Characterizing dynamical behavior in topological open systems with boundary dissipation

We investigate the dynamics of the Su-Schrieffer-Heeger model with boundary dissipations described by Lindblad master equations and unravel distinct dynamical features in the topologically different phases of the underlying Hamiltonian. By examining the long-time damping dynamics, we uncover a dynamical duality phenomenon between the weak and strong dissipation region, which exists only in the topologically non-trivial phase, linked to the structure of the Liouvillian spectra,particularly the stripe closest to the steady state. When dissipation is confined to a single boundary, the dynamical duality phenomenon still exists. Under this condition, the Liouvillian gap fulfills an exponential size scaling relation in the topologically non-trivial phase and a power-law size scaling relation in the topologically trivial phase. Within the topologically non-trivial region, we identify the existence of boundary-localized dark states in the thermodynamical limit, which is responsible for the exponential size decay of Liouvillian gap.