Ideal optical flux lattices

The realization of fractional quantum Hall (FQH) states in cold atomic gases is a long-standing goal in quantum simulation. Established approaches, including rapidly rotating gases and tight-binding lattices, are often hampered by low interaction energies and small many-body energy gaps. While optical flux lattices (OFLs) can achieve higher effective magnetic flux densities, standard two-state configurations generate highly non-uniform fields, and extensions to multi-state systems introduce significant experimental complexity. Here, we present a new paradigm for engineering robust FQH phases in OFLs using only two internal atomic states. We show that the introduction of an additional scalar potential provides a generic mechanism for creating Chern bands that are simultaneously essentially flat and "ideal". These desirable properties arise by tuning lattice parameters to certain $N$-flat manifolds $(N=1,2,\dots)$, where the $1$-flat manifold shares its origin with certain "magic-angle" conditions familiar from moiré materials. A central result is the design of a dark-state OFL whose adiabatic Hamiltonian is exactly of Aharonov-Casher (AC) form. This exact AC equivalence guarantees perfectly flat, exactly vortexable Chern bands in the adiabatic limit. This method allows for precise tuning of band flatness and stabilizes both Abelian and non-Abelian FQH phases. Our scheme is compatible with existing experimental capabilities using vector polarizability, opening practical routes to exploring strongly correlated topological physics with cold atoms.