# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 3 Electron Dressed States: General Formalism  
Let us now consider the changes in the energy dispersions of gapped \(\alpha-\mathcal{T}_{3}\) when an off-resonance high-frequency irradiation is applied. It is known that the effects of such irradiation drastically depend on its polarization type: circularly polarized light modifies the existing energy bandgap, while a linearly polarized field induces an anisotropy of the electron energy dispersions in the direction of the light polarization. We shall examine the most general case of an elliptically-polarized dressing field. If the direction of the light polarization is aligned with the \(x\)-axis, its vector potential is given as  
\[\mathbf{A}(t)=\left[ \begin{array}{c}A_{x}(t)\\ A_{y}(t)\end{array} \right]=\frac{\mathcal{E}_{0}}{\omega}\left[ \begin{array}[] {c}\cos(\omega t)\\ \beta\sin(\omega t)\end{array} \right]\,, \tag{24}\]  
where \(\mathcal{E}_{0}\) is the electric field of the imposed irradiation, so that its intensity is \(I\sim\mathcal{E}_{0}^{2}\), and \(\omega\) is its frequency, meaning that for the high-frequency regime coefficient \(\mathcal{E}_{0}/\omega\) represents only a small correction. Parameter \(\beta=\sin\Theta_{e}\) is the ratio of field strengths (electric field amplitudes) along the two axes of the polarization ellipse. Equation (24) corresponds to the most general type of the light polarization. Thus, \(\beta\longrightarrow 1\) results in a known expression  
\[\mathbf{A}^{(\mathrm{C})}(t)=\left[ \begin{array}{c}A_{x}^{(\mathrm{C})}(t)\\ A_{y}^{(\mathrm{C})}(t)\end{array} \right]=\frac{\mathcal{E}_{0}}{\omega}\left[ \begin{array}{c}\cos(\omega t)\\ \sin(\omega t)\end{array} \right]\,, \tag{25}\]  
for the circularly polarized light, while a limit of \(\beta\longrightarrow 0\) corresponds to the dressing light of a linear polarization  
\[\mathbf{A}^{(\mathrm{L})}(t)=\left[ \begin{array}{c}A_{x}^{(\mathrm{L})}(t)\\ A_{y}^{(\mathrm{L})}(t)\equiv 0\end{array} \right]=\frac{\mathcal{E}_{0}}{ \omega}\left[ \begin{array}{c}\cos(\omega t)\\ 0\end{array} \right]\,. \tag{26}\]  
Thus, analyzing the general elliptical polarization of the imposed irradiation allows us immediately discern the results for these crucial situations as well. It is well-known that the energy dispersions of the electron dress states due to an off-resonant dressing field are obtained by applying the canonical substitution  
\[k_{x,y}\to k_{x,y}-\frac{e}{\hbar}\,A_{x,y}(t) \tag{27}\]  
in the Hamiltonian (1), which corresponds to a non-irradiated material. Consequently, Equation (1)  
\[\hat{\mathcal{H}}_{\tau}^{\phi}(\mathbf{k})\Longrightarrow\hat{\mathcal{H}}^{(e )}(\mathbf{k},t|\,\tau,\phi)\equiv\mathbb{H}_{\tau}^{\phi}(\mathbf{k})+\hat{\mathcal{ H}}_{A}^{(e)}(t|\,\tau,\phi)\, \tag{28}\]  
receives an additional _interaction Hamiltonian term_  
\[\hat{\mathcal{H}}_{A}^{(e)}(t|\,\tau,\phi)=-\frac{\tau c_{0}}{ \sqrt{\cos^{2}\omega t+[\beta\,\sin\omega t]^{2}}}\] \[\times \left\{ \begin{array}{ccc}0&\cos\phi\exp[-i\tau\,\Phi(\beta,t)]& 0\\ \cos\phi\exp[i\tau\,\Phi(\beta,t)]&0&\sin\phi\exp[-i\tau\,\Phi(\beta,t)]\\ 0&\sin\phi\exp[i\tau\,\Phi(\beta,t)]&0\end{array} \right\},\]  
where  
\[\Phi(\beta,t)=\tan^{-1}[\beta\,\tan(\omega t)]\underset{\beta\to 1}{ \underset{\beta\to 1}{\Longrightarrow}}\omega t\, \tag{30}\]  
which corresponds to circularly polarized light. Parameter \(c_{0}=e\mathcal{E}_{0}v_{F}/\omega\), which specifies the strength of the interaction between an electron and a photon, has a unit of energy. This interaction strength parameter is the same for irradiation with different types of polarization, meaning that the irradiation of a specific intensity and frequency affects theelectronic band structure of graphene (and \(\alpha-\mathcal{T}_{3}\) material) at the same level, regardless of its polarization. The way in which the dispersions are affected is obviously very different for the various types of polarizations.