General construction of flat bands with and without band crossings based on wave function singularity

TL;DR

This paper presents a systematic method for constructing flat-band models in lattice systems using the symmetry, shape, and wave function singularities of compact localized states, enabling the study of both gapped and gapless flat bands.

Contribution

The authors introduce a novel, systematic construction scheme for flat-band models based on wave function singularities and symmetry considerations, applicable to any lattice.

Findings

Framework for constructing flat bands with controlled band crossings

Method to determine flat band properties from wave function singularities

Applicable to a wide range of lattice geometries

Abstract

In this work, we develop a systematic method of constructing flat-band models with and without band crossings. Our construction scheme utilizes the symmetry and spatial shape of a compact localized state (CLS) and also the singularity of the flat-band wave function obtained by a Fourier transform of the CLS (FT-CLS). In order to construct a flat-band model systematically using these ingredients, we first choose a CLS with a specific symmetry representation in a given lattice. Then, the singularity of FT-CLS indicates whether the resulting flat band exhibits a band crossing point or not. A tight-binding Hamiltonian with the flat band corresponding to the FT-CLS is obtained by introducing a set of basis molecular orbitals, which are orthogonal to the FT-CLS. Our construction scheme can be systematically applied to any lattice so that it provides a powerful theoretical framework to study exotic properties of both gapped and gapless flat bands arising from their wave function singularities.