  We study bosonic atoms in optical honeycomb lattices with anisotropic
tunneling and find dimerized Mott insulator phases with fractional filling.
These incompressible insulating phases are characterized by an
interaction-driven localization of particles in respect to the individual
dimers and large local particle-number fluctuations within the dimers. We
calculate the ground-state phase diagrams and the excitation spectra using an
accurate cluster mean-field method. The cluster treatment enables us to probe
the fundamental excitations of the dimerized Mott insulator where the
excitation gap is dominated by the intra-dimer tunneling amplitude. This allows
the distinction from normal Mott insulating phases gapped by the on-site
interaction. In addition, we present analytical results for the phase diagram
derived by a higher-order strong-coupling perturbative expansion approach. By
computing finite lattices with large diameters the influence of a harmonic
confinement is discussed in detail. It is shown that a large fraction of atoms
forms the dimerized Mott insulator under experimental conditions. The necessary
anisotropic tunneling can be realized either by periodic driving of the optical
lattice or by engineering directly a dimerized lattice potential. The dimers
can be mapped to to their antisymmetric states creating a lattice with coupled
p-orbitals.
