Eight model systems, chosen because each one gives an instrument a known answer. The instrument is the Criticality Compass: its Detector classifies a system's critical geometry from observations, and its Navigator finds a critical line and moves along it. Every section shows a headline, one live graphic and a cyan “For the Compass” box with its accordions closed; open “Working equations” for the physicist's layer and “Full derivation” inside it for the deepest.
The organising idea is follow the flow: the Detector reads the velocity, accumulation and circulation of the flow over declared scale and control relations. In plain words: the velocity is how much a measured quantity changes from one scale (or one control value) to the next; the accumulation is that change added up along a chain of scales; the circulation is what is left over when you go round a small loop of scale and control steps; and a declared relation is the list of which measurements count as neighbours (this scale and the next, this control value and the next), fixed before the data are looked at. The techniques page teaches the machinery. 21 Quizzes are folded under each section; the final test at the end is drawn from a bank, so a retake gets different items. Scores stay in this browser (Export copies a summary).
Each model is here because it gives the Compass a known answer: a point, a line or a walk, plus one case the alphabet has no letter for yet, a point with two relevant directions (multicritical).
The eight: Ashkin–Teller, Baxter’s eight-vertex model, two Ising layers coupled bilinearly, O(n1)⊕O(n2), Blume–Capel, the Potts model at q = 5 and 6, the XXZ chain with its six-vertex twin, and the self-organising adaptive networks of Sormunen, Gross and Saramäki. Click a miniature to jump to its section. Two of them are the gentlest on-ramps if the topic is new: Blume–Capel (spins that may be missing) and XXZ (a quantum spin chain whose anisotropy you dial). The Ashkin–Teller and coupled-Ising sections are a pair: the same two ingredients, coupled two ways, give two different geometries. The gallery near the end is the one-table summary.
The Compass is tested on geometry, not on model identity. It must pass Ising and Ashkin–Teller and XXZ; if it recognises XY as a line but fails Ashkin–Teller, it has learned a BKT detector, not a critical-manifold detector. ◆ design 21
The validation ladder ◆ design, in the order of the instrument’s specification (its four rungs; §9 gives the build order used inside them): 21
Two Ising layers joined by a four-spin coupling: instead of one critical point there is a whole line of them, and the exponents change continuously as you slide along it.
Every site carries two Ising spins, σ and τ. The coupling K rewards neighbouring σ's that agree, and neighbouring τ's that agree. The four-spin coupling L rewards neighbouring products στ that agree. At L = 0 the layers ignore each other: two copies of the Ising model, critical at the Onsager coupling K = 0.44069. At L = K the four states (σ, τ) = (±1, ±1) are all alike and the model is the four-state Potts modelA lattice model with q states per site, where neighbours gain energy only if they are in the same state. q = 2 is Ising.. In between, a dualityAn exact map between the model at strong and at weak coupling. If the transition is unique, it must sit where the map sends the model to itself: the self-dual locus. pins every critical point to one curve, the self-dual line. Drag along it in the trace above: the specific-heat exponentα/ν: how fast the specific-heat peak grows with system size, C ~ Lα/ν. ratio α/ν climbs from 0 at the Ising end to 1 at the Potts end, while γ/ν = 7/4 never moves. The symbols in the trace: yt is the thermal RG eigenvalue (how fast a temperature deviation grows under coarse-graining, ν = 1/yt); gR is the Coulomb-gas coupling, a number that labels the position on the line; α/ν and γ/ν say how the specific-heat and susceptibility peaks grow with size, C ~ Lα/ν and χ ~ Lγ/ν. The trace starts at s = 0.25 (a quarter of the way along, in the angle coordinate of the working equations). 1 A self-contained tutorial builds this line from two independent Ising copies: Ashkin–Teller for the Working Mathematician.
The live toy: four colours for the four (σ, τ) states. It starts off the line, at K = 0.4, L = 0.05. Tick “lock onto the self-dual line” and slide the position; K and L follow the line. Unlock and move K or L freely to leave it. (Metropolis on a periodic lattice: a toy, not an instrument.)
