Extended Bose-Hubbard model in a shaken optical lattice

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We study an extended Bose-Hubbard model with next-nearest-neighbor (NNN) hopping in a shaken optical lattice. We show how mean-field phase diagram evolves with the change of NNN hopping amplitude t₂, which can be easily tuned via shaking amplitude. As t₂ increases, a Z₂-symmetry-breaking superfluid (Z₂SF) phase emerges at the bottom of the Mott lobs. The tricritical points between normal superfluid, Z₂SF, and Mott insulator (MI) phases are identified. We further demonstrate the tricritical point can be tuned to the tip of the Mott lobe, in which case a new critical behavior has been predicted. Within random-phase approximation, excitation spectra in the three phases are obtained, which indicate how the phase transitions occur.