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.