## 2 Model and Band Structures
### Electron-Electron Interactions  
We now add to \(\hat{H}_{\text{TB}}\) (3) the electron-electron interactions of on-site Coulomb type  
\[\hat{H}_{\text{int}}=\sum_{i}\hat{H}_{\text{int}}^{(i)};\quad\hat{H}_{\text{int}}^ {(i)}=U\sum_{m}\hat{n}_{im\uparrow}\hat{n}_{im\downarrow}+U^{\prime}\sum_{m \
eq m^{\prime}}\hat{n}_{im\uparrow}\hat{n}_{im^{\prime}\downarrow}+(U^{ \prime}-J)\sum_{m<m^{\prime},\sigma}\hat{n}_{im\sigma}\hat{n}_{im^{\prime} \sigma}. \tag{5}\]  
Here, we use the standard choice \(U^{\prime}=U-2J\) and ignore the pair hopping terms [41]. For simplicity, throughout this paper, we focus on the typical case of \(U=3J\), where the last term in Equation (5) vanishes. We use a mean-field approach and approximate \(\hat{H}_{\text{int}}^{(i)}\) by  
\[\hat{H}_{\text{int}}^{(i)}\simeq U\sum_{m}\left(n_{im\downarrow}\hat{n}_{im \uparrow}+n_{im\uparrow}\hat{n}_{im\downarrow}-n_{im\uparrow}n_{im\downarrow} \right)+U^{\prime}\sum_{m\
eq m^{\prime}}\left(n_{im^{\prime}\downarrow}\hat{n} _{im\uparrow}+n_{im\uparrow}\hat{n}_{im^{\prime}\downarrow}-n_{im\uparrow}n _{im^{\prime}\downarrow}\right), \tag{6}\]  
where \(n_{im\sigma}\equiv\langle\hat{n}_{im\sigma}\rangle\) is determined self-consistently and we have neglected the orbital off-diagonal contributions \(\langle\hat{e}_{im\sigma}^{\dagger}\hat{e}_{im^{\prime}\sigma}\rangle\) for simplicity. In the mean-field approximation, we use the unit cell made of two hexagrams, which accommodates 26 electrons, and it is important that the electron number is even. To simulate the charge distribution in the Mott insulator phase, we introduce a fictitious magnetic order in the mean-field approximation. However, this redundant magnetic order is not very harmful when we discuss the charge dynamics driven by strong laser fields. Another drawback is that the translation symmetry is further lowered than a periodic array of hexagrams, but we shall show later that this effect is not very strong. We compare in Figure 2d,e the band structure without and with the electron-electron interaction. Panel (d) is plotted for comparison and shows the same energy bands as in (c) (\(U=3J=0\)), but the bands are now folded corresponding to a two-hexagram unit cell. Figure 2e is the result for \(U=3J=0.816\,\text{eV}\) calculated at zero temperature \(T=0\,\text{K}\). The interactions have relatively weak effects at energies well below the Fermi energy \(E_{F}\). However, near \(E_{F}\), they cause a splitting of the two bands and make the system insulating. We have confirmed that these qualitative features remain unchanged for \(U=3J\leq 2.86\,\text{eV}\). For \(U=3J=0.816\,\text{eV}\), the band gap at the \(\Gamma\) point is \(69.2\,\text{meV}\), which is a few times smaller than the experimental value [15]. While the gap size could be reproduced by tuning \(U\), \(U^{\prime}\), and \(J\), we do not go into further detail in this paper. Figure 2: (**a**) band structure without PLD or Coulomb interaction. The inset schematically shows the Brillouin zone; (**b**) the same result with folded bands obtained by taking each hexagram as a unit cell; (**c**) band structure with PLD (\(x=-1.5\)) and no Coulomb interaction; (**d**) the same result shown with folded bands for two-hexagram unit cell; (**e**) band structure with both PLD (\(x=-1.5\)) and on-site Coulomb interaction (\(U=3J=0.816\,\text{eV}\)) calculated within mean-field approximation (6) at \(T=0\,\text{K}\). Let us now examine charge and spin densities. The results are shown in Figure 3 for the same parameters \(U=3J=0.816\,\)eV at \(T=0\,\)K. All the hexagrams have the identical charge-density distribution: the density increases toward the center of each hexagram, and the 2-fold rotational symmetry around an A site is preserved. The 6-fold rotational symmetry is broken due to the doubled unit cell, but its asymmetry is so small (about 1%) that this effect is ignorable. In the doubled unit cell, the two hexagrams have the spin density distribution with the opposite signs, and this is due to a fictitious Neel order.