Simulating Dirac fermions with Abelian and non-Abelian gauge fields in optical lattices

In this work we present an optical lattice setup to realize a full Dirac Hamiltonian in 2+1 dimensions. We show how all possible external potentials coupled to the Dirac field can arise from perturbations of the existing couplings of the honeycomb lattice model, without the need of additional laser fields. This greatly simplifies the proposed implementations, requiring only spatial modulations of the intensity of the laser beams. We finally suggest several experiments to observe the properties of the Dirac field in the setup.