# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 2 Low-Energy Electronic States of the Considered Materials
### Lieb Lattice  
Flat bands have also been realized in some other types of lattices, mostly optical Lieb and Kagome lattices. A Lieb lattice was observed in a number of existing systems and experimental setups: organic materials, optical lattices, and waveguides  
Figure 1: (Color online) Three-dimensional plots for the energy band structure of irradiated gapped \(\alpha-\mathcal{T}_{3}\) materials. Panel (**a**) represents the energy spectrum for an \(\alpha-\mathcal{T}_{3}\) material with zero bandgap \(\Delta_{0}=0.0\) and a small electron-light coupling parameter \(\lambda_{0}=0.1\); plots (**b,c**) show the dispersions for the \(\alpha-\mathcal{T}_{3}\) model with a finite bandgap \(\Delta_{0}=1.0\) and coupling constants \(\lambda_{0}=0.1\) and phases \(\phi=\pi/7\simeq 25.7^{0}\) and \(\phi=\pi/5\simeq 36^{0}\), correspondingly.  
A Lieb lattice is made of three displaced square sublattices. The Hamiltonian of a Lieb lattice is  
\[\mathcal{H}^{(\mathrm{L})}(\mathbf{k}\,|\,k_{\Delta})=\hbar v_{F}\left[\begin{array} []{ccc}k_{\Delta}&q_{x}&0\\ q_{x}&-k_{\Delta}&q_{y}\\ 0&q_{y}&k_{\Delta}\end{array}\right]\,, \tag{19}\]  
where  
\[q_{x,y}=\frac{\pi}{a_{0}}+k_{x,y}\,, \tag{20}\]  
and \(a_{0}\) is the lattice parameter.  
The three energy dispersions, corresponding to Hamiltonian (19), are \(\varepsilon^{(\mathrm{L})}_{\gamma=\pm 1}(\mathbf{k}\,|\,k_{\Delta})= \gamma\sqrt{k_{\Delta}^{2}+q_{x}^{2}+q_{y}^{2}}\), and \(\varepsilon^{(\mathrm{L})}_{\gamma=0}(\mathbf{k}\,|\,k_{\Delta})=k_{\Delta}\), which could be written using a single unified expression  
\[\varepsilon^{(\mathrm{L})}_{\gamma=\pm 1}(\mathbf{k}\,|\,k_{\Delta})=\hbar v _{F}\left\{\delta_{\gamma,0}\,k_{\Delta}+\gamma\,v_{F}\left(1-\delta_{\gamma,0}\right)\sqrt{q_{x}^{2}+q_{y}^{2}+k_{\Delta}^{2}}\right\}, \tag{21}\]  
where \(\delta_{\gamma,0}\) is the Kronecker symbol. As a result, we have obtained three energy subbands. One of them is a flat band, while the other two have a finite dispersion, such as that shown in Figure 2c. In a Lieb lattice, the flat band is located at the finite energy \(v_{F}k_{\Delta}\) right next to the lowest point of the conduction band, while for the case of a dice lattice with a finite gap, it is located symmetrically between the valence and conduction bands.  
The corresponding wave functions are given by  
\[\boldsymbol{\Psi}^{(\mathrm{L})}_{\gamma=\pm 1}(\mathbf{k}\,|\,k_{ \Delta}) = \left[2\left(k_{\Delta}^{2}+q_{x}^{2}+q_{y}^{2}+\gamma\,k_{\Delta} \sqrt{[k_{\Delta}^{2}+q_{x}^{2}+q_{y}^{2}}\right)\,\right]^{-1/2} \tag{22}\] \[\times \left\{\begin{array}{l}q_{x}\\ -k_{\Delta}-\gamma\sqrt{k_{\Delta}^{2}+q_{x}^{2}+q_{y}^{2}}\\ q_{y}\end{array}\right\}\]  
and  
\[\boldsymbol{\Psi}^{(\mathrm{L})}_{\gamma=0}(\mathbf{k}\,|\,k_{\Delta})= \left(q_{x}^{2}+q_{y}^{2}\right)^{-1/2}\left\{\begin{array}{l}-q_{y}\\ 0\\ q_{x}\end{array}\right\} \tag{23}\]  
for the flat band.  
Figure 2: (Color online) Energy dispersions for the \(\alpha-\mathcal{T}_{3}\) model, a gapped diced lattice, and a Lieb lattice. (**a**–**c**) All three materials exhibit a flat band; however, the location of this flat band is substantially different for each lattice.