## Energy spectrum E=E(k_{y^{prime }}) for λ = 0.05*t*: (a) α = 1, (b) α = 0 for a cylindrical geometry aligned along *x*'

**Figure 3.** Energy spectrum E=E(k_{y^{\prime }}) for λ = 0.05*t*: (a) α = 1, (b) α = 0 for a cylindrical geometry aligned along *x*'. This spectrum shows the projected bulk bands E_{\pm } (\boldsymbol{k}) \rightarrow E_{\pm } (k_{y^{\prime }}) and also reveals the presence of topological edge states inside the bulk gap [4]. In the standard case α = 1, the topological bulk gap is \Delta =6 \sqrt{3} \lambda \approx 0.5. In the absence of the diagonal matrix hopping, i.e. α = 0, the gap is \Delta = 4 \sqrt{3} \lambda \approx 0.34 but survives. The energy unit is given by the tunnelling amplitude *J*.

**Abstract**

We describe a scheme to engineer non-Abelian gauge potentials on a square optical lattice using laser-induced transitions. We emphasize the case of two-electron atoms, where the electronic ground state *g* is laser-coupled to a metastable state *e* within a state-dependent optical lattice. In this scheme, the alternating pattern of lattice sites hosting *g* and *e* states depicts a chequerboard structure, allowing for laser-assisted tunnelling along both spatial directions. In this configuration, the nuclear spin of the atoms can be viewed as a 'flavour' quantum number undergoing non-Abelian tunnelling along nearest-neighbour links. We show that this technique can be useful to simulate the equivalent of the Haldane quantum Hall model using cold atoms trapped in square optical lattices, offering an interesting route to realize Chern insulators. The emblematic Haldane model is particularly suited to investigate the physics of topological insulators, but requires, in its original form, complex hopping terms beyond nearest-neighbouring sites. In general, this drawback inhibits a direct realization with cold atoms, using standard laser-induced tunnelling techniques. We demonstrate that a simple mapping allows us to express this model in terms of *matrix* hopping operators that are defined on a standard square lattice. This mapping is investigated for two models that lead to anomalous quantum Hall phases. We discuss the practical implementation of such models, exploiting laser-induced tunnelling methods applied to the chequerboard optical lattice.