptions converge on $\\{\\rho_{\\rm p,a}(x,\\lambda,t)\\}$ when both are complete.",
          "Agreement is a completeness test; the two inputs are not interchangeable assumptions."
        ]
      },
      "row-64aaa477a718cbc2": {
        "key": "Local root density → dressing and characteristics",
        "cells": [
          "Local root density → dressing and characteristics",
          "Dressing gives $h^{\\rm dr}=h+Kn h^{\\rm dr}/(2\\pi)$ and $v_a^{\\rm eff}=(\\varepsilon_a')^{\\rm dr}/(p_a')^{\\rm dr}$.",
          "The dashed identity $(p_a')^{\\rm dr}=2\\pi\\hbar\\rho_{\\rm s,a}$ must hold in the same convention."
        ]
      },
      "row-bb424225409af7ed": {
        "key": "Dressing → generalized hydrodynamics",
        "cells": [
          "Dressing → generalized hydrodynamics",
          "The dressed characteristics determine Euler currents and Riemann flow, followed by separately controlled diffusion, forces, finite-size effects, and weak-breaking kinetics.",
          "Each later correction has its own gradient, positivity, rate, or platform window."
        ]
      }
    }
  }
}