Fermions, Bosons, Anyons on a 2-manifold

Question

My understanding of fermions, bosons, and anyons is that anyons are disallowed in 3+1 dimensions (or $n+1 | n\geq3$) because of the topology of spacetime. The paths of swapping two particles twice is contractible in 3+1. In 2+1, this isn't the case and so anyons can exist.

However, what is the case for other 2-manifolds than $\boldsymbol{\mathrm{R}}^2$? Can anyons exist on an $\boldsymbol{\mathrm{S}}^2$ manifold for example?

More importantly, what is the appropriate formalism for approaching this problem?

Answer 1

A general formalism to probe whether or not you have anyons/generalised statistics, in $d$ spatial dimensions and arbitrary topology is the following.

Consider $n$-identical particles moving in a $d$-dimensional spatial manifold, $\Sigma_d$. Since the particles are indistinguishable the configuration space for the $n$-particles is the orbifold: $$\mathrm{Sym}^n(\Sigma_d) := \raise{0.2em}{\underset{n\ \text{times}}{\underbrace{\Sigma_d\times\cdots\times\Sigma_d}}}\Big/\raise{-0.2em}{\mathrm{S}_n},$$ where $\mathrm{S}_n$ is the permutation group of $n$ letters. The important space, from which the rest follow is $\mathrm{Sym}^2(\Sigma_d)=\raise{0.13em}{\Sigma_d\times\Sigma_d}\Big/\raise{-0.15em}{\mathbb{Z}_2}$, since this tells you what happens when you exchange two particles.

Quantum mechanically, on a multiply connected space, $\mathrm{Sym}^2(\Sigma_d)$, all observables involving two such particles decompose as $$\mathcal{O}(x,y) = \sum_{\alpha \in \pi_1(\mathrm{Sym}^2(\Sigma_d))} \chi(\alpha)\; \mathcal{O}_\alpha(x,y),$$ where $\mathcal{O}$ stands for observable, and the sum runs over all connected components of $\mathrm{Sym}^2(\Sigma_d)$ (here $\pi_1(\mathrm{Sym}^2(\Sigma_d))$ is the fundamental group of the configuration space). We have further allowed the freedom of a weight $\chi(\alpha)$ for every connected component.

Now, demanding that the observables satisfy the following two physical requirements:

1. Physical observables cannot depend on the mesh used to calculate the homotopy classes
2. Observables need to satisfy the standard convolution property $\mathcal{O}(x,y) = \displaystyle\int \mathrm{d}{z}\ \mathcal{O}(x,z)\,\mathcal{O}(z,y),$

restricts the weights $\chi(\alpha)$ significantly. The weights must satisfy the constraints

1. $\left|\chi(\alpha)\right|^2=1$
2. $\chi(\alpha)\chi(\beta) = \chi(\alpha\;\beta)$.

Therefore the weights provide a one-dimensional unitary representation of $\pi_1(\mathrm{Sym}^2(\Sigma_d))$. The different choices of representations correspond to the different possible statistics. The general strategy is, then:

1. Choose $\Sigma_d$
2. Calculate $\mathrm{Sym}^2(\Sigma_d)$
3. Calculate $\pi_1\!\left(\mathrm{Sym}^2(\Sigma_d)\right)$
4. Look at how many one-dimensional unitary representations it admits.

For example it is easy to check that $\pi_1\!\left(\mathrm{Sym}^2\!\left(\mathbb{R}^{d\geq 3}\right)\right)=\mathbb{Z}_2$, therefore there are two one-dimensional unitary representations. The trivial, corresponding to bosons, and the sign, corresponding to fermions.

On the other hand, $\pi_1\!\left(\mathrm{Sym}^2\!\left(\mathbb{R}^2\right)\right)=\mathbb{Z}$, so there are infinitely many one-dimensional unitary representations, corresponding to all possible statistics interpolating between bosons and fermions.

Finally, for the example of $\mathbb{S}^2$ that you were interested in, we have $$\pi_1\!\left(\mathrm{Sym}^2\!\left(\mathbb{S}^2\right)\right) = \pi_1\!\left(\mathbb{S}^2\times\mathbb{RP}^2\right)=\mathbb{Z}_2,$$ so you cannot have anyons on a sphere.

Answer 2

You can indeed have anyons on a sphere or other 2d manifold. Models which support anyons, such as the Levin-Wen string-net model or Kitaev's Quantum Double model can be defined on a variety of manifolds. The problem with the previous answer given above is that it only allows for two particles to be created and only allows one species of particle. If only two anyons are present on a sphere then they must be particle and anti-particle (because they must be created from the vacuum) which limits their braiding statistics. To treat multiple particles using the method given in the previous answer, I believe that you would need to "mark" or puncture points with additional observer particles (those not undergoing the braiding) which would change the topology of the configuration space.

The crucial difference with the 3d case is that in 3d you can contract the swapping path locally regardless of the presence of other particles. In the 2d sphere case, the contraction can be done only by pulling the path around the entire sphere. However if other particles are present, this deformation is no longer possible because it would cause the path to cross those particles.

To be clear, it is not that the additional anyons themselves change the braiding statistics, but that allowing the possibility of multiple anyons changes which particles you are allowed to create in the first place.