Ultra-cold atoms in optical lattices can nearly ideally realize the simple Hamiltonians that model the behaviour of real condensed-matter systems, but with full control of parameters. Here, we show that Raman coupled ultra-cold fermions in a two-dimensional square optical lattice can have a different behaviour depending on their position in the lattice: at a certain site, the system gains energy if the fermion flips its spin, whereas in the sites around this one, the same process costs energy. This physical system is described by a tight-binding model with a Zeeman coupling that is different for neighbouring sites, thus dividing the lattice into AB sublattices. The single-particle spectrum has four bands; the third of which is shaped like a squarish Mexican hat (Figure). By filling up the energy levels up to the third band, the Fermi surface is squircle-shaped. This squarish-circle favours nesting and the system develops a coupled modulation in the density and spin, analogous to a spin-charge-density wave in solid- state systems.

Using field-theoretical methods, we develop a generalized formalism, which allows us to account for coupled charge and spin degrees of freedom simultaneously. We then determine the critical value of the parameters for the occurrence of the phase transition to this inhomogeneous density and spin state, which occurs at an incommensurate wave vector. Our results could be observed with state-of-the-art spectroscopic techniques. The investigation of spin-dependent optical lattices is an important direction of research in the field of spintronics with ultra-cold atoms, which will further strengthen the bonds between condensed matter and atomic physics.