Tight binding models for ultracold atoms in honeycomb optical lattices
  We discuss how to construct tight-binding models for ultra cold atoms in honeycomb potentials, by means of the maximally localized Wannier functions (MLWFs) for composite bands introduced by Marzari and Vanderbilt [1]. In particular, we work out the model with up to third-nearest neighbors, and provide explicit calculations of the MLWFs and of the tunneling coefficients for the graphene-lyke potential with two degenerate minima per unit cell. Finally, we discuss the degree of accuracy in reproducing the exact Bloch spectrum of different tight-binding approximations, in a range of typical experimental parameters. 
Julen Iba\~nez-Azpiroz, Asier Eiguren, Aitor Bergara, Giulio Pettini,   and Michele Modugno
cond-mat.quant-gas hep-ph