Dirac operators on periodic planar graphs: the right definition advocated by flat bands

Abstract

Massive discrete Dirac-type operators on periodic planar graphs are investigated. Using Floquet--Bloch decomposition, we first show that neglecting the faces leads to spurious flat bands that are unrelated to the graph's geometry. Then, for the correct Dirac--Hodge operator $D_m$, we establish a spectral reduction expressing $\sigma (D_m)$ through the graph Laplacian on vertices $-{\Delta }_V$ and the Laplacian on faces $-{\Delta }_F$. As a consequence, $D_m$ carries a flat band precisely when one of these scalar Laplacians does. We finally apply this reduction to the study of flat bands of the operator $D_m$ on Archimedean lattices and contrast the outcome with that on the Lieb lattice.

arXiv comment21 pages