Dynamical optical conductivity for gapped $\alpha-\mathcal{T}_3$ materials with a curved "flat" band

We have calculated the dynamical optical conductivity for $\alpha-\mathcal{T}_3$ materials in the presence of a finite bandgap in their energy bandstructure. This is a special type of energy dispersions because for all $\alpha-\mathcal{T}_3$ materials with a bandgap, except graphene and a dice lattice limits, the flat band receives a non-zero dispersion and assumes a curved shape. The infinite ${\bf k}$-degeneracy of the flat energy band is also lifted. Such a low-energy bandstructure could be obtained if an $\alpha-\mathcal{T}_3$ material is irradiated off-resonant with circularly polarized light. We have calculated the optical conductivity for the zero and finite temperatures, as well as for the cases of a finite and nearly-zero doping. We have demonstrated that analytical expressions could be in principle obtained for all types of gapped $\alpha-\mathcal{T}_3$ materials and provided the closed-form analytical expressions for a gapped dice lattice. Our numerical results reveal some well-known signatures of the optical conductivity in $\alpha-\mathcal{T}_3$ and silicene with two non-equivalent bandgaps, as well as demonstrate some very specific features which have not been previously found in any existing Dirac materials.