Sample-to-sample width, which is the thing a disorder run already has

Project.toml

@@ -1,6 +1,6 @@
 name = "AbstractQAtlas"
 uuid = "dcea2817-62f8-4a75-b498-1b50a9ed1e4d"
-version = "0.7.13"
+version = "0.7.14"
 authors = ["sota shimozono <shimozono-sota631@g.ecc.u-tokyo.ac.jp>"]
 
 [deps]

docs/references.bib

@@ -858,3 +858,25 @@ @article{Peterson1967
   publisher = {American Physical Society (APS)},
   doi       = {10.1103/RevModPhys.39.69}
 }
+
+@article{AharonyHarris1996,
+  title     = {Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points},
+  author    = {Aharony, Amnon and Harris, A. B.},
+  journal   = {Physical Review Letters},
+  volume    = {77},
+  number    = {18},
+  pages     = {3700--3703},
+  year      = {1996},
+  doi       = {10.1103/PhysRevLett.77.3700}
+}
+
+@article{WisemanDomany1998,
+  title     = {Finite-Size Scaling and Lack of Self-Averaging in Critical Disordered Systems},
+  author    = {Wiseman, Shai and Domany, Eytan},
+  journal   = {Physical Review Letters},
+  volume    = {81},
+  number    = {1},
+  pages     = {22--25},
+  year      = {1998},
+  doi       = {10.1103/PhysRevLett.81.22}
+}

src/AbstractQAtlas.jl

@@ -95,6 +95,7 @@ include("relations/interface.jl")
 "Equilibrium statistical mechanics: occupation statistics → ensembles → thermodynamic potentials → response, FDT & stability."
 module StatisticalMechanics
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     using ..AbstractQAtlas: _beta          # β-or-T normalization (occupation functions)
     import ..AbstractQAtlas: _solve        # extended for a non-affine variable (FreeEnergyFromZ:Z)
     include("relations/thermodynamic.jl")
@@ -107,13 +108,16 @@ end
 "Critical phenomena and conformal field theory: scaling laws, finite-size scaling, Cardy, the c-theorem."
 module Criticality
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
+    import ..AbstractQAtlas: _solve        # extended for a non-affine variable (PseudocriticalWidthScaling:ν)
     include("relations/scaling.jl")
     include("relations/cft.jl")
 end
 
 "Correlations, Green's functions and response: the spectral graph (Dyson, A=−ImG/π), the Keldysh RAK structure + fluctuation–dissipation, Wick / Bloch–De Dominicis (Gaussian factorization), Kramers–Kronig, detailed balance."
 module Correlations
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     using LinearAlgebra: inv, det
     include("relations/spectral.jl")
     include("relations/keldysh.jl")
@@ -123,31 +127,36 @@ end
 "Transport: DC/AC conductivity, thermal & thermoelectric coefficients, the Hall family, Onsager, Wiedemann–Franz, optical sum rule, Johnson–Nyquist."
 module Transport
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     include("relations/transport.jl")
 end
 
 "Quantum information & entanglement: the entropy zoo, its inequalities, multipartite entanglement, measurement and topological entanglement entropy."
 module QuantumInformation
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     using ExperimentalAPI: @experimental
     include("relations/entanglement.jl")
 end
 
 "Quantum-mechanical foundations & bounds: virial, Hellmann–Feynman, Ehrenfest, zero-variance eigenstate, the uncertainty relation and the Lieb–Robinson bound."
 module QuantumFoundations
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     include("relations/quantum.jl")
 end
 
 "Universal bounds stated against a fetched bounding value: Bell (CHSH, Mermin), chaos (MSS), speed limits, fast scrambling, BB84 key rate, optimal cloning, Bekenstein."
 module UniversalBounds
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     include("relations/bounds.jl")
 end
 
 "Topological invariants: Chern number, TKNN, winding, bulk–boundary correspondence."
 module Topology
     using ..AbstractQAtlas
+    import ..AbstractQAtlas: fetch      # unexported, so bare `fetch` here would be Base's
     using LinearAlgebra: det, eigen, Hermitian
     include("relations/topology.jl")
 end

src/relations/quantity_links.jl

@@ -147,6 +147,11 @@ quantities(::HolevoMixingBound) = (VonNeumannEntropy,)
 quantities(::RenyiMonotonicity) = (RenyiEntropy,)
 quantities(::RelativeEntropyNonNegativity) = (RelativeEntropy,)
 
