anyonic

An anyon is a type of quasiparticle that can occur in two-dimensional quantum systems, exhibiting statistical behavior that is neither bosonic nor fermionic. Unlike in three dimensions, exchanging two identical particles in two dimensions can yield a broader range of statistical properties, leading to Abelian or non-Abelian statistics.

In two dimensions the exchange of identical particles is described by the braid group rather than the permutation group. Abelian anyons acquire a phase factor e^{i theta} upon exchange, where theta can take values between 0 and pi (with theta = 0 for bosons and theta = pi for fermions). Non-Abelian anyons, when exchanged, perform a unitary transformation on a degenerate ground-state space, with the result depending on the braiding path.

The concept was introduced in the late 1970s by Leinaas and Myrheim and by Wilczek, who coined the term anyon. Anyons are most commonly discussed in two-dimensional electron systems where strong interactions give rise to topological order and fractional statistics, such as in certain fractional quantum Hall states. Theoretical descriptions often employ Chern-Simons gauge theory and representations of the braid group.

Experimental research focuses on detecting fractional charge and braiding statistics through interferometry and noise measurements in fractional quantum Hall platforms and related systems. Evidence for Abelian anyons has grown, while unambiguous observation of non-Abelian anyons remains an area of active study, with candidates in the ν = 5/2 quantum Hall state and in networks of Majorana modes in topological superconductors.

Applications include topological quantum computation, which aims to store and manipulate information by braiding non-Abelian anyons in a way that is inherently protected from local perturbations.