# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 3 Electron Pumping through SSH Chains
For the one-Gaussian perturbation the couplings between sites locally increase leading to smaller difference between \(V_{i}\) parameters. In this case, the energy gap is not closed. It is the reason that electrons do not flow through the SSH\({}_{0}\) chain for \(\varepsilon_{0}\) smaller than the energy gap width (here the gap appears for \(|\varepsilon-\varepsilon_{0}|\leq 3\), cf. the inset in Figure 3, bottom panel). Note that also for the chain in the nontrivial phase the pumping effect does not occur for these values of \(\varepsilon_{0}\) although there is a topological state at the first chain site (nonzero LDOS at the Fermi level, see the inset, violet curve). In this case, electrons which flow out from the electrode cannot be transferred through the middle part of the chain due to the energy gap at these middle sites. It means that the zero-energy edge states do not play a main role in the electron pumping through 1D systems--this problem will be discussed in more details in the next section. For larger \(\varepsilon_{0}\) (beyond the energy gap) electrons are pumped in the system. In our case for \(\varepsilon_{0}>3\) the bottom sideband peaks which are below the Fermi level (they are occupied) change their positions in the presence of the one-Gaussian perturbation, and thus the electron charge leaks out of the chain (the current is negative). Next, for larger \(\varepsilon_{0}\) (\(\varepsilon_{0}\simeq 4.7\)) the bottom LDOS sideband is symmetrical with respect to the Fermi energy and the current does not flow through the system (similarly as for \(\varepsilon_{0}=0\)). The second upper LDOS sideband lies too far from the Fermi level and does not influence the current. For \(\varepsilon_{0}>4.5\) there are more LDOS peaks above the Fermi level and in the presence of the perturbation they move below the Fermi energy. In that case electrons from the electrode occupy these empty LDOS states and the current is positive. For larger and larger \(\varepsilon_{0}\) there are no electron states near the Fermi level and the net charge cannot be pumped through the system. Note that for the nontrivial chain (SSH\({}_{1}\), violet curve upper panel) and the one-Gaussian impulse the pumping current almost does not flow through the system which is a consequence of relatively low LDOS in the upper and lower sidebands (cf. the violet curve in the inset), and because the sidebands are responsible for the pumping current this effect is marginal in the SSH\({}_{1}\) chain. Thus, the shape of the pumping current as a function of the gate voltage allows us to distinguish between the trivial and nontrivial phases of the topological system. In Figure 3 the results obtained for the SSH\({}_{0}\) and SSH\({}_{1}\) chains are compared also with the normal chain of the same length (black broken curve, upper panel). There are crucial differences between topological and normal systems. First, for the normal chain the pumping current starts to flow for \(\varepsilon_{0}>0\) as there is no energy gap in the system and even small asymmetry in its energetic structure leads to the electron pumping. The maximal pumping current is observed for such \(\varepsilon_{0}\) for which the bottom of chain energy band corresponds to the Fermi energy (here it is for \(\varepsilon_{0}=2\)). For larger \(\varepsilon_{0}\) the pumping current vanishes due to the lack of states at the Fermi level. It is also important that for the  
Figure 3: Charge pumped through the 8-atom SSH chain as a function of \(\varepsilon_{0}\) for \(t_{x}=3\). The upper (lower) panel corresponds to the one-Gaussian (two-Gaussian) perturbation. The broken line corresponds to the normal chain with \(V=1\). The other parameters are the same as in Figure 2. The inset in the bottom panel shows the local DOS at the first site of the SSH chain in the trivial (SSH\({}_{0}\)) and nontrivial (SSH\({}_{1}\)) phases.