Detector: this is the primary LINE control. The exponent varies with the CONTROL (where you are on the line) but not with SCALE. Navigator: the tangent is along the line and the normal points off it; both are known exactly here, on the scoring side, so the Navigator's estimates can be graded. Keep away from the Potts end, where logarithms fake a drifting exponent. ◆ design 21
The Hamiltonian (the inverse temperature is absorbed into the couplings), and Wiseman and Domany's coordinates: 1
H = −Σ⟨ij⟩ [ Kσiσj + Kτiτj + Lσiσjτiτj ], Z = e−4K, X = e−2(K+L)In the unit square of (X, Z): the parabola Z = X2 is L = 0 (decoupled Ising), the diagonal Z = X is L = K (four-state Potts), and duality places the critical line on Z = 1 − 2X. That is the same curve as the form Ginsparg prints, 5
Z = 1 − 2X ⇔ e−4K = 1 − 2e−2K−2L ⇔ 2e−2Ke−2L = 1 − e−4K ⇔ e−2L = (1 − e−4K)/(2e−2K) = ½(e2K − e−2K) = sinh 2K.The critical segment runs from the Ising point X0 = (√2 − 1, 3 − 2√2) (K = 0.44069, L = 0) to the Potts point X4 = (1/3, 1/3) (K = L = ¼ ln 3 = 0.27465). Along it (Coulomb-gas results quoted by Wiseman and Domany): 1
yt = 2 − 2/gR, gR = (8/π) arcsin[½ coth 2K], α/ν = 2yt − 2, γ/ν = 7/4.On the line, coth 2K = (1 + Z)/(1 − Z) = (1 − X)/X, so the argument of the arcsine runs from 1/√2 (Ising: gR = 2, α/ν = 0) to 1 (Potts: gR = 4, α/ν = 1). The trace's slider moves uniformly in θ = arcsin[½ coth 2K], which makes gR = 8θ/π exactly linear (this page's arithmetic). Wiseman and Domany's five equidistant test points, measured with their cluster algorithm up to L = 128:
| point | γ/ν (fit) | α/ν (fit) | α/ν (exact) |
|---|---|---|---|
| X0 (Ising) | 1.751(4) | .23(1) log-log; 0 semi-log | 0 |
| X1 | 1.751(1) | 0.38(2) | 0.2096 |
| X2 | 1.752(3) | 0.542(8) | 0.4182 |
| X3 | 1.763(6) | 0.655(13) | 0.6383 |
| X4 (Potts) | 1.752(4) | 0.747(3) | 1 |
γ/ν = 7/4 holds everywhere; the fitted α/ν misses the exact value, worst at the Potts end. There the marginal dilution field carries multiplicative logarithms: Salas and Sokal derive CH ~ L/(ln L)3/2 and χ ~ L7/4/(ln L)1/8 at the four-state Potts point, plus additive log log corrections, and note that the theory could not have been read off the data. 3 Jin, Sen and Sandvik found a frustrated J1–J2 Ising model to be Ashkin–Teller critical for g ≥ g* = 0.67(1), yet near the Potts end it shows double-peaked energy histograms and a negative Binder minimum up to L = 256: pseudo first-order behaviour at a transition that is continuous. 2
Why duality locates the line. Kramers–Wannier duality maps the Ising model's high-temperature expansion onto the low-temperature expansion of a dual model. On the square lattice Aasen, Mong and Fendley print the self-dual (critical) line of the anisotropic model as 4
(sinh 2Jx)(sinh 2Jy) = 1 (their eq. 2.2).For Jx = Jy = K this gives sinh 2K = 1, i.e. Kc = ½ ln(1 + √2) = 0.4407 (the number is this page's arithmetic; the paper prints only the equation, and its eq. 2.13 drops a factor of 2 relative to 2.2). A duality has a fixed set; if the model has a single transition, it must lie on that set. Ashkin–Teller has two couplings, so its duality map (K, L) → (K*, L*) has a fixed curve, e−2L = sinh 2K, as Wiseman–Domany and Ginsparg print it. The dual map itself is not reproduced here.
Where on the line. Ginsparg gives the position as a compact-boson radius on the orbifold line: sin(πr2/4) = ½ coth 2K, with r = 1 at Ising × Ising and r = √2 at four-state Potts. 5 Put that into Wiseman–Domany's gR: arcsin(sin(πr2/4)) = πr2/4 for r ≤ √2, so gR = 2r2 and yt = 2 − 1/r2. That is this page's combination of the two printed formulas; radius normalisations differ between sources, so treat the factor as convention-dependent.
The marginal operator. At the decoupled point the operator that moves you along the line is ε1ε2, the product of the two Ising energy operators: Ginsparg calls it “the Ashkin–Teller interaction coupling the two Ising energy operators”, the four-fermion term that “simply changes the radius of a free boson”. 5
Put an arrow on every bond of a square lattice and allow only eight kinds of vertex: Baxter solved this exactly, and its exponents vary continuously. It is the textbook critical line, and Ashkin–Teller's exact cousin.
Click a bond to flip its arrow. Each vertex has four arrows round it. The ice ruleTwo arrows point in and two point out at every vertex, like the two near and two far hydrogens round each oxygen in ice. (two arrows in and two out at every vertex) allows six arrangements (types a, b, c); the eight-vertex model also allows the two in which all four arrows point in (a sink) or all point out (a source), called d. Arrangements with one or three arrows in are forbidden (red ×). Each type carries a weight, and the weight of a configuration is the product over its vertices. 6 The “flip a path” button flips a whole chain of bonds from one edge to another. That keeps the number of incoming arrows at every vertex even (its parity), because each vertex on the chain has exactly two of its arrows flipped; in six-vertex mode the chain follows the arrows, so each vertex also keeps exactly two in.
