# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 3 Electron Dressed States: General Formalism
For our future calculations, we also need another type of coupling parameter \(\lambda_{0}=c_{0}/\hbar\omega\), which is dimensionless in contrast to the previously considered \(c_{0}\). For the considered off-resonance regime, we can assume \(\lambda_{0}\ll 1\), which will be used later as a series expansion parameter. To summarize, all our considered parameters could be presented in the following table. \begin{tabular}{l l} \hline \hline Two-dimensional electron wave vector \(\mathbf{k}=(k_{x},k_{y})\), \(k_{\pm}^{\tau}=\tau k_{x}\pm ik_{y}=\tau\,k\,\mathrm{e}^{i\tau\,\theta_{\mathbf{k}}}\), valley index \(\tau=\pm 1\), \(\theta_{\mathbf{k}}=\tan^{-1}(k_{y}/k_{x})\), phase \(\phi\), relative hopping parameter \(\alpha=\tan\phi\), Fermi velocity \(v_{F}\) & Related to the considered material/lattice \\ \hline Electric field amplitude \(\mathcal{E}_{0}\), irradiation frequency \(\omega\) & \\ \hline Strength of the interaction between an electron and irradiation \(c_{0}=e\mathcal{E}_{0}v_{F}/\omega\), dimensionless electron-photon coupling parameter \(\lambda_{0}\) & Combined \\ \hline \hline \end{tabular}   
The key idea for using the Floquet-Magnus perturbation approach is as follows. First, we rewrite the interaction Hamiltonian term \(\mathbb{H}_{A}^{(e)}(t|\,\tau,\phi)\) in the form of two time-independent complimentary conjugate terms:  
\[\hat{\mathbb{H}}_{A}^{(e)}(t|\,\tau,\phi)=\hat{\mathbb{O}}_{\tau}^{(\phi)}\, \mathrm{e}^{i\omega t}\,+\,\mathrm{H.c.}=\hat{\mathbb{O}}_{\tau}^{(\phi)}\, \mathrm{e}^{i\omega t}+\left[\hat{\mathbb{O}}_{\tau}^{(\phi)}\right]^{\dagger }\mathrm{e}^{-i\omega t}\,, \tag{31}\]  
where \(+\mathrm{H.c.}\) means Hermitian conjugate. Operator \(\hat{\mathbb{O}}_{\tau,\phi}\) and its Hermitian conjugate \(\hat{\mathbb{O}}_{\tau,\phi}^{\dagger}\) are _time-independent_. Comparing Hamiltonian (29) and representation (31), we can immediately derive the explicit form of the perturbation operator \(\hat{\mathbb{O}}_{\tau}^{(\phi)}\) as  
\[\hat{\mathbb{O}}_{\tau}^{(\phi)}=-\frac{c_{0}}{2}\left[ \begin{array}{ccc}0& \cos\phi\,(\tau-\beta)&0\\ \cos\phi\,(\tau+\beta)&0&\sin\phi\,(\tau-\beta)\\ 0&\sin\phi\,(\tau+\beta)&0\end{array} \right]\,. \tag{32}\]  
Once this is completed, the effective _time-independent_ Hamiltonian representing our dressed-state system is presented as  
\[\hat{\mathcal{H}}_{\mathrm{eff}}^{(e)}(k,t|\,\tau,\phi)\,=\,\hat{ \mathcal{H}}_{\tau}^{\phi}(k)\] \[+ \frac{1}{\hbar\omega}\big{[}\,\hat{\mathbb{O}}_{\tau}^{(\phi)}, \,\hat{\mathbb{O}}_{\tau,\phi}^{\dagger}\,\big{]}_{-}+\frac{1}{2(\hbar\omega) ^{2}}\bigg{\{}\bigg{[}\big{[}\,\hat{\mathbb{O}}_{\tau}^{(\phi)},\,\hat{ \mathbb{O}}_{\tau}^{\phi}(k)\,\big{]}_{-}\,\hat{\mathbb{O}}_{\tau}^{(\phi)} \,\bigg{]}_{-}\,+\,\mathrm{H.c.}\bigg{\}}\,+\,\cdots\,\,,\]  
where \([\hat{A},\hat{B}]_{-}\equiv\hat{A}\hat{B}-\hat{B}\hat{A}\) is a commutator. Since in our consideration of the off-resonance regime \(\lambda_{0}\ll 1\), it is sufficient to take into consideration only the first few terms of expansion (33). The matrix in Equation (32) is not Hermitian for any \(\beta>1\), which is true for any type of light polarization except the linear one but includes circularly polarized irradiation corresponding to \(\beta=1\). However, it is purely real for all types of Dirac materials with chiral terms \(k_{\tau}^{\pm}=\tau k_{x}-ik_{y}\) in Hamiltonian (1), for both a zero or finite bandgap \(\Delta_{0}\). The zero-order term in expansion (33) must be equivalent to Hamiltonian (1) of an \(\alpha-\mathcal{T}_{3}\) material in the absence of the dressing field. The next \(\sim c_{0}\,\lambda_{0}\) term is evaluated as \[\frac{1}{\hbar\omega}\big{[}\,\mathbb{O}_{\tau}^{(\phi)},\,\mathbb{O}_{\tau,\phi}^ {\dagger}\,\big{]}_{-}=-c_{0}\,\lambda_{0}\,\tau\,\beta\,\Sigma_{z}(\phi)\,=-c_{ 0}\,\lambda_{0}\,\tau\,\beta\,\left[ \begin{array}{ccc}\cos^{2}\phi&0&0\\ 0&-\cos 2\phi&0\\ 0&0&-\sin^{2}\phi\end{array} \right]\,, \tag{34}\]  
whose structure is equivalent to that of the bandgap term (16).