# Electron Pumping and Spectral Density Dynamics in Energy-Gapped Topological Chains
## 4 Spectral Density Dynamics
### Quench Dynamics in SSH Chains  
Topological chains which are characterized by an energy gap along the system can transfer electrons through the induced mid-gap states as shown in the previous section. This effect can be observed only for non-equilibrium processes such as in the presence of a train impulse moving along the chain. Alternatively, after a sudden change of the system parameters (so called the quantum quench) the system needs some time to reach its steady state which opens a new possibility for coexistence of both trivial and nontrivial phases simultaneously in the system. Thus, it is desirable to analyze the mid-gap topological states dynamics between SSH\({}_{1}\) and SSH\({}_{0}\) (or normal) chains, which takes place not just for a while (as during the train impulse) but which lasts for a long time after the quench.  
Thus, in Figure 5 we analyze a segment of energy-dependent LDOS near the Fermi energy (at sites from \(i=1\) to \(i=10\)) as a function of time for topological chain of the length \(N=20\). At a given time (\(t=400\)) the quantum quench switches topological phase from nontrivial to the trivial one (panel a) or to the normal chain (panel b) and we can analyze the mid-gap state dynamics along the chain. As one can see before the transition the system is characterized by wide topological states at the edge sites (\(i=1\), and the same at \(i=20\)) and the energy gap in all middle sites (panels a and b). For very large time after the quench, we can notice that the system reaches its steady state i.e., trivial topological state with energy-gapped LDOS (panel a) or normal nontopological chain with no energy gap (panel b).  
The most interesting physics happens just after the quench. Surprisingly, topological state does not disappear immediately but rather moves into middle parts of the chain. However, in the upper panel this state survives only on the first few boundary sites and then it is blurred in time, but it is visible that at \(t\simeq 410\) it reaches the middle site of the chain. The same signal moves from the last chain site and they pass each other. Thus, the  
Figure 5: LDOS time evolution at first 10 sites in the topologically nontrivial chain, SSH\({}_{1}\) (\(V=0.5\), \(W=2\)) quenched to the trivial chain SSH\({}_{0}\) [\(V=2\), \(W=0.5\), panel (a)] or to the normal chain [\(V=W=2\), panel (b)] at \(t=400\). The chain length is \(N=20\) and the on-site energies \(\varepsilon_{0}=0\).  
X-shaped LDOS is observed and as time evolves these states balance between the edges with vanishing intensities. This process is more evident in the bottom panel where both topological states migrate after the quench through the normal chain and they also form a V-shaped LDOS dynamical structure. Note that in the normal chain the edge states travel much faster and they reach the middle chain site just at \(t\simeq 403\), i.e., only 3 time units after the quench. It means that the travel speed of the mid-gap state strongly depends on the energy-gapped structure in the spectral density function and for nonzero LDOS it moves much faster along the chain. Please note that similar time evolution was observed for the absolute value of the spinor component related to the Majorana mode in 1D topological systems [50].