## (a) Sketch of the spin-dependent trapping potential for atoms with nuclear spin 1/2

**Figure 2.** (a) Sketch of the spin-dependent trapping potential for atoms with nuclear spin 1/2. Atoms in the electronic state *g*, shown as green circles, (resp. *e*, shown as dark red circles) are trapped at the potential minima (resp. maxima) independent of their nuclear spin. This results in a spatial arrangement depicting a chequerboard pattern. A laser resonant on the *g*–*e* transition can induce tunnelling along the diagonals {\bf e}_x^{\prime }, {\bf e}_y^{\prime }. We show for illustration a closed trajectory around a unit cell of the chequerboard lattice. (b) Possible realization using ^{171}Yb atoms with nuclear spin 1/2 in both *g* and *e* manifolds. An applied magnetic field shifts the various transitions between internal states depending on the value of the nuclear spin, allowing independent addressing of each of them (for instance, addressing the π_{1, 2} transitions independently from the σ^{±} transitions). (c) The resulting 'non-Abelian' optical lattice, with *U*(2) hopping operators \hat{U}_{x^{\prime },y^{\prime }} \not\propto \hat{1}_{2 \times 2} acting along nearest-neighbouring sites. Note that we use the notation (*m*, *n*) (resp. (\mathsf {m}, \mathsf {n})) to designate the lattice sites in the *x*'–*y*' (resp. *x*–*y*) axis system.

**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.