  Optical lattice systems offer the possibility of creating and tuning Dirac
points which are present in the tight-binding lattice dispersions. For example,
such a behavior can be achieved in the staggered flux lattice or honeycomb type
of lattices. Here we focus on the strongly correlated bosonic dynamics in the
vicinity of Dirac points. In particular, we investigate bosonic Mott insulator
phase in which quasiparticle excitations have a simple particle-hole
interpretation. We show that linear response dynamics around Dirac points, can
be significantly engineered at least in two ways: by the type of external
perturbation or by changing the lattice properties. The key role is played by
the interband transitions. Moreover, we explain that the behavior of these
transitions is directly connected to different energy scales of the effective
hopping amplitudes for particles and holes. Presented in this work theoretical
study about tunability of linear response dynamics near the Dirac points can be
directly simulated in the optical lattice systems.
