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

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.

Flatband Line States in Photonic Super‐Honeycomb Lattices

“…The missing states are the so‐called noncontractible loop states (NLSs) winding around the entire (infinite) lattices, a manifestation of real‐space topology. [ 37,38 ] NLSs are a new type of flatband eigenstates which cannot be obtained by linear superposition of the conventional CLSs, and they cannot be continuously deformed into the CLSs in a torus geometry representing the periodical boundary conditions. Recently, it was theoretically found that the NLSs are inherent to the singular flatband, where an immovable singularity exists in the band‐crossing of the Bloch wavefunctions.…”

“…Recently, it was theoretically found that the NLSs are inherent to the singular flatband, where an immovable singularity exists in the band‐crossing of the Bloch wavefunctions. [ 38 ] Since an infinite lattice or a torus structure (mimicking the periodic boundary conditions) is difficult to establish in experiment, an alternative approach is to observe the “line states” in truncated lattices with appropriate boundaries. The existence of the line states does not rely on the spatial size of the lattice but rather on the boundary termination.…”

Photonic flat-band lattices and unconventional light localization

“…Like the protection of the Dirac points of graphene, the mechanism for this stability is topological. Very recently, Rhim and Yang [93] showed that flat-bands can be classified into two distinct classes, singular or nonsingular, based on whether the corresponding Bloch functions exhibit a discontinuity at the crossing points with other bands.…”

“…Such missing states are accounted for by noncontractible loops around the torus under periodic boundary conditions (the so-called NLSs). More recently, Rhim et al theoretically found that the NLSs are inherent to the singular flat-band, where an immovable singularity exists in the band-crossing of the Bloch wavefunctions [93]. Once the degeneracy at the band-crossing point is lifted, the flat-band becomes dispersive and may acquire a finite Chern number in general.…”

Photonic zero-energy modes in a metal-based Lieb lattice

“…On the other hand, in the electronic models, the flat bands can have singular character due to the immovable discontinuity of Bloch wave functions, which result in the incompleteness of the CLSs and bring forth the new states at flat band energy. Several previous works have theoretically studied the additional noncompact states in the frustrated lattices and other singular flat band models [25][26][27][28]. In photonic systems, a missing line state has recently been experimentally found in a finite Lieb system under bearded truncations using the photonic writing waveguides [29].…”