How an instrument sees a line.
The models page learns systems. This page learns instruments: the ways a measurement can tell a critical line from an isolated critical point from a transition that only pretends to be critical, and when it must decline to say. Each section names what a technique needs, what it decides, what it is blind to, what it costs, how it reads in the language of the relational kernel, and which side of the firewall it lives on.
The alphabet these pages share (the Criticality Compass’s verdicts ◆ design)1
The organising idea is follow the flow: the velocity, accumulation and circulation of the flow over declared scale and control relations are what the Detector reads. The Detector classifies a system’s critical geometry from observations; the Navigator finds a critical line and moves along it.
How to read: every section shows a headline, one live graphic and a cyan For the Compass box. “Working equations” opens the physicist’s layer; “Full derivation” inside it opens the deepest one. “Check yourself” holds three graded items; the final test at the end is retakeable and draws new items each time. Scores stay in this browser. The mark ◆ design flags a design choice of the instrument, never a physics result.
An instrument never sees a fixed point directly; it sees the flow, and the flow offers three readings: how fast it moves (velocity), how far it has gone (accumulation), and whether it turns (circulation).
Renormalisation replaces a system by a coarser copy of itself, and the couplings of the copy differ from the original’s; repeat, and the couplings trace a path, the RG flowThe path of the effective couplings as the description is coarse-grained step by step. ℓ counts the steps (ℓ = ln b for a scale factor b)..2 The alphabet is a statement about where that path stops. Press the four buttons (the fourth, SPIRAL, is not a verdict; it shows what circulation looks like). The dots on each trajectory mark whole RG levels, and the strip under the flow plots the distance moved per level. At a POINT (started on its critical surfaceThe set of couplings from which the flow runs into the fixed point: every relevant coupling tuned to zero. Here it is the line x = 0.) and on a LINE that distance falls to zero; on a WALKING flow it falls to a small nonzero minimum and rises again; only the spiral carries circulation.
Three numbers per level, three definitions: the step between successive levels is a directed differencex[s] − x[t] over a declared edge s → t. The kernel’s operator A is defined this way; it is not a “gradient”. over the declared scale relation (the kernel’s A); the running sum of steps is an accumulation (the kernel’s B); the work the field does round a small loop is a circulation. That third reading is the FLOW’s; the kernel has no loops and reads it only through R’s antisymmetric response over a chain (§4): R, R(g)[e] = g[e] + ρ (Σsucc g − Σpred g) is an antisymmetric response over a chain of declared relations: the kernel’s discrete analogue of circulation, not a loop integral.12
The Detector will be a total lookup table over a discrete signature of these three readings (their signs, read against a declared margin). REFUSE is one of the table’s cells, not an error message.1
With gn the couplings after n coarse-graining steps:
vn = |gn+1 − gn| (velocity), ΛN = ∑n<N vn (accumulated path length), Γ = ∮ f·dl ≈ (∂fy/∂x − ∂fx/∂y) h² (circulation)The four fields drawn above, with x the direction the page calls relevant (or along the line) and y an irrelevant one: POINT f = (0.6x, −y), fixed point at the origin; LINE f = (0, −y), every point of y = 0 fixed; WALKING f = (−(0.02 + x²), −y), whose x-velocity never falls below 0.02; SPIRAL f = (−0.15x − 0.8y, 0.8x − 0.15y), a complex pair of eigenvalues (§1) and the only field here with nonzero curl. SPIRAL is not a verdict of the alphabet: it is here only to show what circulation looks like. They are illustrations of the four shapes, not models.
The readout under the flow gives, in the plane’s own coupling units: the smallest distance moved in one level and the level where it happens; Λ, the total path length over the levels drawn; and Γ, the circulation round a 0.1 × 0.1 square at that slowest point. The engine adds the start, the end and the RG time ℓ reached.
The two binding constraints on the instrument ◆ design:19
Near a fixed point the flow is a matrix: its eigenvalues are the exponents, its eigenvectors are the directions, and an eigenvalue of exactly 1 (y = 0) that stays exactly 1 beyond linear order makes a whole line of fixed points.
