# Can anyons emerge from momentum-space other than spatial dimensions?

So far in condensed matter physics, I only know anyons(abelian or nonabelian) can emerge as quasiparticles in 2D real-space.

But is there any possibility to construct anyons in momentum-space ? And what about the braiding, fusion rules in momentum-space ? I mean do anyons always live in real-space rather than in momentum-space ?

Maybe this is a trivial question, thank you very much.

I asked my advisor this exact same question a couple years ago. He said that there's no sense of anyonic statistics in momentum space (or in any basis other than real space).

The reason for this is that anyons typically emerge from a microscopic Hamiltonian that is spatially local, and so strictly speaking, anyons are only well-defined when they stay far away from each other in space. If you bring two anyons close together (i.e. on the order of the correlation length or less), then they begin to lose their identity as individual particles (as is always the case in quantum mechanics with identical particles). Strictly speaking, the anyonic statistical phase factor only occurs when when they are braided at long distances - if you try to braid them close together, it becomes ambiguous which particle is which, and therefore how many times they've circled each other. But the correction becomes exponentially small at long distances: the actual phase picked up in a braiding process is

$\theta = \theta_\text{dynamical} + \theta_\text{anyonic} + o(\exp(-r/\xi)),$

where $r$ is the distance between the two anyons and $\xi$ is the correlation length. (Many people are unaware of this subtlety because their intuition is based on Kitaev's toric code, where the correlation length is zero, so anyons remain well-defined even on adjacent lattice sites.)

Anyway, if you tried to localize two anyons into small wave packets in momentum space, then in real space they would be close to plane waves and therefore have large spatial overlap, so everything would get messed up.

This unfortunate asymmetry between real and momentum space originates from the fact that anyons aren't true point particles (because you can't create just one) but are rather connected by strings, so it's very hard to directly second-quantize anyons. By contrast, with a "true" point particle, the canonical commutation relations are preserved under Fourier transforms, so everything is nice and symmetric between real and momentum space.

I was wondering something similar few month ago. Then I concluded that most of the topological staffs appear at the boundary between two different topological sector. A sector being characterised by a Chern number, or if you prefer a topological charge, one needs a boundary / an interface between two systems characterised by different topological charge.

A $k$-space (or momentum, or reciprocal, or Fourier, ...) is well defined only for periodic boundary conditions. The fact that the $x \leftrightarrow k$ is a Fourier transform imposes a periodicity in $x$ or in $k$. That's the stringent condition under which $k$ is a good quantum number. Note that we can still define some quasi-$k$ for disordered media. So we could not in principle define a $k$-space when a system has boundary. Note that infinite system are usually closed by periodic boundary condition, also called Born-von-Karman conditions.

I'm not aware so much about anyons (I'm still learning about that) but I believe they (almost all of them ? all of them ? I don't know) appear due to boundary conditions in condensed matter, for the reason I gave about the topological charge transition. So I believe it should be impossible to define anyons in $k$-space, for the simple reason that the $k$-space is not a correct description of the matter when anyons exist.

I would really appreciate comments/critics about what I said, especially if it's (partially) wrong.