# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 3 Electron Pumping through SSH Chains
In the beginning in Figure 2 we analyze the time-dependent electron occupations (middle panels) and the currents flowing from the left electrode (bottom panels) for the SSH topological chain in the trivial phase (SSH\({}_{0}\), \(V=4,W=1\), yellow curves) and nontrivial phase (SSH\({}_{1}\), \(V=1,W=4\), violet curves) for \(\varepsilon_{0}=3\). Before the train impulse starts (at \(t\simeq 390\)) the system is in its equilibrium state - the occupancies do not change in time and the currents do not flow between electrodes. The structure of LDOS is symmetrical with respect to \(\varepsilon_{0}\) and in the first case (SSH\({}_{0}\) chain) there is an energy gap around the Fermi energy, thus the electron occupancy is almost 0.5 (middle panels) even for \(\varepsilon_{0}=3\). For the one-Gaussian perturbation, in the first stage, the chain-lead coupling strength increases and some charge from the chain leaks to the lead and the chain occupation locally decreases. This is also the reason why in this case the current is negative (it flows from the chain to the left electrode). In the second stage of this train perturbation the couplings between atomic sites inside the chain increase and the lower sideband LDOS states move deeper below the Fermi energy, thus the occupancy locally increases, and the current is positive. On the other hand, in the presence of this perturbation, for the chain with topological states at both edges (nontrivial phase) the empty LDOS states spread below the Fermi level and the occupancy of the chain site increases rapidly (the current flows from the left electrode to the chain). Please note that the perturbation (train impulse) goes through the chain up to \(t=420\) units of time but the occupancies and the currents tend very fast to their equilibrium values. The situation changes for the two-Gaussian perturbation (right panels). In this case, charge oscillations do not vanish rapidly, but they are visible also for larger \(t\), i.e., even for times for which the perturbation has passed through the system. The reason for such long-time oscillations is very low value of the two-Gaussian perturbation for a short period of time, \(f(t)\), which leads to almost vanishing site-site couplings in the system. Thus, the chain sites are almost decoupled for a moment (like in the atomic limit) and then the system needs much more time to reach the equilibrium state. Please note that after and before the perturbation, the occupancies are exactly the same and from the careful inspection of the currents flowing from the left electrode one can find the total charge pumped through the chain. For the SSH\({}_{1}\) chain and two-Gaussian impulse it is evident that most of the violet curve is negative, thus one expects the net pumping charge in the system. For the one-Gaussian impulse (left bottom panel) the currents for SSH\({}_{0}\) and SSH\({}_{1}\) chains oscillate around the zero value, but they are asymmetrical and more detailed study is required to classify these systems as electron pumps. The total charge pumped through the chain is obtained by integrating the time-dependent currents, \(N_{L/R}=\int j_{L/R}(t)dt\), where in our system the condition \(N_{L}=-N_{R}\) is always satisfied. Here, positive values of \(N_{L/R}\) correspond to electrons flowing from the electrode to the chain. In Figure 3 the pumped charge from the left lead is analyzed as a function of \(\varepsilon_{0}\) (which corresponds to the gate voltage) for the one-Gaussian (upper panel) and two-Gaussian perturbations (bottom panel) for both SSH\({}_{0}\) (yellow curves) and SSH\({}_{1}\) (violet curves) chains. As one can see for symmetrical LDOS with respect to the Fermi energy (\(\varepsilon_{0}=0\)) the charge does not flow through the chain, independently on the perturbation shape and topological phase of the chain. For \(\varepsilon_{0}\
eq 0\) the shape of the train impulse spreading through the chain plays a crucial role in the electron pumping effect in topological systems.