anyons

Anyons are quasiparticles in two-dimensional quantum systems that obey exchange statistics that are neither bosonic nor fermionic. In two dimensions the configuration space of identical particles has a richer braid structure, allowing the effect of exchanging particles to be more general than a simple sign. Consequently, exchanging two anyons can multiply the many‑body wavefunction by a nontrivial phase e^{iθ} (Abelian anyons), or, more generally, implement a nontrivial unitary transformation on a degenerate ground-state manifold (non-Abelian anyons).

Abelian anyons: exchanges yield a definite phase e^{iθ} with θ not restricted to 0 or π. Non-Abelian anyons: exchanges correspond to representations of the braid group by unitary operators, so the outcome depends on the history of braiding.

The concept arises in theoretical models and in two-dimensional electronic systems. The fractional quantum Hall effect provides concrete contexts where quasiparticles can carry fractional charge and exhibit fractional statistics. For the Laughlin states at filling factor 1/m, quasiparticles are believed to be Abelian anyons with statistics angle θ = π/m. Non-Abelian anyons are predicted in states such as the ν=5/2 Moore–Read (Pfaffian) state and in certain topological superconductors; Fibonacci anyons are a particularly interesting theoretical example. Some proposed realizations involve Majorana zero modes in topological superconductors, which realize non-Abelian statistics under braiding.

Experimentally, detecting anyonic statistics is challenging; interferometry and tunneling experiments in quantum Hall devices have provided evidence consistent with fractional charge and anyonic behavior, while direct braiding measurements are an active area of research. The non-Abelian case holds promise for fault-tolerant topological quantum computation, since braiding operations can implement quantum gates that are inherently protected from local errors.