  We generalize a proposal by Sorensen et al. [Phys. Rev. Lett. 94, 086803
(2005)] for creating an artificial magnetic field in a cold atom system on a
square optical lattice. This leads us to an effective lattice model with
tunable spatially periodic modulation of the artificial magnetic field and the
hopping amplitude. When there is an average flux of half a flux quantum per
plaquette the spectrum of low-energy excitations can be described by massless
Dirac fermions in which the usually doubly degenerate Dirac cones split into
cones with different "speeds of light" which can be tuned to give a single
Dirac cone and a flat band. These gapless birefringent Dirac fermions arise
because of broken chiral symmetry in the kinetic energy term of the effective
low energy Hamiltonian. We characterize the effects of various perturbations to
the low-energy spectrum, including staggered potentials, interactions, and
domain wall topological defects.
