Anyons and braid statistics
In two spatial dimensions, exchanges can have richer statistics than bosons or fermions.
$$\Psi\mapsto e^{i\theta}\Psi\quad\text{under exchange}$$
What this means
In two spatial dimensions, exchanges can have richer statistics than bosons or fermions.
Deep dive: mathematical and physical mechanism
Particle exchanges trace braids in 2D spacetime. The braid group $B_n$ has more structure than the permutation group. Abelian anyons contribute exchange phases, while non-Abelian anyons act by matrices on a degenerate state space.
How it appears in quantum phenomena
Quasiparticles in fractional quantum Hall liquids are candidate anyons.
Assumptions, caveats and limitations
An exchange phase is not the same as an Aharonov–Bohm phase, although both can contribute in experiments.
Connect the ideas
For context, see the Interacting Topological Phases topic guide and the reading references.