An exactly solved line: the cross-check on the scoring side, never an input to the instrument. It also poses a question the Compass must answer from data: are Ashkin–Teller and the vertex models two line mechanisms, or one family seen twice? The c = 1 picture below suggests one compact boson; which branch each model occupies has to be settled before a representation-invariance test (does the instrument give the same verdict and the same geometry for two models believed to be the same critical theory?) is read. The held XXZ sources list the full (m, n) vertex-operator lattice and no dimension-1/16 twist field, the operator content of the circle line; Ginsparg groups the vertex models with the orbifold. The two may refer to different boundary sectors; this page does not settle it. ◆ design 21
From the literature, not verified locally (Baxter's papers are not held here). The zero-field eight-vertex model with weights a, b, c, d is usually written with the anisotropy
Δ = (a2 + b2 − c2 − d2) / 2(ab + cd),which reduces to the six-vertex Δ = (a2 + b2 − c2)/2ab when d = 0. Baxter (1971–72) solved it and found critical exponents that vary continuously along its critical manifold; Wegner (1972) and Fan & Wu related it to Ashkin–Teller, and Kadanoff & Brown (1979) to the Gaussian model. 20
The c = 1 picture (held, Ginsparg). All the critical theories with central charge c = 1 form a moduli space of two half-lines. The circle line is the Gaussian model; the orbifold line is “equivalent to the critical Ashkin–Teller model (which also encompasses … the 6-vertex model and the 8-vertex model on its critical line)”. They meet at one point, which is the continuum Kosterlitz–Thouless point of the XY model. On the orbifold line Ising × Ising sits at r = 1 and the four-state Potts model at r = √2. Three further c = 1 theories (T, O, I) are isolated points: c = 1 does not by itself imply a line. 5
Redrawn after Ginsparg's Fig. 14; spacing schematic. The highlighted segment is critical Ashkin–Teller from Ising × Ising (r = 1) to four-state Potts (r = √2).
Keep the two Ising layers but couple them spin to spin (gσiτi) instead of product to product: dimension counting says this coupling grows under coarse-graining, so the line is expected (not shown here) to collapse back to a point.
Coarse-graining multiplies each small coupling by by, where b is the blocking factor and the RG eigenvaluey = d − x: the space dimension d minus the scaling dimension x of the operator the coupling multiplies. y > 0 grows (relevant), y < 0 dies (irrelevant), y = 0 is marginal. y = d − x (x is the scaling dimension; this page keeps Δ for the Blume–Capel crystal field and the XXZ anisotropy). An Ising spin has dimension 1/8 and the Ising energy density has dimension 1. The bilinear term gσiτi multiplies two spins: x = 1/4, y = 7/4, strongly relevant. Ashkin–Teller's four-spin term multiplies two energy densities: x = 2, y = 0, marginal. 22 Ginsparg identifies that product, ε1ε2, as the operator that moves the theory along the line. 5
| Ashkin–Teller | bilinear coupled Ising | |
|---|---|---|
| coupling | Lσiσjτiτj (ε1ε2) | gσiτi (σ1σ2) |
| dimension x | 1 + 1 = 2 | 1/8 + 1/8 = 1/4 |
| y = 2 − x | 0: marginal | 7/4: relevant |
| geometry | a LINE of fixed points | leaves the Ising × Ising point; expected: one POINT |
Two-layer toy: colours as in the Ashkin–Teller toy. Start at g = 0 with K just below the Ising value, press Run, then raise g: ⟨στ⟩ climbs to 1 and the two magnetisations rise and move together.
The lesson of the pair: the TYPE of coupling decides LINE versus POINT. Two layers joined by an exactly marginal operator give a line; joined by a relevant one, they give a point. The Compass must read the difference from the flow, not from the model's name. ◆ design 21
The coupled-Ising experiment of the validation ladder ◆ design: 21
H = −J1Σ⟨ij⟩σiσj − J2Σ⟨ij⟩τiτj − gΣiσiτi.At g = 0 the instrument sees two independently understood critical systems (two POINTs). Raise g and ask whether it recovers structure that neither layer has alone. Dimension counting (from the Ising values xσ = 1/8, xε = 1; not checked here against a held source) says g grows as b7/4, so the decoupled point is unstable in the g direction. What it flows to is expected to be a single Ising-like transition with the two order parameters locked, one POINT; that is not shown on this page, and the Compass will be asked to say it from data. 22
When is a marginal operator exactly marginal? Ginsparg's criterion: a dimension-(1,1) operator V generates a one-parameter family of critical theories only if, to leading order, the V×V operator product contains no term CVVV(z − w)−1(z̄ − w̄)−1V. If it does, the two-point function picks up a logarithm and V's weight shifts to (1 − δg πCVVV, 1 − δg πCVVV): marginally relevant on one side, marginally irrelevant on the other. Higher orders require further integrals to vanish. 5
For Ashkin–Teller the check passes for a structural reason: on the critical line ε1ε2 = ψ1ψ̄1ψ2ψ̄2 is the four-fermion term of the massless Thirring model, which only changes the radius of a free boson. Changing a radius is a motion along a line, so no fixed point is left. 5 The bilinear σ1σ2 has no such protection: its y = 7/4 is far from zero, so the question of higher orders never arises.
