{
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.sign-free-qmc-crossing-drift",
  "title": "Sign-free QMC correlation-ratio crossing drift",
  "classification": "schematic finite-size scaling and inference diagram",
  "dominant_point": "Finite-size correlation-ratio crossings are size-pair estimators, not the thermodynamic critical value; a critical estimate requires a conditional covariance-aware drift extrapolation with independent simulation controls.",
  "panels": [
    {
      "id": "A",
      "title": "Size-dependent correlation-ratio crossings",
      "purpose": "Show that successive finite-size curve pairs cross at visibly distinct tuning-parameter estimates.",
      "object_ids": [
        "correlation_ratio_axis",
        "curve_L",
        "curve_sL",
        "curve_s2L",
        "crossing_L_sL",
        "crossing_sL_s2L"
      ]
    },
    {
      "id": "B",
      "title": "Conditional leading-drift extrapolation",
      "purpose": "Show how size-pair crossing estimates can be extrapolated to the x_L = 0 intercept only under a declared continuous-transition scaling ansatz.",
      "object_ids": [
        "drift_axis",
        "drift_point_L_sL",
        "drift_point_sL_s2L",
        "drift_point_s2L_s3L",
        "joint_resampling_error_marks",
        "conditional_leading_fit",
        "critical_intercept"
      ]
    }
  ],
  "objects": [
    {
      "id": "correlation_ratio_axis",
      "panel_id": "A",
      "kind": "coordinate axes",
      "content": "Vertical axis R(g); horizontal axis a generic declared tuning parameter g; neither axis has numerical ticks."
    },
    {
      "id": "curve_L",
      "panel_id": "A",
      "kind": "finite-size correlation-ratio curve",
      "content": "R_L(g), drawn as a directly labeled solid line."
    },
    {
      "id": "curve_sL",
      "panel_id": "A",
      "kind": "finite-size correlation-ratio curve",
      "content": "R_sL(g), drawn as a directly labeled dashed line for fixed size ratio s > 1."
    },
    {
      "id": "curve_s2L",
      "panel_id": "A",
      "kind": "finite-size correlation-ratio curve",
      "content": "R_s^2L(g), drawn as a directly labeled dotted line."
    },
    {
      "id": "crossing_L_sL",
      "panel_id": "A",
      "kind": "finite-size crossing estimator",
      "content": "g^times_(L,sL), the marked equality point of R_L and R_sL; it is not labeled g_c."
    },
    {
      "id": "crossing_sL_s2L",
      "panel_id": "A",
      "kind": "finite-size crossing estimator",
      "content": "g^times_(sL,s^2L), the distinct marked equality point of R_sL and R_s^2L; it is not labeled g_c."
    },
    {
      "id": "drift_axis",
      "panel_id": "B",
      "kind": "finite-size drift coordinates",
      "content": "Vertical axis g^times; horizontal axis x_L = L^[-(1/nu + omega)], with the thermodynamic-limit edge x_L = 0 on the left."
    },
    {
      "id": "drift_point_L_sL",
      "panel_id": "B",
      "kind": "schematic drift point",
      "content": "The panel-A crossing estimate g^times_(L,sL), shown at the largest x_L of the three illustrative size pairs."
    },
    {
      "id": "drift_point_sL_s2L",
      "panel_id": "B",
      "kind": "schematic drift point",
      "content": "The panel-A crossing estimate g^times_(sL,s^2L), shown closer to x_L = 0."
    },
    {
      "id": "drift_point_s2L_s3L",
      "panel_id": "B",
      "kind": "schematic drift point",
      "content": "An illustrative larger-size crossing estimate g^times_(s^2L,s^3L), shown closest to x_L = 0."
    },
    {
      "id": "joint_resampling_error_marks",
      "panel_id": "B",
      "kind": "uncertainty encoding",
      "content": "Vertical capped marks denote covariance-aware uncertainty obtained by jointly resampling S(Q + delta q) and S(Q); their drawn sizes are layout-only and are not model data."
    },
    {
      "id": "conditional_leading_fit",
      "panel_id": "B",
      "kind": "conditional extrapolation",
      "content": "A solid leading-drift line extrapolates the three schematic crossing estimates toward x_L = 0 only if the stated scaling ansatz describes the selected size window."
    },
    {
      "id": "critical_intercept",
      "panel_id": "B",
      "kind": "thermodynamic-limit estimate",
      "content": "g_c is labeled only at the conditional x_L = 0 intercept and nowhere on a finite-size crossing."
    },
    {
      "id": "simulation_controls",
      "scope": "shared",
      "kind": "required controls",
      "content": "Take Delta tau to zero; declare whether g is scanned through a thermal transition, a finite-temperature quantum-critical sequence holds beta/L^z fixed, or beta is independently converged; converge projector length Theta when projection is used; hold aspect ratio and boundary conditions fixed across the size sequence."
