  Two-dimensional lattice models subjected to an external effective magnetic
field can form nontrivial band topologies characterized by nonzero integer band
Chern numbers. In this Letter, we investigate such a lattice model originating
from the Hofstadter model and demonstrate that the band topology transitions
can be realized by simply introducing tunable longer-range hopping. The rich
phase diagram of band Chern numbers is obtained for the simple rational flux
density and a classification of phases is presented. In the presence of
interactions, the existence of fractional quantum Hall states in both |C|=1 and
|C|>1 bands is confirmed, which can reflect the band topologies in different
phases. In contrast, when our model reduces to a one-dimensional lattice, the
ground states are crucially different from fractional quantum Hall states. Our
results may provide insights into the study of new fractional quantum Hall
states and experimental realizations of various topological phases in optical
lattices.
