We review recent progresses in the study of flat band systems, especially focusing on the fundamental physics related to the singularity of the flat bands 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 bands 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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