We investigate the quantum Hall effect in a single Landau level in the presence of a square superlattice of 𝛿-function potentials. The interplay between the superlattice spacing 𝑎𝑠 and the magnetic length ℓ𝐵 in a clean system leads to three interesting characteristic regimes corresponding to 𝑎𝑠<ℓ𝐵, 𝑎𝑠≫ℓ𝐵, and the intermediate one where 𝑎𝑠∼ℓ𝐵. 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 𝑎𝑠=√2⁢𝜋⁢ℓ𝐵. 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.

中文翻译：

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单个 Landau 能级内的超晶格诱导电子渗流


我们研究了在存在 δ 函数电位的方形超晶格的情况下，单个 Landau 能级中的量子霍尔效应。在清洁系统中，超晶格间距 as 和磁长 lB 之间的相互作用导致了三个有趣的特性范围，分别对应于 as<lB、 as≫lB，以及中间一个，其中 as∼lB。在中间状态中，连续磁平移对称性分解为离散晶格对称性。相比之下，我们表明，在其他两种状态中，尽管存在超晶格，但在拓扑带中几乎没有打破相同的情况。在存在弱无序（白噪声）的情况下，由于 Landau 水平的拓扑保护，人们通常期望只有一小部分扩展状态。有趣的是，我们在整个中间状态中获得了很大一部分扩展态，这在特殊点 as=√2πlB 处最大化。我们认为超晶格诱导的渗流现象需要打破时间反转对称性和连续磁平移对称性。它可能对连续量子霍尔系统中的整数平台跃迁和基于晶格的异常量子霍尔效应产生直接影响。