# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 2 Theoretical Description  
The physical model consists of the SSH chain composed of \(N\) sites that may be coupled to the electron reservoir (substrate electrode) underneath or to the left and right leads via the edge sites as shown in Figure 1. This model corresponds to atomic chains on vicinal surfaces (with the insulating or conducting substrate below the chain) or to the linear series of QDs with fully controlled system parameters. The SSH chain is characterized by the two-site primitive cell with two different couplings between sites inside each cell and between cells--\(V\) and \(W\) parameters, respectively. The chain for \(V=W\) represents nontopological normal chain without an energy gap. For the intracell couplings greater than intercell couplings we get a topological SSH chain in the nontrivial phase i.e., with topological mid-gap states at both chain ends (SSH1), otherwise we obtain a chain in the trivial topological phase with an energy gap but without the edge states (SSH0). The Hamiltonian describing the SSH chain composed of \(N\) sites coupled with the electrodes can be written in terms of the second quantization notation as follows:  
\[H = \sum_{i=1}^{N}\varepsilon_{i}a_{i}^{\dagger}a_{i}+\sum_{i=1}^{N} \sum_{\vec{k}_{i}}(\varepsilon_{\vec{k}_{i}}a_{\vec{k}_{i}}^{\dagger}a_{\vec{k }_{i}}+V_{\vec{k}_{i}}a_{\vec{k}_{i}}^{\dagger}a_{i})+\sum_{i,j}^{N}V_{i,j}(t )a_{i}^{\dagger}a_{j}+H.c. \tag{1}\]  
Here \(a_{i}^{\dagger},a_{i}\) are creation/annihilation operators respectively at \(i\)-th site, \(\varepsilon_{i}\) is the on-site energy level and \(\varepsilon_{\vec{k}}\) corresponds to possible electron energies in the leads. \(V_{\vec{k},j}\) stands for the coupling between electrodes and corresponding chain atom and \(V_{i,j}\) is responsible for the couplings between chain sites (note that \(V_{i,j}=V\) within the primitive cell and \(V_{i,j}=W\) between the neighboring cells). These couplings can vary in time during the train-impulse propagation or sudden quenches. For \(V_{\vec{k},j}=0\) except for \(i=1\) and \(i=N\) (i.e., for nonzero \(V_{\vec{k},L}\) and \(V_{\vec{k},R}\) elements) the model corresponds to that from Figure 1.  
In our paper time dynamics of the system is calculated within the interaction picture using the evolution operator method for which the following equation of motion can be written (\(\hbar=1\)):  
\[i\frac{\partial}{\partial t}U(t,t_{0})=\hat{V}(t)U(t,t_{0})\, \tag{2}\]  
where \(\hat{V}(t)=U_{0}(t,t_{0})V(t)U^{\dagger}(t,t_{0})\), \(U_{0}(t,t_{0})=\mathcal{T}\exp\left(i\int_{t_{0}}^{t}dt^{\prime}H_{0}(t^{ \prime})\right)\) and \(\mathcal{T}\) is the time ordering operator. Here \(V(t)\) is the coupling part of the total Hamiltonian and \(H_{0}(t)\) represents the on-site energy part, \(H_{0}=\sum_{\alpha}\varepsilon_{\alpha}a_{\alpha}^{\dagger}a_{\alpha}\). The physical properties of the system are expressed by the evolution operator matrix elements obtained from Equation (2). The local time-dependent charge occupancies, \(n_{i}(t)\), can be found from the relation [48; 49]:  
\[n_{i}(t)=\sum_{j=1}^{N}n_{j}(t_{0})|U_{i,j}(t,t_{0})|^{2}+\sum_{j\vec{k}_{j}} n_{\vec{k}_{j}}(t_{0})|U_{i,\vec{k}_{j}}(t,t_{0})|^{2}\, \tag{3}\]  
where \(n_{i}(t_{0})\) represents the initial filling of the corresponding single-particle state. As we are not interested in the transient effects, which appear only for small \(t\) (\(t\geq t_{0}\)), we assume empty initial occupancies of all chain sites. The evolution operator elements, which  
Figure 1: Model of atomic SSH chain composed of N sites on the insulating surface (which can stand for a gate electrode). The nearest-neighbor couplings between sites are denoted by \(V\) and \(W\) and the couplings with the left and right electrodes represent \(\Gamma_{L/R}\) symbols.