Fusion Rules

Fusion rules are the multiplication table of an anyon theory: they say which anyon types can result when two are brought together. They are the algebraic backbone from which the fusion space, quantum dimensions, and ultimately the computational power of an anyon model all follow.

The fusion algebra

For anyon types labeled $a, b, c, \dots$ , the fusion rule is

a \times b = \sum_c N^{c}_{ab}\, c,

where the non-negative integers $N^{c}_{ab}$ count the number of distinct ways $a$ and $b$ can fuse into $c$ . There is a unique vacuum $1$ with $1 \times a = a$ , and each $a$ has a unique antiparticle $\bar a$ with $a \times \bar a = 1 + \cdots$ . When some $\sum_c N^c_{ab} > 1$ — multiple fusion channels — the theory is non-abelian; when every product is a single anyon, it is abelian.

Quantum dimensions from fusion

The fusion rules fix the quantum dimensions $d_a$ as the largest eigenvalue of the fusion matrix $(N_a)_{bc} = N^c_{ab}$ , equivalently the solution of

d_a\, d_b = \sum_c N^c_{ab}\, d_c .

For Fibonacci anyons $\{1, \tau\}$ with $\tau \times \tau = 1 + \tau$ , this reads $d_\tau^2 = 1 + d_\tau$ , so

d_\tau = \varphi = \frac{1+\sqrt5}{2}.

For Ising anyons $\sigma \times \sigma = 1 + \psi$ , $\psi\times\psi = 1$ , $\sigma\times\psi=\sigma$ , giving $d_\sigma=\sqrt2$ , $d_\psi=1$ .

The fusion space and its dimension

Bring $n$ anyons together and fuse them sequentially. Each fusion that has multiple allowed channels forks the history, and the set of consistent fusion trees with a fixed total charge labels an orthonormal basis of the fusion Hilbert space. Its dimension grows like $d_a^{\,n}$ , which is exactly why a non-abelian $d_a>1$ yields an exponentially large protected memory.

For Fibonacci anyons the bookkeeping is especially clean. Track the running total charge as you fuse $\tau$ 's one at a time. Writing $V_n^c$ for the number of trees of $n$ $\tau$ 's with total charge $c$ , the rule $\tau\times\tau = 1 + \tau$ gives the recursion

V_n^{\,1} = V_{n-1}^{\,\tau}, \qquad V_n^{\,\tau} = V_{n-1}^{\,1} + V_{n-1}^{\,\tau}.

Starting from a single $\tau$ ( $V_1^{\,1}=0$ , $V_1^{\,\tau}=1$ ), the vacuum-channel dimensions are $1,1,2,3,5,8,\dots$ — the Fibonacci numbers, which is where these anyons get their name. For $n=6$ the dimension of the vacuum-total-charge space is $5$ .

Try it

Implement the Fibonacci fusion recursion and return the dimension of the space of $n=6$ $\tau$ anyons that fuse to the vacuum (total charge $0$ ). Track the 2-vector $[\,\#\text{trees}\to 1,\ \#\text{trees}\to\tau\,]$ , start it at $[0,1]$ for a single $\tau$ , and apply the fusion map $n-1$ times. The answer is the Fibonacci number $5$ .

Run your code to see the quantum state.