  We report the theoretical discovery of a large class of 2D tight-binding
models containing nearly-flat bands with nonzero Chern numbers. In contrast
with previous studies, where nonlocal hoppings are usually required, the
Hamiltonians of our models only require short-range hopping and have the
potential to be realized in cold atomic gases. Due to the similarity with 2D
continuum Landau levels, these topologically nontrivial nearly-flat bands may
lead to the realization of fractional anomalous quantum Hall states and
fractional topological insulators in real materials. Among the models we
discover, the most interesting and practical one is a square-lattice three-band
model which has only nearest-neighbor hopping. To understand better the physics
underlying the topological flat band aspects, we also present the studies of a
minimal two-band model on the checkerboard lattice.
