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

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.

18 pages, 9 figures, published version

References in corpus (16)

• High temperature fractional quantum Hall states
• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
• Observation of a localized flat-band state in a photonic Lieb lattice
• Flat bands and Wigner crystallization in the honeycomb optical lattice
• Topological Insulators on the Lieb and Perovskite Lattices
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Bose condensation in flat bands
• Geometric approach to fragile topology beyond symmetry indicators
• Chiral Flat Bands: Existence, Engineering and Stability
• Compactly-supported Wannier functions and algebraic $K$-theory
• Flatband generator in two dimensions
• Stability of fractional Chern insulators in the effective continuum limit of Harper-Hofstadter bands with Chern number $|C|>1$
• Flat bands and nontrivial topological properties in an extended Lieb lattice
• New class of flat-band models on tetragonal and hexagonal lattices: Gapped versus crossing flat bands
• Type-III Dirac Cones from Degenerate Directionally Flat Bands: Viewpoint from Molecular-Orbital Representation