  We consider a one-dimensional Bose-Hubbard model (BHM) with on-site
double-well potentials and study the effect of nearest-neighbor repulsion and
cavity-mediated long-range interactions by calculating the ground-state phase
diagrams with quantum Monte-Carlo simulations. We show that when the intra-well
repulsion is as strong as the on-site repulsion a dimerized Mott insulator
phase appears at the tip of the dimerized Density Wave phase for a density of
one particle per double well. Furthermore, we find a dimerized Haldane
insulator phase in the double-well BHM with nearest-neighbor interaction, which
is identical to a dimerized BHM with repulsive interactions up to the third
neighbor.

Quantum Gases;Statistical Mechanics