Designing flat-band tight-binding models with tunable multifold band touching points

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 >= 2 of bands. Linear flat-band models are rarer: they require N >= 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.

Automated summary

Flat bands on periodic lattices are characterized by macroscopically degenerate eigenstates, and a method to build flat-band tight-binding models with short-range hoppings on any periodic lattice is presented in this work. The resulting models exhibit flat bands as well as multifold band touching points, which can be controlled in terms of number, location, degeneracy, and singularity. Various types of flat-band models can be constructed, including quadratic models from any arbitrary compact localized state and linear models from specific compatibility relations with the lattice structure.