# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 3 Electron Pumping through SSH Chains  
Topological chains seem to be very poor candidates for an effective quantum pump due to their electronic structure i.e., energy gap at the Fermi level. However, for quantum systems which are suddenly quenched the spectral density function at each site changes/evolves and it takes some time for the system to obtain its new equilibrium state. For a system with strong asymmetry in its LDOS this non-equilibrium processes can lead to the pumping effect between unbiased leads. Here we propose a 1D topological chain as an electron pump where the electron current direction can be controlled by means of the gate voltage potential. In such systems a gate electrode can be provided by external auxiliary electrodes or the substrate underneath the atomic chain. To avoid the electron leakage to the substrate we consider the chain on the insulating surface with two electron reservoirs at both chain ends (\(\vec{k}_{1}=\vec{k}L\) and \(\vec{k}_{N}=\vec{k}R\)). It is assumed that the external perturbation applied to the chain does not change the couplings homogeneously but there is a kind of the wire inertia which introduces the phase shift between the couplings (depending on the site position in the chain). In our investigations we consider the one-Gaussian or two-Gaussian perturbations in the form of a train-impulse spreading through the chain, see Figure 2, upper panels, where we show time-dependent perturbations, \(f_{0}\), \(f_{i}\) and \(f_{N}\) which are related to the couplings, \(V_{\tilde{k}L}\), \(V_{i}\) and \(V_{\tilde{k}R}\), respectively and \(V_{i}(t)=V_{i}f_{i}(t)\). The analytical relations for these perturbations can be written in the following form \(f_{j}(t)=1+\exp\left(\frac{t^{2}}{\sigma}\right)\) for the one-Gaussian pulse and \(f_{j}(t)=1+\exp\left(\frac{t^{2}}{\sigma}\right)-\exp\left(\frac{\tilde{I}-t _{0}\|^{2}}{\sigma}\right)\) for the two-Gaussian perturbation, where \(\tilde{I}=t-(t_{00}+\mu t_{x})\) and \(j=0,1,2,\cdots,N\). Here \(t_{00}\) stands for the time for the first Gaussian maximum, \(t_{x}\) is responsible for the time shift of the external perturbation between the nearest-neighbor couplings (for \(t_{x}=0\) all couplings change simultaneously in time in the same way), \(t_{1}\) stands for the time shift between the Gaussian functions, and \(\sigma\) corresponds to the half-width of the Gaussian signal. Such train pulses locally modify the site-site couplings and can change the system topology (depending on the coupling differences between chain sites). This perturbation concerns only small part of the whole chain, but it has a huge impact on the electron current flowing through the system. The time-dependent occupancies are obtained from Equation (3) and the current flowing through the system from Equation (6), which in the wide-band approximation we calculate from the relation:  
\[j_{L}(t) = -\Gamma^{L}f_{0}^{2}(t)n_{1}(t)-2\text{Im}\left\{\sum_{\tilde{k}L }n_{\tilde{k}L}(0)V_{\tilde{k}L}f_{0}(t)e^{i(\varepsilon_{\tilde{k}L}- \varepsilon_{0})t}U_{1,\tilde{k}L}(t)\right\}, \tag{8}\]  
Figure 2: One-Gaussian (left panel) and two-Gaussian (right panel) perturbations of the time-dependent couplings: \(f_{0}\) (between the left electrode and first wire site, red lines), \(f_{i}\) (between i-th and \((i+1)\)-th sites, broken curves) and \(f_{N}\) (between the last chain site and the right electrode, back solid curves)—upper panels. Charge occupation of the first wire site (middle panels) and time-dependent current flowing from the left electrode (bottom panels) for the SSH chain in the trivial phase (yellow lines, \(V=4,W=1,\text{SSH}_{0}\)) and for nontrivial phase (violet lines, \(V=1,W=4,\text{SSH}_{1}\)). The other parameters are: \(\varepsilon_{0}=3\), \(t_{x}=3\), \(\sigma=10\), \(V_{L}=V_{R}=4\), \(t_{00}=400\), \(t_{1}=4\), \(N=8\). where the evolution operator matrix elements \(U_{1,\bar{k},\bar{L}}\) are obtained from Equation (5).