Kadanoff’s idea was to throw detail away on purpose by blocking spins; Wilson made the blocking a map on couplings and found the exponents in its fixed points.2 Close to a fixed point the map is linear, δK′ = M δK, and everything is in the eigenvalues, written λ = by for a scale factor b: a relevanty > 0 (λ > 1): the deviation grows under coarse-graining; it must be tuned to zero to sit on the critical point. direction grows, an irrelevanty < 0 (λ < 1): the deviation shrinks and is forgotten; this is universality. one shrinks, and a marginaly = 0 (λ = 1): at linear order the deviation neither grows nor shrinks. If it stays exactly marginal at every order, the fixed point is one of a line. one does neither. Drag the four entries of M. The arrows are the step δK′ − δK; the dotted orbit is repeated application of M. Set an eigenvalue to 1 and the fixed line appears; give M a complex pair and the orbit turns.
Navigator: the marginal eigenvector is the tangent to the line; the relevant one is the normal, the direction that must be tuned. Detector: a POINT has one eigenvalue above 1 and the rest below; a LINE has one that stays at exactly 1, level after level.
| y | λ | name | under coarse-graining |
|---|---|---|---|
| y > 0 | λ > 1 | relevant | grows; one knob to tune per relevant direction |
| y < 0 | λ < 1 | irrelevant | shrinks; the microscopic details live here |
| y = 0 | λ = 1 | marginal | undecided at linear order: exactly marginal to all orders gives a line; marginal only at linear order can give logarithms (BKT) |
The table is the RG page’s.2 The exponents follow from the leading thermal and magnetic eigenvalue exponents, yT (the temperature-like, spin-flip-even direction) and yH (the field-like, odd direction): ν = 1/yT and η = d + 2 − 2yH. Ron, Brandt & Swendsen use exactly these (their yT1 = 1.591 gives ν = 0.6285 and their yH1 = 2.4829 gives η = 0.0342 in d = 3), though their paper does not print λ = by itself.4 A complex pair λ = |λ|e±iθ rotates the deviation by θ per step: the orbit spirals in (|λ| < 1) or out (|λ| > 1). The classifier in the graphic is a total table over the signs of y1, y2 (each +, 0 or −) and a flag for a complex pair; no chain of conditions decides it.
The worked example: 1D Ising decimation.2 Sum out every second spin of a chain with coupling K:
Σσ2=±1 eK(σ1σ2 + σ2σ3) = 2 cosh(K(σ1 + σ3)) = eg + K′σ1σ3Taking σ1σ3 = +1 gives 2 cosh 2K = eg + K′; taking −1 gives 2 = eg − K′. Divide: e2K′ = cosh 2K, so
K′ = ½ ln cosh 2K, b = 2.Fixed points: K* = 0 and K* = ∞, nothing between. Linearise at K* = 0: dK′/dK = tanh 2K → 0, so λ = 0 < 1: the high-temperature fixed point is fully attractive (a phase, not a transition). At large K, K′ ≈ K − ½ ln 2: the flow creeps down by a constant each step, so K* = ∞ is unstable, and the chain has no finite-temperature transition. A 1 × 1 “matrix” already shows the whole logic: fixed points, eigenvalues, stability.
Coarse-grain the same snapshots again and again: the change from one level to the next is the flow, and a single lattice gives several RG steps.
Swendsen (1979) turned Kadanoff’s picture into a measurement: simulate at the candidate critical point, block the snapshots, block the blocks, and compare levels.3 The toy is a 64×64 Ising lattice (32×32 on a narrow screen) with Metropolis updates. Each smaller panel is a 2×2 majority-ruleA block spin takes the sign of the sum of its four spins; a tie takes the block’s top-left spin (a fixed rule, so the map is deterministic). blocking of the one before, labelled with its nearest-neighbour correlation, and the chart above plots those four numbers (nn = the nearest-neighbour correlation ⟨sisj⟩, averaged over neighbouring pairs). Run it and move T. Near Tc = 2/ln(1 + √2) ≈ 2.269 (the arithmetic is in §5) the levels look alike and their correlations nearly agree: a fixed point. Below it the blocks order level by level; above it they disorder. That drift is the flow, read from one lattice.
The trace below turns the same snapshots into the linearised matrix of §1. It is equations (2)–(5) of Ron, Brandt & Swendsen, lit term by term while the toy’s own correlations fill them in.4 With one tracked coupling (nearest neighbours) the “matrix” is a single number you can watch settle as samples accumulate.
