Emergence and transition of incompressible phases in modulated Landau levels

We show how the interplay between electron interactions and disorder superlattices drives topological phase transitions in Chern bands. Projecting local delta potentials arranged in a superlattice onto a Chern band defines a ``doped Chern band'', a new topologically non-trivial band that can host topological phases. As the doped and normal Chern bands (or multiple doped Chern bands) cross, the system undergoes a sequence of transitions: from an incompressible fractional quantum Hall (FQH) phase (or analogous FCI phase), through a compressible metallic phase, and into another incompressible topological phase. For the FQH–IQH transition, we show that less uniform potentials broaden the compressible regime. Crucially, on the lowest Landau level (LLL), whether transitions between different FQH phases can occur is determined by the null space of the Hamiltonian, and FQH phases remain robust under superlattice disorder only when this null space forms a conformal Hilbert space. Finally, we demonstrate that these disorder-induced transitions also occur in realistic moiré MoTe₂ systems, providing a route to experimentally control topological phases.