  In this paper, we study Bose-Hubbard models on the square and honeycomb
lattices with complex hopping amplitudes, which are feasible by recent
experiments of cold atomic gases in optical lattices. To clarify phase
diagrams, we use an extended quantum Monte-Carlo simulations (eQMC). For the
system on the square lattice, the complex hopping is realized by an artificial
magnetic field. We found that vortex-solid states form for certain set of
magnetic field, i.e., the magnetic field with the flux quanta per plaquette
$f=p/q$, where $p$ and $q$ are co-prime natural numbers. For the system on the
honeycomb lattice, we add the next-nearest neighbor complex hopping. The model
is a bosonic analog of the Haldane-Hubbard model. By means of the eQMC, we
study the model with both weak and strong on-site repulsions. Numerical study
shows that the model has a rich phase diagram. We also found that in the system
defined on the honeycomb lattice of the cylinder geometry, an interesting edge
state appears.
