ABSTRACT

We review recent progresses in the study of flat band systems, especially focusing on the fundamental physics related to the singularity of the flat band’s Bloch wave functions. We first explain that the flat bands can be classified into two classes: singular and non-singular flat bands, based on the presence or absence of the singularity in the flat band’s Bloch wave functions. The singularity is generated by the band crossing of the flat band with another dispersive band. In the singular flat band, one can find a special kind of eigenmodes, called the non-contractible loop states and the robust boundary modes, which exhibit nontrivial real-space topology. Then, we review the experimental realization of these topological eigenmodes of the flat band in the photonic lattices. While the singularity of the flat band is topologically trivial, we show that the maximum quantum distance around the singularity is a bulk invariant representing the strength of the singularity which protects the robust boundary modes. Finally, we discuss how the maximum quantum distance or the strength of the singularity manifests itself in the anomalous Landau level spreading of the singular flat band when it has a quadratic band-crossing with another band.

摘要

我们回顾了平带系统研究的最新进展，特别是关注与平带Bloch波函数奇异性相关的基本物理学。我们首先解释，根据平坦带的Bloch波函数中是否存在奇异性，可以将平坦带分为两类：奇异平坦带和非奇异平坦带。奇异性是通过平坦带与另一个色散带的带交叉而产生的。在奇异的平坦频带中，可以找到一种特殊的本征模式，称为不可收缩环态和鲁棒边界模态，它们表现出非平凡的实空间拓扑。然后，我们回顾了光子晶格中平坦带的这些拓扑本征模式的实验实现。扁平带的奇异性在拓扑上是微不足道的，我们表明，奇异点附近的最大量子距离是一个整体不变式，代表了奇异点的强度，从而保护了稳健的边界模态。最后，我们讨论了最大量子距离或奇异强度如何在奇异平坦带与另一个谱带发生二次交叉的情况下，在奇异平坦带的Landau能级反常扩展中表现出来。