This is the instrument. Kernel reading: A is the level-n versus level-(n+1) difference over the declared blocking relation; B is the sum of each such difference and its immediate successor down the chain of levels; R adds the antisymmetric term ρ·(Σsucc − Σpred), the page’s stand-in for circulation in coupling space ◆ design. It passes the design bound with room to spare. Blind spots: a walking minimum that sits under the noise at reachable levels (then REFUSE), and too few tracked couplings, where truncation can manufacture a drift.19
Swendsen’s construction, as Ron, Brandt & Swendsen print it (p. 2).4 Level-n configurations carry unknown couplings Kα(n) conjugate to operators Sα(n) (nearest-neighbour sum, next-nearest, four-spin, …). Then
T(n+1,n)αβ = ∂Kα(n+1) / ∂Kβ(n) (2) ∂⟨Sγ(n+1)⟩/∂Kβ(n) = Σα [∂⟨Sγ(n+1)⟩/∂Kα(n+1)] [∂Kα(n+1)/∂Kβ(n)] (3) ∂⟨Sγ(n+1)⟩/∂Kβ(n) = ⟨Sγ(n+1)Sβ(n)⟩ − ⟨Sγ(n+1)⟩⟨Sβ(n)⟩ (4) ∂⟨Sγ(n+1)⟩/∂Kα(n+1) = ⟨Sγ(n+1)Sα(n+1)⟩ − ⟨Sγ(n+1)⟩⟨Sα(n+1)⟩ (5)Everything on the right of (4) and (5) is a correlation measured in the simulation, so (3) is a linear system for T. Blöte and co-workers write the same thing as 𝔅T = ℭ (their eq. 5; matching it to (3)–(5) makes 𝔅 the level-(n+1) auto-correlations and ℭ the cross-correlations, our reading), and solve it after truncating to a finite set of couplings; the number of couplings used in the literature rose from 7 (1979) to 99 (1992).5 The eigenvalues of T give y = ln λ / ln b with b = 2.2
The block rule is a choice. Ron, Brandt & Swendsen use a tunable rule, equation (6):
P(σ′l) = exp(w σ′l Σj∈l σj) / [exp(w Σj∈l σj) + exp(−w Σj∈l σj)] (6)which is the majority rule at w = ∞. Tuned to w = 0.4314 (even sector) and 0.555 (odd sector), with 30 even and 20 odd couplings on the 3D Ising model, it gives yT1 = 1.591(1) and yH1 = 2.4829(2), converged by the second or third iteration.4 (The even sector holds couplings unchanged when every spin is flipped, such as nearest-neighbour products, and gives yT; the odd sector holds those that change sign, such as the field, and gives yH.)
| n (iteration) | yT1, majority rule (w = ∞) | yT1, tuned (w = 0.4314) |
|---|---|---|
| 1 | 1.4224(7) | 1.5930(8) |
| 2 | 1.5072(6) | 1.5919(5) |
| 3 | 1.5504(5) | 1.5912(5) |
| 4 | 1.5718(7) | 1.5906(6) |
| 5 | 1.5788(11) | 1.5908(7) |
3D Ising, 256³ lattice, 30 even couplings (Tables II and III of Ron, Brandt & Swendsen).4
The surprise. The eigenvalues converge while the renormalised correlation functions do not: “there is no evidence that the tuned RG transformation brings the renormalized Hamiltonians closer to the fixed point.”4 Blöte et al. (1996) had already shown a nearly flat approach for a tuned Hamiltonian and rule: yT = 1.5885 → 1.5852 → 1.5829 and yH = 2.48492 → 2.48309 → 2.48219 over n = 1–3 at L = 32.5 In two dimensions the plain majority rule already converges by the second iteration.4 Why it matters for the instrument: the page says “watch the flow”, and the surprise says the eigenvalues (the flow’s local rates) can settle while the position on the trajectory has not. The Detector therefore never trusts a settled eigenvalue: it reads the per-level differences, each with its declared band, through the five-valued sign (§3), level by level, and a zero is claimed only where the sign reads ZERO! at every declared level.
How MCRG sees the alphabet ◆ design:19 POINT: one eigenvalue above 1, the rest below, and levels n and n+1 agree. LINE: one eigenvalue that stays at exactly 1 across levels, and fixed-level statistics that depend on where the run started. WALKING: an eigenvalue near 1 that drifts; level differences reach a small nonzero minimum, then grow.
