Orthogonal flatbands in Hamiltonians with local symmetry

Abstract

We derive symmetry-based conditions for tight-binding Hamiltonians with flat- bands to have compact localized eigenstates (CLS) occupying a single unit cell. The conditions rely on unitary operators commuting with the Hamiltonian and are associated with local symmetries that ensure the existence of orthogonal compact localized states and a flatband. We also demonstrate the inverse: orthogonal flatbands, whose CLS can be made to occupy a single unit cell, always have associated local symmetries that satisfy our conditions.We illustrate our analytical result with several model Hamiltonians with given local symmetries, including examples with finite-range hopping and higher-dimensional ones.