Tutorial 3 · Verlinde, fibres, and the Lee–Yang answer
Braiding determines fusion. Fusion does not determine braiding. The relation between the two runs one way, there is a 1988 theorem for it, and the asymmetry turns out to be more useful than a symmetry would have been.
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: tutorial 1 for the four categories over one ring, and the Verlinde by Hand lab for the formula itself. Sections 1, 2 and 4 also want Derivation 1 (Galois conjugation); Section 3 assumes Derivation 4, the two-spin Lee–Yang calculation, from the paper course. No Berry phase anywhere in this one.
Start from a number. The Fibonacci fusion ring has largest eigenvalue φ. So does the Yang–Lee one. Same number, two theories — one a universal quantum computer, the other a critical point. This tutorial is about what that coincidence actually is: not a failure of the arithmetic, but a map that runs in one direction only, with a four-element fibre sitting over the ring.
"You cannot derive an argument from a modulus." True — and it is only half of a statement. The other half is a 1988 theorem.
You can derive the modulus from the argument. Fusion is computable from braiding. The map runs one way, it is not symmetric, and the asymmetry is the structure.
The modular data of a topological order is a pair of matrices, S and T, indexed by anyon types. T is diagonal and carries the twists θa. S encodes the braiding — the amplitude for one anyon's worldline to link another's.
Verlinde's theorem (1988) says the fusion coefficients are not independent input. They are computed from S:
The mechanism underneath is clean: S simultaneously diagonalises all the fusion matrices. The fusion ring's structure constants are recovered from the eigenvalue data of the braiding. The integers are shadows of the phases.
Braiding determines fusion. Fusion does not determine braiding.
The magnitude is a function of the phase. The phase is not a function of the magnitude. Both halves are theorems — Verlinde for the forward direction, and the four-categories-one-ring fact from tutorial 1 for the failure of the converse.
Set the map up properly. Let 𝔐 be the set of modular data and 𝔉 the set of fusion rings. Verlinde gives a map
It is tempting to ask whether V has a point where the two sides coincide. It cannot: they live in different sets. Categories are not rings.
The well-posed version of the question is about the fibre:
RSW classify unitary modular tensor categories. Yang–Lee fails their Frobenius–Perron criterion and does not appear in their tables at all. So the count of four over this fusion ring is not an RSW result — it includes non-unitary members their classification excludes by construction. Citing RSW §5.3 for it would be reaching into a table for a number the table was built to leave out.
What RSW does give, in its own introduction, is the pairing: "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 holding the ring rigid while everything else moves. That is stronger than a fibre count, because it is a mechanism rather than a tally.
A fibre with one point would be a fusion ring that does determine its braiding — a place where the modulus forgets nothing. Those exist. Finding which rings have singleton fibres is a sharper question, and it is answerable from the same paper.
So the object you were reaching for is real, but it is a set, and its size is the measure of how much the modulus forgets. Fibre size 1 means no information lost. Fibre size 4, as here, means the ring throws away exactly two binary choices: which root, and which chirality.
Freedman's Galois Conjugates of Topological Phases, Theorem IV.5, proves that no gapped Hermitian Lieb–Robinson Hamiltonian has a Yang–Lee phase as its ground state, in any gauge. The mechanism is a signature obstruction: the Galois-conjugated braid representation is dense in a noncompact SU(p,q) preserving a mixed-signature form, while a unitary theory preserves a definite one — and no conjugation bridges them, because eigenvalue moduli are conjugation-invariant.
The authors signpost the exit themselves: §V says the Lieb–Robinson tools simply are not available for non-Hermitian Hamiltonians, and Sanno 2022 takes that route with an open dissipative system. ⇒ Any claim that a Yang–Lee phase could be physically realised must carry "in an open / non-Hermitian system", or it is refuted. Narrower than "Yang–Lee is impossible" — and narrower than how the result is usually repeated.
There is a place where phase data turns into magnitude data, and it is on the criticality side.
Lee–Yang zeros. Take a finite system and treat the partition function as a polynomial in fugacity or in the magnetic field. It has zeros. For real physical parameters the partition function is strictly positive, so the zeros live at complex values — pure argument data, with no direct physical meaning at finite size.
Yang and Lee's insight (1952): as the system size goes to infinity, the zeros migrate. A phase transition occurs precisely where the zeros pinch the real axis. The free energy is analytic everywhere the zeros are not, and singular exactly where they arrive.
Complex zeros are argument. A divergent susceptibility is modulus — a real, measurable magnitude. The critical point is the locus where the first becomes the second. Phase information is converted into magnitude information at, and only at, criticality.
And the endpoint of the arc of zeros is itself a critical point with its own exponents: the Yang–Lee edge singularity, described by a non-unitary conformal field theory with c = −22/5.
Which is the fourth row of tutorial 1's table. The two tutorials close a loop. The non-unitary partner of Fibonacci is not an exotic curiosity dragged in for symmetry — it is the field theory of the place where complex zeros touch the real axis.
Tutorial 1 made c− the weld between magnitude and phase, via the Gauss sum. Do not promote it to the carrier of computational power.
Doubled Fibonacci — the Levin–Wen string-net built from Fib ⊗ Fib-bar — has c− = 14/5 − 14/5 = 0. It is non-chiral. It has no chiral edge mode at all. And it is still universal for quantum computation.
So c− = 0 with full computational power. The Gauss sum can return a trivial phase while the braid representation stays dense. If you were forming an intuition that the weld measures the power, break it now rather than at 2 a.m. on Sunday.
Could you deform continuously from Fibonacci to Yang–Lee, passing through a coincidence? No — and the obstruction is worth naming precisely.
The two dimensions are the roots of d² = d + 1, whose discriminant is 5. Not zero, so the roots never collide. To make them collide you would have to deform the polynomial, which means deforming the fusion ring itself — but fusion coefficients are non-negative integers. Integers do not deform continuously.
Fusion is discrete and rigid — integer structure constants, no continuous deformations. Braiding lives on continuous groups. That is a real structural difference between the two, and it may be more load-bearing than the phase/magnitude framing that leads people to it.
It also explains why the fibre is finite. A rigid base with a continuous total space would give infinite fibres; the finiteness says the modular data is quantised too.
Plainly: a multicritical point is where two or more independent tuning conditions are met simultaneously — two order parameters going critical together, or in the topological setting two distinct gap-closings coinciding. Those are real objects and they are worth chasing.
But that is not the same as fusion and braiding "becoming one." Multicriticality is about codimension in a parameter space. The fusion–braiding relation is about a map between two different kinds of mathematical object. Conflating them will cost you a weekend.
The honest version of the question: is there a system where tuning to a multicritical point causes the fibre to degenerate — where two of the four categories become indistinguishable in the observable data? That is well posed, and it is not obviously answered.
Checked against the paper itself: Rowell, Stong & Wang (arXiv:0712.1377) for the Galois pairing and for what their classification does and does not cover; Freedman et al., Galois Conjugates of Topological Phases, for Theorem IV.5 and the signature obstruction; Sanno et al. (2022) for the open-system route; Fisher (1978) for the edge singularity.
Stated on reputation, not checked here: Verlinde (1988); Yang & Lee (1952); Levin–Wen for the doubled theories; the Jones index theorem. None of the mathematics here is new, and the only assembly is reading V as a map and asking about its fibres — which is a restatement, not a result.
Section 1's directional claim is the solid part. Section 3's loop-closing and Section 5's reframing were reconstructed rather than read from a source — read the papers before building on either.