Stackings and effective models of bilayer dice lattices

We introduce and classify nonequivalent commensurate stackings for bilayer dice or lattice. For each of the four possible stackings, an effective low-energy model is derived. Although the energy spectrum remains always gapless with three bands interesting at the same point, depending on the stacking, different types of quasiparticle spectra arise. They include those with flat, tilted, anisotropic semi-Dirac, and three-fold-corrugated energy bands. We use the derived models to calculate the density of states and the spectral function. The corresponding results reveal drastic redistribution of the spectral weight due to the inter-layer coupling that is unique for each of the stackings.