# (a) The Haldane model on the honeycomb lattice

**Figure 1.** (a) The Haldane model on the honeycomb lattice. The unit cells of the honeycomb 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 hopping, with amplitude *J* and denoted by dotted lines, takes place between the *A* (red) and *B* (blue) sites. The complex NNN hoppings, introduced to open a QH gap, are represented by thick blue and red lines inside one honeycomb cell. The NNN tunnelling factors are +iλ according to the orientation designated by the circular arrow, and −iλ otherwise, i.e., they introduce a chirality in the system. (b) The same Haldane model translated into a non-Abelian square lattice, with matrix hopping operators U_{x^{\prime },y^{\prime }}. The 'undesired' diagonal hoppings \hat{D} are depicted by red dotted arrows and disappear in the limit α = 0 (see the text). Note that when α = 0, this model reduces to the non-Abelian optical lattice illustrated in figure 2(c). The modified Haldane 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.