NonAbelian anyons are hypothesized particles that, if found, could form the basis of a faulttolerant quantum computer. The theoretical finding that they may turn up in three dimensions comes as a surprise.
In an article published in Physical Review Letters, Teo and Kane describe^{1} a theoretical model for systems in which an exotic state of matter, termed a topological insulator, coexists with superconductivity. They found, surprisingly, that the model predicts hypothesized particles called nonAbelian anyons, and allows these particles to move around in three dimensions, thus breaking free of the bonds that were thought to chain them down to two dimensions.
To appreciate how radical this is, it is useful to recall what nonAbelian anyons are and why they are so interesting. Electrons and photons can be viewed as pointlike excitations of the vacuum, or 'empty' space. When space is not empty but instead filled with matter — like that which we encounter in everyday life — the excitations can be quite different. The most surprising and, perhaps, important discovery in physics of the past 30 years is the existence of fractional quantum Hall states: under certain circumstances, the pointlike excitations of a material may behave as though they are a fraction of an electron^{2}. These excitations, called quasiparticles, are seen in extremely pure gallium–arsenide semiconductor devices in which the electrons are confined to move in a twodimensional plane at temperatures close to absolute zero (below 1 kelvin) and in high magnetic fields (about 10 tesla, which is 3–10 times stronger than the magnetic field in a clinical magnetic resonance imaging machine). Their electrical charge is onethird that of an electron^{3}. Even more shocking, these excitations are anyons^{4} — neither bosons (such as photons) nor fermions (such as electrons).
Bosons obey Bose–Einstein statistics: when two identical bosons are exchanged, the quantummechanical wavefunction of the system is unchanged. Fermions obey Fermi–Dirac statistics: when two fermions are swapped, the wavefunction changes sign. In both cases, it is unimportant whether the exchange is clockwise or anticlockwise. Abelian anyons are particles that, when swapped, cause the wavefunction to be multiplied by a complex number e^{iθ} , if they are exchanged in an anticlockwise fashion, and by e^{−iθ} if the exchange is clockwise^{4}; θ is a parameter that varies from one type of anyon to another, and is 0 for bosons and π for fermions; i is the imaginary unit of complex numbers. In three dimensions, values of θ other than 0 and π are impossible because an anticlockwise exchange can be continuously deformed into a clockwise exchange. Thus, anyons can occur only in situations in which they are confined to move in a twodimensional plane. Quasiparticles such as those that emerge in the fractional quantum Hall state in gallium–arsenide semiconductors are anyons with θ = π/3. A phase of matter supporting such quasiparticles is called a topological phase of matter or a topologically ordered phase^{5}.
NonAbelian anyons^{6} are an exotic variant of anyons. In an ensemble of n such particles, there are many degenerate states (that is, states with the same energy), even when the positions of the particles are fixed. The degenerate states form a subspace of the space of all possible states of the system; the degenerate subspace has a dimension that is exponentially large in n. When two particles are swapped, the system can transform from one state into another in the degenerate subspace.
One particular type of nonAbelian anyon, called an Ising anyon, has an nparticle degenerate subspace of dimension 2^{[n/2]−1}, where [n/2] is the greatest integer less than or equal to n/2. Ising anyons have been conjectured^{6} to exist in one of the fractional quantum Hall states, the '5/2 state', and experimental evidence has mounted in favour of the conjecture^{7,8,9}. This has caused great excitement in the physics community. NonAbelian anyons, if found, would not only be a type of particle never before seen in nature, but could also become the basis for a faulttolerant quantum computer^{6,10}, a machine that could solve problems beyond the reach of today's computers. The transformations implemented by exchanging quasiparticles would be the logic 'gates' of such a computer, and the irrelevance of small deviations in the exchange route leads to builtin fault tolerance.
Previously, Fu and Kane proposed^{11} a device in which a thin superconducting film is grown on top of a special kind of insulating material called a threedimensional topological insulator^{12,13}. Despite sharing the word topological, a topological insulator is quite distinct from a topological phase. A topological insulator is a material whose bulk excitations are ordinary electrons, not anyons, but whose surface excitations are massless electrons. Fu and Kane showed^{11} that at the interface between a superconductor and a topological insulator, a topological phase of matter forms that supports Ising anyons. This^{11} and other theoretical work^{14,15,16} emphasizes the point that nonAbelian anyons are not intrinsically limited to high magnetic fields, ultralow temperatures and ultrapure materials, but could occur, in principle, anywhere in nature. What's more, the proposed interfaces are technologically feasible, and experimental efforts are under way to try to realize them.
One puzzling feature of Fu and Kane's proposal is that it relies on a special property of a threedimensional topological insulator, the existence of a single branch of massless electron states that occurs at a topological insulator's surface. There is no way for a purely twodimensional system to have such states. Thus, their model is threedimensional in an essential way. But one would expect a system supporting anyons to be twodimensional. There is some tension between these two requirements, and it comes to the fore when we consider situations in which the superconductivity is not confined to a thin layer at the interface, as Teo and Kane consider in their study^{1}.
Suppose that the superconductivity occurs in a threedimensional region. For instance, if the topological insulator is doped with copper^{17}, superconductivity can occur in the bulk of the material. Alternatively, the superconducting film could be thickened into a slab. Further suppose that we can dynamically change the geometry of the superconducting region by making the superconductivity penetrate deeper into the topological insulator or retreat farther away. Fu and Kane's nonAbelian anyons can thereby be moved around in three dimensions. But what happens to their nonAbelian anyonic properties?
To answer this question, Teo and Kane^{1} constructed a theoretical model that generalizes Fu and Kane's setup. In this model, excitations can move around in three dimensions and yet, when there are n such excitations, there are once again many degenerate states that form a 2^{[n/2]−1}dimensional subspace of the space of states of the system, and the effect of swapping excitations is the same as in the twodimensional Isinganyon case. This is a surprising result, because there can be no pointlike excitations that are anyons, let alone nonAbelian anyons, in three dimensions.
The resolution of this apparent paradox is that Teo and Kane's excitations are not pointlike. Therefore, the quasiparticles and their exchange routes alone are not a complete description of their model, as they would be for anyons in two dimensions. Instead, one must consider the quasiparticles' surroundings as well, such as the strength of the superconductivity in the region between the quasiparticles. As Teo and Kane's 'quasiparticles' are exchanged, they 'drag' their surroundings around with them. Thus, the set of operations characterizing their model is somewhat richer than simply the set of exchanges. For instance, Teo and Kane found some operations, which they call 'braidless operations', in which the quasiparticles are not moved at all; only their surroundings are changed. However, even accounting for the quasiparticles' surroundings is not enough to explain how nonAbelian anyons can arise.
In fact, the transformations generated by exchanges of Teo and Kane's quasiparticles have the following property (M. H. Freedman et al., personal communication), which mathematicians call a projective representation: even though the braidless operations are physically equivalent regardless of the order in which they are done, their action on the system's wavefunction may differ by a minus sign, depending on the order. It is only through quantummechanical 'projection' that threedimensional excitations — even those that are not pointlike — can be an incarnation or avatar of twodimensional nonAbelian anyons. Once again, quantum mechanics is making a mockery of our classical intuition and showing that any bizarre possibility that is mathematically allowed can and, apparently, will occur.
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Nayak, C. Another dimension for anyons. Nature 464, 693–694 (2010). https://doi.org/10.1038/464693a
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