+# The three self-averaging relations name no quantity: `R_X = Var(X)/[X]²` leaves `X`
+# generic, and a width is not a mean, so neither `Typical` nor `DisorderAveraged`
+# names it and `CriticalTemperature` would name the thing `δT_c` is the spread of.
+# Same abstention as the two below, for the same reason.
+
 # ── Disorder statistics ──
 # Both are fully symbol-keyed: `X_typ`/`X_avg` and `F_quenched`/`F_annealed` are
 # REDUCTIONS over an ensemble, and no single quantity names a reduction, so there

src/relations/scaling.jl

@@ -702,6 +702,72 @@ autocorrelation `G(t) ∼ exp(−A|ln t|^d)` of Eq. (A.34) (= Eq. (9.7), §9.1.3
 @relation :scaling OrderedGriffithsEnergyScale(dloglogΩ_dloglogL, d::SpatialDimension) =
     dloglogΩ_dloglogL * d - 1
 
+"""
+    StrongSelfAveraging <: AbstractRelation
+
+`R_X ∼ (L/ξ)^{-d}` for `L ≫ ξ`, so `d ln R_X/d ln(L/ξ) = -d`, where
+`R_X = Var(X)/[X]²` over an ensemble of samples (Aharony & Harris,
+[AharonyHarris1996](@cite)).
+
+Off criticality only, and the abscissa is `L/ξ` rather than `L`, which is also
+what keeps this slope a different variable from
+[`CriticalSelfAveraging`](@ref)'s: one number cannot be both, and a graph that
+shared the name would read an off-critical law off a critical measurement.
+
+Variables: `dlogR_dlogLξ` (caller-computed), `d`.
+"""
+@relation :scaling StrongSelfAveraging(dlogR_dlogLξ, d::SpatialDimension) = dlogR_dlogLξ + d
+
+"""
+    CriticalSelfAveraging <: AbstractRelation
+
+`R_X ∼ L^{α/ν}` at criticality when the randomness is IRRELEVANT, so
+`d ln R_X/d ln L = α/ν` ([AharonyHarris1996](@cite)).
+
+The exponents are the PURE system's, so they are `α_pure` and `ν_pure` and stay
+untyped: [`SpecificHeatExponent`](@ref) elsewhere in the registry means the system
+under study, and at a random fixed point that is a different number. Wiseman and
+Domany conjectured this form with the random fixed point's exponents;
+[AharonyHarris1996](@cite) and the simulations in [WisemanDomany1998](@cite) both
+contradict it, the width going to a constant there instead.
+
+At `α = 0`, the marginal case this literature is largely about, the residual does
+not depend on `ν` and a passing check says nothing about it.
+
+Variables: `dlogR_dlogL` (caller-computed), `α_pure`, `ν_pure`.
+"""
+@relation :scaling CriticalSelfAveraging(dlogR_dlogL, α_pure, ν_pure) =
+    dlogR_dlogL - α_pure / ν_pure
+
+"""
+    PseudocriticalWidthScaling <: AbstractRelation
+
+`δT_c(L) ∼ L^{-1/ν}`, so `d ln δT_c/d ln L = -1/ν`, where `δT_c` is the width of
+the sample-to-sample distribution of pseudocritical temperatures
+([WisemanDomany1998](@cite), measured directly).
+
+Not `L^{-d/2}`, the central-limit answer, which holds only where the Harris
+criterion does. The two are close and the measurement has to be good enough to
+tell them apart: the site-dilute Ising model in `d = 3` gives `1.449(8)` against
+`d/2 = 1.5`, six of its own errors away, agreeing with that model's separately
+fitted `1/ν = 1.467(5)`.
+
+Variables: `dlogδTc_dlogL` (caller-computed), `ν`.
+"""
+@relation :scaling PseudocriticalWidthScaling(dlogδTc_dlogL, ν::CorrelationLengthExponent) =
+    dlogδTc_dlogL + 1 / ν
+
+# `1/ν` is not affine in `ν`, so the generic three-probe inverter refuses it. That
+# inverter also refuses a non-finite probe; this one has to refuse its own pole, or
+# a flat slope returns an infinity whose SIGN comes from the caller's zero.
+function _solve(::PseudocriticalWidthScaling, ::Val{:ν}; dlogδTc_dlogL, _extra...)
+    iszero(dlogδTc_dlogL) && error(
+        "solve: PseudocriticalWidthScaling has no ν at dlogδTc_dlogL = 0. A width " *
+        "that does not shift with L does not identify a correlation-length exponent.",
+    )
+    return -1 / dlogδTc_dlogL
+end
+
 """
     WeinribHalperinExponent <: AbstractRelation
 

