Synthetic non-Abelian anyons in fractional Chern insulators and beyond

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An exciting new prospect in condensed matter physics is
the possibility of realizing fractional quantum Hall
states in simple lattice models without a large external magnetic
field, which are called fractional Chern insulators. A fundamental question is whether qualitatively new
states can be realized on the lattice as compared with ordinary
fractional quantum Hall states. Here we propose new symmetry-enriched topological
states, topological nematic states, which are a dramatic consequence of
the interplay between the lattice translational symmetry and
topological properties of these fractional Chern insulators. The
topological nematic states are realized in a partially filled flat band with a
Chern number N, which can be mapped to an N-layer quantum Hall
system on a regular lattice. However, in the topological nematic states
the lattice dislocations become non-Abelian defects which create
"worm holes" connecting the effective layers, and effectively change
the topology of the space. Such topology-changing defects, which
we name as "genons", can also be defined in other physical systems. We develop methods to compute the projective
non-abelian braiding statistics of the genons, and we find the braiding
is given by  adiabatic modular transformations, or Dehn twists,
of the topological state on the effective genus g surface. We find
situations where the

> genons have quantum dimension 2 and can be used for
universal topological quantum computing (TQC), while the host
topological state is by itself non-universal for TQC.