  We present an ab initio analysis of a continuous Hamiltonian that maps into
the celebrated Haldane model. The tunnelling coefficients of the tight-binding
model are computed by means of two independent methods - one based on the
maximally localized Wannier functions, the other through analytic expressions
in terms of gauge-invariant properties of the spectrum - that provide a
remarkable agreement and allow to accurately reproduce the exact spectrum of
the continuous Hamiltonian. By combining these results with the numerical
calculation of the Chern number, we are able to draw the phase diagram in terms
of the physical parameters of the microscopic model. Remarkably, we find that
only a small fraction of the original phase diagram of the Haldane model can be
accessed, and that the topological insulator phase is suppressed in the deep
tight-binding regime.
