# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 4 Spectral Density Dynamics
### Train Impulses  
Here, we are going to explain more precisely the electron transfer process through the energy-gapped structure between unbiased leads. Thus, in Figure 4 we show the spectral density functions related to four sites of the SSH\({}_{0}\) chain during the propagation of the train impulse (one-Gaussian perturbation, panel a, or two-Gaussian impulse, panel b). As one can see the chain is characterized by the energy gap at the Fermi level, independent on the site number (the total chain length is \(N=8\)). The perturbation starts at \(t=400\) and it leads to wider LDOS - the sidebands are showed outside the Fermi energy due to larger values of the coupling parameters. After some time, all LDOS peaks return to their initial positions and the induced sideband structure follows the perturbation. When the train impulse reaches the edge site the system evolves to its previous equilibrium state. In this case, electrons are transferred through the LDOS sidebands which are outside the energy gap of the chain.  
The situation changes for the two-Gaussian perturbation (panel b) as now some site-site couplings decrease in time and at the same time some of them increase. It leads to the induced sidebands outside the energy gap (like in the one-Gaussian impulse) but also  
Figure 4: LDOS time evolution at \(i=1,3,5\) and \(8\) sites of the SSH trivial chain [\(V=4,W=1\), panels (**a,b**)] and nontrivial chain [\(V=1,W=4\), panel (**c**)] composed of \(N=8\) sites. Panel (**a**) represents the system disturbed by the one-Gaussian perturbation, panels (**b,c**) correspond to the two-Gaussian impulse, the same as in Figure 2.  
dynamical nonzero LDOS structure in the energy gap region is observed. Thus, for this perturbation electrons can be pumped through these temporary mid-gap states. For the nontrivial topological chain, SSH\({}_{1}\), the LDOS dynamics during the two-Gaussian impulse is analyzed in panel c. In this case, the sidebands outside the energy gap are slightly visible and the main pumping process is realized only inside the energy gap region. Thus, one expects lower values of the net pumping current in comparison with the trivial chain, as one can see in Figure 3. The inside structure of LDOS during the perturbation is formed from the mid-gap topological state localized at the first site--this state oscillates in time and we observe a periodic emission of this state to other chain sites. These induced states also follow the train-impulse propagation along the chain. Please note that sudden local change of the site-site couplings disturbs both topological states related to the first and the last chain sites for a moment (the difference between the nearest-neighbor couplings decreases rapidly which influences the chain periodicity) but after some time the mid-gap states are rebuilt again. It is also interesting that for the two-Gaussian pulse both topological states have very high intensity for short period of time. This results from the fact that for a moment the coupling between the first (last) chain site and its neighboring site decreases and the energetic structure of LDOS tends to the atomic limit in this case.  
From the detailed studies of time-dependent LDOS characteristics we have found that electrons are transferred through the mid-gap temporary states in nontrivial topological chains or through the induced sidebands outside the energy gap region in trivial chains. The pumped current direction depends only on the relative position of the LDOS peaks and the leads Fermi energy, cf. Figure 3. We have also checked that the spectral density dynamics of the SSH chain coupled with a noninsulating surface underneath does not reveal significant differences. In that case the chain-substrate coupling is responsible for the half-width of the spectral density peaks and for larger couplings these LDOS peaks are wider and smoother. It is important from the practical point of view, because it opens more perspectives for experimental studies of such structures.