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In this paper, we study Bose-Hubbard models on the square and honeycomblattices with complex hopping amplitudes, which are feasible by recentexperiments of cold atomic gases in optical lattices. To clarify phasediagrams, we use an extended quantum Monte-Carlo simulations (eQMC). For thesystem on the square lattice, the complex hopping is realized by an artificialmagnetic field. We found that vortex-solid states form for certain set ofmagnetic 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 thehoneycomb lattice, we add the next-nearest neighbor complex hopping. The modelis a bosonic analog of the Haldane-Hubbard model. By means of the eQMC, westudy the model with both weak and strong on-site repulsions. Numerical studyshows that the model has a rich phase diagram. We also found that in the systemdefined on the honeycomb lattice of the cylinder geometry, an interesting edgestate appears.

Phase diagrams of Bose-Hubbard model and Haldane-Bose-Hubbard model with complex hopping amplitudes