# How can I derive fusion rules for anyons?

I am reading Pachos "Introduction to Topological Quantum Computation". Pachos writes that a model for anyons consists of a list of all anyons and a fusion rule for them.

Given a model with anyonic excitations, how do I deduce which anyons I have? Given a set of fusion rules, how do I construct a model with anyonic excitations (if there is one) obeying these relations?

Later in the book he talks about Ising and Fibonacci Anyons and states their fusion rules, but he does not say how one would derive them.

• Basically, you can just make them up, as long as you satisfy certain consistency conditions (pentagon/hexagon equation etc.) – Norbert Schuch Sep 22 '15 at 10:06
• Ok, but how do I then try to guess where they might appear? For example, Ising Anyons are probably at the fractional quantum hall effect for a filling factor of 5/2. Fib. Anyons are at the FQHE for a filling factor for 12/5. How I do make these connections from my (made up) rules to the physical world? – Thomas Elliot Sep 22 '15 at 13:46
• Wouldn't you usually rather pick a specific model of interest and try to understand the properties of its excitations? – Norbert Schuch Sep 22 '15 at 13:52
• I am just wondering how one can make the connection between the theory (making up some fusion rules) and specific physical systems (FQHE). – Thomas Elliot Sep 22 '15 at 14:32

Naively, the collection of data which describes a model of anyons is not its fusion rules, but rather its modular data - that is, a pair of matrices $S$ and $T$ which generate a representation of the (modular) group $SL(2,\mathbb Z)$.

The (diagonal) matrix $T$ encodes the mutual statistics of quasi-particles - that is what happens when you exchange two identical excitations. The entries of this matrix are called twists. Vafa's Theorem says that all of these must be roots of unity. An excitation is anyonic if and only if it has twist not equal to $1$ or $-1$. The $S$ matrix encodes evaluations of colored Hopf links when braiding pairs of different excitations, but that's less important at the moment. It is however important that the $S$ matrix be non-degenerate.

A very important thing is that the Verlinde formula can be used to derive the fusion rules from the $(S,T)$ matrices. The converse is not true - there are plenty of examples (Ising, Fibonacci) where a set of fusion rules may have give rise to more than one set of $S$ and $T$ matrices. Taking everything together through the magic of polynomials, these give rise to mathematical objects called modular tensor categories, and it is really here that the story begins. MTCs are the objects which describe the algebraic properties for a given model of anyons.

Important note: I take issue with the phrase "you can just make them up" because it makes a set of very big problems seem trivial. It also takes as fact something which is only at this point conjecture - that every pair $(S,T)$ is uniquely realized by some MTC.

We can now give a partial answer the parenthetical to the first question: If a set of fusion rules does not give rise to a set of modular data, there does not exist a model (though we've still not said what a model is) with anyonic excitations realizing those fusion rules.

Another note: This statement sweeps several things under the rug like $G$-crossed modular categories and minimal modularizations of ribbon categories. Let's not complicate matters.

The next important question then is: What is a model with anyonic excitations? Beyond their exchange properties we also know some other physical things our model with anyons should do. The big ones are that there is degeneracy in the ground state and that our anyonic excitations should be robust against local perturbations. Add all of the bits and pieces up and you get that the lowest energy Hilbert space should be described by a topological quantum field theory or TQFT.

Usually, the information for a physical model is contained in some Hamiltonian $H$ or Lagrangian $L$, and here the situation will be no different. To say that I have a a model with anyonic excitations will be to say that I have some $H$ or $L$ whose lowest energy Hilbert space is described by a TQFT. A choice of TQFT uniquely determines the MTC (i.e. the anyonic excitations), and given an MTC and a manifold there is a way to construct a TQFT.

