# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 2 Low-Energy Electronic States of the Considered Materials
### The \(\alpha-\mathcal{T}_{3}\) Model with a Finite Gap  
Importantly, an \(\alpha-\mathcal{T}_{3}\) material could also acquire a finite gap in a complete analogy to graphene [83]. It could be induced either by a special type of a dielectric substrate, or by a circularly polarized dressing field.  
In particular, we are very interested in investigating how an existing bandgap in an \(\alpha-\mathcal{T}_{3}\) lattice created by a substrate would be modified or renationalized if a curricular polarized irradiation is also added. Therefore, we put a gapped \(\alpha-\mathcal{T}_{3}\) model in the focus of our study.  
The Hamiltonian of the gapped \(\alpha-\mathcal{T}_{3}\) model is explicitly given as  
\[\hat{\mathcal{H}}_{\tau}^{(\phi)}(\boldsymbol{k}\,|\,\Delta_{0})=\left\{ \begin{array}{ccc}\Delta_{0}\,\cos^{2}\phi&\hbar v_{F}\,k_{\tau}^{-}\,\cos \phi&0\\ \hbar v_{F}\,k_{\tau}^{+}\,\cos\phi&-\Delta_{0}\,\cos(2\phi)&\hbar v_{F}\,k_{ \tau}^{-}\,\sin\phi\\ 0&\hbar v_{F}\,k_{\tau}^{+}\,\sin\phi&-\Delta_{0}\,\sin^{2}\phi\end{array} \right\}\,, \tag{13}\]  
where \(\Delta_{0}\) is the induced gap parameter.  
Similarly to graphene, the Hamiltonian for the gapped \(\alpha-\mathcal{T}_{3}\) model (13) is modified as  
\[\hat{\mathcal{H}}_{\tau}^{(\phi)}(\boldsymbol{k})\longrightarrow\hat{ \mathcal{H}}_{\tau}^{(\phi)}(\boldsymbol{k}\,|\,\Delta_{0})=\hat{\mathcal{H}} _{\tau}^{(\phi)}(\boldsymbol{k})+\hat{\mathcal{H}}_{\alpha}^{(\phi)}(\Delta _{0})\,, \tag{14}\]  
meaning that Hamiltonian (1) receives an additional term  
\[\hat{\mathcal{H}}_{\alpha}^{(\phi)}(\Delta_{0})=\frac{\Delta_{0}} {2}\,\hat{\Sigma}_{z}(\phi)\,, \tag{15}\] \[\hat{\Sigma}_{z}^{(\phi)}=-i\,\big{[}\hat{\Sigma}_{x}(\phi),\hat{ \Sigma}_{y}(\phi)\big{]}\]which is explicitly given as  
\[\hat{\mathcal{H}}_{a}^{(\phi)}(\Delta_{0})=\frac{\Delta_{0}}{2}\,\hat{\Sigma}_{z} (\phi)\,=\,\Delta_{0}\left\{\begin{array}{ccc}\cos^{2}\phi&0&0\\ 0&-\cos 2\phi&0\\ 0&0&-\sin^{2}\phi\end{array}\right\}\,. \tag{16}\]  
Here, \(\hat{\Sigma}_{z}(\phi)\) is the \(\phi\)-dependent generalization of the remaining Pauli matrix \(\hat{\Sigma}_{z}\).  
We immediately discern that in the presence of a finite bandgap \(\Delta_{0}\), the energy spectrum for the flat band of an \(\alpha-\mathcal{T}_{3}\) material with \(0<\alpha<1\) undergoes a special type of a deformation which results in a finite \(k\)-dispersion, i.e., the _flat band is no longer flat_. Finding the complete energy dispersions for the gapped \(\alpha-\mathcal{T}_{3}\) model is a challenging problem, which involves solving a cubic equation [56; 73]. However, it is relatively straightforward to verify that Hamiltonian (13) results in two inequivalent gaps between the conduction and flat, and the flat and the valence bands which depend on parameter \(\alpha\) (or phase \(\phi\)) and valley index \(\tau\).  
For a dice lattice with \(\phi\to\pi/4\), it is simplified as  
\[\hat{\Sigma}_{z}^{(1)}=\left[\begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&0&-1\end{array}\right], \tag{17}\]  
and Hamiltonian (13) is presented in the following form  
\[\hat{\mathcal{H}}_{d}(\mathbf{k}\,|\,\tau)=\frac{\hbar\
u_{F}}{\sqrt{2}}\left[ \begin{array}{ccc}\Delta_{0}/\sqrt{2}&k_{-}^{\tau}&0\\ k_{+}^{\tau}&0&k_{-}^{\tau}\\ 0&k_{+}^{\tau}&-\Delta_{0}/\sqrt{2}\end{array}\right]. \tag{18}\]  
In contrast to the Hamiltonian (1) for the general \(\alpha-\mathcal{T}_{3}\) model, the eigenvalue equation corresponding to Equation (18) results in a symmetric band structure with two equal bandgaps and an unaffected flat band right in the middle between the valence and conduction bands (see Figure 1). Therefore, the flat band is deformed and receives a finite \(k\)-dispersion for all \(\alpha-\mathcal{T}_{3}\) materials with a finite gap, expect for a dice lattice with \(\phi=\pi/4\).