Write the level-n weight as exp(Σβ Kβ(n)Sβ(n)) (Ron, Brandt & Swendsen put no minus sign in the exponent, and neither do we here).4 For any quantity Q computed from a configuration, ∂⟨Q⟩/∂Kβ = ⟨Q Sβ⟩ − ⟨Q⟩⟨Sβ⟩, because ⟨Q⟩ = Z−1Σ Q eΣKS and differentiating the numerator brings down Sβ, differentiating Z−1 brings down −⟨Sβ⟩.
Take Q = Sγ(n+1), the block operator computed from the blocked configuration. Averaged over the level-n ensemble it is a function of the K(n); averaged over the level-(n+1) ensemble it is a function of the K(n+1). The blocking rule makes the two averages the same number, so ⟨Sγ(n+1)⟩ depends on K(n) only through K(n+1)(K(n)). The chain rule is (3). Applying the identity above in each ensemble gives (4) (Q = S′, derivative with respect to a level-n coupling, so the partner operator is S(n)) and (5) (derivative with respect to a level-(n+1) coupling, partner S(n+1)). In matrix form (3) is (4) = (5) T, so T = (5)−1(4); with one coupling, T is the ratio of two numbers, which is what the trace computes live.
At one size LINE, WALKING and BKT look alike; how an exponent moves tells them apart.
They differ in how an effective exponent changes with SCALE (the system size, or the RG level) and with CONTROL (a knob such as the temperature or a coupling, varied between runs), and reading that difference honestly needs a margin ε declared before the data.
Finite-size scaling reads exponents from how observables change with the system size L. Iino, Morita, Kawashima and Sandvik showed how that goes wrong: the q = 5 Potts transition is first order, yet sizes L = 24–256 collapse beautifully with β = 0.070(8), ν = 0.61(5) and error bars smaller than the symbols.7 The cure is to follow the exponent: from crossings of size pairs (L/2, L) they read a size-dependent ν(L) and β(L), whose power-law extrapolation gives 1/ν(∞) ≈ 2.23, larger than 2, which is impossible for a continuous transition in two dimensions (by hyperscaling, α = 2 − dν, it would mean α > 1, beyond even the first-order value ν = 1/d).17 Gorbenko, Rychkov and Zan explain the drift: along a walking flow scaling dimensions drift with scale in a tangent-shaped way.8 At a BKT point the drift is different again: finite-size values approach their limits only as 1/(ln L + C).10
The four sequences above are synthetic, built with those shapes. Each panel shows an effective exponent against log2 L at two control values. Sort them: which varies with the control but not with scale, which varies with neither, which drifts faster and faster, which slows but never settles?
Reading a drift needs a verdict per step, and that is where a five-valued sign comes in ◆ design.1 Let v be a velocity estimate (a per-level difference) and w the half-width of its band, so the band is [v − w, v + w]; it is read against a declared margin ±ε:
The cell is looked up in a table with a row for every combination of four yes/no tests: a = the whole band lies above +ε (v − w > ε); b = it lies below −ε (v + w < −ε); c = it lies inside ±ε (|v| + w < ε); r = it excludes zero. The four answers form the key abcr (1 = yes). Some keys cannot occur for w, ε ≥ 0 (a and b together would need a negative band); they are given the cell ? so that the table is total and no input falls through it. In words: c without r (0010) is ZERO!; c with r (0011) is SLOW by the overlap rule; r alone (0001) is SLOW; a with r (1001) is SIG+; b with r (0101) is SIG−; every other key is ?. No chain of conditions decides the cell. Where both c and r hold (key 0011: the band lies inside ±ε and excludes zero), the ZERO! test and the SLOW test are both met, and the cell is SLOW: the overlap resolves toward the weaker claim, and a LINE is never asserted on an ambiguous reading. With ε = 0 the table collapses to the three-valued sign (SIG+, SIG−, ?). The margin plays the role of the equivalence bounds of a TOSTTwo one-sided tests: reject “the effect is at or below −Δ” and reject “it is at or above +Δ”; together they support “inside the bounds”. The bounds are declared first. procedure, which Lakens insists are declared before the data are seen: “no significant difference” is not equivalence.11