Now then, the questions can be rephrased as follows:

1. Given a modular tensor category $\mathcal C$, how does one construct a Hamiltonian $H$ or Lagrangian $L$ realizing a TQFT whose quasi-particle excitations are described by $\mathcal C$?
2. Given a Hamiltonian $H$ or Lagrangian $L$ realizing a TQFT, how does one determine the MTC $\mathcal C$ describing its anyonic excitations?

As far as I know, the answer to both questions is incomplete. I do not know a general recipe for producing a Hamiltonian/Lagrangian which realizes an arbitrary MTC. Conversely, my understanding of the opposite direction is that given $H$/$L$, one must first show that the ground state is degenerate and then go through and construct the appropriate operators describing the anyonic excitations. I don't know much about that, but it seems as if it could be a tricky proposition.

Because of this, what I've included below are classes of examples in which both questions have known answers.

1. For MTCs which can be realized as centers of some unitary fusion categories, one can construct a Levin-Wen Hamiltonian on a trivalent lattice. In fact, Levin-Wen models take as input the UFC and lattice, and there is a formula to follow for constructing the quasi-particle operators which correspond to objects in the center. Levin's thesis has of the details of this.
2. A particularly nice subclass of these are Kitaev's double models. What one does in that case is play the game.
3. To provide a partial example which starts in the other direction, one could start with a manifold $M$, Lie group $G$ and "level" (essentially an integer) $k$ and construct its Chern-Simons action. After some work - which I won't try to describe because I'm not terribly familiar with it - one obtains a TQFT whose quasiparticle excitations are described by the MTC for the category of representations for the affine Lie algebra associated to $G$ with highest integral weight $k$. Both Ising and Fibonacci are examples of this. Ising is the category $SU(2)_4$ and Fibonacci is the even half of the category $SU(2)_3$.
• 'Ising is the category $SU(2)_4$' Do you mean $SU(2)_2$? – Ruben Verresen Jul 19 '17 at 17:06
• Apologies for the activity on an old question! Could you provide any references for the construction described in example 3? – S Valera Oct 26 '17 at 0:46
• I don't remember exactly at the moment but I believe these should come from Witten's original papers on TQFTs and the Jones Polynomial. – Matthew Titsworth Oct 29 '17 at 17:46

In my understanding, you cannot make up such rules. Those are uniquely determined (calculated) from consistency rules (pentagon, hexagon, etc.).

From Nayak, C., Simon, S. H., Stern, A., Freedman, M., & Sarma, S. D. (2008). Non-Abelian anyons and topological quantum computation. Reviews of Modern Physics, 80(3), 1083.:

The $$F$$ and $$R$$ matrices play particularly important roles. The $$F$$ matrix can essentially be viewed an associativity relation for fusion: we could first fuse $$i$$ with $$j$$, and then fuse the result with $$k$$; or we could fuse $$i$$ with the result of fusing $$j$$ with $$k$$. The consistency of this property leads to a constraint on the $$F$$-matrices called the pentagon equation. Consistency between $$F$$ and $$R$$ leads to a constraint called the hexagon equation. Modularity is the condition that the $$S$$-matrix be invertible. These self-consistency conditions are sufficiently strong that a solution to them completely defines a topological phase

Thus, such rules can be calculated solely from the hexagon and pentagon equations. see Trebst, S., Troyer, M., Wang, Z., & Ludwig, A. W. (2008). A short introduction to Fibonacci anyon models. Progress of Theoretical Physics Supplement, 176, 384-407. for example.

I wrote on the subject on my blog with examples too long to put it here.

As for how you make such connections to an actual physical system, you don't. All the system provides is the fusion rule

$$a \times b = \sum_c N_{ab}^c c.$$

Once you have rules, you can start filling in the super- and subscripts of $$F$$-matrices and start solving for the equations. Again, read Trebst, S., Troyer, M., Wang, Z., & Ludwig, A. W. (2008). A short introduction to Fibonacci anyon models. Progress of Theoretical Physics Supplement, 176, 384-407. Or a follow-up note on Ising anyons.