Revealing discontinuous and continuous quantum phase transitions in shaken optical lattices

We perform the cluster mean-field with density matrix renormalization (CMFT+DMRG) calculation on the two-band extended Bose-Hubbard model to uncover the physics behind the discontinuous transitions observed in the one-dimensional shaken optical lattice with hybridized two lowest Bloch bands. We determine the superfluid, Mott insulator, and staggered superfluid phases associated with this model using the appropriate order parameters. We obtained the phase diagrams for two values of shaking amplitudes and illustrated the phases and nature of phase transitions. The transition from Mott insulator and superfluid to staggered superfluid is discontinuous for small shaking amplitudes. We analyse how the shaking frequency controls the effective chemical potential in the model and, consequently, the ground state energy that drives the discontinuous transition. Finally, we compare our results with those of earlier works.