Fusion of Anyon
8. Fusion and Structure of Hilbert Space
8.1 Basics of Particles and Fusion - The Abelian Case
Particle types:
There should be a finite set of labels which we call particle types. For now, let us call them $a$, $b$, $c$, etc.
Fusion:
World line can merge which we call fusion, or do the reverse, which we call splitting. If an $a$ particle merges with $b$ to give $c$, we write $a\times b=b\times a=c$.
In out abelian anyone model of charges and fluxes if the statistical angle is $\theta=\pi p/m$ then we have species $a=(aq,a\Phi)$ for $a=0,\cdots,m-1$, where $q\Phi=\pi p/m$. The fusion rules are simply abelian modulo $m$. That is $a\times b=(a+b)\mbox{mod } m$.
Identity:
Exactly one of the particle should be called the identity or vaccum. We write this as $1$ or $0$ or $I$ or $e$. The identity fuses trivially $$\begin{align*}a\times I=a\end{align*}$$ for any particles type $a$. In the charge-flux model we should think of the identity as being no charge and no flux.
Antiparticles:
Each particle $a$ should have a unique antiparticle which we denote as $\bar{a}$. The antiparticle is defined by $a\times \bar{a}=I$. A particle going forward in time should be equivalent to an antiparticle going backwards in time.
8.2 Multiple Fusion Channels - the Nonabelian Case
For the nonabelian theories, the dimension of the Hilbert space must increase with the number of particles present. In nonabelian models we have multiple possible orthogonal fusion cahnnels $$\begin{align} a\times b=c+d+\cdots\end{align}$$ meaning that $a$ and $b$ can come together to form either $c$ or $d$ or $\cdots$. A theory is nonabelian if any two particles fuse in such a way that are multiple possible fusion channels.
Locality:
The principle of locality is an predominant theme of anyon physics.
The quantum number of a particle is locally conserved in space.
Antiparticles in the Case of Multiple Fusion Channels:
When we have multiple fusion channels we define antiparticles via the principle that a particle can fuse with its antiparticles to give the identity, although other fusion channels may be possible. $$\begin{align*}a\times \bar{a}=I+\mbox{other fusion channels}\end{align*}$$
8.2.1 Example: Fibonacci Anyons
$$\begin{align*} \mbox{Particle types}=\{I,\tau\}\end{align*}$$
The fusion rules are $$\begin{align*} I\times I &=& I\\ I\times \tau &=& \tau \\ \tau\times \tau &=& I+\tau\end{align*}$$
With three Fibonacci anyones the Hilbert space is $3$ dimensional ($\tau\times\tau\times\tau$). There is a single state in the Hilbert space of three anyons with overall fusion channel $I$. This state is labeled as $\left| N\right\rangle$. There are two states in Hilbert space of three anyons with overall fusion channel $\tau$. These labeled $\left| 1\right\rangle$ and $\left| 0\right\rangle$.
The dimension of Hilbert space follows Fibonacci sequence. Since the $N^{th}$ element of the Fibonacci sequence for large $N$ is approximately $$\begin{align}\mbox{dim}(\mathcal{H}_N)\sim \left( \frac{1+\sqrt{5}}{2}\right)^N.\end{align}$$
8.2.2 Example: Ising Anyons
$$\begin{align*}\mbox{Particle types}=\{I,\sigma,\psi\}\end{align*}$$
The nontrivial fusion rules are $$\begin{align*} \psi\times \psi &=& I\\ \psi\times \sigma &=& \sigma\\ \sigma\times \sigma &=& I+\psi\end{align*}$$
8.3 Fusion and the $N$ matrices
The general fusion rules can be written as $$\begin{align}a\times b=\sum_c N^c_{ab}c\end{align}$$ where the $N^c_{ab}$ are nonnegative integers known as the fusion multiplets. $N^c_{ab}$ is zero if $a$ and $b$ cannot fuse to $c$.
Elementary properties of the fusion multiplicity matrices
$$\begin{align}N^c_{ab}=N^c_{ba}\end{align}$$
$$\begin{align}N^c_{ab}=N^\bar{c}_{\bar{a}\bar{b}}\end{align}$$
$$\begin{align}N^b_{aI}=\delta_{ab}\end{align}$$
$$\begin{align}N^I_{ab}=\delta_{b\bar{a}}\end{align}$$
$$\begin{align}N_{ab\bar{c}}=N^c_{ab}\end{align}$$
$N_{abc}$ is fully symmetric.
$$\begin{align}N^c_{ab}=N^\bar{b}_{a\bar{c}}=N^c_{\bar{a}b}.\end{align}$$
Fusing Multiple Anyons
$$\begin{align}\mbox{dim}(\mathcal{H}_a)=\mbox{largest eigenvalue of }[N_a]\end{align}$$
8.3.1 Associativity
It should be noted that the fusion multiplicity matrices $N$ are very special matrices since the outcome of a fusion should not depend on the order of fusion. $(a\times b)\times c=a\times (b\times c)$
$$\begin{align} \sum_d N^d_{ab}N^e_{cd}=\sum_f N^f_{cd}N^e_{af}\end{align}$$
Again, thinking of $N^c_{ab}$ as a matrix labeled $N_a$ with indices $b$ and $c$, this tells us that $$\begin{align}[N_a,N_c]=0\end{align}$$ Therefore all of the $N$ matrices commute with each other. In addition the $N$'s are normal matrices, meaning that they commute with their own transpose. A set of normal matrices that all commute can be simultaneously diagonalized, thus $$\begin{align}[U^\dagger N_aU]_{xy}=\delta_{xy}\lambda_x{(a)}\end{align}$$ and all $N_a$'s get diagonalized with the same unitary matrix $U$. Surprisingly for well behaved (so-called modular anyon theories) the matrix $U$ is precisely the modular $S$-matrix.