Two different order parameters, two critical lines that meet at one point: the sign of D = u1u2 − w2 decides whether the two orders coexist (tetracritical) or exclude each other (bicritical).
The plane shows the two “temperatures” r1 and r2, one per order parameter; each orders when its r turns negative. The self-interactions u1, u2 keep each order finite; the cross-coupling w decides how they meet. Slide w: with w2 < u1u2 a green mixed phase, where both orders coexist, opens between two critical lines (four lines meet: tetracritical); with w2 > u1u2 the orders repel and a first-order flop line separates them (bicritical). This is mean field, the Landau picture, computed live as the lowest of the candidate minima at every point. 10
A multicritical point has at least two relevant directions. The alphabet has no cell for that yet: POINT allows one escape direction. That is a design gap, stated here rather than hidden. Its finite-size shadow is known: in the fully frustrated XY model the effective exponent sits on a plateau, then moves on. ◆ design 21
The Landau–Ginzburg–Wilson Hamiltonian of Calabrese, Pelissetto and Vicari (φ1 has n1 components, φ2 has n2): 10
H = ∫ddx { ½[(∂φ1)2 + (∂φ2)2] + ½[r1φ12 + r2φ22] + (1/4!)[u1(φ12)2 + u2(φ22)2 + 2wφ12φ22] }Mean field: D = u1u2 − w2 > 0 tetracritical, < 0 bicritical (D is this page’s letter for the discriminant, to keep Δ for the Blume–Capel crystal field and the XXZ anisotropy). Tetracritical: four continuous lines meet, bounding a mixed phase where both orders coexist. Bicritical: two continuous lines meet a first-order line, the flop line, across which the system jumps from one order to the other. The biconical fixed point is a third, non-symmetric fixed point where both couplings stay finite; the decoupled one has w = 0. Beyond mean field, which fixed point controls the meeting depends on N = n1 + n2 (d = 3):
| N | stable fixed point | phase diagram if continuous |
|---|---|---|
| 2 (Ising + Ising) | symmetric O(2) | bicritical or tetracritical |
| 3 | biconical (exponents within errors of Heisenberg) | tetracritical in mean field |
| ≥ 4 | decoupled | tetracritical |
| outside the attraction domain | none | first-order multicritical point |
Crossover exponents measure how slowly a nearly symmetric point is left. P4,4 is the spin-4 (quartic, fully anisotropic) perturbation of the O(N) point and φ4,4 its crossover exponent: at the O(N) point φ4,4 ≈ 0.01 for N = 3 (the crossover is practically invisible), ≈ 0.08 for N = 4 and ≈ 0.15 for N = 5. 10
What a real multicritical crossover looks like at finite size. In the fully frustrated XY model on the square lattice the chiral (Ising) and spin (KT) transitions are 1.6% apart. Hasenbusch, Pelissetto and Vicari, with L up to 1000, find the chiral exponent ν on a plateau near 0.8 for L ≲ ξs ≈ 118 and the Ising value 1 beyond it: “by using only data with L ≲ ξs one would obtain ν = 0.8 instead of the Ising value.” 11 Granato, Kosterlitz, Lee and Nightingale's coupled XY–Ising systems (1991) had reported exponents varying along one transition line. 12
The mean-field computation behind the graphic. Write a = φ12, b = φ22 ≥ 0 and F = ½(r1a + r2b) + (1/24)(u1a2 + u2b2 + 2wab). The four candidate minima are disordered (0, 0) with F = 0; phase 1 (a = −6r1/u1, b = 0) with F = −3r12/2u1; phase 2 likewise; and the mixed phase solving [u1 w; w u2](a, b) = −6(r1, r2), with F = ¼(r1a + r2b). A candidate that is not a minimum (a negative a or b, or a mixed stationary point that is a saddle because D < 0) has F = +∞; the phase is the candidate of lowest F. The phase-1 state is unstable to φ2 when r2 < (w/u1)r1, and phase 2 to φ1 when r1 < (w/u2)r2; the wedge between those lines exists exactly when w2 < u1u2. When D < 0, phases 1 and 2 have equal F on r1/√u1 = r2/√u2: the flop line. (This page's algebra on the printed Hamiltonian.)
The ε-expansion reversal. To low order (Nelson, Kosterlitz and Fisher; Fisher and Nelson) the symmetric O(N) point is the stable one for N < Nc = 4 − 2ε + O(ε2); above Nc the biconical and then the decoupled point take over as N grows. Naively continued to ε = 1 the leading term gives Nc ≈ 2, so O(3) would already be unstable; the low-order series cannot settle it. Five-loop ε and six-loop fixed-dimension series do: the spin-4 perturbation P4,4 is relevant for N ≥ 3, with Nc = 2.87(5) (< 2.9), so O(N) is stable only for N = 2, the biconical point takes over at N = 3 and the decoupled point at N ≥ 4. 10
Ising spins that may also be missing (σ = 0), and a knob Δ that charges rent for every spin that is present: turn Δ up and the ordering temperature falls, until at one special place, the tricritical point, the transition turns abruptly first order.