One machine for every case: five-valued signs over declared relations, with ε and the null declared before the data. LINE versus WALKING is a question of power (how small an ε the data can close), not of a null hypothesis. The verdict is typed LINE(ε, nlevels), and the blind spot “walking slower than ε per level” is declared in advance.1
Effective exponents from crossings (Iino et al., eqs 5–6), with R4 = ⟨m4⟩/⟨m2⟩2 the Binder ratio and t the reduced temperature at the crossing of the (L/2, L) pair:7
1/ν(L) = log2 [ (d ln(R4(t,L) − 1)/dt) / (d ln(R4(t,L/2) − 1)/dt) ], (1 − 2β(L))/ν(L) = log2 [ (d⟨m2⟩(t,L)/dt) / (d⟨m2⟩(t,L/2)/dt) ]The drifting dimension and the drifting exponent (Gorbenko, Rychkov & Zan, (4.33) and (4.34)):8
δφ(r) = Re ΔCφ − (√3 ε / 2π) Cφφε′ tan(ε ln(r/r0) / π), ν(r) = 1 / (d − δε(r))(Their ε is the walking parameter, not the margin of this page.) The walking correlation length is huge for q = 5: ξ ≈ 2512 lattice spacings, 159 for q = 6, 48 for q = 7, so at L ≪ ξ the transition imitates a critical point.9
The logarithmic approach at BKT (Hasenbusch 2005, eq. 31): the helicity modulus at the transition on an L × L lattice is
Υ(L) = 0.63650817819… + 0.318899454… / (ln L + C) + …so it is still 5 % above its limit at L = 2048 for C of order 1 (an illustration of the form, not a fitted C).10
The five-valued sign ◆ design. Four predicates: a = (v − w > ε), b = (v + w < −ε), c = (|v| + w < ε), r = (v − w > 0 or v + w < 0). The key abcr indexes a 16-row table (shown under the graphic). SIG+ = the whole band beyond +ε; SIG− = beyond −ε; ZERO! = c without r (0010): the whole band inside the margin and not excluding zero; SLOW = resolved nonzero but not beyond ε; ? = everything else, including keys that cannot occur and NaN.1 TOST: equivalence holds when the 90 % confidence interval (two one-sided tests at α = 5 %) lies inside the declared bounds [−ΔL, ΔU].11
Why the doubling chain fails the design bound. A finite-size-scaling chain simulates at L, 2L, 4L, …. The work per independent sample grows like Ld+z, with z the dynamic exponent of the update: Ld sites per sweep, and about Lz sweeps between independent samples. Each doubling therefore multiplies the cost by 2d+z; for d = 2 and an illustrative z = 2 that is 16, and five doublings cost about 165 ≈ 106 times the first size. Separating LINE from WALKING needs L to approach ξ, which is 2512 for q = 5. That is exponential cost in the number of doublings, and it is why the instrument follows the flow within one lattice (§2) rather than across lattices.19
The ε = 0 limit. With ε = 0, c needs |v| + w < 0, impossible for w ≥ 0, so ZERO! is empty; SLOW needs r without a or b, and with ε = 0 r implies a or b, so SLOW is empty too. What remains is SIG+, SIG− and ?: the three-valued sign exactly.1
Declare the relations first; then three definitions turn exponent estimates into signs a table can read.
The three are a directed difference (A), an accumulation over immediate successors (B) and an antisymmetric response (R). R(g)[e] = g[e] + ρ (Σsucc g − Σpred g) is an antisymmetric response over a chain of declared relations: the kernel’s discrete analogue of circulation, not a loop integral (the circulation of §0 is the continuum picture it is an analogue of). An interior locus is a node with both a declared predecessor and a declared successor, s → t → u. B needs one, and so does the full A → B → R path; without one, the primary kernel Σk (the antisymmetric response applied without B) is the fallback.
The relational kernel is a small set of operator definitions with one rule above them: nothing enters an operator undeclared.12 The chain is observable → M → a numerical projection (numbers) → declared relations → operators → result → comparison with observables, where M is the declared mapping from what was observed to the numbers the operators are applied to. Here the nodes are exponent estimates at RG levels 0–4; the edges are declared relations between them. Click an edge to declare or undeclare it, press a preset, or type your own values.
