Interplay of spectral topology and band topology in open quantum systems

We investigate how non-Hermitian skin effect (NHSE) physics reshapes dynamics when it coexists with topological edge states (TES). Using the dissipative Hofstadter lattice as a minimal, experimentally relevant setting, we track the real-time evolution of particle density and uncover two robust signatures: (i) chiral damping, a directional relaxation driven by the NHSE, and (ii) edge-selective extremal damping, in which persistent TES enforce maximal or minimal decay rates at specific edges. Crucially, we show that these damping processes occur on distinct, parametrically decoupled time scales set by a boundary-induced spectral topology rather than bulk dispersion alone, enabling their simultaneous observation under time dynamics. We map the parameter space and identify intermediate magnetic flux as the optimal regime: NHSE-enabled chiral damping is partially restored while TES remain long-lived enough to control edge-resolved decay. Our results provide a unified dynamical picture of how spectral topology (skin modes and non-Bloch spectra) and band topology (protected edge modes) interact and compete in open quantum systems, and how their interplay can be probed directly.