    },
    {
      "id": "schematic_status",
      "scope": "shared",
      "kind": "status statement",
      "content": "The curves, crossing positions, drift points, fit, and error-mark sizes are schematic layout geometry and contain no model data."
    }
  ],
  "relations": [
    {
      "from": "curve_L",
      "to": "curve_sL",
      "through": "crossing_L_sL",
      "type": "equal_at_pair_crossing",
      "meaning": "R_L(g^times_(L,sL)) = R_sL(g^times_(L,sL))."
    },
    {
      "from": "curve_sL",
      "to": "curve_s2L",
      "through": "crossing_sL_s2L",
      "type": "equal_at_pair_crossing",
      "meaning": "R_sL(g^times_(sL,s^2L)) = R_s^2L(g^times_(sL,s^2L))."
    },
    {
      "from": "crossing_L_sL",
      "to": "drift_point_L_sL",
      "type": "replotted_as",
      "meaning": "The first panel-A size-pair crossing supplies the first labeled crossing estimate in panel B."
    },
    {
      "from": "crossing_sL_s2L",
      "to": "drift_point_sL_s2L",
      "type": "replotted_as",
      "meaning": "The second panel-A size-pair crossing supplies the second labeled crossing estimate in panel B."
    },
    {
      "from": "joint_resampling_error_marks",
      "to": "drift_point_L_sL",
      "type": "quantifies_uncertainty_of",
      "meaning": "The ratio uncertainty must retain covariance between numerator and denominator."
    },
    {
      "from": "joint_resampling_error_marks",
      "to": "drift_point_sL_s2L",
      "type": "quantifies_uncertainty_of",
      "meaning": "The ratio uncertainty must retain covariance between numerator and denominator."
    },
    {
      "from": "joint_resampling_error_marks",
      "to": "drift_point_s2L_s3L",
      "type": "quantifies_uncertainty_of",
      "meaning": "The ratio uncertainty must retain covariance between numerator and denominator."
    },
    {
      "from": "conditional_leading_fit",
      "to": "critical_intercept",
      "type": "extrapolates_conditionally_to",
      "meaning": "Under the declared leading-correction ansatz, x_L -> 0 gives the fitted thermodynamic critical value g_c."
    },
    {
      "from": "simulation_controls",
      "to": "conditional_leading_fit",
      "type": "required_before_interpretation",
      "meaning": "Discretization, thermal or projector, aspect-ratio, and boundary-condition limits must be controlled before interpreting the drift fit."
    },
    {
      "from": "schematic_status",
      "to": "conditional_leading_fit",
      "type": "bounds_claim",
      "meaning": "The drawn line illustrates the logic of extrapolation and is not evidence for a numerical critical value."
    }
  ],
  "assumptions": [
    "The scaling discussion is restricted to an ordinary continuous transition with a dimensionless correlation ratio and one declared tuning parameter g.",
    "The size sequence uses a fixed ratio s > 1, fixed lattice shape and aspect ratio, and identical boundary conditions and observable definitions.",
    "Q is the declared ordering-wavevector peak and delta q is the smallest nonzero momentum in a fixed periodic direction; delta q = 2 pi / L assumes unit lattice spacing.",
    "For a thermal transition, every scanned temperature and beta = 1/T is declared. For a quantum critical point studied with finite-temperature QMC, simulations hold beta/L^z fixed with a declared z or independently converge beta; projector simulations independently converge projector length Theta and trial-state effects.",
    "Delta tau is extrapolated to zero when a discrete-time formulation is used.",
    "The leading irrelevant correction with exponent omega > 0 describes the chosen size window, higher corrections are negligible at the claimed precision, and the leading crossing-drift amplitude is not accidentally zero.",
    "A = S(Q + delta q) and B = S(Q) are jointly resampled so their covariance propagates into R = 1 - A/B and then into every crossing estimate."
  ],
  "limits": [
    {
      "condition": "finite L",
      "statement": "g^times_(L,sL) is a size-pair estimator and generally differs from g_c."
    },
    {
      "condition": "x_L -> 0 with the declared ansatz valid",
      "statement": "The conditional leading-drift intercept estimates g_c."
    },
    {
      "condition": "subleading corrections are visible or the fitted size window is unstable",
      "statement": "The leading straight-line extrapolation is insufficient and must be extended or abandoned."
    },
    {
      "condition": "first-order, Berezinskii-Kosterlitz-Thouless, multicritical, anisotropic, or dangerously irrelevant scaling",
      "statement": "The displayed ordinary continuous-transition form need not apply and a problem-specific scaling form is required."
    },
    {
      "condition": "equal-time structure factor",
      "statement": "No analytic continuation is needed for the displayed correlation ratio, although other phase diagnostics may carry continuation errors."
