A model that has shown some interest in recent years is the golden chain. In the golden chain you deal with a one-dimensional chain of spin-like particle, similar to the <PERSON> (or <PERSON>) model. But in this model the spin degrees of freedom are replaced by (non-Abelian) anyons (see e.g. this thread). The type of anyon used in this model are the Fibonacci anyons.
To see how the golden ratio enters this model we have to look at the Hilbert space, specifically its dimension. In an ordinary spin chain each spin carries a degree of freedom to which we can assign a Hilbert space of dimension 2, $\mathcal{H}$. The tensor product of two spins is spanned by the singlet and triplet combination. The total Hilbert space of a chain with $n$ spins is the tensor product, $\otimes \mathcal{H}$ and it has dimension $2^n$.
Non-Abelian anyons on the other hand carry a different kind of spin. When two anyons combine they will form what is known as a fusion product. The fusion product of two anyons depends on the type of anyon you are dealing with. Fibonacci anyons satisfy the relation
$\tau \times \tau = 1 + \tau$
We can think of this analogous to spin (with a fundamental difference). When we bring two anyons together they will fuse together and form a composite-like particle. 976
This is similar to the spin of two (s=1/2) particles which combine into a singlet (S=0) or doublet (S=1). In the case of the Fibonacci anyons the particle can form two type of composites: the vacuum particle $1$ ("zero spin") and the Fibonacci particle $\tau$.
What happens when we bring another $\tau$ \particle to this composite? It will fuse with the composite particle to form another composite. However, the allowed particles which can be formed depend on the fusion channel of the first two anyons:
If $\tau_1 \times \tau_2 \rightarrow 1$, then $(\tau_1 \times \tau_2)\times \tau_3 \rightarrow \tau$.
If $\tau_1 \times \tau_2 \rightarrow \tau$, then $(\tau_1 \times \tau_2)\times \tau_3 \rightarrow 1+\tau$
There are two ways in which a $\tau$ particle is formed in the end, while there is only one way in which a vacuum particle is formed. All in all:
$\tau \times \tau \times \tau = 1+ 2\tau$
Where the factor of two on the right hand side refers to the number of ways in which a $\tau$ particle can be formed. The dimension of the Hilbert space of three tau particles is therefore 3 dimensional.
This gives the following conclusion:
We set the dimension of the Hilbert space of zero particles equal to 1.
The dimension of H for 1 Fib. anyon is also 1.
The dimension of H for 2 Fib. anyons is 2.
The dimension of H for 3 Fib. anyons is 3.
Any guess what the dimension of the 4 anyons will be? It's 5 dimensional. You can derive it yourself: just count the number of ways in which you can fuse the anyons together.
The sequence of the dimension of the Hilbert spaces for $n$ anyons is:
$1,1,2,3,5,8,13,\ldots$
Yes, this is the <PERSON> sequence! And the <PERSON> sequence has a very nice feature: It grows roughly as $\phi^n$ where $\phi$ is the golden ratio! The dimension of the Hilbert space of $n$ Fibonacci anyons grows roughly as $\phi^n$!
One way to think of this is that the Fibonacci anyons carry a spin of dimension $\phi$. This statement is wrong though: the Hilbert space always has an integer dimension. It is therefore not referred to as a spin but rather as the quantum dimension of the anyons. The rule is that on average the Hilbert space grows by a factor of $\phi$ every time you add an anyon to the chain (just like the H-space for an Ising chain grows with a factor of two every time you add a spin to the system).
On a last note: Fibonacci anyons might be realized in certain quantum Hall systems and they are useful for topological quantum computing, if they are ever found. 976