New class of flat-band models on tetragonal and hexagonal lattices: Gapped versus crossing flat bands

TL;DR

This paper introduces a new class of tight-binding models on tetragonal and honeycomb lattices with controllable flat bands that can be gapped or crossing dispersive bands, expanding the understanding of flat-band physics and potential superconductivity.

Contribution

The paper develops a novel class of flat-band models with adjustable gap properties, including models with nonzero flat-band energy and the ability to extend to higher dimensions.

Findings

Models include gapped and crossing flat bands.

Flat bands have controllable energy gaps.

Extension to three or more dimensions is possible.

Abstract

We propose a new class of tight-binding models where a flat band is either gapped from or crossing right through a dispersive band on two-band (i.e., two sites/unit cell) tetragonal and honeycomb lattices. By imposing a condition on the hopping parameters for generic models with up to third-neighbor hoppings, we first obtain models having a rigorously flat band isolated from a dispersive band with a gap, which accommodate both rank-reducing and non-rank-reducing of the Hamiltonian. The class of models include Tasaki's flat-band models, but the present model has a nonzero flat-band energy, whose gap from the dispersive band is controllable as well. We then modify the models by appropriately changing the second- or third-neighbor hoppings, leading to a new class of two-dimensional lattices where a (slightly warped) flat band pierces a dispersive one. As with the known flat-band models, the connectivity condition is satisfied in the present models, so that we have unusual, unorthogonalizable Wannier orbitals. We have also shown that the present flat-band model can be extended to three (or higher) dimensions. Implications on possible high-$T_{C}$ superconductivity are discussed when a repulsive electron-electron interaction is introduced, where the flat band is envisaged to be utilized as intermediate states in pair scattering processes.