  Inspired by the recent creation of the honeycomb optical lattice and the
realization of the Mott insulating state in a square lattice by shaking, we
study here the shaken honeycomb optical lattice. For a periodic shaking of the
lattice, a Floquet theory may be applied to derive a time-independent
Hamiltonian. In this effective description, the hopping parameters are
renormalized by a Bessel function, which depends on the shaking direction,
amplitude and frequency. Consequently, the hopping parameters can vanish and
even change sign, in an anisotropic manner, thus yielding different band
structures. Here, we study the merging and the alignment of Dirac points and
dimensional crossovers from the two dimensional system to one dimensional
chains and zero dimensional dimers. We also consider next-nearest-neighbor
hopping, which breaks the particle-hole symmetry and leads to a metallic phase
when it becomes dominant over the nearest-neighbor hopping. Furthermore, we
include weak repulsive on-site interactions and find the density profiles for
different values of the hopping parameters and interactions, both in a
homogeneous system and in the presence of a trapping potential. Our results may
be experimentally observed by using momentum-resolved Raman spectroscopy.