test/relations/test_interface.jl

@@ -18,9 +18,9 @@ AbstractQAtlas.domain(::_NonAffineDemo) = :test_only
     # domain that has no line here shows up as a mismatch in this number alone.
     # Universal-only: the model-specific relations (spin glass, Drude mobility,
     # single-band Hall) live in QAtlas.
-    @test length(rels) == 170
+    @test length(rels) == 173
     @test allunique(typeof.(rels))
-    @test length(all_relations(; domain=:scaling)) == 30   # +ActivatedDynamicalScaling, ActivatedFiniteSizeScaling, +14 Appendix-A scaling types, +WeinribHalperinExponent, +QuantumHyperscaling; +TypicalCorrelationLength, GriffithsExponentDivergence, GriffithsSusceptibility, GriffithsSpecificHeat, ActivatedMomentGrowth; +CriticalCorrelationDecay, ActivatedCriticalCorrelation (#158)
+    @test length(all_relations(; domain=:scaling)) == 33   # +ActivatedDynamicalScaling, ActivatedFiniteSizeScaling, +14 Appendix-A scaling types, +WeinribHalperinExponent, +QuantumHyperscaling; +TypicalCorrelationLength, GriffithsExponentDivergence, GriffithsSusceptibility, GriffithsSpecificHeat, ActivatedMomentGrowth; +CriticalCorrelationDecay, ActivatedCriticalCorrelation (#158); +3 self-averaging
     @test length(all_relations(; domain=:thermodynamic)) == 15
     @test length(all_relations(; domain=:fundamental)) == 9   # +GrandPotentialLegendre, ParticleNumberResponse (grand-canonical); +ElectricCurrentResponse (j = −∂H/∂A)
     @test length(all_relations(; domain=:topology)) == 3

test/relations/test_scaling.jl

@@ -6,7 +6,7 @@
 # pins the no-float-promotion contract.
 
 using AbstractQAtlas
-using AbstractQAtlas: residual, check, solve
+using AbstractQAtlas: residual, check, solve, derive
 using LinearAlgebra
 