A missing declaration does not become zero: the operator is NOT EVALUATED, and that is a value the verdict table can read. Declare a closing edge (a ring) and the whole relation set becomes inadmissible: B’s accumulation needs a well-founded successor order, and on a ring a node would be its own predecessor. Tick “second control value” and the two relation families appear: Continuation over scale (along each row) and Coupling over control (between the rows). The trace below walks B through term by term on row 1’s chain.
The Detector’s verdict will be a lookup over the signs of A, B and R (the table in the graphic is ◆ design, illustrative; the instrument’s own table does not exist yet). Statistics enters only as the declared band w and margin ε; every “if σ > x” hidden in code is a review finding. Kernel-conformant is not the same as physically correct: they are two separate gates.121
| operator | definition (Kernel V8) | type |
|---|---|---|
| A / Δ | x[s] − x[t] over each declared edge | NodeField → EdgeField (the only node→edge step) |
| B | g[e] + Σ over immediate successors; terminal edges accumulate nothing | EdgeField → EdgeField |
| R | g[e] + ρ[src(e)]·(Σ succ − Σ pred) (plus a declared cross-coupling between components) | EdgeField → EdgeField |
| ρ | ρbase·χ/(χ0 + χ), χ = max |A| at the node | M supplies ρbase, χ0 |
| Σk (the primary kernel; a named operator, not a summation) | Δ then the antisymmetric response applied WITHOUT B | B needs an interior locus; Σk applies the antisymmetric response without B, which is why it is the form used when no interior locus is established |
| C | a declared projection (states what it keeps and discards) | application layer, not the kernel |
As the kernel reference states them.12
Two gates. The kernel reference names four verification stages: declaration (the domain, M, the relations and their provenance), mathematical (only declared operators, no imported constructs), observable validation (against independent observables, with no tuning) and repeatability.12 Passing them says the computation is traceable: every number follows from declared observables through declared operators. It does not say the physics is right. A block rule that throws away the relevant operator can be declared perfectly and still hide a LINE; a physically sound estimate built with an undeclared smoothing step can be correct and still fail conformance. So the instrument is checked twice: once for conformance and once against the exact routes of §5, and a pass at one gate says nothing about the other.
Why B needs an interior locus. B adds each edge’s immediate successors. On a chain s → t with nothing after t, the only edge is terminal and B(g) = g: nothing is accumulated, so B carries no information beyond A. Only when some node has both a declared incoming and a declared outgoing edge does B differ from A, which is why the reference makes that declaration the condition for the A → B → R path.
When the Hamiltonian is known, three exact tools, duality, the transfer matrix and partition-function zeros, locate and classify a transition; they generate truth for scoring and never touch the instrument’s inputs.
The transfer matrix. Build the matrix that adds one row to an infinitely long Ising strip of width W, periodic across the width (2W states). Its largest eigenvalue gives the free energy per site f(W). At a conformally invariant critical point the leading finite-width correction is universal, f(W) = f∞ − πc/(6W2), where c is the central charge (Blöte, Cardy and Nightingale; Affleck).15 The demo computes W = 2 to 10 in your browser. Step through the trace and read c off the slope; then move K away from Kc and watch the estimate stop converging.
Duality. Kramers and Wannier found that the square-lattice Ising model maps onto itself with high and low temperature exchanged; the self-dual point is the transition. Aasen, Mong and Fendley print the anisotropic version as a whole critical, self-dual line, (sinh 2Jx)(sinh 2Jy) = 1.13 Slide Jx along it, then find the isotropic point: sinh 2K = 1 gives Kc = ½ ln(1 + √2) ≈ 0.4407. (That number is this page’s arithmetic from their equation (2.2); they do not print it.)
A line of critical points is not a LINE of fixed points. Every point of that curve is critical, and every point is the same Ising class (Onsager’s exact solution of the anisotropic model, not held here): a line of critical points with no varying exponent. Strictly, a blocking that keeps the anisotropy has a fixed point for each ratio Jy/Jx, but they differ only by rescaling one axis (a redundant direction: a change of the couplings that only rescales a coordinate and changes no universal quantity), so an eigenvalue of 1 along it is not a LINE. The primer’s LINE is an exactly marginal direction along which universal quantities vary; a redundant direction changes none, so the anisotropic Ising curve is not a LINE.
