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

TL;DR

This paper classifies flat bands into singular and nonsingular types based on the behavior of their Bloch wave functions, revealing new topological properties and providing a scheme for designing flat band models.

Contribution

It introduces a novel classification of flat bands using Bloch wave function singularities and offers a general construction scheme for flat band Hamiltonians.

Findings

Singular flat bands have immovable discontinuities in Bloch wave functions.

Nonsingular flat bands form a complete vector bundle and can be isolated.

Singular flat bands exhibit a unique bulk-boundary correspondence.

Abstract

We show that flat bands can be categorized into two distinct classes, that is, singular and nonsingular 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 immovable discontinuities generated by the band-crossing with other bands, and thus the vector bundle associated with the flat band cannot be defined. This singularity precludes the compact localized states from forming a complete set spanning the flat band. Once the degeneracy at the band crossing point is lifted, the singular flat band becomes dispersive and can acquire a finite Chern number in general, suggesting a new route for obtaining a nearly flat Chern band. On the other hand, the Bloch wave function of a nonsingular flat band has no singularity, and thus forms a vector bundle. A nonsingular flat band can be completely isolated from other bands while preserving the perfect flatness. All one-dimensional flat bands belong to the nonsingular 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 nonsingular 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.