(3+1) massive Dirac fermions with ultracold atoms in frustrated cubic optical lattices

We propose the experimental realization of (3+1) relativistic Dirac fermions using ultracold atoms in a cubic optical lattice in a frustrating magnetic field which can be realized by rotating the lattice or, alternatively, using a synthetic gauge field. We show that it is possible to give mass to the Dirac fermions by coupling the ultracold atoms to a Bragg pulse: the method relies on the peculiar position of the Dirac points in the (magnetic) Brillouin zone, and it would not generally work for other lattices (e.g., for honeycomb lattices). A dimensional crossover from (3+1) to (2+1) Dirac fermions can be obtained by varying the anisotropy of the lattice. Finally, we also discuss under which conditions the interatomic potentials give rise to relativistically invariant interactions among the Dirac fermions.