statistical mechanics - 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?

Answer

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 $$ \Psi(z_i) = [\chi_k(z_i)]^n $$ was constructed, where $\chi_k(z_i)$ is the IQH wave function with $k$ filled Landau levels. Such a state has a low energy effective theory which is the $SU(n)$ level $k$ non-Abelian Chern-Simons theory. When $k >1,\ n>1$, it 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.