The Haldane model [1] is a paradigmatic two-dimensional tight-binding model describing a topological insulator, characterized by the breaking of time-reversal symmetry due the presence of a magnetic field with vanishing flux across each unit cell. In its original formulation, the model is constructed by means of the so-called Peierls substitution (PS) [2], a widely employed approach that amounts to adding a phase factor proportional to the line integral of the vector gauge field to the ‘bare’ tunneling coefficients. In recent works [3], we have pointed out that the conditions for the applicability of the PS are explicitly violated in the Haldane model (and whenever the vector potential varies on the same scale of the underlying lattice). Nevertheless, the values of the tunneling coefficients can be obtained from simple closed expressions in terms of gauge invariant, measurable properties of the spectrum (namely, the gap at the Dirac point and the bandwidths), matching their ab-initio values (by means of the maximally localized Wannier functions [4]) with great accuracy. In any case, the general structure of its Hamiltonian is in fact preserved, as a direct consequence of the symmetries, but the accessibility of the topological phase diagram is dramatically suppressed.
Remarkably, the whole topological phase diagram of the Haldane model can be explored by using a different strategy, as recently demonstrated in a seminal experiment with ultracold atoms in a shaken optical lattice [5]. Motivated by this result, we have investigated the correspondence between the tight-binding Floquet Hamiltonian of a periodically modulated honeycomb lattice as in [5], and the original Haldane model. We find that – though the two systems share the same topological phase diagram – the corresponding Hamiltonians are not equivalent, the one of the shaken lattice presenting a much richer structure[6].

[1] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
[2] B. A. Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, 2013).
[3] J. Ibañez-Azpiroz, A. Eiguren, A. Bergara, G. Pettini, and M. Modugno, Phys. Rev. A 90, 033609 (2014); Phys. Rev. B
92, 195132 (2015); M. Modugno, J. Ibañez-Azpiroz, and G. Pettini, Sci. China-Phys. Mech. Astron. 59, 660001 (2016).
[4] N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997).
[5] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, 237 (2014).
[6] M. Modugno and G. Pettini, Phys. Rev. A 96, 053603 (2017).