  We design an interaction-driven topological insulator for fermionic cold
atoms in an optical lattice, that is, we pose the question of whether we can
realize in a continuous space a spontaneous symmetry breaking induced by the
inter-atom interaction into a topological Chern insulator. Such a state,
sometimes called a "topological Mott insulator", has yet to be realized in
solid-state systems, since this requires, in the tight-binding model, large
offsite interactions on top of a small onsite interaction. Here we overcome the
difficulty by introducing a spin-dependent potential, where a spin-selective
occupation of fermions in $A$ and $B$ sublattices makes the onsite interaction
Pauli-forbidden, while a sizeable inter-site interaction is achieved by a
shallow optical potential with a large overlap between neighboring Wannier
orbitals. This puts the system away from the tight-binding model, so that we
adopt the density functional theory for cold-atoms, here extended to
accommodate non-collinear spin structures emerging in the topological regime,
to quantitatively demonstrate the phase transition to the topological Mott
insulator.
