Synthetic non-Abelian anyons in fractional Chern insulators and beyond

Abstract

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.