Fibonacci Anyons: FibonacciAnyon
FibonacciAnyon represents the Fibonacci modular fusion category. It has a trivial anyon :I and a non-trivial anyon :τ (:tau is accepted as an ASCII constructor alias).
Sector type
TensorKitSectors.FibonacciAnyon — Type
struct FibonacciAnyon <: Sector
FibonacciAnyon(s::Symbol)Represents the anyons of the Fibonacci modular fusion category. It can take two values, corresponding to the trivial sector FibonacciAnyon(:I) and the non-trivial sector FibonacciAnyon(:τ) with fusion rules $τ ⊗ τ = 1 ⊕ τ$.
Fields
isunit::Bool: indicates whether the sector corresponds to the trivial anyon:I(true), or the non-trivial anyon:τ(false).
Both sectors are self-dual, and the unit is FibonacciAnyon(:I).
Fusion Rules
The only non-trivial fusion rule is
\[\tau \otimes \tau = I \oplus \tau.\]
Thus FusionStyle(FibonacciAnyon) = SimpleFusion(). The quantum dimensions are
\[d_I = 1,\qquad d_\tau = \varphi = \frac{1 + \sqrt{5}}{2}.\]
Topological Data
The non-trivial associator appears when all external anyons are $\tau$:
\[F^{\tau\tau\tau}_{\tau} = \begin{pmatrix} \varphi^{-1} & \varphi^{-1/2}\\ \varphi^{-1/2} & -\varphi^{-1} \end{pmatrix},\]
in the intermediate basis $(I,\tau)$. All other allowed Fsymbol values are 1.
The braiding style is Anyonic(). For $\tau \otimes \tau$,
\[R^{\tau\tau}_I = e^{4\pi i/5},\qquad R^{\tau\tau}_\tau = e^{-3\pi i/5}.\]
There is no fusion tensor as the fusion category does not originate from a group or its representations.
Iteration and basis conventions
values(FibonacciAnyon) iterates the labels in the order :I, :τ.