Band structure and optical response of Kekulé-modulated $\alpha-\mathcal{T}_3$ model

This work studies the electronic band structure and optical response of a hybrid $\alpha-\mathcal{T}_3$ model featuring a $\sqrt{3}\times\sqrt{3}$ Kekulé-pattern modulation. Such a hybrid system may result from depositing adatoms in a hexagonal lattice, with the two sublattices displaced in the perpendicular direction, like germanene and silicene. We derive an analytical expression for the energy dispersion and eigenfunctions using a tight-binding approximation of nearest-neighbor hopping electrons. The energy spectrum presents a double-cone structure with Dirac points at zero momentum caused by the Brillouin zone folding and a flat band due to destructive quantum interference effects. Furthermore, we study the spectrum of intraband and interband transitions through the joint density of states, the optical conductivity, and the Drude spectral weight. We found new conductivity terms resulting from the opening of intervalley channels that are absent in the $\alpha-\mathcal{T}_3$ model and manifest as van Hove singularities in the optical response. Additionally, we identify an absorption resonance peak related to intervalley transport, which serves as a viable signature for detecting Kekulé periodicity in these systems.