Ising Anyons: IsingAnyon
IsingAnyon represents the Ising modular fusion category. It has labels :I, :σ, and :ψ; the ASCII aliases :sigma and :psi are accepted by the constructor.
Sector type
TensorKitSectors.IsingAnyon — Type
struct IsingAnyon <: Sector
IsingAnyon(s::Symbol)Represents the anyons of the Ising modular fusion category. It can take three values, corresponding to the trivial sector IsingAnyon(:I) and the non-trivial sectors IsingAnyon(:σ) and IsingAnyon(:ψ), with fusion rules $ψ ⊗ ψ = 1$, $σ ⊗ ψ = σ$, and $σ ⊗ σ = 1 ⊕ ψ$.
Fields
s::Symbol: the label of the represented anyon, which can be:I,:σ, or:ψ.
All three sectors are self-dual, and the unit is IsingAnyon(:I).
Fusion Rules
The non-trivial fusion rules are
\[\psi \otimes \psi = I,\qquad \sigma \otimes \psi = \psi \otimes \sigma = \sigma,\qquad \sigma \otimes \sigma = I \oplus \psi.\]
Thus FusionStyle(IsingAnyon) = SimpleFusion(). The quantum dimensions are
\[d_I = 1,\qquad d_\psi = 1,\qquad d_\sigma = \sqrt{2}.\]
Topological Data
The non-trivial associator for four $\sigma$ anyons is
\[F^{\sigma\sigma\sigma}_{\sigma} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix},\]
in the intermediate basis $(I,\psi)$. Additional signs involving the fermion $\psi$ are encoded by Fsymbol.
The braiding style is Anyonic(). Important braiding phases are
\[R^{\sigma\sigma}_I = e^{-\pi i/8},\qquad R^{\sigma\sigma}_\psi = e^{3\pi i/8},\qquad R^{\psi\psi}_I = -1.\]
There is no fusion tensor as the fusion category does not originate from a group or its representations.
Iteration and basis conventions
values(IsingAnyon) iterates the labels in the order :I, :σ, :ψ.