Two dimensional sub-wavelength topological dark state lattices

We present a general framework for engineering two-dimensional (2D) sub-wavelength topological optical lattices using spatially dependent atomic dark states in a $Λ$-type configuration of the atom-light coupling. By properly designing the spatial profiles of the laser fields inducing coupling between the atomic internal states, we show how to generate sub-wavelength Kronig-Penney-like geometric scalar potential accompanied by narrow and strong patches of the synthetic magnetic field localized in the same areas as the scalar potential. These sharply peaked magnetic fluxes are compensated by a smooth background magnetic field of opposite sign, resulting in zero net flux per unit cell while still enabling topologically nontrivial band structures. Specifically, for sufficiently narrow peaks, their influence is minimum, and the behavior of the system in a remaining smooth background magnetic field resembles the Landau problem, allowing for the formation of nearly flat energy bands with unit Chern numbers. Numerical analysis confirms the existence of ideal Chern bands and the robustness of the topological phases against non-adiabatic effects and losses. This makes the scheme well-suited for simulating quantum Hall systems and fractional Chern insulators in ultracold atomic gases, offering a new platform for exploring strongly correlated topological phases with high tunability.