Adiabatic pumping in the quasi-one-dimensional triangle lattice

We analyze the properties of the quasi-one-dimensional triangle lattice emphasizing the occurrence of flat bands and band touching via the tuning of the lattice hopping parameters and on-site energies. The spectral properties of the infinite system will be compared with the transmission through a finite piece of the lattice with attached semi-infinite leads. Furthermore, we investigate the adiabatic pumping properties of such a system: Depending on the transmission through the lattice, this results in nonzero integer charge transfers or transfers that increase linearly with the lattice size.

Figure 1

Tight-binding triangle lattice with semi-infinite leads. A number of unit cells ($N$) with the two sites A and B are connected with hopping parameters $t_1$, $t_2$, and $t_3$ to form the triangle lattice of length $L=Na$. Note that an additional $(N+1)$th A site is required to complete the finite triangle chain.

Figure 2

Illustration of the band gap for the infinite triangle lattice. Brightness visualizes the value of the energy gap. White dashed curves correspond to the appearance of a flat band. The crosses (green) are located at band-touching (a) or band-crossing points (b). Corresponding parameters in (a): ${\tau }_2=8$; (b) $\Delta \epsilon =0.5$.

Figure 3

Collection of energy bands $\epsilon \left(\kappa \right)$ of the infinite lattice, transmission probability $T_N\left(\epsilon \right)$ and $\alpha \left(\epsilon \right)$ for the finite open lattice ($N=$ number of unit-cells). (a)–(d) Band touching with ${\tau }_1={\tau }_2=\Delta \epsilon =1$; (e),(f) band crossing with ${\tau }_1={\tau }_2=0$, $\Delta \epsilon =0.5$; (g),(h) splitting with ${\tau }_1={\tau }_2=0.1$, $\Delta \epsilon =0.5$.

Figure 4

Varying parameters ${\tau }_1$ and $\Delta \epsilon$ (path $\mathcal{A}$) around two band-touching points (marked with black crosses) for different Fermi energies ${\epsilon }_{\mathrm{F}}$ and $N$. The bottom panels show the parameter cycle (blue) on the density plot of transmission $T$ (black, $T=1$; white, $T=0$). In the top panels the charge transfer is plotted versus ${\tau }_1^{\left(\max \right)}$. Vertical dashed lines (cyan) indicate the boundary between insulating and transmitting regions. For all panels, ${\tau }_2=8$.

Figure 5

Demonstration of different charge transfer results with pumping parameters ${\tau }_1$ and ${\tau }_2$ (path $\mathcal{B}$). Panels (a) and (b): Symmetric pumping cycle with ${\tau }_i^{\left(\max \right)}=-{\tau }_i^{\left(\min \right)}$, $\Delta \epsilon =0.5$, ${\epsilon }_{\mathrm{F}}=0$, and $N=11$ around crossing point. Panels (c) and (d): Charge transfer of pumping in transmitting region increases linearly with lattice length $N$. $\Delta \epsilon =1.8$, ${\epsilon }_{\mathrm{F}}=1$. The brightness of panels (a) and (c) corresponds to transmission $T$ (black, $T=1$; white, $T=0$).