# Hidden mechanism for embedding the flat bands of Lieb, kagome, and checkerboard lattices in other structures

## TL;DR

This paper reveals a new mechanism allowing flat bands from well-known lattices like Lieb, kagome, and checkerboard to be embedded into more flexible structures, expanding potential material applications.

## Contribution

It introduces a novel mechanism for embedding flat bands into diverse structures beyond traditional line graph lattices, enhancing design options for quantum and photonic devices.

## Key findings

- Flat bands can be embedded into new structures not related by unitary transformation.
- The mechanism broadens the range of materials for electronic and photonic applications.
- Understanding of localized quantum states is enriched by this embedding approach.

## Abstract

The interplay of hopping parameters that can give rise to flat bands in consequence of quantum interference in electronic, photonic, and other interesting materials has become an extensively studied topic. Most of the recognized structures having flat bands are the lattices that can be understood by the mathematical theory of line graphs, such as the Lieb, kagome, and checkerboard lattices. Here, we demonstrate that the structures that can realize the same kind of flat bands given by those well-known lattices hosting exotic quantum phases are more flexible. The flat bands belonging to the recognized structures can be ideally embedded into the new structures that cannot be considered as the original ones in terms of a unitary transformation. The uncovered mechanism enriches the understanding of physics behind the localized quantum states and broadens the choice of materials that can be used for designing electronic and photonic devices from the zero band dispersion.