Start here if this model is new. Picture a lattice where each site is either empty (σ = 0) or holds a little magnet pointing up or down (σ = ±1). Neighbouring magnets that agree lower the energy by J, as in the Ising model. Every magnet that is present costs Δ, the crystal fieldHere: a single-site energy Δσ2. With the plus sign used on this page, Δ > 0 makes nonzero spins expensive and so controls how many sites are empty.. Click the little lattice above to change sites, and drag Δ: at T = 0 a fully ordered site gains zJ/2 = 2 of bond energy (four neighbours, each bond shared) and pays Δ, so emptiness wins beyond Δ0 = 2. 13 The same competition at finite temperature produces the phase diagram below, the kind of diagram seen in 3He–4He mixtures.
The cursor on the diagram is the toy's (Δ, T); drag either. Empty sites are dark. Along the solid line the transition is continuous and Ising-like; along the dotted piece near Δ = 2 it is first order; they meet at the tricritical point (◆). 13
A point with two relevant directions: tuning T alone does not reach it. The first-order line is a decoy: a line in the phase diagram, not a line of fixed points, which the Detector must not read as LINE. Near the tricritical point the crossover can look like WALKING at finite L; the flow's minimum velocity is what tells them apart. ◆ design 21
Zierenberg, Fytas, Weigel, Janke and Malakis (2017) mapped the boundary by crossing it at fixed T while varying Δ, with multicanonical simulations, and used ξ/L at the size crossings as the universality test (it must reach the exact Ising value 0.905 on the continuous part). Their Hamiltonian (J = kB = 1; here Δ is the crystal field, not the XXZ anisotropy of §7): 13
H = −JΣ⟨ij⟩σiσj + ΔΣiσi2 = EJ + ΔEΔ, σi ∈ {−1, 0, +1}.Δ → −∞ removes the vacancies and gives the Ising model; Δ = 0 orders at T0 ≈ 1.69; the boundary reaches T = 0 at Δ0 = zJ/2 = 2. On the second-order part the exponents are Ising: α = 0, β = 1/8, γ = 7/4, ν = 1; at T = 1.2 the paper measures ν = 0.97(10) and γ/ν = 1.750(3), and a logarithmic specific-heat peak. At T = 0.5 (first order) the pseudo-critical fields shift as L−2 and the peaks grow as L2, the volume.
The tricritical point. Quoted literature values (Δt, Tt): Beale (1986) (1.9655(50), 0.610(5)); Wilding and Nielaba (1996) (1.9665(3), 0.608(1)); Kwak et al. (2015) (1.9660(1), 0.6080(1)), the most precise. The tricritical exponents are not printed in the held source, and this page does not quote them.
The approach. Far above Tt the universal ratio ξ/L at the crossings extrapolates to the exact Ising value 0.905; approaching Tt “additional corrections emerge”, the crossings depart from 0.905, and a fit at T = 0.65 with L ≥ 64 still has poor quality. The paper reads this as the crossover to tricritical scaling; it does not measure a crossover exponent. Finite-size forms used:
first order: Δ*L = Δ* + aL−D; second order: Δ*L = Δc + aL−1/ν, C*L ~ ln L, χ*L ~ Lγ/ν.Why a tricritical point needs two knobs (schematic Landau argument, no numbers). Expand the free energy in the order parameter m: F = a m2 + b m4 + c m6, with c > 0. If b > 0 the minimum leaves m = 0 continuously when a changes sign: a second-order line, one condition (a = 0). If b < 0 the m6 term holds the order finite and m jumps: first order. The two behaviours meet where a = 0 and b = 0: two conditions, so two knobs, here T and Δ. The vacancies are what make b change sign: diluting the magnets weakens the effective quartic stiffness. (Standard Landau theory; not checked here against a held source.)
With five or more states per site the 2D Potts transition is first order, but so weakly that any lattice smaller than the correlation length looks critical: the flow walks slowly past a pair of fixed points that have moved into the complex plane.
Slide your lattice size L. A first-order transition shows itself (two coexisting phases, a latent heat) only once L is larger than the correlation length ξ. For q = 10, ξ is about ten lattice spacings; for q = 5 it is about 2512. 14 16 Below that size the system imitates a critical point, and a standard analysis returns exponents with small error bars that are simply wrong. Gorbenko, Rychkov and Zan explain why: as q passes 4, the critical and tricritical fixed points collide and move off into the complex plane as a conjugate pair. The real flow passes between them; it slows down there without stopping, and then leaves.
