  We review the derivation of the effective Dirac equation for ultracold atoms
in one-dimensional bichromatic optical lattices, following the proposal by
Witthaut et al. Phys. Rev. A 84, 033601 (2011). We discuss how such a
derivation - based on a suitable rotation of the Bloch basis and on a coarse
graining approximation - is affected by the choice of the Wannier functions
entering the coarsening procedure. We show that in general the Wannier
functions obtained by rotating the maximally localized Wannier functions for
the original Bloch bands can be sufficiently localized for justifying the
coarse graining approximation. We also comment on the relation between the
rotation needed to achieve the Dirac form and the standard Foldy-Wouthuysen
transformation. Our results provide a solid ground for the interpretation of
the experimental results by Salger et al. Phys. Rev. Lett. 107, 240401 (2011)
in terms of an effective Dirac dynamics.
