The possibility of non-abelian statistics
Statistics of the particles is related to the representation of braid group (1925). https://en.wikipedia.org/wiki/Braid_group. The 1-dimensional representations correspond to abelian (fractional) statistics (1977). https://en.wikipedia.org/wiki/Anyon The braid group also has higher dimensional representations, such as Burau representations (1936). https://en.wikipedia.org/wiki/Burau_representation . Those higher dimensional representations correspond to non-abelian statistics . However, not all higher dimensional representations are realizable by well defined physical systems. Thus, we should not use representation of braid group to define non-abelian statisics. (Question: can Burau representations be realized by physical systems of atoms or electrons?)
Non-Abelian statistics in condensed matter systems
In physics, abelian fractional statistics can be realized by fractional quantum Hall states (1984), based on Laughlin wave function. https://en.wikipedia.org/wiki/Fractional_quantum_Hall_effect. The realization of non-Abelian statistics in condensed matter systems ( https://en.wikipedia.org/wiki/Anyon#Non-abelian_anyons ) 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.
Moore-Read's approach is based on the conformal field theory construction, and Wen's approach is based on projective (slave-particle, see figure below) construction. The CFT construction realizes a Pfaffien state and the projective construction realizes a SU(2) level 2 state. In fact, the two states produce the same type of non-abelian statistics (an interesting coincidence, since the two constructions are totally different), and differ only by anelian statistics.
The realization of non-abelian statistics in condensed matter system is surprising since particle with non-abelian statistics must carry a fractional degrees of freedom (ie the quantum dimension is non-integer).
Projective (slave) particle construction is obtained by cutting an electron into two halves.
(this picture is adapted from one in Tony Zee's book)
What is non-abelian statistics
The key property of non-abelian is the emergence of topological degeneracy: 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 (say by trapping Hamiltonians), 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 approaches 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? ) Such topological degeneracies for various numbers of particles are described by fusion category.
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 rotate into each other. People usually refer such a non-Abelian geometric phase as non-Abelian statistics. But the appearance of quasi-particle-induced topological degeneracyis more important, and is the precondition that a non-Abelian geometric phase can even exist. The topological degeneracies for various numbers of particles plus the action of braidings are described by non-degenerate braided fusion category or modular tensor category.
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, where the flat connection of gauge fields 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.
Once we have quasi-particle-induced-topological-degeneracy, we have non-abelian statistics as a direct consequence.
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