Birefringent break up of Dirac fermions in a square optical lattice

We introduce a model of spinless fermions on a square lattice in a spatially periodic magnetic field. This model has Dirac points in its spectrum when there is an average flux of half a flux quantum per plaquette. The dispersion in the vicinity of these Dirac points has the unusual feature that the double degeneracy of Dirac cones is broken. This corresponds to a situation in which the low energy excitations have two different ``speeds of light.'' This is a consequence of broken chiral symmetry in the model, which occurs in the kinetic energy term, and hence leaves the spectrum gapless in the vicinity of the Dirac points. This chiral symmetry breaking is fundamentally different from spontaneous chiral symmetry breaking that leads to mass generation in field theoretic models. We investigate the effects of several perturbations on the spectrum such as staggered potentials, nearest neighbor interactions, and domain wall topological defects. We provide a physical setting in which this model might be realized, namely for fermions in an optical lattice in an artificial magnetic field.