# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 3 Electron Dressed States: General Formalism
This unexpected result reveals a lot of interesting features of the electron dressed states for the circular and elliptical polarizations of the dressing field. Most importantly, we know that the term in Equation (34) mainly determines the bandgap of the radiated \(\alpha-\mathcal{T}_{3}\) materials since it does not depend on the wave vector components \(k_{x}\) and \(k_{y}\). It represents the key contribution to the energy dispersions at \(\mathbf{k}=0\) since the next term \(\backsim\lambda_{0}^{2}\) would only result in a much smaller correction. Therefore, we find that for linearly polarized light with \(\beta=0\), the bandgap remains completely unaffected by the dressing field. The gap due to circular polarized light is mainly formed in the same way as the initial bandgap \(\Delta_{0}\), which is induced by a specific substrate. However, there is a very important difference: the irradiation-induced bandgap in Equation (34) is proportional to the valley index \(\tau=1\). Therefore, the total bandgap of the irradiated \(\alpha-\mathcal{T}_{3}\) materials could be either increased or decreased, depending on whether we are considering the electronic states in the \(K\) or \(K^{\prime}\) valley (\(\tau=\pm 1\)). After carrying out the commutation relation in Equation (33), we arrive at the following expression for the effective perturbation Hamiltonian up to the order of \(\mathcal{O}(\lambda_{0}^{2})\)  
\[\frac{1}{2(\hbar\omega)^{2}}\Bigg{\{}\bigg{[}\,\big{[}\,\mathbb{O }_{\tau}^{(\phi)},\,\hat{\mathcal{H}}_{\tau}^{\phi}(\mathbf{k})\,\big{]}_{-}, \,\mathbb{O}_{\tau}^{(\phi)}\,\bigg{]}_{-}\,+\,\text{H.c.}\,\bigg{\}}\] \[= -\frac{\tau}{4}\,\hbar v_{F}\lambda_{0}^{2}\left[ \begin{array}[] {ccc}d_{1}^{(\beta)}(\Delta_{0}\,|\,\phi)&h_{12}(\mathbf{k}\,|\,\tau,\phi)&o_ {\beta}(\Delta_{0}\,|\,\phi)\\ h_{12}^{*}(\mathbf{k}\,|\,\tau,\phi)&d_{2}^{(\beta)}(\Delta_{0}\,|\,\phi)&h_{ 23}(\mathbf{k}\,|\,\tau,\phi)\\ o_{\beta}^{*}(\Delta_{0}\,|\,\phi)&h_{23}^{*}(\mathbf{k}\,|\,\tau,\phi)&d_{3}^ {(\beta)}(\Delta_{0}\,|\,\phi)\end{array} \right]\,,\]  
where  
\[h_{12}(\mathbf{k}\,|\,\tau,\phi)=\cos\phi[1+3\cos(2\phi)](\beta^{2}k_{x}-i \tau k_{y})\,, \tag{36}\]  
\[h_{23}(\mathbf{k}\,|\,\tau,\phi)=\sin\phi[1-3\cos(2\phi)](\beta^{2}k_{x}-i \tau k_{y})\,, \tag{37}\]  
\[d_{1}^{(\beta)}(\Delta_{0}\,|\,\phi)=\left(1+\beta^{2}\right)\Delta_{0}\,\cos ^{2}\phi\left[1+3\cos(2\phi)\right], \tag{38}\]  
\[d_{2}^{(\beta)}(\Delta_{0}\,|\,\phi)=-4\Big{(}1+\beta^{2}\Big{)}\,\Delta_{0} \,\cos(2\phi)\,, \tag{39}\]  
\[d_{3}^{(\beta)}(\Delta_{0}\,|\,\phi)=-\Big{(}1+\beta^{2}\Big{)}\,\Delta_{0}\, \sin^{2}\phi\left[1-3\cos(2\phi)\right], \tag{40}\]  
and, finally,  
\[o_{\beta}(\Delta_{0}\,|\,\phi)=-\frac{3}{4}\left(1-\beta^{2}\right)\Delta_{0} \,\sin(4\phi)\,, \tag{41}\]  
which would vanish for a circularly polarized light (\(\beta=1\)), as well as for graphene (\(\phi=0\)) and a dice lattice (\(\phi=\frac{\tau}{4}\)). Dressed States for a Lieb Lattice  
We apply canonical substitution (27) so that Hamiltonian (19) has an additional interaction term  
\[\hat{\mathcal{H}}_{A}^{(e)}(t|\,\tau,\phi)=-c_{0}\left[ \begin{array}{ccc}0& \cos(\omega t)&0\\ \cos(\omega t)&0&\sin(\omega t)\\ 0&\sin(\omega t)&0\end{array} \right], \tag{42}\]Comparing the Hamiltonian in Equation (19) with the representation in Equation (31), we immediately derive an explicit form for the perturbation operator \(\hat{\mathbb{O}}_{\tau}^{(\phi)}\) as  
\[\hat{\mathbb{O}}_{\tau}^{(\phi)}=-\frac{c_{0}}{2}\left[ \begin{array}{ccc}0&1&0 \\ 1&0&-i\\ 0&-i&0\end{array} \right]\,. \tag{43}\]  
\[\frac{1}{\hbar\omega}\Big{[}\,\hat{\mathbb{O}}_{\tau}^{(\phi)},\,\hat{\mathbb{O }}_{\tau,\phi}^{\dagger}\Big{]}_{-}=c_{0}\,\lambda_{0}\,\beta\left[ \begin{array} []{ccc}0&0&2i\\ 0&0&0\\ -2i&0&0\end{array} \right]\,.