  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 localized 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.
