We present two independent calculations of the tight-binding parameters for a specific realization of the Haldane model with ultracold atoms. The tunneling coefficients up to next-to-nearest neighbors are computed ab-initio by using the maximally loc
alized Wannier functions, and compared to analytical expressions written in terms of gauge invariant, measurable properties of the spectrum. The two approaches present a remarkable agreement and evidence the breakdown of the Peierls substitution: (i) the phase acquired by the next-to-nearest tunneling amplitude $t_{1}$ presents quantitative and qualitative differences with respect to that obtained by the integral of the vector field A, as considered in the Peierls substitution, even in the regime of low amplitudes of A; (ii) for larger values, also $|t_{1}|$ and the nearest-neighbor tunneling $t_{0}$ have a marked dependence on A. The origin of this behavior and its implications are discussed.
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 suit
able 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.
We present an accurate ab initio tight-binding model, capable of describing the dynamics of Dirac points in tunable honeycomb optical lattices following a recent experimental realization [L. Tarruell et al., Nature 483, 302 (2012)]. Our scheme is bas
ed on first-principle maximally localized Wannier functions for composite bands. The tunneling coefficients are calculated for different lattice configurations, and the spectrum properties are well reproduced with high accuracy. In particular, we show which tight binding description is needed in order to accurately reproduce the position of Dirac points and the dispersion law close to their merging, for different laser intensities.