    }
  ],
  "nonclaims": [
    "No displayed curve is a universal scaling function.",
    "No value of nu, omega, z, or any other exponent is supplied or inferred.",
    "The displayed direction of crossing drift is not universal.",
    "No numerical critical value g_c is supplied or inferred.",
    "No phase, order, universality class, material, or Hamiltonian is identified.",
    "The figure contains no model data, simulation output, benchmark values, or quantitative uncertainty estimates.",
    "A sign-free configuration weight does not remove Trotter, projection, autocorrelation, covariance, finite-size, geometry, or model errors."
  ],
  "visual_encodings": {
    "canvas": "explicit white background for light, dark, monochrome, and print contexts",
    "panel_order": "A above B gives a phone-first vertical reading order",
    "curve_L": "directly labeled solid black line",
    "curve_sL": "directly labeled dashed gray line",
    "curve_s2L": "directly labeled dotted black line",
    "finite_crossings": "open circular markers with dashed projections to distinct directly labeled g-axis positions",
    "drift_points": "open circular markers labeled by their size pairs",
    "uncertainty": "gray vertical capped marks; sizes are schematic and encode no numerical values",
    "conditional_fit": "solid black line directly labeled conditional leading fit",
    "critical_intercept": "filled black point at the left-hand x_L = 0 axis, directly labeled g_c and conditional intercept",
    "color_dependency": "none; line style, marker fill, position, arrows, and direct labels redundantly carry every distinction"
  },
  "source_equations": [
    {
      "id": "correlation_ratio",
      "equation": "R_L = 1 - S(Q + delta q) / S(Q)",
      "meaning": "dimensionless comparison of the ordering-wavevector peak with the nearest declared momentum"
    },
    {
      "id": "pair_crossing",
      "equation": "R_L(g^times_(L,sL)) = R_sL(g^times_(L,sL))",
      "meaning": "definition of a finite size-pair crossing estimator"
    },
    {
      "id": "continuous_transition_scaling",
      "equation": "R_L(g) = F((g - g_c) L^(1/nu)) + L^(-omega) G((g - g_c) L^(1/nu)) + ...",
      "meaning": "ordinary continuous-transition scaling at fixed thermal geometry or after ground-state convergence"
    },
    {
      "id": "leading_crossing_drift",
      "equation": "g^times_(L,sL) - g_c proportional to L^[-(1/nu + omega)]",
      "meaning": "leading conditional drift for fixed s when the stated expansion and nonzero leading amplitude apply"
    },
    {
      "id": "ratio_covariance",
      "definitions": "A = S(Q + delta q), B = S(Q), R = 1 - A/B",
      "equation": "Var(R) = Var(A)/B^2 + A^2 Var(B)/B^4 - 2 A Cov(A,B)/B^3",
      "meaning": "first-order covariance propagation; joint resampling should be used in the analysis"
    }
  ],
  "caption": "Schematic correlation-ratio finite-size analysis for an ordinary continuous transition. Curves for successive sizes cross at distinct size-pair estimates; the thermodynamic critical value follows only from a declared correction-to-scaling extrapolation with joint covariance and independent control of time step, thermal-transition or finite-temperature quantum-critical protocol, projection, aspect ratio, and boundaries. Curve shapes, error-mark sizes, drift direction, and exponents are illustrative, not universal, and no model data are shown.",
  "alt_text": "Two vertically stacked panels separate finite-size crossings from a thermodynamic extrapolation. The upper panel shows solid R_L, dashed R_sL, and dotted R_s-squared-L curves with two distinct pairwise crossings, neither labeled g_c. The lower panel replots those crossings and one larger-size estimate with covariance-aware error marks against x_L tending to zero; a conditional leading-drift line reaches g_c only at the left-hand zero-size-correction intercept. Notes require time-step, thermal-transition or finite-temperature quantum-critical, projector, aspect-ratio, and boundary controls and state that the drawing contains no model data.",
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      "url": "https://doi.org/10.1007/s100510070120",
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    },
    {
      "citation": "Steven R. White, Douglas J. Scalapino, Robert L. Sugar, E. Y. Loh Jr., James E. Gubernatis, and Richard T. Scalettar, Numerical Study of the Two-Dimensional Hubbard Model, Physical Review B 40 (1989) 506-516",
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      "use": "sign-free half-filled repulsive-Hubbard auxiliary-field QMC, antiferromagnetic structure factors, and finite-size interpretation"
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    {
      "citation": "Chuang Chen, Xiao Yan Xu, Zi Yang Meng, and Martin Hohenadler, Charge-Density-Wave Transitions of Dirac Fermions Coupled to Phonons, Physical Review Letters 122 (2019) 077601",
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  ]
}