Breakdown of the Peierls substitution for the Haldane model with ultracold atoms

We present two independent approaches for calculating the tight-binding parameters of the Haldane model with ultracold atoms. The tunneling coefficients up to next-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-nearest tunneling amplitude t1 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 ; and (ii) for larger values, also |t1| and the nearest-neighbor tunneling t0 have a marked dependence on A . The origin of this behavior and its implications are discussed