For Ashkin–Teller, duality answers with a genuine LINE: the self-dual line Z = 1 − 2X, along which the exponents vary.14
Partition-function zeros. Z of a finite system is a polynomial in e−β or in the field, so it has zeros, none of them on the positive real axis (the physical values), because every term of Z is positive. As L grows they pinch the axis, and the rate carries the order of the transition: the first zero approaches like L−1/ν at a continuous transition and like L−d at a first-order one (Janke and Kenna).17 Lee and Yang proved that for a ferromagnet the zeros in the complex plane of z = e−2βh (h the field) lie exactly on the unit circle.18
These are truth generators for the controls. None of them may touch the instrument’s inputs (the firewall): they need the Hamiltonian. The transfer matrix scores LINE (c constant and a marginal operator), duality locates lines exactly, and zeros at one control value decide the ORDER of a transition, not whether it sits on a LINE (that needs ν at several control values).19
Kramers–Wannier as Aasen, Mong & Fendley print it.13 The critical, self-dual line is (2.2) (sinh 2Jx)(sinh 2Jy) = 1. With the weights e2Jy = cot uV and e2Jx = cot(π/4 − uH), the critical line is uH = uV and the isotropic critical point is uH = uV = π/8. The duality relation printed as (2.13), sinh J sinh Ĵ = 1 (Ĵ the coupling on the corresponding bond of the dual lattice), lacks the factor 2 that (2.2) carries: a slip in the paper, not in this page. In plain words: duality can be done on part of the lattice only, and the seam between the two parts is a line-like object, a defect, that can be moved freely without changing Z. In symbols it is Dσ with Dσσzσz = μxDσ and Dσσx = μzμzDσ, which fuses as Dσ2 = 1 + Dψ and is not invertible.
Ashkin–Teller. With Z = e−4K and X = e−2(K+L), the self-dual line Z = 1 − 2X runs from the Ising end to the 4-state Potts end, and the thermal exponent yt = 2 − 2/gR varies along it.14 That line has c = 1: it is the orbifold branch of the c = 1 compact boson.25
Walking in the strip. For q = 5 Potts the complex fixed point has c ≈ 1.138 ± 0.021i (Gorbenko, Rychkov & Zan, (3.2)),8 Re c = 1.13755. Ma and He measure c(L) on quantum Potts chains: at q = 5 it approaches Re c (1.149 at L = 20, 1.139 at L = 70), and the drift fits a tangent form c(L) = cR − α tan(γ ln(L/L0)) better than a polynomial in 1/L.16 A strip free-energy estimate of c is expected to show the same plateau near Re c at W ≪ ξ (Ma and He’s c is from the entanglement entropy; the strip version is not checked here).
Zeros, as Janke & Kenna write them.17
second order: rj(L) ~ ((2j − 1)/Ld)1/(νd) ~ L−1/ν (2.14); first order: rj(L) ~ L−d (2.17) GL(rj) = (2j − 1)/(2Ld) = a1 rja2 + a3 (3.1–3.2)Here rj is the distance of the j-th zero from the transition point, GL is the fraction of zeros closer than rj (the integrated density of zeros), and a1, a2, a3 are fit constants: a1 an amplitude, a2 the power, a3 an offset. Read the fit: a2 ≈ 1 means first order, with the latent heat set by the slope a1; a2 > 1 means second order with α = 2 − a2; a3 ≠ 0 means no transition. The first-order rate L−d is “the formal identification of ν with 1/d”. Fisher zeros live in the complex temperature plane at zero field; Lee–Yang zeros in the complex field plane. Whether the zeros of a walking system approach at a drifting rate is reasoned, not found in a source here.19
The strip transfer matrix, symmetrised. Let s = (s1…sW) be one row, periodic across the width. Split the Boltzmann weight into a row part D(s) = exp(K Σi sisi+1) and a between-row part V(s, s′) = exp(K Σi sis′i). Then T = D1/2 V D1/2 is symmetric, and V is a tensor product of W copies of the 2×2 matrix [[eK, e−K], [e−K, eK]], so applying T to a vector costs W·2W operations rather than 4W. For an M-row strip Z = tr TM → λmaxM, so the free energy per site in units of kBT is f(W) = −ln λmax/W. Power iteration finds λmax. Two widths give c: c = (6/π) [f(W) − f(W′)] / [1/W′2 − 1/W2]. This construction is standard; the code on this page is its direct transcription.
