Flat bands with band crossings enforced by symmetry representation

Abstract

Flat bands have band crossing points with other dispersive bands in many systems including the canonical flat band models in the Lieb and kagome lattices. Here we show that some of such band degeneracy points are unavoidable because of the symmetry representation (SR) of the flat band under unitary symmetry. We refer to such a band degeneracy point of flat bands as a SR-enforced band crossing. SR-enforced band crossing is distinct from the conventional band degeneracy protected by symmetry eigenvalues or topological charges in that its protection requires both specific symmetry representation and band flatness of the flat band, simultaneously. Even $n$-fold rotation $C_n$ ($n=2,3,4,6$) symmetry, which cannot protect band degeneracy without additional symmetries due to its abelian nature, can protect SR-enforced band crossings in flat band systems. In two-dimensional flat band systems with $C_n$ symmetry, when the degeneracy of a SR-enforced band crossing is lifted by a $C_n$ symmetry-preserving perturbation, we obtain a nearly flat Chern band. Our theory not only explains the origin of the band crossing points of FBs existing in various models, but also gives a strict no-go theorem for isolated FBs in a given lattice arising from the SR.

8+14 pages, 4+6 figures, published version

References in corpus (19)

• 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
• Bulk Topological Invariants in Noninteracting Point Group Symmetric Insulators
• Topological Magnon Insulator in Insulating Ferromagnet
• Topological Insulators on the Lieb and Perovskite Lattices
• Building Blocks of Topological Quantum Chemistry: Elementary Band Representations
• Bose condensation in flat bands
• Topological flat band models with arbitrary Chern numbers
• Geometric approach to fragile topology beyond symmetry indicators
• Chiral Flat Bands: Existence, Engineering and Stability
• Flat bands with higher Chern number in pyrochlore slabs
• Compactly-supported Wannier functions and algebraic $K$-theory
• Dirac Cones, Topological Edge States, and Nontrivial Flat Bands in Two-Dimensional Semiconductors with a Honeycomb Nanogeometry
• Flatband generator in two dimensions
• 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

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• Ferromagnetism in armchair graphene nanoribbon heterostructures
• Confined states and topological phases in two-dimensional quasicrystalline $\pi$-flux model
• Pinch-point spectral singularity from the interference of topological loop states
• Interorbital Antisymmetric Hopping Generated Flat Bands on Kagome and Pyrochlore Lattices
• Unconventional gapless semiconductor in an extended martini lattice in covalent honeycomb materials
• Spinor ice correlation in flat-band electronic states on kagome and pyrochlore lattices with spin-orbit coupling
• Singular flat bands in three dimensions: Landau level spreading, quantum geometry, and Weyl reconstruction
• Symmetry-Based Real-Space Framework for Realizing Flat Bands and Unveiling Nodal-Line Touchings