Robustness of translational invariance in topological bands against lattice potentials and disorders

We theoretically show that the continuous magnetic translational invariance within the Hilbert (sub)space of a single Landau level (LL) can persist even in the presence of a superlattice electrostatic potential modulation, while such invariance is broken in the full real-space Hilbert space. This is due to the interplay of the superlattice constant and the fundamental length scale of the quantum Hall fluids. In particular for the lowest LL (LLL), when the spacing of superlattice is below the magnetic length, continuous magnetic translational symmetry is very robust. For the fractional quantum Hall phases, the continuous translational symmetry is preserved when the superlattice spacing is below the corresponding fundamental lengths which we can now quantitatively define, which is different from the length scale from the quantum metric. Moreover, our analysis implies that the dynamics of the anyonic excitations can be robust against the short wavelength part of the disorder potentials, and we discuss the related experimental ramifications.