Polarizability and plasmons in pseudospin-1 gapped materials with a flat band

The collective electronic properties of various types of pseudospin-$1$ Dirac-cone materials with a flat band and finite bangaps in their energy spectra are the subject of our reported investigation. Specifically, we have calculated the dynamical polarization, plasmon dispersions as well as their decay rates due to Landau damping. Additionally, we present closed-form analytical expressions for the wave function overlaps for both the gapped dice lattice and the Lieb lattice. The gapped dice lattice is a special case of the more general $\alpha$-${\cal T}_3$ model since its band structure is symmetric and the flat band remains dispersionless. On the other hand, the Lieb lattice has a flat band which appears at the lowest point of its conduction band. Our results for these two cases exhibit unique features in the plasmon spectra and their damping regions, which have never been reported in previous studies. For example, the particle-hole modes of a Lieb lattice appear as finite-size regions, while the plasmon modes exist only in a region with small wave numbers but an extended range of frequencies.