Anyon


In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons. In general, the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables. Anyons are generally classified as abelian or non-abelian. Abelian anyons have been detected and play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected, although this is an active area of research.

Abelian anyons

In quantum mechanics, and some classical stochastic systems, indistinguishable particles have the property that exchanging the states of particle with particle does not lead to a measurably different many-body state.
In a quantum mechanical system, for example, a system with two indistinguishable particles, with particle 1 in state and particle 2 in state, has state in Dirac notation. Now suppose we exchange the states of the two particles, then the state of the system would be. These two states should not have a measurable difference, so they should be the same vector, up to a phase factor:
In space of three or more dimensions, elementary particles are either fermions or bosons, according to their statistical behaviour. Fermions obey Fermi–Dirac statistics, while bosons obey Bose–Einstein statistics. For bosons, the phase factor is, and for fermions, it is. In particular, this is why fermions obey Pauli exclusion principle: If two fermions are in the same state, then we have
The state vector must be zero, which means it's not normalizable, thus unphysical.
In two-dimensional systems, however, quasiparticles can be observed that obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo in 1977. In the case of two particles this can be expressed as
where can be other values than just or. It is important to note that there is a slight abuse of notation in this shorthand expression, as in reality this wave function can be and usually is multi-valued. This expression actually means that when particle 1 and particle 2 are interchanged in a process where each of them makes a counterclockwise half-revolution about the other, the two-particle system returns to its original quantum wave function except multiplied by the complex unit-norm phase factor. Conversely, a clockwise half-revolution results in multiplying the wave function by. Such a theory obviously only makes sense in two-dimensions, where clockwise and counterclockwise are clearly defined directions.
In the case θ = π we recover the Fermi–Dirac statistics and in the case the Bose–Einstein statistics. In between we have something different. Frank Wilczek coined the term "anyon" to describe such particles, since they can have any phase when particles are interchanged. Unlike bosons and fermions, anyons have the peculiar property that when they are interchanged twice in the same way, the wave function is not necessarily the same but rather generally multiplied by some complex phase.
We may also use with particle spin quantum number s, with s being integer for bosons, half-integer for fermions, so that
At an edge, fractional quantum Hall effect anyons are confined to move in one space dimension. Mathematical models of one-dimensional anyons provide a base of the commutation relations shown above.
In a three-dimensional position space, the fermion and boson statistics operators are just 1-dimensional representations of the permutation group acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic statistics operators are just 1-dimensional representations of the braid group acting on the space of wave functions. Non-abelian anyonic statistics are higher-dimensional representations of the braid group. Anyonic statistics must not be confused with parastatistics, which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group.

Topological equivalence

The fact that the homotopy classes of paths are relevant hints at a more subtle insight. It arises from the Feynman path integral, in which all paths from an initial to final point in spacetime contribute with an appropriate phase factor. Recall that the Feynman path integral can be motivated from expanding the propagator using a method called time-slicing, in which time is discretized.
In non-homotopic paths, one cannot get from any point at one time slice to any other point at the next time slice. This means that we can consider homotopic equivalence class of paths to have different weighting factors.
So it can be seen that the topological notion of equivalence comes from a study of the Feynman path integral.
For a more transparent way of seeing that the homotopic notion of equivalence is the "right" one to use, see Aharonov–Bohm effect.

Experiment

A group of theoretical physicists working at the University of Oslo, led by Jon Leinaas and Jan Myrheim, calculated in 1977 that the traditional division between fermions and bosons would not apply to theoretical particles existing in two dimensions. Such particles would be expected to exhibit a diverse range of previously unexpected properties. They were given the name anyons by Frank Wilczek in 1982.
. The four light-grey regions are Au/Ti gates of undepleted electrons; the blue curves are the edge channels from the equipotentials of these undepleted electrons. The dark-grey curves are etched trenches depleted of electrons, the blue dots are the tunneling junctions, the yellow dots are Ohmic contacts. The electrons in the device are confined to a 2d plane.
Daniel Tsui and Horst Störmer discovered the fractional quantum Hall effect in 1982. The mathematics developed by Leinaas and Myrheim proved to be useful to Bertrand Halperin at Harvard University in explaining aspects of it. Frank Wilczek, Dan Arovas, and Robert Schrieffer verified this statement in 1985 with an explicit calculation that predicted that particles existing in these systems are in fact anyons.
In 2005 a group of physicists at Stony Brook University constructed a quasiparticle interferometer, detecting the patterns caused by interference of anyons, which were interpreted to suggest that anyons are real, rather than just a mathematical construct. However, these experiments remain controversial and are not fully accepted by the community.
In 2020, H. Bartolomei and co-authors from the École normale supérieure from an experiment in two-dimensional the heterostructure GaAs/AlGaAs was determined intermediate anyon statistics by electrical correlation measurements currents through the third contact in anyon collisions in electronic gas from two point contacts
With developments in semiconductor technology meaning that the deposition of thin two-dimensional layers is possible – for example, in sheets of graphene – the long-term potential to use the properties of anyons in electronics is being explored.

