Anyons are particles which quantum statistics is neither bosonic or fermionic one. They are proved to exist only in 2 dimensions and can have some quantum numbers of fractional values in respect to other elementary particles such as for example electron. An anyon can be charged and the charge can be a fracture of 1e. (quarks have a charge of value $\frac 1 3$ but they are not anyons because their statistics is not fractional). After exchanging two identical particles the quantum mechanics predicts that the wave function gain a phase factor:

Ψ → *e**i**θ*Ψ

- For bosons
*θ*= 0 - For fermions
*θ*=*π* - For anyons
*θ*= 2*π**ν*− 1

The structure of the excitation spectrum is efficiently described in terms of a gauge theory and Aharanov-Bohm gauge interactions.

### History

These particles were predicted for the first time in 1977 by J. M. Leinaas and J. Myrheim and studied independently in more details by F. Wilczek in 1982 who gave them the name “anyons”. In 1983 R. B. Laughlin proposted a model where anyons can be found. There was however for many years no idea how to observe them directly. In recent investigation of F. E. Camino, Wei Zhou, and V. J. Goldman show how to design such an experiment using interferometry methods. Nowdays the most of interest is focused on so called non-Abelian anyons which are believed to provide a tool to introduce fault-tolerant topological quantum computing.

### fractionalization of statistics

In order to understand how this can be possible let's see how the phase of wave functiona changes during the evolution of a system. Let's take a particle moving slowly along a loop $\mathcal(C)$ in parameter space *R*(*t*). It's phase changes like:

$\Psi_{R(t_f)}(t_f)=exp\left(-\frac i \hbar\int^{t_f}_{t_i}dt' E(R(t'))+i\gamma(\mathcal C)\right)\Psi_{R(t_i)}(t_i)$

The first term is just the dynamical phase and the second is independent of *t**i* and*t**f* and depend only on the trajectory $\mathcal C$. $\gamma(\mathcal C) = i \oint_{\mathcal C}\langle\psi_{R(t)}|\nabla_{R(t)}\Psi_R(t)\rangle dR(t).$When the particle is an electron moving in magnetic field this reduces to the Ahoronov-Bohm effect and:$\gamma(\mathcal C)=e\frac \Phi \hbar$where Φ is the flux passing through the contour $\mathcal C$. Now if we exchange two electrons their wave function changes sing. If we, however could construct two particles consisting of an electorn and a magnetic flux Ψ attached to it we would have an extra phase of geometric origin. We would be able to tune this term to change the statistics from fermionic to bosonic or even to an in-between one. This new particles would have anyonic statistics.

### Braiding

The characteristic feature of anyons is that their movements are best described by the braid group. This is due to the fact that while braiding their world lines they can gain non-trivial phase factor or even, in non-Abelian the process of braiding can be equivalent to multiplication by an unitary matrix. In the latter case the final state can be an superposition.

### Fusion rules

The other process that plays a key role in anyons life is the fusion. It is well know that a fusion of two fermions gives a boson. The generalization of this phenomena is the fusion of anyons. In general the fuison rule look like:$a\times b =\sum_c \mathcal N_{ab}^c c$And if$\sum_c \mathcal N ^c_{ab}\geq 2$then the particles a and b are non-Abelian.

### Abelian anyons

The description of anyons usually uses the braid group representation. The braid group is infinite so it has infinite number of representations. The one dimensional representation describes the Abelian anyons. Its generators are then just the multiplication by a phase factor: *σ**j* = *e**i**ϕ**j* The first and so far the only system where the existence of Abelian anyons is convincing is the system with FQHE (Fractional Quantum Hall Effect) described by Lauglin model. Lately several lattice models with anyonic excitations have been proposed. In quantum computation using them one could store some information but not process it. In order to do so the non-Abelian anons are needed.

### Non-Abelian anyons

The phase that a wave function gains after moving an anyon around a loop, with or without other anyons inside, in general can be trajectory dependent. That's why we say that non-Abelian anyons follow the braiding-statistics. Using this concept one can build a universal set of computational gates. The unitary gate operations are carried out by braiding quasi-particles, and then measuring the multi-quasi-particle states. The fault-tolerance arises from the non-local encoding of the states of the quasi-particles.

### literature

J. M. Leinaas and J. Myrheim Nuevo Cimento B **37** (1977) p.1 F. Wilczek, Phys. Rev. Lett. **48**, 1144 (1982).R. B. Laughlin, Phys. Rev. Lett. **50**, 1395 (1983).