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Two-dimensional lattice models subjected to an external effective magneticfield can form nontrivial band topologies characterized by nonzero integer bandChern numbers. In this Letter, we investigate such a lattice model originatingfrom the Hofstadter model and demonstrate that the band topology transitionscan be realized by simply introducing tunable longer-range hopping. The richphase diagram of band Chern numbers is obtained for the simple rational fluxdensity and a classification of phases is presented. In the presence ofinteractions, the existence of fractional quantum Hall states in both |C|=1 and|C|>1 bands is confirmed, which can reflect the band topologies in differentphases. In contrast, when our model reduces to a one-dimensional lattice, theground states are crucially different from fractional quantum Hall states. Ourresults may provide insights into the study of new fractional quantum Hallstates and experimental realizations of various topological phases in opticallattices.

Tunable Band Topology Reflected by Fractional Quantum Hall States in Two-Dimensional Lattices