Potts toy with the energy histogram below it. Press “go to Tc(q)” (Tc = 1/ln(1 + √q)) and run. When the lattice is larger than ξ(q), a long run near Tc can show two energy peaks; at q = 5 no lattice you can run here is large enough. 16
The drifting exponent: the effective thermal dimension δε changes like a tangent in ln r, so the “exponent” ν(r) you measure depends on the scale you measure it at. The amplitude is read off Gorbenko–Rychkov–Zan's Fig. 12 for q = 5 and is illustrative. 15
WALKING: at a fixed control value the exponent drifts with SCALE, and the flow's minimum velocity is small but not zero. At L ≪ ξ that minimum can sit below the noise; the honest verdict is then REFUSE, not LINE and not POINT. Potts q = 5 and 6 are the walking controls. ◆ design 21
The model: H = −JΣ⟨ij⟩δ(σi, σj), σ = 1…q, with kBTc = J/ln(1 + √q) on the square lattice (the self-dual point; consistent with Iino et al.’s Tc = 0.851(5) at q = 5). The transition is continuous for q = 2, 3, 4 (q = 4 with logarithms) and first order for q > 4 (Baxter 1973). 16 Correlation lengths at the transition (Buffenoir–Wallon values, as tabulated by Gorbenko–Rychkov–Zan and by Iino et al.):
| q | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|
| ξ | ∞ | 2512.2 | 158.9 | 48.1 | 23.9 | 14.9 | 10.6 |
Walking. With λ the coupling of the nearly marginal operator ε′ and t = ln(scale), Gorbenko, Rychkov and Zan write 14
dλ/dt = −y − λ2 + …, y > 0: λ(t) = −√y tan[√y (t − t0)], ΛUV/ΛIR ~ eπ/√y,with complex fixed points λ* = ±i√y. For Potts, y = (q − 4)/π2 at leading order, which gives ξ ~ ξ0 exp(π2/√(q − 4)): faster than any power. The fixed points are complex CFTs; at q = 5 the central charge is c ≈ 1.138 ± 0.021i. 15 Along the real flow two-point functions carry a drifting dimension (their eq. 4.33) and a drifting exponent
δφ(r) = ReΔφ − (√3ε/2π) Cφφε′ tan(ε ln(r/r0)/π), ν(r) = 1/(d − δε(r)), ε = √(q − 4).The deceptive collapse. Iino, Morita, Kawashima and Sandvik simulated q = 5 up to L = 256. The best data collapse gives β = 0.070(8), ν = 0.61(5), with statistical errors smaller than the symbols; the true first-order values are β = 0, ν = 1/d = 0.5. From (L/2, L) crossings, ν(L) and β(L) drift with L; a power-law extrapolation of 1/ν(L) returns 1/ν(∞) ≈ 2.23 > d, impossible for a continuous transition. Their diagnostic: the Binder ratio's ln(R4 − 1) collapses only inside a linear window. 16
The shadow. Ma and He measured the central charge of the quantum Potts chain from entanglement entropy: c(L) drifts in the tan shape c(L) = cR − α tan(γ ln(L/L0)), with cR near Re c = 1.13755 at q = 5. 17 The Kosterlitz flow's E > 0 branch has the same tangent shape, x(t) = ε tan(arctan(x0/ε) − εt); 8 the shape is shared, but the two ε's are not the same parameter.
The correlation length, exactly. In the second paper Gorbenko, Rychkov and Zan find that the one-loop walking answer matches the exact Bethe-ansatz result of Buffenoir and Wallon to all orders in 1/|m|: 15
ξPotts = (√2/16) eπ|m|/2 + O(e−π|m|/2), |m(q)| = 2π / ln[(q − 2 + √(q(q − 4)))/2].At q = 5, |m| = 6.5285 and ξ = 2512; at q = 6, |m| = 4.7710 and ξ = 158.9 (the table above). Walking needs only that the imaginary part of the nearly marginal operator's dimension be small, not that q be close to 4; the expansion parameter (ε2/π2 with ε = √(q − 4), or more naturally 4/|m|2) stays below about 0.6 up to q = 10, which is why the picture works over that whole range. 15
A chain of quantum spins whose z-coupling Δ you can dial: for every Δ between −1 and +1 the chain is critical, so a whole segment of Δ is one critical line. The same physics can be drawn as arrows on a 2D lattice, the six-vertex model.
Start here if this model is new. Line up spin-½ particles, each up or down. The Hamiltonian has two kinds of term. The XX + YY part takes a neighbouring up–down pair and swaps it: think of an up spin as a particle that hops one site (the arcs above). The ΔZZ part only looks: it adds an energy for each neighbouring pair depending on whether the two agree. Hopping wants the spins to spread and fluctuate; Δ wants them to line up (one sign) or alternate (the other). For |Δ| < 1 neither wins: the chain stays gaplessExcitations cost arbitrarily little energy in a long chain, and correlations decay as powers of distance: the chain is critical. for every value in the range. With this page’s sign, Δ > 0 favours neighbours that agree along z (toward the ferromagnet at Δ = +1) and Δ < 0 favours neighbours that alternate (toward the Néel antiferromagnet at Δ = −1). 7 ⚠ sign convention: H = −JΣ[…], so Δ = +1 is the ferromagnet, Δ = −1 the antiferromagnetThis page follows Franchini's H = −JΣ[…]: Δ = +1 is the ferromagnet and Δ = −1 the antiferromagnet. Many texts write H = +JΣ[…], which reverses the sign of Δ and swaps the names of the two ends.
