Tutorial 1 · fusion rings, modular categories, and the edge

One Modulus,
Two Arguments

Two different physical theories share the Fibonacci fusion rule. One is a universal quantum computer. The other is a critical point. The fusion algebra cannot tell them apart — and that is not a defect, it is the content.

What this is

A worksheet, not a result. The mathematics on this page is standard topological quantum computation, roughly 1984–2009. None of it is new. “It was simply new to my hands. I had read it for years and never computed with it, and those turn out to be different states of knowing.” These sheets exist to move a reader from the first state to the second, and they make no claim of their own.

The method is predict–then–check. Every input box, drill and gate here is built for one move: write down what you think the number is before you reveal it. Numbers are sourced to a paper where a paper says them, and marked unchecked where none does — a sheet whose whole method is check it yourself cannot ask you to take its own arithmetic on trust.

If you are coming from an operator formalism, you may want to ask of every quantity here what the declared observable is, and what mapping puts it into the domain before the algebra acts. That is a fair question, and these sheets sit upstream of it — the arithmetic of d² = d + 1 is the same arithmetic whichever way it is answered. They take no position on it. The full note is on the index.

Prerequisites: the Golden Arithmetic lab (φ, and the identity φ² = φ + 1). Derivation 1 (Galois conjugation) opens Section 3 and Derivation 3 (the central charge) opens Section 6 — both are in the paper course. No Berry phase anywhere in this one.

Start from a number. The rule τ ⊗ τ = 1 ⊕ τ gives the matrix [[0, 1], [1, 1]], whose largest eigenvalue is φ = 1.618… — the same φ the Golden Arithmetic lab reduces by hand. This tutorial is about exactly one thing: what that single number does, and does not, tell you about the theory it came out of. The answer is that four different theories hand you the same φ, and separating them takes data the matrix does not contain.

Section 1A fusion ring is just a matrix of non-negative integers

Start with the thing that is not in dispute. Anyon types combine, and the rule for combining them is bookkeeping: if you bring type a next to type b, what can you get?

The Fibonacci rule has two objects — the vacuum 1 and one nontrivial anyon τ — and exactly one interesting line.

Fig. 1 — the only rule there is
τ τ 1 annihilate or τ τ τ survive
τ ⊗ τ = 1 ⊕ τ. Two τ's fuse either to nothing or back to a τ — two channels, and no others. Every number in this tutorial descends from this one line. Nothing here is continuous; nothing here is a phase.

Write that as a matrix whose entries count the ways each outcome can occur:

Nτ = [[0, 1], [1, 1]]Rows and columns indexed by (1, τ). Every entry is a non-negative integer — it counts channels.

This matrix is the fusion ring. It is combinatorics. It contains no phases, no complex numbers, no notion of unitarity, and no physics beyond "what can fuse to what."

The modulus falls out immediately

A non-negative integer matrix has a largest real eigenvalue — its Perron–Frobenius eigenvalue. Apply Nτ over and over and watch what the ratio does.

Fig. 2 — the ring hands you φ, whether you ask or not
φ = 1.6180 2.0 1.0 iterations of N τ applied to (1, 0) 1/12/1 3/25/3 8/513/8 21/1334/21
Ratios of consecutive Fibonacci numbers, oscillating around φ and closing on it. The Perron–Frobenius eigenvalue is forced — fixed by the integer matrix alone, with no further input. This is the magnitude. And knowing φ tells you nothing else about the theory. That last clause is the whole tutorial.

Section 2Two roots, one polynomial

A fusion ring is not yet a physical theory. To get one you must assign each object an actual quantum dimension da, behaving multiplicatively under fusion. Consistency forces dτ to satisfy the same quadratic the matrix does — and a quadratic has two roots.

Fig. 3 — d² = d + 1 has two solutions, and both are legal
−1/φ −0.618 φ +1.618 0 d
The two roots are φ (positive — a real count of states, so a unitary theory) and −1/φ (negative — not a count of anything, so non-unitary). The negative branch is not a curiosity: it is the Yang–Lee edge singularity, a real critical point that appears when the Ising model is given an imaginary magnetic field.
The refinement worth taking away

Perron–Frobenius dimension and quantum dimension are two different things, and unitarity is precisely the condition that they agree. The PF dimension is φ in both theories — a property of the ring. The quantum dimension is φ or −1/φ — a property of the category. The slogan "you cannot derive an argument from a modulus" is sharper than it first sounds: the modulus does not even fix the dimension, only the polynomial the dimension solves.

