In this Thesis we focus on topological systems out of equilibrium, where the nonequilibrium state is established by quenching parameters of the Hamiltonian between phases with different topological invariants. We study how the system, which is initially in the ground state of the one-dimensional Su-Schrieffer-Heeger (SSH) model, responds to slow quenches. The topological invariant that distinguishes between the trivial and the topological insulator is the winding number: it is zero in the trivial phase and one in the topological phase. In the ground state, the number of edge states equals the winding number, namely there are no edge states in the trivial phase and there is one edge state at each end of the chain in the topological phase. We are interested in the evolution of the topological invariant and the dynamics of the edge states. In the literature, both sudden and slow quenches between topologically nonequivalent regimes have been studied in Chern insulators. The Hall conductivity was found to change although the Chern number stayed the same. The central finding of this Thesis is that in the SSH model the winding number is not preserved if the parameters are quenched between topologically nonequivalent regimes. We explain this in terms of pseudospins precessing in pseudo-magnetic field. The edge states are occupied after a quench from the trivial to the topological regime, and the slower the quench, the closer the state of the propagated system is to the ground state. Depending on how fast the quench is performed we distinguish two regimes: if (with respect to the chosen system size) the quench is slow enough, the system stays in the ground state of the instantaneous Hamiltonian, otherwise transitions into unoccupied one-particle states occur. The characteristic time scales with the system size squared. We also consider the time dependence of the following quantities: energy, overlap of the state of the propagated system with the ground state, matrix elements of the reduced density matrix in the subspace spanned by the two-particle edge states and the trace of its square. The characteristic time estimated from the final trace also scales with the system size squared. In the bulk, the problem is equal to the Landau-Zener transition in a two-level system. In a bounded system the problem is analogous to the multi-state Landau-Zener transition, which by reducing the Hilbert space translates into the Landau-Zener transition between two states. The latter predicts scaling of the characteristic time that matches numerical results.
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