Classification of flat bands according to the band-crossing singularity of Bloch wave functions

By Jun‐Won Rhim, Bohm-Jung Yang. Physical review B, 2018

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We show that flat bands can be categorized into two distinct classes, that is, singular and non-singular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space. In the case of a singular flat band, its Bloch wave function possesses irremovable discontinuities generated by the band crossing with other bands. Once the degeneracy at the band crossing point is lifted, the flat band becomes dispersive and may acquire a finite Chern number in general. On the other hand, the Bloch wave function of a non-singular flat band has no singularity, and thus it can be completely isolated from other bands while preserving the perfect flatness. All one dimensional flat bands belong to the non-singular class. We show that a singular flat band displays a novel bulk-boundary correspondence such that the presence of the robust boundary mode is guaranteed by the singularity of the Bloch wave function. Moreover, we develop a general scheme to construct a flat band model Hamiltonian in which one can freely design its singular or non-singular nature. Finally, we propose a general formula for the compact localized state spanning the flat band, which can be easily implemented in numerics and offer a basis set useful in analyzing correlation effects in flat bands.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/0169941a6e2ed5ad3ed948cd48725cf40e34cf07