Superlattice induced electron percolation within a single Landau level

We investigate the quantum Hall effect in a single Landau level in the presence of a square superlattice of $\delta$-function potentials. The interplay between the superlattice spacing $a_s$ and the magnetic length $\ell_B$ in clean system leads to three interesting characteristic regimes corresponding to $a_s \lt \ell_B$, $a_s \gg \ell_B$ and the intermediate one where $a_s \sim \ell_B$ . In the intermediate regime, the continuous magnetic translation symmetry breaks down to discrete lattice symmetry. In contrast, we show that in the other two regimes, the same is hardly broken in the topological band despite the presence of the superlattice. In the presence of weak disorder (white-noise) one typically expects a tiny fraction of extended states due to topological protection of the Landau level. Interestingly, we obtain a large fraction of extended states throughout the intermediate regime which maximizes at the special point $a_s = \sqrt{2\pi} \ell_B$. We argue the superlattice induced percolation phenomenon requires both the breaking of the time reversal symmetry and the continuous magnetic translational symmetry. It could have a direct implication on the integer plateau transitions in both continuous quantum Hall systems and the lattice based anomalous quantum Hall effect.