# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 5 Summary and Remarks  
In this paper, we carried out a rigorous theoretical and numerical investigation into the effects arising from the electron-photon interaction. In particular, we examine the resulting electron dressed states for some specific types of Dirac cone materials. It is worth noting that their low-energy spectrum exhibits a flat band and a finite bandgap. Figure 8: (Color online) Three-dimensional plots for the energy band structure \(\varepsilon_{d}(\lambda_{0}|k_{x},k_{y}=0)\) for a Lieb lattice. Each panel describes a material with bandgap \(\Delta_{0}=1.0\) and a specific value of the coupling parameter \(\lambda_{0}=0.5\) (**a**–**c**). Figure 7: (Color online) The \(k_{x}\)-dependence of the energy dispersions \(\varepsilon_{d}(\lambda_{0}|k_{x},k_{y}=0)\) for a Lieb lattice. Each panel describes a material with bandgap \(\Delta_{0}=1.0\) and a selected value of the coupling parameter \(\lambda_{0}\), as labeled. Specifically, we demonstrate the modification and the finite dispersions of the flat band \(\mathcal{E}=\hbar v_{F}k_{\Delta}\) due to the irradiation. A circularly polarized dressing field with \(\beta=1\) was selected for (**a**–**f**). We consider that an electron in a two-dimensional material is irradiated with a high-frequency off-resonance dressing field. In this case, its interaction with the irradiation results in a specific unified quantum state with modified energy dispersions and bandgaps. The properties of the obtained dressed states strongly depend on the type of polarization of the imposed radiation. Whereas a linearly polarized dressing field is known to induce finite anisotropy, circularly polarized light is considered a tool to open or modify the existing energy gap. Creating and analyzing new electronic states which appear as a consequence of electron-light interaction have significant applications in several areas of physics. Specifically, these include electron conductance and quantum transport. It is evident that a new type of electronic band structures and a new structure of electron transition between the valence, conduction, and middle bands lead to significantly novel electronic properties of a lattice and create several opportunities for device applications. It is of interest to examine the thermal dependence of the electronic conductivity and Boltzmann transport over certain temperature ranges. A novel electronic band structure may potentially lead to a smaller increase in the resistivity due to the temperature which is crucial for electronic nanodevices. Our main goal was to consider the modification or renormalization of the existing bandgaps of pseudospin-1 Dirac cone materials with a flat band. Specifically, we focused on the gapped \(\alpha-\mathcal{T}_{3}\) model, a dice lattice, and a Lieb lattice. Each of these materials has a very distinguished energy band structure. However, their common feature is the existence of a dispersionless flat band, as well as finite gaps between the valence, flat, and conduction bands. We emphasize that our work is the suject of a theoretical and numerical study of the dressed states in a class of materials. We also present the derivation of several crucial analytical relations for the energy dispersions in the presence of circularly and elliptically polarized light. We have demonstrated that the existing bandgap could be either increased or decreased depending on several material parameters of the specific lattice and the valley index. By applying circularly and elliptically polarized irradiation to various pseudospin-1 lattices with a flat band and finite bandgap, we achieved several very specific types of electron band structures. We found that the flat band could receive a finite dispersion, and its location also changes. The two remaining valence and conduction bands are also affected differently for each type of material. Either one or both bandgaps could be increased or decreased. Finite anisotropy could also be induced by elliptically polarized light.