The six-vertex builder: only ice-rule vertices are allowed. Set the weights a, b, c and watch Δ = (a2 + b2 − c2)/2ab move the point across Reshetikhin's three regimes; the circle a2 + b2 = c2 is the free-fermion point Δ = 0. 6
Exponents along the line. One number, the Luttinger parameter K, sets them all: xm,n = m2/(4K) + n2K. 9 As Δ moves, K moves and so do the exponents; the marginal operator ∂X∂̄X (the one that changes the radius) keeps dimension 2.
An exactly solved LINE. Whether it is a second line mechanism or the same c = 1 family as Ashkin–Teller seen again is the open question §2 poses; either way, use the INTERIOR. One end (Δ = −1 here) is a BKT-type point, with all the treachery of the KT page's finite-size section; the other is a ferromagnet. c = 1 all along the line, so a transfer-matrix measurement of c cannot say WHERE on the line you are; the exponents can. ◆ design 21
The chain in Franchini's convention (S = σ/2, periodic): 7
H = −JΣn[SxnSxn+1 + SynSyn+1 + ΔSznSzn+1] − 2hΣnSzn (4.1)| Δ (h = 0) | phase |
|---|---|
| Δ > 1 | gapped ferromagnet (along z) |
| Δ = +1 | isotropic ferromagnet: ground-state energy continuous, but “elements similar to a first order” transition (massively degenerate ground state) |
| −1 < Δ < 1 | gapless: a c = 1 theory with varying compactification radius; Δ = 0 is free fermions |
| Δ = −1 | isotropic antiferromagnet: an infinite-order BKT-type transition |
| Δ < −1 | gapped Néel antiferromagnet |
In the critical regime, Δ = −cos γ (0 < γ < π), the sound velocity is vS = π sin γ/(2γ) and the Luttinger parameter is K = π/(2(π − γ)) = π/(2 arccos Δ): K = 1/2 at Δ = −1, 1 at Δ = 0, and → ∞ as Δ → +1. The staggered Sz correlation decays as r−2K. The saturation field is hs = (J/2)(1 − Δ). 7 In Ueda and Oshikawa's notation the vertex operator with charges (m, n) has dimension xm,n = m2/(4K) + n2K, 9 so the staggered Sz (n = 1) has 2x = 2K, matching Franchini, and the transverse spin (m = 1) has 2x = 1/(2K). Ginsparg writes the compact-boson dimensions in the same shape with a radius, m2/(4r2) + n2r2 (the r–K dictionary depends on normalisation; not checked here). 5
The six-vertex side. Weights a (a1, a2), b (b1, b2), c (c1, c2); R-matrix [[a,0,0,0],[0,b,c,0],[0,c,b,0],[0,0,0,a]]; Δ = (a2 + b2 − c2)/2ab. Transfer matrices with equal Δ commute, and the transfer matrix is “the generating function for [the] commuting family of local Hamiltonians”: H1 = Σ(σxσx + σyσy + Δσzσz), printed with an overall plus sign. 6 At zero field Reshetikhin’s regimes are “ferromagnetic” (frozen, Δ > 1), “disordered” (|Δ| < 1, power-law correlations) and “antiferromagnetic” (Δ < −1), with the free-fermion point at Δ = 0. (In nonzero fields frozen phases occur for every Δ, and a disordered region also for Δ > 1.) Franchini states the dictionary explicitly (his App. B): Δ = (a2 + b2 − c2)/2ab (B.31); a = ρ sinh(λ + φ), b = ρ sinh λ, c = ρ sinh φ, Δ = cosh φ (B.33); and d ln T/dλ at λ = 0 is the XXZ chain plus a constant (B.45), up to the sign conventions noted above. 7 A special six-vertex model, the F-model (= the BCSOS surface model), is rigorously of KT type and served Hasenbusch, Marcu and Pinn as their exact reference flow. 8
The Bethe ansatz in one screen (Franchini, ch. 4 and App. C). 7 Start from the all-up state and add M down spins. Bethe's guess: the eigenstate is a sum over permutations of plane waves, eiΣkjnj, with an amplitude that changes only by a two-body phase when two magnons pass. Parametrising the momenta by rapidities, eik = sin((φ/2)(λ + i))/sin((φ/2)(λ − i)), the phase depends only on rapidity differences provided cosh φ = Δ. Periodicity gives M coupled Bethe equations. In the planar regime φ becomes imaginary, φ = iγ′, so Δ = cos γ′; Franchini’s Table 4.1 writes the same regime with γ = π − γ′, which is the Δ = −cos γ used in the working equations, and in the thermodynamic limit the ground state is a filled sea of real rapidities with a smooth density. The finite-size spectrum then has the form of a c = 1 Gaussian theory, ΔE = (2πvS/L)[(ΔN)2/(4Z2) + (Zd)2 + N+ + N−], where Z is the dressed charge at the edge of the sea. Comparing with the Gaussian form identifies K = Z2; a Wiener–Hopf evaluation at vanishing field gives K = π/(2(π − γ)).
