Emergence and transition of incompressible phases in decorated Landau levels

A single Landau level (LL) dressed with periodic electrostatic potentials can realize a plethora of interacting topological phases where the Hall conductivity generally does not equal to the LL filling factor. Their physics can be captured by a new family of flat topological bands: decorated Landau levels (dLL) from imposing an electrostatic delta potential lattice within a single LL. With $p/q$ magnetic fluxes per unit cell, there are $q$ dispersive bands and $p-q$ zero energy bands forming the dLL. When the electrostatic potential strength dominates the electron-electron interaction, band mixing is suppressed and the dispersion bands consist of ``localized states" with vanishing total Chern number. Nevertheless these dispersive bands can have highly nontrivial Berry curvature distribution, and even non-zero Chern numbers when $q>1$. Interestingly even in the limit of large short range interaction, band mixing between dLL and dispersion bands can be strongly suppressed at low filling factor, leading to robust topological phases within the dLL stabilized by the one-body potential. The dLL and the associated dispersive bands can serve as minimal theoretical models for correlated physics in lattice or moiré systems; they are also highly tunable experimental platforms for realizing rich phase diagrams of exotic 2D quantum fluids.