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

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 $-{\mathrm{\Delta }}_V$ and the Laplacian on faces $-{\mathrm{\Delta }}_F$. 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.