  The recent theoretical discovery of fractional Chern insulators (FCIs) has
provided an important new way to realize topologically ordered states in
lattice models. In earlier works, on-site and nearest neighbor Hubbard-like
interactions have been used extensively to stabilize Abelian FCIs in systems
with nearly flat, topologically nontrivial bands. However, attempts to use
two-body interactions to stabilize non-Abelian FCIs, where the ground state in
the presence of impurities can be massively degenerate and manipulated through
anyon braiding, have proven very difficult in uniform lattice systems. Here, we
study the remarkable effect of long-range interactions in a lattice model that
possesses an exactly flat lowest band with a unit Chern number. When spinless
bosons with two-body long-range interactions partially fill the lowest Chern
band, we find convincing evidence of gapped, bosonic Read-Rezayi (RR) phases
with non-Abelian anyon statistics. We characterize these states through
studying topological degeneracies, the overlap between the ground states of
two-body interactions and the exact RR ground states of three- and four-body
interactions, and state counting in the particle-cut entanglement spectrum.
Moreover, we demonstrate how an approximate lattice form of Haldane's
pseudopotentials, analogous to that in the continuum, can be used as an
efficient guiding principle in the search for lattice models with stable
non-Abelian phases.
