Simulating the Haldane model with ultracold atoms: different perspectives & unexpected results*

The Haldane model is a paradigmatic two-dimensional tight-binding model describing a topological insulator, characterized by the breaking of the time-reversal symmetry due the presence of a microscopic magnetic field with vanishing flux across the unit cell. Recently, its phase diagram has been experimentally realized by means of ultracold atoms in a shaken optical lattice [Jotzu et al., Nature 515, 237 (2014)]. In its original formulation the Haldane model was constructed by using the so-called Peierls substitution (PS), a widely employed approach that consists in adding a phase factor proportional to the line integral of the vector gauge field to the ‘bare’ tunneling coefficients. However, the conditions of applicability of the PS are explicitly violated in the Haldane model, as in many other cases considered in the literature, that I will review in this talk from an historical perspective. Despite the failure of the PS, we have shown that the general structure of the Haldane Hamiltonian is protected by the symmetries of the system and that the values of the tunneling coefficients can be obtained from simple closed expressions in terms of gauge invariant, measurable properties of the spectrum. Indeed, they match with great accuracy the ab-initio values obtained by means of the maximally localized Wannier functions. We have also investigated the correspondence between the tight-binding Floquet Hamiltonian of a periodically modulated honeycomb lattice and the original Haldane model. Remarkably, 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.