Flat bands of periodic graphs

Mostafa Sabri, Pierre Youssef

Published 2023-04-13Version 1

We study flat bands of periodic graphs in a Euclidean space. These are infinitely degenerate eigenvalues of the corresponding adjacency matrix, with eigenvectors of compact support. We provide some optimal recipes to generate desired bands, some sufficient conditions for a graph to have flat bands, we characterize the set of flat bands whose eigenvectors occupy a single cell and we compute the list of such bands for small cells. We next prove that flat bands are rare and vanish under arbitrarily small perturbations by periodic potentials. Additional folklore results are proved and many questions are still open.