Extended Haldane Model in The Dice Lattice: Multiple Flat-Band-Induced topological Transitions Revealed

In this study, we examine the introduction of the Haldane model into the dice lattice by altering the flow between the next-nearest-neighbour sites. This breaks the lattice's inversion and time-reversal symmetries. We demonstrate the presence of point-charge particle symmetries at $\phi^c=\pi/6$ and $5\pi/6$ and derive the analytical expression for quasi-energies. We demonstrate that a gap closure occurs at these critical points, inducing a topological transition. This is confirmed by calculating the Berry curvature and orbital magnetic moment. A topological analysis shows that the Chern numbers of the valence band $(\nu=0)$, the flat band $(\nu=1)$ and the conduction band $(\nu=2)$ depend strongly on the relationship between the fluxes $\phi^a$ and $\phi^c$. When $\phi^c = \phi^a$, the Chern numbers are $(C_0, C_1, C_2) = (2, -2, 0)$ in the region $\phi^c \in [0, \pi/6[$, and (0, 2, -2) in the region $\phi^c\in ]5\pi/6, \pi]$. Conversely, when $\phi^c \neq \phi^a$, the topological invariants become $(C_1, C_2) = (-1, -1)$ for $\phi^c \in [0, \pi/6[$, and $(C_0, C_1, )= (1, 1)$ for $\phi^c\in ]5\pi/6, \pi]$. These variations reflect topological phase transitions at the critical points $\phi^c=\pi/6$ and $5\pi/6$, affecting all of the system's bands. Furthermore, the anomalous Hall conductivity exhibits a quantized plateau of 2$\sigma_{0}$, as well as an unquantized tilted plateau evolving from 1.50$\sigma_{0}$ to 1.25$\sigma_{0}$ at the same transition points. Controlling the flux allows topological transitions to be engineered and quantum transport in the dice lattice to be optimised, offering promising prospects for reconfigurable topological devices with low dissipation and robust quantum transport.