奇异平坦带
Singular flat bands
摘要：
本文综述了近年来平带系统的研究进展，特别是与平带布洛赫波函数奇异性有关的基础物理。我们首先解释了平带可以分为两类：奇异和非奇异的平带，根据在平带的布洛赫波函数的奇异性的存在或不存在。奇异性是由平坦带与另一个色散带的带交叉产生的。在奇异平坦带中，我们可以发现一类特殊的本征模，称为不可收缩环态和鲁棒边界模，它们具有非平凡的实空间拓扑结构。然后，我们回顾了光子晶格中平带拓扑本征模的实验实现。虽然奇异性的平带是拓扑平凡的，我们表明，周围的奇异性的最大量子距离是一个体积不变的奇异性，保护强大的边界模式的强度。最后，我们讨论了当奇异平带与另一个带有二次带交叉时，奇异平带的最大量子距离或奇异强度如何表现在奇异平带的反常朗道能级扩展中。
英文摘要：
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 nonsingular 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 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.
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