 const ISING2D = (α=0//1, β=1//8, γ=7//4, δ=15//1, ν=1//1, η=1//4)
@@ -604,7 +604,7 @@ end
     rs = relations_constraining(SpatialDimension())
     # Every relation that takes a d, named. Listing a subset leaves the rest
     # resting on the soft coverage ratio, which cannot see one missing entry.
-    @test length(rs) == 13
+    @test length(rs) == 14
     for r in (
         Josephson(),
         QuantumHyperscaling(),
@@ -619,6 +619,7 @@ end
         FixedPointDisorderStrength(),
         ConventionalFieldSusceptibility(),
         ConventionalFieldSpecificHeat(),
+        StrongSelfAveraging(),
     )
         @test r in rs
     end
@@ -684,3 +685,63 @@ end
     @test CriticalCorrelationDecay() in
         relations_constraining(DisorderAveraged{ConnectedSpinCorrelation})
 end
+
+@testset "how the sample-to-sample width scales says which fixed point it is" begin
+    # `R_X = Var(X)/[X]²` over an ensemble of samples, which is what a disorder
+    # calculation already produces per size.
+    d, α, ν = 3, -0.1, 0.7
+    @test solve(StrongSelfAveraging(), Val(:d); dlogR_dlogLξ=(-d)) ≈ d
+    @test solve(CriticalSelfAveraging(), Val(:α_pure); dlogR_dlogL=α / ν, ν_pure=ν) ≈ α
+    @test check(StrongSelfAveraging(); dlogR_dlogLξ=(-d), d=d, atol=1e-12)
+    @test check(CriticalSelfAveraging(); dlogR_dlogL=α / ν, α_pure=α, ν_pure=ν, atol=1e-12)
+    @test !check(StrongSelfAveraging(); dlogR_dlogLξ=(+d), d=d, atol=1e-9)   # sign
+    @test !check(
+        CriticalSelfAveraging(); dlogR_dlogL=(-α / ν), α_pure=α, ν_pure=ν, atol=1e-9
+    )
+
+    # Relevant randomness means no decay at all, so both decay laws are refused.
+    @test !check(StrongSelfAveraging(); dlogR_dlogLξ=0.0, d=d, atol=1e-6)
+    @test !check(CriticalSelfAveraging(); dlogR_dlogL=0.0, α_pure=α, ν_pure=ν, atol=1e-6)
+
+    # The two abscissas are different variables and must stay so. Sharing the name let
+    # an off-critical law be read off a critical slope and, through `d`, put a wrong ν
+    # on the far side of Josephson with no error.
+    @test isdisjoint(variables(StrongSelfAveraging()), variables(CriticalSelfAveraging()))
+    @test_throws "not reachable" derive(:ν; dlogR_dlogL=α / ν, α=α)
+
+    # At α = 0 the residual does not depend on ν, so a pass says nothing about it.
+    # Documented rather than refused: α = 0 is the marginal case, not a bad input.
+    @test all(
+        x -> check(CriticalSelfAveraging(); dlogR_dlogL=0.0, α_pure=0.0, ν_pure=x, atol=0),
+        (0.001, 0.63, 999.0),
+    )
+
+    # `δT_c ∼ L^{-1/ν}`, not `L^{-d/2}`. The site-dilute Ising model in d = 3 measures
+    # 1.449(8), which is six of its own errors from d/2 = 1.5, so the discriminator
+    # discriminates. Asserted in units of that error, the two being 3.4% apart.
+    ρ, σρ = 1.449, 0.008
+    @test abs(ρ - d / 2) / σρ > 5
+    # Round-tripped through `check` rather than compared to the hand-written inverse,
+    # which would be that formula against itself.
+    ν_solved = solve(PseudocriticalWidthScaling(), Val(:ν); dlogδTc_dlogL=(-ρ))
+    @test check(PseudocriticalWidthScaling(); dlogδTc_dlogL=(-ρ), ν=ν_solved, atol=1e-12)
+    # Not a test of this package: a guard on the two numbers the docstring quotes, so
+    # editing one without the other is caught. They are separate measurement channels
+    # on the same model and agree at 1.9 of their combined error.
+    @test abs(ρ - 1.467) < 2.5 * sqrt(σρ^2 + 0.005^2)
+    @test !check(
+        PseudocriticalWidthScaling(); dlogδTc_dlogL=(-d / 2), ν=ν_solved, atol=1e-3
+    )
+
+    # A flat width identifies no exponent, so the specialized inverse refuses its own
+    # pole rather than returning an infinity signed by the caller's zero.
+    @test_throws "no ν at" solve(PseudocriticalWidthScaling(), Val(:ν); dlogδTc_dlogL=0.0)
+    @test_throws "no ν at" solve(PseudocriticalWidthScaling(), Val(:ν); dlogδTc_dlogL=-0.0)
+
+    @test variable_types(StrongSelfAveraging()) == (SpatialDimension,)
+    @test variable_types(PseudocriticalWidthScaling()) == (CorrelationLengthExponent,)
+    # Untyped on purpose: these are the PURE system's exponents, and the registry's
+    # exponent types mean the system under study. `WeinribHalperinExponent` leaves
+    # `ν_dis` bare for the same reason.
+    @test isempty(variable_types(CriticalSelfAveraging()))
+end