# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 1 Introduction  
Atomic wires are the thinnest electrical conductors, and they can be used in plenty of applications [1; 2; 3; 4]. Many new quantum effects were observed in such systems as spin-charge separation [5], charge-density waves [6] or Majorana topological states [7; 8]. Non-equilibrium one-dimensional (1D) systems reveal much richer physics and in the presence of time-dependent perturbations the turnstile effect, photon-assisted tunneling, or coherent destruction of tunneling [9; 10; 11; 12; 13; 14] can appear. Dynamical phenomena in 1D structures can also lead to novel solid-state phases such as the Floquet topological insulators [15], time crystals [16; 17; 18] or transient crystals [19]. For a quantum system between unbiased leads the electron current does not flow in the stationary case. However, in the presence of non-equilibrium phenomena (e.g., time-dependent forces or perturbations) electrons can be transferred between leads even without the source-drain voltage which leads to the electron pumping effect. This effect has been the subject of many theoretical and experimental works [18; 20; 21; 22; 23; 24; 25]--it was observed in single and double quantum dot (QD) systems for periodical changes of two or more device-control parameters [20; 26] which were responsible for the left-right spatial symmetry breaking of the whole system. Please note that in the presence of the spatial symmetry the probability of electron tunneling from the central system to the left or to the right lead is exactly the same. The pumping current can be also generated for breaking the time-reversal symmetry in the system e.g., by adding the second harmonic to the driving potential or in the presence of time-dependent dipole-like forces [25]. In particular, single-parametric pumping or train-impulse pumps are especially interesting due to their potential applications in nanoelectronics [27; 28; 29]. For larger one-dimensional systems electron pumping effects were studied for different time-periodic perturbations (delta-like,harmonic or pulsed) [30; 31; 32; 33; 34], where the net electron current flows between unbiased leads. From the local electronic properties point of view this current can appear if the system is characterized by finite spectral density function (Local Density of States, LDOS) near the Fermi energy. In such a case an electron can flow from the lead to the unoccupied LDOS state of the central system and then pass to the second lead. On the other hand, the presence of an energy gap in the structure of the central system strongly restricts the pumping effect. However, this conclusion can be invalid for the new state of matter i.e., for topological insulators where energy gaps appear inside the system. These structures are also characterized by collective edge states (topological states) at the system boundaries. Please note that the quantization of particle transport in such structures were studied for Thouless pumping [35], Moire pumping [36], for quantum spin-Hall insulator [37], for twisted bilayer graphene [38], coupled fermionic chains [39], nonadiabatic Floquet structures [40] or others. Topological phases in one-dimensional chains can be obtained within a Su-Schrieffer-Heeger (SSH) model [41; 42; 43]. The model due its \(\mathcal{CT}\) symmetry possesses two different topological phases: the trivial phase with an energy gap along the whole system, and the nontrivial one for which there exist spectrally isolated mid-gap states localized at the system boundaries which are protected against local perturbations [41; 42; 44]. In this paper, we explore the possibility of electron pumping in the SSH chain under a time-periodic driving in the form of external one-Gaussian or two-Gaussian perturbations moving along the system (so called the train impulse) which in the nonadiabatic regime can generate the pumping current between unbiased leads.