A network that rewires itself finds the edge between silence and runaway activity, then keeps changing its structure while staying on that edge (2023); with two kinds of transition, it can drift along the Hopf manifold into the Turing one (2025).
Each node is inactive, firing or refractory. A firing node excites each inactive out-neighbour at rate β, stops firing at rate δ and recovers at rate γ; now and then an inactive node fires on its own. The adaptive rule is homeostatic: a firing node loses an incoming link at rate l, and new links appear between random nodes at rate g. Too many links and activity runs away, so links are cut; too few and activity dies, so links accumulate. The system settles where the network's leading eigenvalue λ1The largest eigenvalue of the adjacency matrix: how strongly the network amplifies activity. Activity persists when λ1 exceeds the threshold λ1*. sits at the threshold: the transition in 2023 is the onset of activity. The toy runs N = 200 nodes with sped-up adaptation, so its trace is noisier than the paper's N = 105; watch λ1 climb and then hover near the cyan line while ⟨k⟩ rises past it and the in/out-degree correlation ρ turns negative. 18
2025, schematic (a different model from the toy above: here the eigenvalues are of the network Laplacian; the toy’s λ1 is an adjacency eigenvalue): two local plasticity rules act on different parts of the network (the Hopf rule on well-connected nodes, the Turing rule on weakly connected ones), so the system can hold one condition while it drifts into the other. 19
The Navigator's test bed: blind the instrument to the adaptation rule, compare the direction it infers with the network's own drift, then switch the adaptation off. The hard lesson: this manifold is one universality class. The avalanche exponents stay put while the network moves, so no exponent says where along the manifold you are; the Navigator needs structural observables. ◆ design 21
2023 (Sormunen, Gross & Saramäki, Phys. Rev. Lett. 130, 188401). Three-state stochastic neurons on a directed Erdős–Rényi network, the model of Droste, Do and Gross. A pair approximation about the all-inactive state gives the threshold on the leading adjacency eigenvalue 18
λ1* = δ/β + (δ + γ/2)/(δ + γ) (eq. 1)With β = 0.7, δ = 0.95, γ = 0.4, λ1* = 2.21. The paper's runs: N = 105, l = 10−3, g = 10−6, spontaneous rate η = 1/(100N). Result, in three stages: λ1 rises and pins near the threshold; then, with λ1 held, ⟨k⟩ keeps rising (about 2.3 → 2.5), the mean excess degree mean excess degree ⟨q⟩ (the average out-degree of the node you reach by following a link) creeps up and the in/out-degree correlation ρ falls to about −0.15; finally ⟨k⟩ settles. Freezing the network and perturbing ⟨k⟩ shows it sitting on the avalanche peak (the ⟨k⟩ at which the largest avalanche size and duration of the frozen network peak, i.e. the critical ⟨k⟩), and the peak itself moves over time. The avalanche size and duration distributions keep the critical slopes −3/2 and −2 throughout. Mechanism: firing probability tracks eigenvector centrality, so the links cut are the ones that lower λ1 most, and more links must be added than removed to hold λ1 fixed.
2025 (arXiv:2506.04275, preprint). N = 50 FitzHugh–Nagumo oscillators on a weighted network with Laplacian L: 19
U̇ = −U3 + U − V − C00LU − C01LV, V̇ = b(U − aV) − C10LU − C11LV.A master-stability analysis separates node dynamics from topology: the Hopf bifurcation is governed by the leading (largest) Laplacian eigenvalue λ1, the Turing bifurcation by the second smallest, λn−1 (eigenvalues ordered from largest, λ1, down to λn = 0; the ordering is this page’s reading of the indices). The authors define multicriticality as “a system sitting at two or more bifurcations at once, or, phrased differently, at a bifurcation of higher codimension”, and argue that two critical surfaces generically intersect, so rules that separately find each bifurcation can combine to find both. (λ means an adjacency eigenvalue in 2023 and a Laplacian one in 2025.)
The toy's settings. β, δ, γ as in the paper; N = 200 starting from ⟨k⟩ = 1; l = 0.05, g = 2×10−4 per node, η = 2.5×10−4 per node, time step 0.05; λ1 by power iteration. The adaptation is about 50 times faster than the paper's so that the drift fits in a minute; the paper notes that self-organisation is exact only as l and g/l go to zero.
One table to keep: what each model gives the Compass, what exact truth it offers, and what breaks it.
The build order of work on the Detector and Navigator tracks (inside the four rungs of §0): Ising (POINT) → Ashkin–Teller (LINE, on both the Detector and the Navigator tracks) → the eight-vertex cross-check → Potts (WALKING) → BKT last, as the breaker that tests honesty rather than skill. ◆ design 21
Ten lines to keep, the names to recognise, and a 25-item test drawn fresh on every attempt.