Section 3What extra data turns a ring into a category

To pin down a full modular tensor category — the object that is a topological order — you need the fusion ring, the quantum dimensions, the twists θa = e2πi ha, and the S and T matrices. The ring is the magnitude. The twists are the argument. Four categories sit above one ring:

Fig. 4 — one fusion ring, four theories
CHIRALITY + CHIRALITY − UNITARY NON-UNITARY Fibonacci d τ = φ h τ = 2/5 c₋ = 14/5 universal quantum computer Fibonacci-bar d τ = φ h τ = 3/5 c₋ = −14/5 the mirror image Yang–Lee d τ = −1/φ h τ = −1/5 c₋ = −22/5 a critical point Yang–Lee-bar d τ = −1/φ h τ = 1/5 c₋ = 22/5 its mirror
Read the cells: hτ takes the four values 1/5, 2/5, 3/5, 4/5 — all distinct — and the four central charges are distinct too. The theories are separated on every invariant except the fusion ring itself. That is the sense in which the modulus forgets.
Falsifier check — the paper arrived, and the answer SPLITS

✓ The claim survives, and gets a better name. Rowell–Stong–Wang say it in their own introduction: "the Galois conjugate of the Fibonacci theory is the Yang-Lee theory, which is non-unitary." Galois action provably fixes the fusion matrices — a named group action that holds the ring rigid while moving everything else. That is "one modulus, two arguments" as a theorem rather than an observation. The two dimensions are the roots of RSW's own Theorem 3.1 at m = 1.

⚠ But the twist column below is NOT sourced. No source consulted for this page supports hτ = −1/5, θτ or c− = −22/5 for Yang–Lee. Freedman never writes a central charge or the word twist; Sanno 2022 gives the numbers but flags the topological spin as a conjecture; and — the surprise — Cardy 1985 never prints h = −1/5, and Cardy 2023 prints neither c nor h. Both are routinely cited for exactly those values. The numbers are folklore-correct and unverified here.

Section 4What Rowell–Stong–Wang actually does

The paper is Rowell, Stong & Wang, "On classification of modular tensor categories" (Comm. Math. Phys. 292, 2009; arXiv:0712.1377). It gives a complete classification of MTCs of low rank — rank meaning the number of anyon types.

The method is what matters. Rather than constructing categories and hoping to find them all, they turn the consistency conditions into Diophantine constraints and solve them. Number theory does the classifying, and the output is a finite explicit list at each low rank together with a proof that nothing else exists.

Admissible quantum dimensions below 2 are exactly 2cos(π/n): 1, √2, φ, √3, 1.8019, … There is nothing between √2 and φ. The spectrum is discrete, so the gap is forbidden by representation theory — which is precisely the Gell-Mann situation, where the classification itself forbids the intermediate case.a note made before reading Rowell–Stong–Wang

That instinct was right, and RSW is its stronger form: the Jones index theorem constrains the dimensions, RSW classifies the categories. Both make an empty slot rigorous instead of rhetorical.

⚠⚠ The trap in citing RSW — read this before quoting a number from it

RSW classify UNITARY modular tensor categories. Yang–Lee fails their Frobenius–Perron criterion and does not appear in their tables at all.

So RSW cannot be cited for a Yang–Lee value, and it cannot be cited for the count of four categories over this fusion ring — that count includes non-unitary members its classification excludes by construction. Reaching into a table for the number you want, when the table was built to leave that case out, is a citation error and not a shortcut.

What RSW does give you here is the Galois statement, quoted above, which is in the introduction and is about the pair rather than about either member's twist.

A hard constraint that changes what to look for

Freedman's Galois Conjugates of Topological Phases, Theorem IV.5, is a proof that no gapped Hermitian Lieb–Robinson Hamiltonian has a Yang–Lee phase as its ground state — in any gauge. Sanno 2022 cites it and routes around it with an open dissipative system.

