Being dispersionless, flat bands on periodic lattices are solely characterized by their macroscopically degenerate eigenstates: compact localized states (CLSs) in real space and Bloch states in reciprocal space. Based on this property, this work presents a straightforward method to build flat-band tight-binding models with short-range hoppings on any periodic lattice. The method consists in starting from a CLS and engineering families of Bloch Hamiltonians as quadratic (or linear) functions of the associated Bloch state. The resulting tight-binding models not only exhibit a flat band, but also multifold quadratic (or linear) band touching points (BTPs) whose number, location, degeneracy, and (non)singularity can be controlled to a large extent. Quadratic flat-band models are ubiquitous: they can be built from any arbitrary CLS, on any lattice, in any dimension and with any number $N\ge 2$ of bands. Linear flat-band models are rarer: they require $N\ge 3$ and can only be built from CLSs that fulfill certain compatibility relations with the underlying lattice. Most flat-band models from the literature can be classified according to this scheme: Mielke's and Tasaki's models belong to the quadratic class, while the Lieb, dice, and breathing kagome models belong to the linear class. Many novel flat-band models are introduced, among which an $N=4$ bilayer honeycomb model with fourfold quadratic BTPs, an $N=5$ dice model with fivefold linear BTPs, and an $N=3$ kagome model with BTPs that can be smoothly tuned from linear to quadratic.

• Received 25 June 2021
• Accepted 1 November 2021

DOI:https://doi.org/10.1103/PhysRevB.104.195128