Extended Bose-Hubbard Model with dipolar and contact interactions

We study the phase diagram of the one-dimensional boson gas trapped inside an optical lattice with contact and dipolar interaction taking into account next-nearest terms for both tunneling and interaction. Using the density matrix renormalization group, we calculate how the locations of phase transitions change with increasing dipolar interaction strength for average density $\rho = 1$. Furthermore, we show an emergence of pair-correlated phases for a large dipolar interaction strength and $\rho \geq 2$, including a supersolid phase with an incommensurate density wave ordering manifesting the corresponding spontaneous breaking of the translational symmetry.