The Lee–Yang circle theorem, Theorem 3 of their Paper II (verbatim): “If the interaction u between two gas atoms is such that u = +∞ if the two atoms occupy the same lattice [site] and u ≤ 0 otherwise, then all the roots of the polynomial 𝒫 lie on the unit circle in the complex z-plane.” In the Ising language the variable of the theorem is z = e−2βh (proportional to the lattice-gas fugacity), so the ferromagnet’s field-plane zeros sit on a circle, and the transition is where they pinch the positive real axis. The theorem assumes nothing about the range of u, the dimension, or the lattice.18
Fourteen techniques side by side: what each needs, what it decides, whether it fits the design bound, how well it reduces to A, B and R, and which side of the firewall it lives on.
Two scales before the table. Design bound: PASS = fits a 486-class machine (16 MB, ~107 operations/s) comfortably; MARGINAL = fits only in a reduced form or at the edge; FAIL = does not fit. For example MCRG on a 64² lattice at 104 sweeps is ~108 operations, about 10 s at 107 operations/s: a PASS. Kernel fit, in the notes’ own words and ordered roughly from worst to best: none < weak < indirect < moderate < fine < good < strong < direct, meaning how completely the method reduces to directed differences, accumulations and the antisymmetric response over declared relations; plus one separate category, projection: the method’s output is a declared projection (a loop sum, such as a winding number), not an operator.
The map places each technique by how well it reduces to the kernel’s operators (left to right) and by the side of the firewall it lives on (bottom to top). Click a dot or a filter. Instrument-side methods take observations: snapshots or time series. Scoring-side methods need the Hamiltonian and generate truth. Tensor-network RG (Levin–Nave; Evenbly–Vidal),20 information-based RG (Koch-Janusz & Ringel),22 and time-series early warning (Scheffer and co-workers; Bury’s spectral variant)21 each get one row.
Pick by geometry, not by fashion. The instrument side holds MCRG on snapshots, the drifting exponent from one snapshot set, winding counts and early-warning screens; everything else is scoring-side truth for the controls.19
What the columns mean. Design bound: could it run on a 486-class machine (16 MB, ~107 operations/s)? The costs are order-of-magnitude arithmetic, not timings: MCRG on 64² Ising at 104 sweeps is ~108 operations; a 212-state sparse transfer-matrix step ~105; tensor RG ~D6 per step, which fits 16 MB at bond dimension D = 16–24. Kernel fit: does it reduce to directed differences over a declared relation, their accumulations over a declared chain, and the antisymmetric response R over it? (The kernel has no loops; a loop sum such as a winding number is a declared projection, not R.)19
Claims in this table that are not verified here: that Fisher zeros of a walking system approach the axis at a drifting rate; that a histogram version of information-based RG fits the design bound; that Migdal–Kadanoff mishandles marginality on Ashkin–Teller specifically (the general weakness is known). Treat them as leads.
How each technique sees POINT, LINE, WALKING and BKT, and what its honest failure looks like.
The strip at the top is the whole distinction between the last two columns in one picture. (ℓ is the RG time, the number of coarse-graining steps, ln b in general.) A walking flow’s per-level velocity reads SIG → SLOW → SIG: it turns and leaves. A BKT flow near the end of its line approaches it only as 1/ℓ (the critical branch y = 1/(1/y0 + ℓ)),24 so its per-level velocity y² falls only as 1/ℓ², and reads SIG → SLOW → ? with no turn, because the band widens before the velocity settles. That sequence is a named cell of the table: REFUSE(“slow”).1 Click a column head to light one geometry.
Print it. It is the Detector’s table before it is code, and the last column is as much a part of the specification as the first four: a GOOD failure on BKT is REFUSE(“slow”); a BAD failure is a confident WALKING or POINT.1
The rows restate §2 (MCRG), §3 (drift and the five-valued sign), §4 (kernel signs) and §5 (exact routes), plus early-warning signals: variance and lag-1 autocorrelation rise near a bifurcation because recovery from kicks slows,21 which, by the same slowing-down argument (reasoned here, not sourced), holds for a line, a walking flow and a plain bifurcation alike. The BKT column’s strip entries come from the Kosterlitz flow’s critical branch, whose approach is 1/ℓ with no power law,24 and the transfer-matrix entry from the c = 1 moduli space, where the KT point sits.25
Ten lines to keep, a 25-item final test that draws new items on every retake, a name drill, and your scores.