Non-unitary is not merely unusual here. It is excluded from gapped Hermitian ground states by theorem. If one hoped to find a naturally occurring Yang–Lee medium, the obvious route is closed, and the open-system route is the one Sanno had to take.

Section 5Reidemeister isn't one thing, and the split is the physics

Knot theory treats the Reidemeister moves as the invariance group — what survives a wiggle. That is right for knots. It is not right for anyons, because anyon worldlines are not curves but ribbons, carrying a framing.

Fig. 5 — two moves hold, one charges you
R1 — CURL × θ τ NOT free the twist is an observable R2 — POKE free R3 — SLIDE free
R2 and R3 hold in any ribbon category — which is what makes the braid group representation well defined, and that representation is the phase. R1 does not hold. Removing a curl multiplies the amplitude by the twist θa. In the crossings above, the under-strand is drawn as two paths with a gap and the over-strand whole on top.
The point

R1's failure is not a defect in the classification. It is an observable. The amount by which R1 fails is the topological spin — the same θτ that distinguishes the four cells of Fig. 4.

R2/R3-invariance gives the braid representation, the phase. The fusion matrix's PF eigenvalue gives φ, the magnitude. R1 is the move that couples them.

Section 6The Gauss sum is the weld

Magnitude and phase do not stay separate. In any modular tensor category one identity ties them together:

Σa da² θa = 𝒟 · e2πi c−/8𝒟 = total quantum dimension. Sum over all anyon types.

Dimensions weighted by twists on the left; a single number — the chiral central charge, a criticality quantity living on the gapless boundary — on the right. For Fibonacci the sum has just two terms, and you can watch them land.

Fig. 6 — two arrows that compute a central charge
Re Im d₁²θ₁ = 1 d τ²θ τ = φ²e ⁱ ¹⁴⁴° 𝒟 ∠126° = 1.902 126° = 2πc₋/8  ⇒  c₋ = 14/5 the central charge, read off two arrows
Add the two terms tip to tail. The resultant lands exactly on the dashed circle of radius 𝒟 = √(1+φ²) = 1.902, at 126° — and 126° is 2πc₋/8 for c₋ = 14/5. Two arrows, drawn from braiding and fusion data, produce the central charge of a conformal field theory living on the boundary. Every number in this figure was computed and checked while drawing it.

So the bulk's braiding data determines the boundary's critical theory. "The edge between topology and criticality" reads literally.

Fig. 7 — one motif, two base spaces
BASE SPACE = PARAMETER ν = 0 gapped ν = 1 gapped gap closes critical tune a coupling → seam / gate — criticality only at the transition BASE SPACE = REAL gapped bulk { d a , θ a } no transition anywhere gapless edge CFT Σ d²θ fixes c₋ bulk–boundary → criticality in a phase that never transitions
One motif, two base spaces. Left: an invariant changes only by passing through a gap closing (watch the red band pinch shut). Right, a second instance: a gapped topological order carries a gapless critical theory on its physical boundary, with the bulk data fixing the boundary's central charge, while nothing transitions at all. Same coupling motif, different base space.
Sources

Checked against the paper itself: Rowell, Stong & Wang, On classification of modular tensor categories (Comm. Math. Phys. 292, 2009; arXiv:0712.1377) — for the low-rank classification, for the 2cos(π/n) spectrum, and for the Galois pairing quoted in Section 4, which is in their introduction. Freedman et al., Galois Conjugates of Topological Phases, for Theorem IV.5. Sanno et al. (2022) for the open-system route, including its own flag that the topological spin there is a conjecture. Cardy (1985) and Cardy (2023) were opened specifically to check the Yang–Lee twist values, and neither contains them — which is why Section 3 marks that column unsourced rather than citing Cardy for it, as is commonly done.

Stated on reputation, not checked here: Kitaev, Wen, Read–Green and Verlinde for bulk–boundary and the Gauss sum; Fisher for the Yang–Lee edge; Freedman–Larsen–Wang for density ⇒ universality; the Jones index theorem. The mathematics is standard; the attributions are the soft part.

No DOIs appear anywhere in this document, deliberately. Citations get opened, not remembered — which is the same discipline the sheet asks of you with its numbers.