Over the past few months, anyons hit the news for two important discoveries. Here I try to give a short introduction to anyons and explain why this is interesting.

## About anyons

In condensed matter physics, we usually define quasiparticles, that are not elementary particles, but excitations of collective electrons.
This concept has been introduced by Landau and Kapitza to treat helium superfluidity, where they suggested that these excitations could be treated as particles, with a different mass and momentum than the electron itself.
Since then, a lot of problems have been treated with quasiparticles, as Brown-Zak fermions we introduced recently, but there are a large range of quasiparticles, with exotic behaviours.
Usually, quasiparticles fall into two possible categories: *bosons* or *fermions*.

Let’s focus on position, for example (but any quantum number associated with a given quasiparticle would work), and describe a two-particle system with a wavefunction $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > $, where $\mathbf{x}_1$ is the position of the first quasiparticle, and $\mathbf{x}_2$ is the position of the second one. If we swap the positions of the two quasiparticles, the resulting state should be $ \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $.

In the case of bosons - that obey the Bose-Enstein statistics - switching their position doesn’t change the wavefunction: $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $.

For the case of fermions, that obey the Fermi-Dirac statistics, the Pauli principle forces the wavefunction to be rotated by 180°: $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = - \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $, i.e. $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = e^{i\pi} \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $.

Anyons are neither fermions nor bosons. Switching anyons two different places would induce a rotation of some intermediate angle: $ \vert \psi (\mathbf{x}_1,\mathbf{x}_2) > = e^{i\alpha} \vert \psi (\mathbf{x}_2,\mathbf{x}_1) > $. The way to pick up a phase angle that is neither 0 or $\pi$ is to use the Aharonov-Bohm effect to switch particles in a 2D-electron gas. This would induce a phase shift. Importantly, anyons can only exist in 2D systems, as they would violate the standard model otherwise.

## Fractional statistics and the evidence of the phase factor

In April, a team at ENS and Sorbonne Université (Paris) have realised anyons and observed their phases. The results have been reported in Science. The authors used a 2D electron gas made with a GaAs/AlGaAs heterostructure, cool it to about 10mK and study it in the presence of a magnetic and electric field. Experimental observation of anyon statistics is challenging as it is based on interferometry, for which theoretical interpretation is difficult.

The interferometer was studied in the fractional quantum hall regime, where quasiparticles have a charge that is 1/3 of the electron charge (Laughlin quasiparticles), for long suspected to be anyons. In this system, chiral conducting channels form along the edges of the device, and tunneling of Laughlin quasiparticles can occur between quantum point contacts (QPC). Ideally, a quasiparticle would be emitted from QPC1 and reach QPC2. It becomes interesting if the quasiparticle reaches a third one: QPC3. As the two anyons reach the QPC from opposite sides. In the case of fermions, they would block eachother rom tunneling through QPC3 because of the Pauli principle. In the case of bosons, correlations between drain currents would appear when the quasiparticles combine together. For abelian anyons, that is different: some correlation is expected, but specific details depend upon the anyon statistics. In a nutchell, the author basically swap particles from a scattering experiment and look at the correlations where particles arrive. This observation of expected anyonic properties provides experimental support to this quasiparticle.

## Anyon braiding to build a quantum computer

Much recently, this work has led to further developments. A team at Purdue university interfered anyons in a similar experimental setup and posted it last week on arXiv.org (2006.14115). In their experiment, they create and destroy anyonic states on the bulk of the 2DEG, and produce anyons running on the edges. Between two QPCs, there are two posible paths anyons can undergo. At the end of the journey, an interference pattern would appear. This interference pattern shows the relative amount of rotation between the two paths, with discontinuities that show evidence of the creation and annihilation of anyons in the bulk of the material.

The ability to make anyons appear or disappear raises promises for the building of a topological quantum computer. In fact, pairs of anyons could encode information with the angle phase. The basic interpretation is using the number of circles they made around one another to store data. As the path or perturbations do not alter them, this offers a kind of topological protection to the computation.

## References:

- L. Saminadayar
*et al*1997, Observation of the e/3 Fractionnally Charged Laughlin Quasiparticle, Phys. Rev. Lett. - H. Bartolomei
*et al*2020, Fractional statistics in anyon collisions, Science - J. Nakamura
*et al*2020, Direct observation of anyonic braiding statistics at the ν = 1/3 fractional quantum hall state