# How Non-abelian anyons arise in solid-state systems?

Recently it has been studied non-abelian anyons in some solid-state systems. These states are being studied for the creation and manipulation of qubits in quantum computing.

But, how these non-abelian anyons can arise if all particles involved are fermions?

In other words, how electronic states can have different statistics than the fermionic if all electrons are fermions?

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The realization of non-Abelian statistics in condensed matter systems was first proposed in the following two papers. G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991) X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991)

Zhenghan Wang and I wrote a review article to explain FQH state (include non-Abelian FQH state) to mathematicians, which include the explanations of some basic but important concepts, such as gapped state, phase of matter, universality, etc. It also explains topological quasiparticle, quantum dimension, non-Abelian statistics, topological order etc.

The key point is the following: consider a non-Abelian FQH state that contains quasi-particles (which are topological defects in the FQH state), even when all the positions of the quasi-particles are fixed, the FQH state still has nearly degenerate ground states. The energy splitting between those nearly degenerate ground states approaches zero as the quasi-particle separation approach infinity. The degeneracy is topological as there is no local perturbation near or away from the quasi-particles that can lift the degeneracy. The appearance of such quasi-particle induced topological degeneracy is the key for non-Abelian statistics. (for more details, see direct sum of anyons? )

When there is the quasi-particle induced topological degeneracy, as we exchange the quasi-particles, a non-Abelian geometric phase will be induced which describes how those topologically degenerate ground states rotated into each other. People usually refer such a non-Abelian geometric phase as non-Abelian statistics. But the appearance of quasi-particle induced topological degeneracy is more important, and is the precondition that a non-Abelian geometric phase can even exist.

How quasi-particle-induced-topological-degeneracy arise in solid-state systems? To make a long story short, in "X.-G. Wen, Phys. Rev. Lett. 66, 802 (1991)", a particular FQH state was constructed whose low energy effective theory is the non-Abelian Chern-Simons theory, which leads to quasi-particle-induced-topological-degeneracy and non-Abelian statistics. In "G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)", the FQH wave function is constructed as a correlation in a CFT. The conformal blocks correspond to quasi-particle-induced-topological-degeneracy.

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The point is that the anyons are not electronic states at all. As you've rightly noted, the electrons are fermions, and there's nothing that's going to make them forget that, but very rarely are condensed matter systems so simple that electrons are appropriate degrees of freedom to work with. Instead, the fractional quantum Hall states conjectured to give rise to non-abelian anyonic degrees of freedom are complex many-body states, and the anyons are emergent, rather than fundamental degrees of freedom, arising from the collective behavior of the electrons. For a more transparent example of this phenomena, look up spin-charge separation in 1D electronic systems.

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The dichotomy of bosonic and fermionic behaviour essentially arises because of the nature of the rotation group in dimensions greater than 2+1.

The exchange of 2 particles (which determines the statistics) introduces the same phase that you get when you rotate a particle by 2pi - this really comes from the spin-statistics theorem that tells you that exchanges (that determine statistics) and rotations (that generate spin) are in some sense identical. So if we can understand how rotations differ in 2D space and 3D space, we can see qualitatively how the statistics in 2D and 3D should differ too.

I will not go into the details but you can see in Sternberg's excellent book the topological differences between the rotation groups SO(2) and SO(3) in 2D and 3D space respectively that lead to the restriction of phase changes in 3D to +1 or -1, but produce no such restriction in 2D space.

So planar systems can produce "anyonic" statistics (that interpolate between fermionic and bosonic). The answer to your question of how fermions can display this behaviour is (as pointed out in another answer) that the anyonic states are collective excitations of a large number of fermions.

For example, in a planar Quantum Hall system, the wavefunctions of all the electrons in the system rearrange and overlap and conspire to produce an overall collective wavefunction (best approximated by the Laughlin wavefunction) for quasiparticles and quasiholes. These wavefunctions have a form that produces an Aharanov-Bohm phase change on exchanging two quasiparticles/holes (taking one around another "half times") that differs from +1 or -1. So these are systems whose quasiparticle excitations (which are collective "emergent" excitations of the underlying fermionic electrons) produce abelian anyonic statistics. Similar Quantum Hall systems with degenerate ground states produce non-abelian anyonic statistics because you no longer have just one phase but a whole matrix of phases transforming your degenerate subspace as it picks up statistical phases on exchange.

And I suppose similar things happen in other condensed matter systems were anyons are expected, but I wouldn't know the details.

To summarize: you need 1) a planar (2D) system to exploit the topological subtleties of the SO(2) rotation group 2) a large number of fermions who conspire to have collective/emergent anyonic wavefunctions, like in QHall systems 3) and a degenerate ground state if you want non-abelian and not just abelian statistics

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You started with it, but you didn't restate it in your conclusion, so I insist on it in this comment: You can have anyons in 1D systems as well (since it has dimension smaller than 2+1 (+1 for time)). The best example is the anyonic statistic of Majorana fermions in topological wire. Thanks for this good answer. – FraSchelle Mar 30 '13 at 20:08