## (a) The π-flux model on the square lattice

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**Figure 5.** (a) The π-flux model on the square lattice. The unit cells of the π-flux lattice, with inequivalent sites *A*, *B*, are labelled by the coordinates (m,n) \in \mathbb {Z}. Two inequivalent sites *A*, *B* belonging to the same unit cell are connected by a full black line. Standard NN hoppings are characterized by the tunnelling factors ±*J*. The complex NNN hoppings, with the tunnelling factor +iλ, are represented by purple dotted arrows (the hoppings with the opposite factor, −iλ, correspond to the reversed paths). The chirality introduced by the NNN terms potentially results in anomalous QH phases. (b) The same π-flux model translated into a non-Abelian square lattice, with matrix hopping operators U_{x^{\prime },y^{\prime }}. The 'undesired' diagonal hoppings \hat{D}_{1,2} are depicted by red dotted arrows and disappear in the limit α = 0 (cf the text). Note that when α = 0, this model reduces to the non-Abelian optical lattice illustrated in figure 1(c). The modified π-flux model, corresponding to α = 0, is represented in the appendix.

**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.