8.4 Application of Fusion: Dimension of Hilbert Space on $2$-Manifolds
The structure of fusion rules can be used to calculate the ground state degeneracy of wavefunction on $2$-dimensional manifolds.
Let us start by considering the sphere $S^2$, and assume that there are no anyons on the surface of the sphere. There is unique ground state in this situation because there are no non-contractivle loops. The dimension of the Hilbert space is just $1$, $$\begin{align*} \mbox{dim} V(S^2)=1.\end{align*}$$
Consider the possibility of having a single anyone on the sphere. In fact such a thing is not possible because you can only create particles in a way that conserves that overall quantum number. $$\begin{align}\mbox{dim} V(S^2\mbox{ with one anyon})=0\end{align}$$
Consider the possibility of two anyons on a sphere. $$\begin{align*} \mbox{dim} V(S^2\mbox{ with one }a\mbox{ and one }\bar{a})=1\end{align*}$$ Think particle as a hole, and identifying $a$ and $\bar{a}$ become torus, and consider every kind of $a$ became torus state. $$\begin{align*} \mbox{dim} V(T^2)=\mbox{Number of particle types}\end{align*}$$
Consider a sphere with three particles. three holes make 'pants' and gluing this makes two handed torus. $$\begin{align*} \mbox{dim} V(\mbox{Two handled torus})=\sum_{abc}N_{abc}N_{\bar{a}\bar{b}\bar{c}}\end{align*}$$
Consider a torus $T^2$ with a single anyone on it labelled $a$. $$\begin{align}\mbox{dim} V(T^2\mbox{ with one }a)=\sum_bN_{b\bar{b}a}\equiv L_a\end{align}$$
Consider ground state degeneracy of a three handled torus.
$$\begin{align}\mbox{dim} V(\mbox{Three handled torus})=\sum_{abc} L_aL_bL_c N_{\bar{a}\bar{b}\bar{c}}\end{align}$$
8.5 Product Theories
Consider the product of two anyon theories. Given two theories with particle types $$\begin{align*}a,b,c,\cdots &\in & t\\ A,B,C,\cdots &\in & T\end{align*}$$ we consider the product theory $T\times t$. $$\begin{align*} \alpha\in T\times t \Rightarrow \alpha=(Y,x)\mbox{ with } Y\in T \mbox{ and } x\in t\end{align*}$$ The fusion multiplicity matrices $N$ for the product theories are just the product of the $N$ matrices for the constituent theories $$\begin{align*} N^{(C,c)}_{(A,a),(B,b)}=N^C_{A,B}N^c_{a,b}\end{align*}$$
8.6 Appendix: Tensor Description of Fusion and Splitting Spaces
For each fusion $N^c_{ab}$ we define a space $V^c_{ab}$ known as a fusion space and a space $V^{ab}_c$ known as a splitting space. Both of these spaces have dimension $N^c_{ab}$ $$\begin{align*} \mbox{dim}V^c_{ab}=\mbox{dim} V^{ab}_c=N^c_{ab}\end{align*}$$ Each of these spaces can be given an orthonormal basis, which we label with an index $\mu$. We can write states in this space as kets like $$\begin{align}\left| a,b;c,\mu\right\rangle \in V^{ab}_c\end{align}$$ describes the splitting space. The Hermitian conjugates, the corresponding bras, are $$\begin{align}\left\langle a,b;c,\mu\right| \in V^c_{ab}\end{align}$$ which describes the fusion space.
Three anyons fusing into one is $$\begin{align}\left|(a,b);d,\mu\right\rangle \otimes \left| d,c;\epsilon,\nu\right\rangle \in V^{ab}_d\otimes V^{dc}_e\subseteq V^{abc}_e\end{align}$$ The full splitting space $V^{abc}_e$ can thus be describled as $$\begin{align}V^{abc}_e\simeq \bigoplus_d V^{ab}_d\otimes V^{dc}_e\end{align}$$ with a corresponding dimension of this space $$\begin{align}\mbox{dim} V^{abc}_e=\sum_d N^{ab}_dN^{dc}_e\end{align}$$
One the other hand, we could just as well have described a state in this space as $$\begin{align}\left| a,f;e,\lambda\right\rangle \otimes \left| (b,c);f.\eta\right\rangle \in V^{af}_e\otimes V^{bc}_f \in V^{abc}_e\end{align}$$ In this language the full splitting space $V^{abc}_e$ can be described as $$\begin{align} V^{abc}_e\simeq \bigoplus_f V^{af}_e\otimes V^{bc}_f\end{align}$$ with corresponding dimension of this space $$\begin{align}\mbox{dim} V^{abc}_e=\sum_d N&{af}_eN^{bc}_f\end{align}$$ We thus have $$\begin{align} V^{abc}_e\simeq \bigoplus_d V^{ab}_d\otimes V^{dc}_e \simeq \bigoplus_f V^{af}_e\otimes V^{bc}_f\end{align}$$
Reference
Steven H. Simons - Topological Quantum: Lecture Notes and Proto-Book