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