Non-abelian anyons

In 1988, Jürg Fröhlich showed that it was valid under the spin–statistics theorem for the particle exchange to be monoidal. In particular, this can be achieved when the system exhibits some degeneracy, so that multiple distinct states of the system have the same configuration of particles. Then an exchange of particles can contribute not just a phase change, but can send the system into a different state with the same particle configuration. Particle exchange then corresponds to a linear transformation on this subspace of degenerate states. When there is no degeneracy, this subspace is one-dimensional and so all such linear transformations commute. When there is degeneracy and this subspace has higher dimension, then these linear transformations need not commute.
Gregory Moore, Nicholas Read, and Xiao-Gang Wen pointed out that non-Abelian statistics can be realized in the fractional quantum Hall effect. While at first non-abelian anyons were generally considered a mathematical curiosity, physicists began pushing toward their discovery when Alexei Kitaev showed that non-abelian anyons could be used to construct a topological quantum computer. As of 2012, no experiment has conclusively demonstrated the existence of non-abelian anyons although promising hints are emerging in the study of the ν = 5/2 FQHE state. Experimental evidence of non-abelian anyons, although not yet conclusive and currently contested, was presented in October, 2013.

Fusion of anyons

In much the same way that two fermions can be looked at together as a composite boson, two or more anyons together make up a composite anyon. The composite anyon is said to be the result of the fusion of its components.
If identical abelian anyons each with individual statistics all fuse together, they together have statistics. This can be seen by noting that upon counterclockwise rotation of two composite anyons about each other, there are pairs of individual anyons that each contribute a phase. An analogous analysis applies to the fusion of non-identical abelian anyons. The statistics of the composite anyon is uniquely determined by the statistics of its components.
Non-abelian anyons have more complicated fusion relations. As a rule, in a system with non-abelian anyons, there is a composite particle whose statistics label is not uniquely determined by the statistics labels of its components, but rather exists as a quantum superposition. If the overall statistics of the fusion of all of several anyons is known, there is still ambiguity in the fusion of some subsets of those anyons, and each possibility is a unique quantum state. These multiple states provide a Hilbert space on which quantum computation can be done.

Topological basis

In more than two dimensions, the spin–statistics theorem states that any multiparticle state of indistinguishable particles has to obey either Bose–Einstein or Fermi–Dirac statistics. For any d > 2, the Lie groups SO and Poincaré have Z2 as their first homotopy group. Because the cyclic group Z2 is composed of two elements, only two possibilities remain.
The situation changes in two dimensions. Here the first homotopy group of SO, and also Poincaré, is Z. This means that Spin is not the universal cover: it is not simply connected. In detail, there are projective representations of the special orthogonal group SO which do not arise from linear representations of SO, or of its double cover, the spin group Spin. Anyons are evenly complementary representations of spin polarization by a charged particle.
This concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group SO has an infinite first homotopy group.
This fact is also related to the braid groups well known in knot theory. The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the symmetric group S2 but rather the braid group B2. The essential point is that one braid can wind around the other one, an operation that can be performed infinitely often, and clockwise as well as counterclockwise.
A very different approach to the stability-decoherence problem in quantum computing is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

Higher dimensional generalization of anyons

Fractionalized excitations as point particles can be bosons, fermions or anyons in 2+1 spacetime dimensions.
It is known that point particles can be only either bosons or fermions in 3+1 and higher spacetime dimensions.
However, the loop or membrane like excitations are extended objects can have fractionalized statistics.
Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the key signatures for identifying 3+1 dimensional topological orders.
The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular topological quantum field theories in 4 spacetime dimensions. Explained in a colloquial manner, the extended objects can be potentially anyonic in 3+1 and higher spacetime dimensions in the long-range entangled systems.