Flatband generator in two dimensions

TL;DR

This paper extends the systematic flatband generator approach from one to two dimensions, enabling analytical solutions and classification of flatband networks, including well-known lattices like kagome and Lieb.

Contribution

It introduces a two-dimensional flatband generator that incorporates the shape of localized states for classifying and deriving flatband networks.

Findings

Analytical solutions for 2D flatband networks obtained.

Reclassification of known lattices such as kagome and Lieb.

Generator can be extended to three dimensions.

Abstract

Dispersionless bands -- \emph{flatbands} -- provide an excellent testbed for novel physical phases due to the fine-tuned character of flatband tight-binding Hamiltonians. The accompanying macroscopic degeneracy makes any perturbation relevant, no matter how small. For short-range hoppings flatbands support compact localized states, which allowed to develop systematic flatband generators in $d=1$ dimension in Phys. Rev. B {\bf 95} 115135 (2017) and Phys. Rev. B {\bf 99} 125129 (2019). Here we extend this generator approach to $d=2$ dimensions. The \emph{shape} of a compact localized state turns into an important additional flatband classifier. This allows us to obtain analytical solutions for classes of $d=2$ flatband networks and to re-classify and re-obtain known ones, such as the checkerboard, kagome, Lieb and Tasaki lattices. Our generator can be straightforwardly generalized to three lattice dimensions as well.